Study of Transient Flow and Particle Transport in ContinuousSteel Caster Molds Part I Fluid Flow

QUAN YUAN BRIAN G THOMAS and SP VANKA

Unsteady three-dimensional flow in the mold region of the liquid pool during continuous casting of steelslabs has been computed using realistic geometries starting from the submerged inlet nozzle Three large-eddy simulations (LES) have been validated with measurements and used to compare results betweenfull-pool and symmetric half-pool domains and between a full-scale water model and actual behavior ina thin-slab steel caster First time-dependent turbulent flow in the submerged nozzle is computed Thetime-dependent velocities exiting the nozzle ports are then used as inlet conditions for the flow inthe liquid pool Complex time-varying flow structures are observed in the simulation results in spite ofthe nominally steady casting conditions Flow in the mold region is seen to switch between a ldquodouble-rollrdquo recirculation zone and a complex flow pattern with multiple vortices The computed time-averagedflow pattern agrees well with measurements obtained by hot-wire anemometry and dye injection in full-scale water models Full-pool simulations show asymmetries between the left and right sides of the flowespecially in the lower recirculation zone These asymmetries caused by interactions between twohalves of the liquid pool are not present in the half-pool simulation This work also quantifies differ-ences between flow in the water model and the corresponding steel caster The top-surface liquid profileand fluctuations are predicted in both systems and agree favorably with measurements The flow fieldin the water model is predicted to differ from that in the steel caster in having higher upward velocitiesin the lower-mold region and a more uniform top-surface liquid profile A spectral analysis of the com-puted velocities shows characteristics similar to previous measurements The flow results presented hereare later used (in Part II of this article) to investigate the transport of inclusion particles

I INTRODUCTION

TURBULENT flow during the continuous casting of steelis important because it influences critical phenomena thataffect steel quality These include inclusionbubble transportand entrapment[123] the transport and dissipation of super-heat[4] the shape and fluctuations of the top-surface level[56]

and the entrainment of mold flux from velocity variationsacross the top surface The continuous casting process isschematically shown in Figure 1 during which the molten steelflows into the liquid pool from the tundish through the sub-merged entry nozzle (SEN) The flow rate is controlled usingeither a stopper rod (shown in Figure 1) or a slide gate thatrestricts the opening area The bifurcated or trifurcated nozzleports direct molten steel jets into the mold cavity at the desiredangle with various levels of turbulence and swirl The water-cooled mold freezes a thin solid shell which is continuouslywithdrawn at the casting speed The shell contains a largeliquid pool in which steel circulates as shown in Figure 1

The flow in both the SEN and the liquid pool is highly tur-bulent with Reynolds numbers in excess of 10000 Thuschaotic turbulent structures exist in the liquid pool even fornominally steady-state casting conditions This might lead toquality problems The effect of unsteady flow on quality prob-lems has not been investigated even though many experi-mental studies have confirmed that transient flow conditions

(involving changes in flow patterns) are associated with qual-ity problems[7] Flow transients influence transport of inclu-sion particles (eg alumina) and bubbles carried by the jetsinto the liquid pool and might be the cause for intermittentand asymmetrical defects observed in plants[7] The turbulentjets traverse the liquid pool and impinge on the narrow faceto build up an unsteady heat flux which might cause shell-thinning breakouts if the instantaneous flux is too high[8] Thetop surface especially near the meniscus is responsible forsurface quality problems and other defects A liquid slag layer(Figure 1) covers the molten steel top surface to prevent itfrom being reoxidized by the air The liquid flux also creepsinto the interfacial gap between the mold and the shell to actas a lubricant to prevent surface defects Excessive fluctua-tions of the liquid level at the top surface interrupt the steadysupply of the liquid slag into the interfacial gap and causeheat-transfer variations resulting in longitudinal cracks[9]

transverse depressions[10] and other defects[11] The velocityacross the top surface also varies with time Excessive localsurface velocity can shear off liquid slag into the liquid poolto form harmful mold-slag inclusions[12] causing skin delam-inations slivers and other defects in rolled sheet product[7]

Heat transfer at the top surface is also important If the sur-face temperature is too cold and the flow is too slow themeniscus might solidify to form hooks or deep oscillationmarks Plant observations have found that these defects areintermittent[13] suggesting that they are related to flow tran-sients The present work aims to generate fundamental insightsinto the transient flow in the mold region as an essential steptoward minimizing defects in the final steel product

Many previous studies have used Reynolds-averaged turbu-lence models (mainly the k- model) to study flow in the liq-uid pool[14ndash22] These models predict time-averaged velocities

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash685

QUAN YUAN PhD Candidate BRIAN G THOMAS W Grafton andLillian B Wilkins Professor and SP VANKA Professor are withthe Department of Mechanical and Industrial Engineering University ofIllinois at Urbana-Champaign Urbana IL 61801 Contact e-mail bgthomasuiucedu

Manuscript submitted July 14 2003

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with reasonable accuracy[21] and at a reasonable computationalcost However they poorly estimate turbulent dynamics suchas quantified by root-mean-square (rms) values[21] A transientReynolds-averaged approach which is more computationallydemanding was used previously to study transient changes inflow pattern resulting from drastic changes in inlet conditions[23]

However this approach was unable to capture inherent flowtransients during steady casting in several test cases This maybe due to the large numerical dissipation associated with thesemodels or due to the assumption of isotropic turbulence

The present work employs the large-eddy simulation (LES)approach This approach was shown in recent studies toaccurately predict the dynamics of large-scale turbulencestructures during continuous casting of steel[212224ndash26] How-ever LES entails many challenges when applied to this com-plex flow problem including the prescription of the transientinlet velocities resolution of the complex domain geometrythe moving solidification front and computing long-termtransients Therefore the application of LES to these aspectsof continuous casting requires large amounts of computermemory and central processing unit (CPU) time As a resultthis model may be considered as a way to generate funda-mental understanding and benchmarks for simple modelsrather than as an engineering tool for parametric study

Due to the high operating temperature it is difficult andexpensive to conduct flow measurements in continuous steelcasters in order to validate model predictions[21] However

because of the nearly equal kinematic viscosities of moltensteel and water water models with transparent plastic wallscan be used to study single-phase flow in steel castingprocesses[2728529ndash32] where Froude similarity is usuallyemployed Thus measurements using dye injection hot-wireanemometry and particle-image velocimetry (PIV) to quan-tify the flow velocities in water models are used to validatethe predictions of the LES code in this work

Although very valuable for validation of computationalmodels the water model differs from the actual steel casterin several aspects important to the flow field First its side-walls which represent the moving solidifying shell frontare nonporous and stationary Further the water model hasa flat bottom with outlet ports instead of the long taperingmolten-steel pool These two major differences give rise todifferent flow phenomena as this work will show

Three simulations are presented in this article to under-stand the differences between full-pool and symmetric half-pool domains and between a full-scale water model and realsteel-caster behavior The first simulation (case 1) is on afull-scale water model corresponding to a standard slabcaster A symmetry condition was imposed at the centerplane between narrow faces so only half of the physicalliquid pool (Figure 2(a)) is simulated (Figure 2(b)) The nexttwo cases simulate the full pool of a thin-slab caster watermodel (case 2-W) and its corresponding steel caster (case2-S) Detailed information on the computational model andthe three simulations are given in the next section This isfollowed by results from the three simulations Comparisonswith measurements are made where available includingvelocities along the jet and across the top surface the pro-file of the top-surface liquid level and a spectral analysisThe turbulent flow structures in the pool are presented alongwith the transient flow asymmetries The flow field is quanti-fied with the time-averaged and root-mean-square velocitiesalong the jets across the top surface and in the lower recir-culation zone This article also quantifies the differencesbetween the velocity field in the water model and the corres-ponding steel caster including the top-surface liquid profile

II MODEL DESCRIPTION

The unsteady three-dimensional NavierndashStokes equationsgoverning the flow field are given by

[1]

[2]

where

[3]

[4]

In the context of an LES these velocities represent the large-scale eddies with the influence of small-scale eddies beingrepresented by subgrid-scale (SGS) models In the previousequations vi is the three-dimensional time-dependent velocityvector representing the motion of large eddies and x y

vt 001(xyz)23 A vi

xj vi

xj

vi

xj vj

xi

veff v0 vt

Dvi

Dt

1r

p

xi

xjv eff avi

xj

vj

xib

vi

xi 0

Fig 1mdashSchematics of the process of continuous casting of steel

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and z are the size of each cell in the Cartesian computa-tional grid The eddy viscosity (vt) can be computed usingSGS models to represent the dissipative effect of the unre-solved small eddies It vanishes if the grid resolution iscapable of resolving the smallest eddies (Kolmogorov scale)Equation [4] describes the evolution of the traditional

Smagorinsky SGS model[33] which was used in the firstcomputation as a preliminary exercise In the other two com-putations the eddy viscosity was set to zero These no-SGSmodel simulations could be interpreted as coarse-grid directnumerical simulations as the effect of the SGS model wasconsidered to be small as discussed elsewhere[34]

(a) (b)

(c) (d)

Fig 2mdashSchematics of (a) the physical water model of case 1 and (b) its computational domain and the thin slab caster (c) case 2-W and (d) case 2-S

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A Boundary Conditions

1 InletThe liquid pool is fed by a bifurcated (case 1) or trifurcated

(case 2) nozzle which has an important influence on the flowpattern[21] Thus nozzle simulations were first conducted toacquire accurate inlet conditions Unsteady flow velocities leav-ing the nozzle ports were collected every 0001 seconds Thetime-dependent velocities were stored and then recycled peri-odically as the inlet conditions for the liquid-pool simulationsMore details of the inlet-nozzle simulations are given later

2 OutletOne of the important differences between water models

and steel casters is that water models have outlet ports ontheir plastic bottom or side walls while in steel castersthe liquid pool gradually tapers to a solid end In this workthe computational domain of the water model (case 1) is thesame as the physical model However the domain extentof the thin-slab caster (case 2) is limited by computationalresources For computational efficiency simulations of case 2only compute flow in the mold liquid pool from the topsurface to a depth of 12 m (water model) or 24 m (steelcaster) below the top surface This creates an artificial out-let plane where flow leaves the computational domain Incurrent simulations a constant-pressure boundary conditionwith zero gradient of other variables was used at this planewhere the flow becomes nearly uniform

3 Top-surface and symmetry-center planeThe symmetry plane condition was only imposed in the half-

pool simulation (case 1) A free-slip condition was imposedalong both the top surface and the center plane Specificallyalong these two boundaries the normal velocity was constrainedto zero and the normal gradients of pressure and the other twovelocity components were set to zero The predictions andmeasurements of this work (presented in section III-H) sug-gest that the top surface is relatively quiescent so no modelof free-surface deformation is necessary to accurately modelthe flow The effect of the symmetry plane condition is inves-tigated by comparing full-pool and half-pool simulations

4 Narrow-face and wide-face wallsWater models and steel casters are very different at the

narrow- and wide-face walls Water models have stationarystraight plastic side walls representing the solidification frontThus all three velocity components were set to zero at thewall boundary However flow in the steel caster was mod-

eled up to but not including the front of the downward-moving mushy zone[20] where solidification occurs to takeaway mass from the molten steel pool This required taper-ing the domain walls to include only the liquid pool To fur-ther account for the solidification and downward motion avelocity boundary condition given by Eq [5] was used

[5a]

[5b]

Derivation of Eq [5] can be found in the Appendix In bothsystems no wall functions were used to represent near-wallturbulence because of a relatively fine mesh near the wall

B Solution Procedure

The time-dependent three-dimensional NavierndashStokesequations are discretized using the HarlowndashWelch fractional-step procedure[35] Second-order central differencing is usedfor the convection terms and the CrankndashNicolson scheme[36]

is used for the diffusion terms The AdamsndashBashforthscheme[37] is used to discretize in time with second-orderaccuracy The pressure Poisson equation is solved usingeither a fast-Fourier-transform (FFT) solver[38] in case 1 witha structured grid or an algebraic multigrid (AMG) solver[39]

in case 2 with unstructured Cartesian grids

C Computational Details

The three computational domains are presented in Fig-ure 2 The geometry casting conditions material propertiesand computational parameters are given in Table I For com-putational efficiency the domain was divided into the nozzleand liquid-pool regions Flow in each region was computedseparately Nozzle simulations were first performed fromwhich transient velocities exiting the nozzle ports were col-lected and used as conditions at the pool inlet

1 Case 1Figure 2(a) shows the physical water model of a standard

slab caster (case 1) Water enters the liquid pool from thebifurcated nozzle ports at a downward angle of approximately25 deg Water exits the model from four outlet pipes nearthe bottom of the wide face as shown in Figure 2(a) Detailedgeometry and operation conditions are published elsewhere[128]

vz Vn sin u Vt cos u ars

rl sin2 u cos2 ub Vcasting

vx Vn cos u Vt sin u ars

rl 1b sin u cos u Vcasting

Table I Properties and Conditions of the Simulations

ParameterProperty Case 1 Case 2-W Case 2-S

Mold width (mm) 1830 984 984Mold thickness (mm) 238 132 132Water model length (mm) 2152 2600 mdashMold length (mm) mdash mdash 1200Domain width (mm) 238 984 984 (top) 93404 (domain bottom)Domain thickness (mm) 238 132 132 (top) 7948 (domain bottom)Domain length (mm) 2152 1200 2400Nozzle port height thickness (mm mm) 51 56 (Fig 2) 75 32 (inner bore) 75 32 (inner bore)Bottom nozzle port diameter (mm) mdash 32 32SEN submergence depth (mm) 150 127 127Casting speed (mms) 152 254 254Fluid kinematic viscosity (m2s) 10 106 10 106 798 107

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This half-pool simulation adopted velocity profiles froma prior turbulent-pipe flow simulation as the inlet velocitiesfrom the small elliptical nozzle ports The pipe was 38 pctopen at its entry where a constant velocity was imposedTransient velocities were collected 75 pipe diameters awayfrom the pipe entry for 16 seconds and then rotated 25 degdownward to feed the pool simulation

Figure 2(b) shows the half-pool computational domain witha symmetry condition at the pool center The nozzle portwas modeled as an opening on the symmetry plane The out-let ports which are far from our flow region of interest wereapproximated as square openings with the same area as thephysical ports This computational domain was discretizedwith a structured Cartesian grid consisting of 16 million cellsThe time step of the simulation was 00008 seconds

2 Case 2Domains of a thin-slab steel caster (case 2-S) and its water

model (case 2-W) are presented in Figures 2(d) and (c) respec-tively Both systems used the same 11-m-long inlet nozzlestarting from the tundish bottom and fed through the annulusformed by a 644 pct open stopper rod down a 70-mm-diameterround-bore upper nozzle that tapered into a thin trifurcatedoutlet region A prior nozzle simulation with realistic geome-try was performed with a 06-million-cell mesh using anunstructured Cartesian cell code Time-dependent velocitiesleaving the trifurcated nozzle ports were stored every 0025seconds for 945 seconds (10-day computation on a Pentium IV17 GHz CPU) to be used as inflow to the liquid pool

Figure 2(d) shows that the domain of the steel caster hasa curved side boundary which represents the solidifying frontat the liquidus temperature The boundary shape was obtainedfrom the prediction of an in-house code CON1D[4041] whichis shown to agree with measurements (Figure 3) on a break-out shell[42] It should be noted that the symmetry-planeassumption was not needed in the two full-pool simulationsThere is no argon gas in any of the simulations in order tomatch the real caster where calcium treatment was used toavoid nozzle clogging

Unstructured Cartesian grids consisting of 07 and 13 mil-lion cells were employed for the water-model and the steelcaster simulations respectively The latter grid features cellscentered 025 mm from the wall in the upper mold includingthe impingement region This half-cell size gradually increases

to a maximum of 71 mm in the lower-gradient interior of thedomain The time step of 0001 seconds was used in both simu-lations The simulation took 295 CPU seconds per time stepon a Pentium IV 17 GHz CPU for the 13-million-cell gridor 24 days for 70000 time steps (70 seconds of real time)with the AMG solver Compared to the AMG the FFT solveris about 4 times faster and takes about one-quarter of the mem-ory However it is not suitable for the complex geometries thatare necessary to represent the actual caster

III RESULTS

A The Standard Slab-Caster Water Model (Case 1)

Case 1 is a half-pool simulation to model a typical slabcaster with bifurcated nozzles Its transient inflow velocitiesobtained from the prior pipe simulation are averaged tempo-rally and shown in Figure 4(a) Higher velocities are revealed

Fig 3mdashPredicted steel shell thickness of case 2-S using CON1D[4041]

compared with measurements[42]

(a)

(b)

Fig 4mdashInlet velocities of the standard slab caster water model (a) time-averaged velocity vectors at the inlet port and (b) transverse (x) and down-ward (z) velocity components along nozzle port centerline

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in the lower portion This is consistent with previous meas-urements and predictions on similar nozzles[4344] The outwardand downward velocity components along the port centerlineare depicted in Figure 4(b) with maximum values at approxi-mately one-third of the distance from the port bottom

A typical instantaneous velocity field in the liquid pool isshown in Figure 5 Water emerges from the inlet port as ajet diffuses as it traverses across the liquid pool impinges onthe narrow face and splits into two recirculation zones con-sisting of complex structures The predicted flow patternmatches experimental observations[128] A closer view of theturbulent structures in the upper roll is presented in Figures 6(a)and (b) for two time instants The figures show that the upperroll consists of a relatively simple vortex at the first instantwhich evolves to a pattern involving more complex multiplevortices at the second instant The upper roll alternates irreg-ularly between the two extremes in the simulation It is alsofound that only close to the top surface is the velocity directionconsistently horizontal This is important in understanding theaccuracy of the indirect measurement of the flow velocity in

steel casters using electromagnetic sensors[24] which requiresa consistent flow direction passing the sensors

Figure 7 compares the results of the simulation with meas-urements The time-averaged speed ((vx

2 vy2)12) of the

fluid was measured using hot-wire anemometers in a previouswork[28] The measurement was made along four vertical linesin the center plane at specified distances from the SEN Thecomputation agrees reasonably well with the measurementsThe biggest discrepancy occurs along the line 460 mm fromthe SEN where the predicted maximum-speed location isapproximately 100 mm deeper than the measurement in set 1This might be due to uncertainties in the measurements Itshould be noted that significant differences exist between themeasured time averages taken at different times likely due toinsufficient time for calculating statistics More validations ofour simulations were made on a 04-scale water model andare published elsewhere[2124] They show good agreementbetween prediction and PIV measurements

B The Trifurcated Inlet Nozzle of Thin-Slab Caster (Case 2)

A realistic nozzle simulation was conducted to generateaccurate unsteady inlet velocities for the computations in the

Fig 5mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 1) obtained from simulation

(a)

(b)

Fig 6mdashPredicted chaotic flow patterns in the upper recirculation zone (case 1)(a) simple vortices and (b) complex multiple vortices

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thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

686mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

with reasonable accuracy[21] and at a reasonable computationalcost However they poorly estimate turbulent dynamics suchas quantified by root-mean-square (rms) values[21] A transientReynolds-averaged approach which is more computationallydemanding was used previously to study transient changes inflow pattern resulting from drastic changes in inlet conditions[23]

However this approach was unable to capture inherent flowtransients during steady casting in several test cases This maybe due to the large numerical dissipation associated with thesemodels or due to the assumption of isotropic turbulence

The present work employs the large-eddy simulation (LES)approach This approach was shown in recent studies toaccurately predict the dynamics of large-scale turbulencestructures during continuous casting of steel[212224ndash26] How-ever LES entails many challenges when applied to this com-plex flow problem including the prescription of the transientinlet velocities resolution of the complex domain geometrythe moving solidification front and computing long-termtransients Therefore the application of LES to these aspectsof continuous casting requires large amounts of computermemory and central processing unit (CPU) time As a resultthis model may be considered as a way to generate funda-mental understanding and benchmarks for simple modelsrather than as an engineering tool for parametric study

Due to the high operating temperature it is difficult andexpensive to conduct flow measurements in continuous steelcasters in order to validate model predictions[21] However

because of the nearly equal kinematic viscosities of moltensteel and water water models with transparent plastic wallscan be used to study single-phase flow in steel castingprocesses[2728529ndash32] where Froude similarity is usuallyemployed Thus measurements using dye injection hot-wireanemometry and particle-image velocimetry (PIV) to quan-tify the flow velocities in water models are used to validatethe predictions of the LES code in this work

Although very valuable for validation of computationalmodels the water model differs from the actual steel casterin several aspects important to the flow field First its side-walls which represent the moving solidifying shell frontare nonporous and stationary Further the water model hasa flat bottom with outlet ports instead of the long taperingmolten-steel pool These two major differences give rise todifferent flow phenomena as this work will show

Three simulations are presented in this article to under-stand the differences between full-pool and symmetric half-pool domains and between a full-scale water model and realsteel-caster behavior The first simulation (case 1) is on afull-scale water model corresponding to a standard slabcaster A symmetry condition was imposed at the centerplane between narrow faces so only half of the physicalliquid pool (Figure 2(a)) is simulated (Figure 2(b)) The nexttwo cases simulate the full pool of a thin-slab caster watermodel (case 2-W) and its corresponding steel caster (case2-S) Detailed information on the computational model andthe three simulations are given in the next section This isfollowed by results from the three simulations Comparisonswith measurements are made where available includingvelocities along the jet and across the top surface the pro-file of the top-surface liquid level and a spectral analysisThe turbulent flow structures in the pool are presented alongwith the transient flow asymmetries The flow field is quanti-fied with the time-averaged and root-mean-square velocitiesalong the jets across the top surface and in the lower recir-culation zone This article also quantifies the differencesbetween the velocity field in the water model and the corres-ponding steel caster including the top-surface liquid profile

II MODEL DESCRIPTION

The unsteady three-dimensional NavierndashStokes equationsgoverning the flow field are given by

[1]

[2]

where

[3]

[4]

In the context of an LES these velocities represent the large-scale eddies with the influence of small-scale eddies beingrepresented by subgrid-scale (SGS) models In the previousequations vi is the three-dimensional time-dependent velocityvector representing the motion of large eddies and x y

vt 001(xyz)23 A vi

xj vi

xj

vi

xj vj

xi

veff v0 vt

Dvi

Dt

1r

p

xi

xjv eff avi

xj

vj

xib

vi

xi 0

Fig 1mdashSchematics of the process of continuous casting of steel

09-03-364B-Eqxd 92604 840 AM Page 686

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash687

and z are the size of each cell in the Cartesian computa-tional grid The eddy viscosity (vt) can be computed usingSGS models to represent the dissipative effect of the unre-solved small eddies It vanishes if the grid resolution iscapable of resolving the smallest eddies (Kolmogorov scale)Equation [4] describes the evolution of the traditional

Smagorinsky SGS model[33] which was used in the firstcomputation as a preliminary exercise In the other two com-putations the eddy viscosity was set to zero These no-SGSmodel simulations could be interpreted as coarse-grid directnumerical simulations as the effect of the SGS model wasconsidered to be small as discussed elsewhere[34]

(a) (b)

(c) (d)

Fig 2mdashSchematics of (a) the physical water model of case 1 and (b) its computational domain and the thin slab caster (c) case 2-W and (d) case 2-S

09-03-364B-Eqxd 92604 840 AM Page 687

688mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

A Boundary Conditions

1 InletThe liquid pool is fed by a bifurcated (case 1) or trifurcated

(case 2) nozzle which has an important influence on the flowpattern[21] Thus nozzle simulations were first conducted toacquire accurate inlet conditions Unsteady flow velocities leav-ing the nozzle ports were collected every 0001 seconds Thetime-dependent velocities were stored and then recycled peri-odically as the inlet conditions for the liquid-pool simulationsMore details of the inlet-nozzle simulations are given later

2 OutletOne of the important differences between water models

and steel casters is that water models have outlet ports ontheir plastic bottom or side walls while in steel castersthe liquid pool gradually tapers to a solid end In this workthe computational domain of the water model (case 1) is thesame as the physical model However the domain extentof the thin-slab caster (case 2) is limited by computationalresources For computational efficiency simulations of case 2only compute flow in the mold liquid pool from the topsurface to a depth of 12 m (water model) or 24 m (steelcaster) below the top surface This creates an artificial out-let plane where flow leaves the computational domain Incurrent simulations a constant-pressure boundary conditionwith zero gradient of other variables was used at this planewhere the flow becomes nearly uniform

3 Top-surface and symmetry-center planeThe symmetry plane condition was only imposed in the half-

pool simulation (case 1) A free-slip condition was imposedalong both the top surface and the center plane Specificallyalong these two boundaries the normal velocity was constrainedto zero and the normal gradients of pressure and the other twovelocity components were set to zero The predictions andmeasurements of this work (presented in section III-H) sug-gest that the top surface is relatively quiescent so no modelof free-surface deformation is necessary to accurately modelthe flow The effect of the symmetry plane condition is inves-tigated by comparing full-pool and half-pool simulations

4 Narrow-face and wide-face wallsWater models and steel casters are very different at the

narrow- and wide-face walls Water models have stationarystraight plastic side walls representing the solidification frontThus all three velocity components were set to zero at thewall boundary However flow in the steel caster was mod-

eled up to but not including the front of the downward-moving mushy zone[20] where solidification occurs to takeaway mass from the molten steel pool This required taper-ing the domain walls to include only the liquid pool To fur-ther account for the solidification and downward motion avelocity boundary condition given by Eq [5] was used

[5a]

[5b]

Derivation of Eq [5] can be found in the Appendix In bothsystems no wall functions were used to represent near-wallturbulence because of a relatively fine mesh near the wall

B Solution Procedure

The time-dependent three-dimensional NavierndashStokesequations are discretized using the HarlowndashWelch fractional-step procedure[35] Second-order central differencing is usedfor the convection terms and the CrankndashNicolson scheme[36]

is used for the diffusion terms The AdamsndashBashforthscheme[37] is used to discretize in time with second-orderaccuracy The pressure Poisson equation is solved usingeither a fast-Fourier-transform (FFT) solver[38] in case 1 witha structured grid or an algebraic multigrid (AMG) solver[39]

in case 2 with unstructured Cartesian grids

C Computational Details

The three computational domains are presented in Fig-ure 2 The geometry casting conditions material propertiesand computational parameters are given in Table I For com-putational efficiency the domain was divided into the nozzleand liquid-pool regions Flow in each region was computedseparately Nozzle simulations were first performed fromwhich transient velocities exiting the nozzle ports were col-lected and used as conditions at the pool inlet

1 Case 1Figure 2(a) shows the physical water model of a standard

slab caster (case 1) Water enters the liquid pool from thebifurcated nozzle ports at a downward angle of approximately25 deg Water exits the model from four outlet pipes nearthe bottom of the wide face as shown in Figure 2(a) Detailedgeometry and operation conditions are published elsewhere[128]

vz Vn sin u Vt cos u ars

rl sin2 u cos2 ub Vcasting

vx Vn cos u Vt sin u ars

rl 1b sin u cos u Vcasting

Table I Properties and Conditions of the Simulations

ParameterProperty Case 1 Case 2-W Case 2-S

Mold width (mm) 1830 984 984Mold thickness (mm) 238 132 132Water model length (mm) 2152 2600 mdashMold length (mm) mdash mdash 1200Domain width (mm) 238 984 984 (top) 93404 (domain bottom)Domain thickness (mm) 238 132 132 (top) 7948 (domain bottom)Domain length (mm) 2152 1200 2400Nozzle port height thickness (mm mm) 51 56 (Fig 2) 75 32 (inner bore) 75 32 (inner bore)Bottom nozzle port diameter (mm) mdash 32 32SEN submergence depth (mm) 150 127 127Casting speed (mms) 152 254 254Fluid kinematic viscosity (m2s) 10 106 10 106 798 107

09-03-364B-Eqxd 92604 840 AM Page 688

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash689

This half-pool simulation adopted velocity profiles froma prior turbulent-pipe flow simulation as the inlet velocitiesfrom the small elliptical nozzle ports The pipe was 38 pctopen at its entry where a constant velocity was imposedTransient velocities were collected 75 pipe diameters awayfrom the pipe entry for 16 seconds and then rotated 25 degdownward to feed the pool simulation

Figure 2(b) shows the half-pool computational domain witha symmetry condition at the pool center The nozzle portwas modeled as an opening on the symmetry plane The out-let ports which are far from our flow region of interest wereapproximated as square openings with the same area as thephysical ports This computational domain was discretizedwith a structured Cartesian grid consisting of 16 million cellsThe time step of the simulation was 00008 seconds

2 Case 2Domains of a thin-slab steel caster (case 2-S) and its water

model (case 2-W) are presented in Figures 2(d) and (c) respec-tively Both systems used the same 11-m-long inlet nozzlestarting from the tundish bottom and fed through the annulusformed by a 644 pct open stopper rod down a 70-mm-diameterround-bore upper nozzle that tapered into a thin trifurcatedoutlet region A prior nozzle simulation with realistic geome-try was performed with a 06-million-cell mesh using anunstructured Cartesian cell code Time-dependent velocitiesleaving the trifurcated nozzle ports were stored every 0025seconds for 945 seconds (10-day computation on a Pentium IV17 GHz CPU) to be used as inflow to the liquid pool

Figure 2(d) shows that the domain of the steel caster hasa curved side boundary which represents the solidifying frontat the liquidus temperature The boundary shape was obtainedfrom the prediction of an in-house code CON1D[4041] whichis shown to agree with measurements (Figure 3) on a break-out shell[42] It should be noted that the symmetry-planeassumption was not needed in the two full-pool simulationsThere is no argon gas in any of the simulations in order tomatch the real caster where calcium treatment was used toavoid nozzle clogging

Unstructured Cartesian grids consisting of 07 and 13 mil-lion cells were employed for the water-model and the steelcaster simulations respectively The latter grid features cellscentered 025 mm from the wall in the upper mold includingthe impingement region This half-cell size gradually increases

to a maximum of 71 mm in the lower-gradient interior of thedomain The time step of 0001 seconds was used in both simu-lations The simulation took 295 CPU seconds per time stepon a Pentium IV 17 GHz CPU for the 13-million-cell gridor 24 days for 70000 time steps (70 seconds of real time)with the AMG solver Compared to the AMG the FFT solveris about 4 times faster and takes about one-quarter of the mem-ory However it is not suitable for the complex geometries thatare necessary to represent the actual caster

III RESULTS

A The Standard Slab-Caster Water Model (Case 1)

Case 1 is a half-pool simulation to model a typical slabcaster with bifurcated nozzles Its transient inflow velocitiesobtained from the prior pipe simulation are averaged tempo-rally and shown in Figure 4(a) Higher velocities are revealed

Fig 3mdashPredicted steel shell thickness of case 2-S using CON1D[4041]

compared with measurements[42]

(a)

(b)

Fig 4mdashInlet velocities of the standard slab caster water model (a) time-averaged velocity vectors at the inlet port and (b) transverse (x) and down-ward (z) velocity components along nozzle port centerline

09-03-364B-Eqxd 92604 840 AM Page 689

690mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

in the lower portion This is consistent with previous meas-urements and predictions on similar nozzles[4344] The outwardand downward velocity components along the port centerlineare depicted in Figure 4(b) with maximum values at approxi-mately one-third of the distance from the port bottom

A typical instantaneous velocity field in the liquid pool isshown in Figure 5 Water emerges from the inlet port as ajet diffuses as it traverses across the liquid pool impinges onthe narrow face and splits into two recirculation zones con-sisting of complex structures The predicted flow patternmatches experimental observations[128] A closer view of theturbulent structures in the upper roll is presented in Figures 6(a)and (b) for two time instants The figures show that the upperroll consists of a relatively simple vortex at the first instantwhich evolves to a pattern involving more complex multiplevortices at the second instant The upper roll alternates irreg-ularly between the two extremes in the simulation It is alsofound that only close to the top surface is the velocity directionconsistently horizontal This is important in understanding theaccuracy of the indirect measurement of the flow velocity in

steel casters using electromagnetic sensors[24] which requiresa consistent flow direction passing the sensors

Figure 7 compares the results of the simulation with meas-urements The time-averaged speed ((vx

2 vy2)12) of the

fluid was measured using hot-wire anemometers in a previouswork[28] The measurement was made along four vertical linesin the center plane at specified distances from the SEN Thecomputation agrees reasonably well with the measurementsThe biggest discrepancy occurs along the line 460 mm fromthe SEN where the predicted maximum-speed location isapproximately 100 mm deeper than the measurement in set 1This might be due to uncertainties in the measurements Itshould be noted that significant differences exist between themeasured time averages taken at different times likely due toinsufficient time for calculating statistics More validations ofour simulations were made on a 04-scale water model andare published elsewhere[2124] They show good agreementbetween prediction and PIV measurements

B The Trifurcated Inlet Nozzle of Thin-Slab Caster (Case 2)

A realistic nozzle simulation was conducted to generateaccurate unsteady inlet velocities for the computations in the

Fig 5mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 1) obtained from simulation

(a)

(b)

Fig 6mdashPredicted chaotic flow patterns in the upper recirculation zone (case 1)(a) simple vortices and (b) complex multiple vortices

09-03-364B-Eqxd 92604 840 AM Page 690

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash691

thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash687

and z are the size of each cell in the Cartesian computa-tional grid The eddy viscosity (vt) can be computed usingSGS models to represent the dissipative effect of the unre-solved small eddies It vanishes if the grid resolution iscapable of resolving the smallest eddies (Kolmogorov scale)Equation [4] describes the evolution of the traditional

Smagorinsky SGS model[33] which was used in the firstcomputation as a preliminary exercise In the other two com-putations the eddy viscosity was set to zero These no-SGSmodel simulations could be interpreted as coarse-grid directnumerical simulations as the effect of the SGS model wasconsidered to be small as discussed elsewhere[34]

(a) (b)

(c) (d)

Fig 2mdashSchematics of (a) the physical water model of case 1 and (b) its computational domain and the thin slab caster (c) case 2-W and (d) case 2-S

09-03-364B-Eqxd 92604 840 AM Page 687

688mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

A Boundary Conditions

1 InletThe liquid pool is fed by a bifurcated (case 1) or trifurcated

(case 2) nozzle which has an important influence on the flowpattern[21] Thus nozzle simulations were first conducted toacquire accurate inlet conditions Unsteady flow velocities leav-ing the nozzle ports were collected every 0001 seconds Thetime-dependent velocities were stored and then recycled peri-odically as the inlet conditions for the liquid-pool simulationsMore details of the inlet-nozzle simulations are given later

2 OutletOne of the important differences between water models

and steel casters is that water models have outlet ports ontheir plastic bottom or side walls while in steel castersthe liquid pool gradually tapers to a solid end In this workthe computational domain of the water model (case 1) is thesame as the physical model However the domain extentof the thin-slab caster (case 2) is limited by computationalresources For computational efficiency simulations of case 2only compute flow in the mold liquid pool from the topsurface to a depth of 12 m (water model) or 24 m (steelcaster) below the top surface This creates an artificial out-let plane where flow leaves the computational domain Incurrent simulations a constant-pressure boundary conditionwith zero gradient of other variables was used at this planewhere the flow becomes nearly uniform

3 Top-surface and symmetry-center planeThe symmetry plane condition was only imposed in the half-

pool simulation (case 1) A free-slip condition was imposedalong both the top surface and the center plane Specificallyalong these two boundaries the normal velocity was constrainedto zero and the normal gradients of pressure and the other twovelocity components were set to zero The predictions andmeasurements of this work (presented in section III-H) sug-gest that the top surface is relatively quiescent so no modelof free-surface deformation is necessary to accurately modelthe flow The effect of the symmetry plane condition is inves-tigated by comparing full-pool and half-pool simulations

4 Narrow-face and wide-face wallsWater models and steel casters are very different at the

narrow- and wide-face walls Water models have stationarystraight plastic side walls representing the solidification frontThus all three velocity components were set to zero at thewall boundary However flow in the steel caster was mod-

eled up to but not including the front of the downward-moving mushy zone[20] where solidification occurs to takeaway mass from the molten steel pool This required taper-ing the domain walls to include only the liquid pool To fur-ther account for the solidification and downward motion avelocity boundary condition given by Eq [5] was used

[5a]

[5b]

Derivation of Eq [5] can be found in the Appendix In bothsystems no wall functions were used to represent near-wallturbulence because of a relatively fine mesh near the wall

B Solution Procedure

The time-dependent three-dimensional NavierndashStokesequations are discretized using the HarlowndashWelch fractional-step procedure[35] Second-order central differencing is usedfor the convection terms and the CrankndashNicolson scheme[36]

is used for the diffusion terms The AdamsndashBashforthscheme[37] is used to discretize in time with second-orderaccuracy The pressure Poisson equation is solved usingeither a fast-Fourier-transform (FFT) solver[38] in case 1 witha structured grid or an algebraic multigrid (AMG) solver[39]

in case 2 with unstructured Cartesian grids

C Computational Details

The three computational domains are presented in Fig-ure 2 The geometry casting conditions material propertiesand computational parameters are given in Table I For com-putational efficiency the domain was divided into the nozzleand liquid-pool regions Flow in each region was computedseparately Nozzle simulations were first performed fromwhich transient velocities exiting the nozzle ports were col-lected and used as conditions at the pool inlet

1 Case 1Figure 2(a) shows the physical water model of a standard

slab caster (case 1) Water enters the liquid pool from thebifurcated nozzle ports at a downward angle of approximately25 deg Water exits the model from four outlet pipes nearthe bottom of the wide face as shown in Figure 2(a) Detailedgeometry and operation conditions are published elsewhere[128]

vz Vn sin u Vt cos u ars

rl sin2 u cos2 ub Vcasting

vx Vn cos u Vt sin u ars

rl 1b sin u cos u Vcasting

Table I Properties and Conditions of the Simulations

ParameterProperty Case 1 Case 2-W Case 2-S

Mold width (mm) 1830 984 984Mold thickness (mm) 238 132 132Water model length (mm) 2152 2600 mdashMold length (mm) mdash mdash 1200Domain width (mm) 238 984 984 (top) 93404 (domain bottom)Domain thickness (mm) 238 132 132 (top) 7948 (domain bottom)Domain length (mm) 2152 1200 2400Nozzle port height thickness (mm mm) 51 56 (Fig 2) 75 32 (inner bore) 75 32 (inner bore)Bottom nozzle port diameter (mm) mdash 32 32SEN submergence depth (mm) 150 127 127Casting speed (mms) 152 254 254Fluid kinematic viscosity (m2s) 10 106 10 106 798 107

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash689

This half-pool simulation adopted velocity profiles froma prior turbulent-pipe flow simulation as the inlet velocitiesfrom the small elliptical nozzle ports The pipe was 38 pctopen at its entry where a constant velocity was imposedTransient velocities were collected 75 pipe diameters awayfrom the pipe entry for 16 seconds and then rotated 25 degdownward to feed the pool simulation

Figure 2(b) shows the half-pool computational domain witha symmetry condition at the pool center The nozzle portwas modeled as an opening on the symmetry plane The out-let ports which are far from our flow region of interest wereapproximated as square openings with the same area as thephysical ports This computational domain was discretizedwith a structured Cartesian grid consisting of 16 million cellsThe time step of the simulation was 00008 seconds

2 Case 2Domains of a thin-slab steel caster (case 2-S) and its water

model (case 2-W) are presented in Figures 2(d) and (c) respec-tively Both systems used the same 11-m-long inlet nozzlestarting from the tundish bottom and fed through the annulusformed by a 644 pct open stopper rod down a 70-mm-diameterround-bore upper nozzle that tapered into a thin trifurcatedoutlet region A prior nozzle simulation with realistic geome-try was performed with a 06-million-cell mesh using anunstructured Cartesian cell code Time-dependent velocitiesleaving the trifurcated nozzle ports were stored every 0025seconds for 945 seconds (10-day computation on a Pentium IV17 GHz CPU) to be used as inflow to the liquid pool

Figure 2(d) shows that the domain of the steel caster hasa curved side boundary which represents the solidifying frontat the liquidus temperature The boundary shape was obtainedfrom the prediction of an in-house code CON1D[4041] whichis shown to agree with measurements (Figure 3) on a break-out shell[42] It should be noted that the symmetry-planeassumption was not needed in the two full-pool simulationsThere is no argon gas in any of the simulations in order tomatch the real caster where calcium treatment was used toavoid nozzle clogging

Unstructured Cartesian grids consisting of 07 and 13 mil-lion cells were employed for the water-model and the steelcaster simulations respectively The latter grid features cellscentered 025 mm from the wall in the upper mold includingthe impingement region This half-cell size gradually increases

to a maximum of 71 mm in the lower-gradient interior of thedomain The time step of 0001 seconds was used in both simu-lations The simulation took 295 CPU seconds per time stepon a Pentium IV 17 GHz CPU for the 13-million-cell gridor 24 days for 70000 time steps (70 seconds of real time)with the AMG solver Compared to the AMG the FFT solveris about 4 times faster and takes about one-quarter of the mem-ory However it is not suitable for the complex geometries thatare necessary to represent the actual caster

III RESULTS

A The Standard Slab-Caster Water Model (Case 1)

Case 1 is a half-pool simulation to model a typical slabcaster with bifurcated nozzles Its transient inflow velocitiesobtained from the prior pipe simulation are averaged tempo-rally and shown in Figure 4(a) Higher velocities are revealed

Fig 3mdashPredicted steel shell thickness of case 2-S using CON1D[4041]

compared with measurements[42]

(a)

(b)

Fig 4mdashInlet velocities of the standard slab caster water model (a) time-averaged velocity vectors at the inlet port and (b) transverse (x) and down-ward (z) velocity components along nozzle port centerline

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690mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

in the lower portion This is consistent with previous meas-urements and predictions on similar nozzles[4344] The outwardand downward velocity components along the port centerlineare depicted in Figure 4(b) with maximum values at approxi-mately one-third of the distance from the port bottom

A typical instantaneous velocity field in the liquid pool isshown in Figure 5 Water emerges from the inlet port as ajet diffuses as it traverses across the liquid pool impinges onthe narrow face and splits into two recirculation zones con-sisting of complex structures The predicted flow patternmatches experimental observations[128] A closer view of theturbulent structures in the upper roll is presented in Figures 6(a)and (b) for two time instants The figures show that the upperroll consists of a relatively simple vortex at the first instantwhich evolves to a pattern involving more complex multiplevortices at the second instant The upper roll alternates irreg-ularly between the two extremes in the simulation It is alsofound that only close to the top surface is the velocity directionconsistently horizontal This is important in understanding theaccuracy of the indirect measurement of the flow velocity in

steel casters using electromagnetic sensors[24] which requiresa consistent flow direction passing the sensors

Figure 7 compares the results of the simulation with meas-urements The time-averaged speed ((vx

2 vy2)12) of the

fluid was measured using hot-wire anemometers in a previouswork[28] The measurement was made along four vertical linesin the center plane at specified distances from the SEN Thecomputation agrees reasonably well with the measurementsThe biggest discrepancy occurs along the line 460 mm fromthe SEN where the predicted maximum-speed location isapproximately 100 mm deeper than the measurement in set 1This might be due to uncertainties in the measurements Itshould be noted that significant differences exist between themeasured time averages taken at different times likely due toinsufficient time for calculating statistics More validations ofour simulations were made on a 04-scale water model andare published elsewhere[2124] They show good agreementbetween prediction and PIV measurements

B The Trifurcated Inlet Nozzle of Thin-Slab Caster (Case 2)

A realistic nozzle simulation was conducted to generateaccurate unsteady inlet velocities for the computations in the

Fig 5mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 1) obtained from simulation

(a)

(b)

Fig 6mdashPredicted chaotic flow patterns in the upper recirculation zone (case 1)(a) simple vortices and (b) complex multiple vortices

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash691

thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

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700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

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688mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

A Boundary Conditions

1 InletThe liquid pool is fed by a bifurcated (case 1) or trifurcated

(case 2) nozzle which has an important influence on the flowpattern[21] Thus nozzle simulations were first conducted toacquire accurate inlet conditions Unsteady flow velocities leav-ing the nozzle ports were collected every 0001 seconds Thetime-dependent velocities were stored and then recycled peri-odically as the inlet conditions for the liquid-pool simulationsMore details of the inlet-nozzle simulations are given later

2 OutletOne of the important differences between water models

and steel casters is that water models have outlet ports ontheir plastic bottom or side walls while in steel castersthe liquid pool gradually tapers to a solid end In this workthe computational domain of the water model (case 1) is thesame as the physical model However the domain extentof the thin-slab caster (case 2) is limited by computationalresources For computational efficiency simulations of case 2only compute flow in the mold liquid pool from the topsurface to a depth of 12 m (water model) or 24 m (steelcaster) below the top surface This creates an artificial out-let plane where flow leaves the computational domain Incurrent simulations a constant-pressure boundary conditionwith zero gradient of other variables was used at this planewhere the flow becomes nearly uniform

3 Top-surface and symmetry-center planeThe symmetry plane condition was only imposed in the half-

pool simulation (case 1) A free-slip condition was imposedalong both the top surface and the center plane Specificallyalong these two boundaries the normal velocity was constrainedto zero and the normal gradients of pressure and the other twovelocity components were set to zero The predictions andmeasurements of this work (presented in section III-H) sug-gest that the top surface is relatively quiescent so no modelof free-surface deformation is necessary to accurately modelthe flow The effect of the symmetry plane condition is inves-tigated by comparing full-pool and half-pool simulations

4 Narrow-face and wide-face wallsWater models and steel casters are very different at the

narrow- and wide-face walls Water models have stationarystraight plastic side walls representing the solidification frontThus all three velocity components were set to zero at thewall boundary However flow in the steel caster was mod-

eled up to but not including the front of the downward-moving mushy zone[20] where solidification occurs to takeaway mass from the molten steel pool This required taper-ing the domain walls to include only the liquid pool To fur-ther account for the solidification and downward motion avelocity boundary condition given by Eq [5] was used

[5a]

[5b]

Derivation of Eq [5] can be found in the Appendix In bothsystems no wall functions were used to represent near-wallturbulence because of a relatively fine mesh near the wall

B Solution Procedure

The time-dependent three-dimensional NavierndashStokesequations are discretized using the HarlowndashWelch fractional-step procedure[35] Second-order central differencing is usedfor the convection terms and the CrankndashNicolson scheme[36]

is used for the diffusion terms The AdamsndashBashforthscheme[37] is used to discretize in time with second-orderaccuracy The pressure Poisson equation is solved usingeither a fast-Fourier-transform (FFT) solver[38] in case 1 witha structured grid or an algebraic multigrid (AMG) solver[39]

in case 2 with unstructured Cartesian grids

C Computational Details

The three computational domains are presented in Fig-ure 2 The geometry casting conditions material propertiesand computational parameters are given in Table I For com-putational efficiency the domain was divided into the nozzleand liquid-pool regions Flow in each region was computedseparately Nozzle simulations were first performed fromwhich transient velocities exiting the nozzle ports were col-lected and used as conditions at the pool inlet

1 Case 1Figure 2(a) shows the physical water model of a standard

slab caster (case 1) Water enters the liquid pool from thebifurcated nozzle ports at a downward angle of approximately25 deg Water exits the model from four outlet pipes nearthe bottom of the wide face as shown in Figure 2(a) Detailedgeometry and operation conditions are published elsewhere[128]

vz Vn sin u Vt cos u ars

rl sin2 u cos2 ub Vcasting

vx Vn cos u Vt sin u ars

rl 1b sin u cos u Vcasting

Table I Properties and Conditions of the Simulations

ParameterProperty Case 1 Case 2-W Case 2-S

Mold width (mm) 1830 984 984Mold thickness (mm) 238 132 132Water model length (mm) 2152 2600 mdashMold length (mm) mdash mdash 1200Domain width (mm) 238 984 984 (top) 93404 (domain bottom)Domain thickness (mm) 238 132 132 (top) 7948 (domain bottom)Domain length (mm) 2152 1200 2400Nozzle port height thickness (mm mm) 51 56 (Fig 2) 75 32 (inner bore) 75 32 (inner bore)Bottom nozzle port diameter (mm) mdash 32 32SEN submergence depth (mm) 150 127 127Casting speed (mms) 152 254 254Fluid kinematic viscosity (m2s) 10 106 10 106 798 107

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash689

This half-pool simulation adopted velocity profiles froma prior turbulent-pipe flow simulation as the inlet velocitiesfrom the small elliptical nozzle ports The pipe was 38 pctopen at its entry where a constant velocity was imposedTransient velocities were collected 75 pipe diameters awayfrom the pipe entry for 16 seconds and then rotated 25 degdownward to feed the pool simulation

Figure 2(b) shows the half-pool computational domain witha symmetry condition at the pool center The nozzle portwas modeled as an opening on the symmetry plane The out-let ports which are far from our flow region of interest wereapproximated as square openings with the same area as thephysical ports This computational domain was discretizedwith a structured Cartesian grid consisting of 16 million cellsThe time step of the simulation was 00008 seconds

2 Case 2Domains of a thin-slab steel caster (case 2-S) and its water

model (case 2-W) are presented in Figures 2(d) and (c) respec-tively Both systems used the same 11-m-long inlet nozzlestarting from the tundish bottom and fed through the annulusformed by a 644 pct open stopper rod down a 70-mm-diameterround-bore upper nozzle that tapered into a thin trifurcatedoutlet region A prior nozzle simulation with realistic geome-try was performed with a 06-million-cell mesh using anunstructured Cartesian cell code Time-dependent velocitiesleaving the trifurcated nozzle ports were stored every 0025seconds for 945 seconds (10-day computation on a Pentium IV17 GHz CPU) to be used as inflow to the liquid pool

Figure 2(d) shows that the domain of the steel caster hasa curved side boundary which represents the solidifying frontat the liquidus temperature The boundary shape was obtainedfrom the prediction of an in-house code CON1D[4041] whichis shown to agree with measurements (Figure 3) on a break-out shell[42] It should be noted that the symmetry-planeassumption was not needed in the two full-pool simulationsThere is no argon gas in any of the simulations in order tomatch the real caster where calcium treatment was used toavoid nozzle clogging

Unstructured Cartesian grids consisting of 07 and 13 mil-lion cells were employed for the water-model and the steelcaster simulations respectively The latter grid features cellscentered 025 mm from the wall in the upper mold includingthe impingement region This half-cell size gradually increases

to a maximum of 71 mm in the lower-gradient interior of thedomain The time step of 0001 seconds was used in both simu-lations The simulation took 295 CPU seconds per time stepon a Pentium IV 17 GHz CPU for the 13-million-cell gridor 24 days for 70000 time steps (70 seconds of real time)with the AMG solver Compared to the AMG the FFT solveris about 4 times faster and takes about one-quarter of the mem-ory However it is not suitable for the complex geometries thatare necessary to represent the actual caster

III RESULTS

A The Standard Slab-Caster Water Model (Case 1)

Case 1 is a half-pool simulation to model a typical slabcaster with bifurcated nozzles Its transient inflow velocitiesobtained from the prior pipe simulation are averaged tempo-rally and shown in Figure 4(a) Higher velocities are revealed

Fig 3mdashPredicted steel shell thickness of case 2-S using CON1D[4041]

compared with measurements[42]

(a)

(b)

Fig 4mdashInlet velocities of the standard slab caster water model (a) time-averaged velocity vectors at the inlet port and (b) transverse (x) and down-ward (z) velocity components along nozzle port centerline

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690mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

in the lower portion This is consistent with previous meas-urements and predictions on similar nozzles[4344] The outwardand downward velocity components along the port centerlineare depicted in Figure 4(b) with maximum values at approxi-mately one-third of the distance from the port bottom

A typical instantaneous velocity field in the liquid pool isshown in Figure 5 Water emerges from the inlet port as ajet diffuses as it traverses across the liquid pool impinges onthe narrow face and splits into two recirculation zones con-sisting of complex structures The predicted flow patternmatches experimental observations[128] A closer view of theturbulent structures in the upper roll is presented in Figures 6(a)and (b) for two time instants The figures show that the upperroll consists of a relatively simple vortex at the first instantwhich evolves to a pattern involving more complex multiplevortices at the second instant The upper roll alternates irreg-ularly between the two extremes in the simulation It is alsofound that only close to the top surface is the velocity directionconsistently horizontal This is important in understanding theaccuracy of the indirect measurement of the flow velocity in

steel casters using electromagnetic sensors[24] which requiresa consistent flow direction passing the sensors

Figure 7 compares the results of the simulation with meas-urements The time-averaged speed ((vx

2 vy2)12) of the

fluid was measured using hot-wire anemometers in a previouswork[28] The measurement was made along four vertical linesin the center plane at specified distances from the SEN Thecomputation agrees reasonably well with the measurementsThe biggest discrepancy occurs along the line 460 mm fromthe SEN where the predicted maximum-speed location isapproximately 100 mm deeper than the measurement in set 1This might be due to uncertainties in the measurements Itshould be noted that significant differences exist between themeasured time averages taken at different times likely due toinsufficient time for calculating statistics More validations ofour simulations were made on a 04-scale water model andare published elsewhere[2124] They show good agreementbetween prediction and PIV measurements

B The Trifurcated Inlet Nozzle of Thin-Slab Caster (Case 2)

A realistic nozzle simulation was conducted to generateaccurate unsteady inlet velocities for the computations in the

Fig 5mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 1) obtained from simulation

(a)

(b)

Fig 6mdashPredicted chaotic flow patterns in the upper recirculation zone (case 1)(a) simple vortices and (b) complex multiple vortices

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash691

thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

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700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash689

This half-pool simulation adopted velocity profiles froma prior turbulent-pipe flow simulation as the inlet velocitiesfrom the small elliptical nozzle ports The pipe was 38 pctopen at its entry where a constant velocity was imposedTransient velocities were collected 75 pipe diameters awayfrom the pipe entry for 16 seconds and then rotated 25 degdownward to feed the pool simulation

Figure 2(b) shows the half-pool computational domain witha symmetry condition at the pool center The nozzle portwas modeled as an opening on the symmetry plane The out-let ports which are far from our flow region of interest wereapproximated as square openings with the same area as thephysical ports This computational domain was discretizedwith a structured Cartesian grid consisting of 16 million cellsThe time step of the simulation was 00008 seconds

2 Case 2Domains of a thin-slab steel caster (case 2-S) and its water

model (case 2-W) are presented in Figures 2(d) and (c) respec-tively Both systems used the same 11-m-long inlet nozzlestarting from the tundish bottom and fed through the annulusformed by a 644 pct open stopper rod down a 70-mm-diameterround-bore upper nozzle that tapered into a thin trifurcatedoutlet region A prior nozzle simulation with realistic geome-try was performed with a 06-million-cell mesh using anunstructured Cartesian cell code Time-dependent velocitiesleaving the trifurcated nozzle ports were stored every 0025seconds for 945 seconds (10-day computation on a Pentium IV17 GHz CPU) to be used as inflow to the liquid pool

Figure 2(d) shows that the domain of the steel caster hasa curved side boundary which represents the solidifying frontat the liquidus temperature The boundary shape was obtainedfrom the prediction of an in-house code CON1D[4041] whichis shown to agree with measurements (Figure 3) on a break-out shell[42] It should be noted that the symmetry-planeassumption was not needed in the two full-pool simulationsThere is no argon gas in any of the simulations in order tomatch the real caster where calcium treatment was used toavoid nozzle clogging

Unstructured Cartesian grids consisting of 07 and 13 mil-lion cells were employed for the water-model and the steelcaster simulations respectively The latter grid features cellscentered 025 mm from the wall in the upper mold includingthe impingement region This half-cell size gradually increases

to a maximum of 71 mm in the lower-gradient interior of thedomain The time step of 0001 seconds was used in both simu-lations The simulation took 295 CPU seconds per time stepon a Pentium IV 17 GHz CPU for the 13-million-cell gridor 24 days for 70000 time steps (70 seconds of real time)with the AMG solver Compared to the AMG the FFT solveris about 4 times faster and takes about one-quarter of the mem-ory However it is not suitable for the complex geometries thatare necessary to represent the actual caster

III RESULTS

A The Standard Slab-Caster Water Model (Case 1)

Case 1 is a half-pool simulation to model a typical slabcaster with bifurcated nozzles Its transient inflow velocitiesobtained from the prior pipe simulation are averaged tempo-rally and shown in Figure 4(a) Higher velocities are revealed

Fig 3mdashPredicted steel shell thickness of case 2-S using CON1D[4041]

compared with measurements[42]

(a)

(b)

Fig 4mdashInlet velocities of the standard slab caster water model (a) time-averaged velocity vectors at the inlet port and (b) transverse (x) and down-ward (z) velocity components along nozzle port centerline

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690mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

in the lower portion This is consistent with previous meas-urements and predictions on similar nozzles[4344] The outwardand downward velocity components along the port centerlineare depicted in Figure 4(b) with maximum values at approxi-mately one-third of the distance from the port bottom

A typical instantaneous velocity field in the liquid pool isshown in Figure 5 Water emerges from the inlet port as ajet diffuses as it traverses across the liquid pool impinges onthe narrow face and splits into two recirculation zones con-sisting of complex structures The predicted flow patternmatches experimental observations[128] A closer view of theturbulent structures in the upper roll is presented in Figures 6(a)and (b) for two time instants The figures show that the upperroll consists of a relatively simple vortex at the first instantwhich evolves to a pattern involving more complex multiplevortices at the second instant The upper roll alternates irreg-ularly between the two extremes in the simulation It is alsofound that only close to the top surface is the velocity directionconsistently horizontal This is important in understanding theaccuracy of the indirect measurement of the flow velocity in

steel casters using electromagnetic sensors[24] which requiresa consistent flow direction passing the sensors

Figure 7 compares the results of the simulation with meas-urements The time-averaged speed ((vx

2 vy2)12) of the

fluid was measured using hot-wire anemometers in a previouswork[28] The measurement was made along four vertical linesin the center plane at specified distances from the SEN Thecomputation agrees reasonably well with the measurementsThe biggest discrepancy occurs along the line 460 mm fromthe SEN where the predicted maximum-speed location isapproximately 100 mm deeper than the measurement in set 1This might be due to uncertainties in the measurements Itshould be noted that significant differences exist between themeasured time averages taken at different times likely due toinsufficient time for calculating statistics More validations ofour simulations were made on a 04-scale water model andare published elsewhere[2124] They show good agreementbetween prediction and PIV measurements

B The Trifurcated Inlet Nozzle of Thin-Slab Caster (Case 2)

A realistic nozzle simulation was conducted to generateaccurate unsteady inlet velocities for the computations in the

Fig 5mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 1) obtained from simulation

(a)

(b)

Fig 6mdashPredicted chaotic flow patterns in the upper recirculation zone (case 1)(a) simple vortices and (b) complex multiple vortices

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash691

thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

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Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

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700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

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690mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

in the lower portion This is consistent with previous meas-urements and predictions on similar nozzles[4344] The outwardand downward velocity components along the port centerlineare depicted in Figure 4(b) with maximum values at approxi-mately one-third of the distance from the port bottom

A typical instantaneous velocity field in the liquid pool isshown in Figure 5 Water emerges from the inlet port as ajet diffuses as it traverses across the liquid pool impinges onthe narrow face and splits into two recirculation zones con-sisting of complex structures The predicted flow patternmatches experimental observations[128] A closer view of theturbulent structures in the upper roll is presented in Figures 6(a)and (b) for two time instants The figures show that the upperroll consists of a relatively simple vortex at the first instantwhich evolves to a pattern involving more complex multiplevortices at the second instant The upper roll alternates irreg-ularly between the two extremes in the simulation It is alsofound that only close to the top surface is the velocity directionconsistently horizontal This is important in understanding theaccuracy of the indirect measurement of the flow velocity in

steel casters using electromagnetic sensors[24] which requiresa consistent flow direction passing the sensors

Figure 7 compares the results of the simulation with meas-urements The time-averaged speed ((vx

2 vy2)12) of the

fluid was measured using hot-wire anemometers in a previouswork[28] The measurement was made along four vertical linesin the center plane at specified distances from the SEN Thecomputation agrees reasonably well with the measurementsThe biggest discrepancy occurs along the line 460 mm fromthe SEN where the predicted maximum-speed location isapproximately 100 mm deeper than the measurement in set 1This might be due to uncertainties in the measurements Itshould be noted that significant differences exist between themeasured time averages taken at different times likely due toinsufficient time for calculating statistics More validations ofour simulations were made on a 04-scale water model andare published elsewhere[2124] They show good agreementbetween prediction and PIV measurements

B The Trifurcated Inlet Nozzle of Thin-Slab Caster (Case 2)

A realistic nozzle simulation was conducted to generateaccurate unsteady inlet velocities for the computations in the

Fig 5mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 1) obtained from simulation

(a)

(b)

Fig 6mdashPredicted chaotic flow patterns in the upper recirculation zone (case 1)(a) simple vortices and (b) complex multiple vortices

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash691

thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

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700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash691

thin-slab caster The computed results are presented in Figures 8through 11 Two typical transient flow patterns are plotted inFigure 8 showing flow exiting the nozzle ports at differentinstants Supplementary to the vector length the darkness ofthe arrow also represents the velocity magnitude In Figure8(a) a symmetrical flow pattern is observed between the sideports while it is apparently asymmetrical in Figure 8(b)The downward angle of the two side jets varies in time from30 to 45 deg The two side jets switch between the twoextremes The jet angle is important because it greatly affectsthe transport of harmful inclusions carried by the jet entering

Fig 7mdashComparison of the prediction and measurement of the time-averaged speed (vx2 vz

2)12 along four vertical lines at different distances from SEN (case 1)

(a) (b)

Fig 8mdashTypical instantaneous velocity fields near nozzle ports at the center plane between wide faces obtained from a large eddy simulation of the noz-zle (cases 2-W and 2-S)

the liquid pool Jets at a deeper angle tend to transport moreinclusions into the lower roll encouraging the formation ofinternal defects such as slivers and blisters (discussed in moredetail in Part II of this article) The jet angle is also impor-tant as it influences the velocity and the profile of the top-surface liquid level Jets at smaller downward angles are likelyto increase the velocity and the liquid-level fluctuations alongthe top surface by carrying more fluid and momentum intothe upper roll This can cause quality problems as discussedin the previous sections Accurate prediction of this angle isessential for optimizing the nozzle design The center-jet

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692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

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696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

692mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

velocity is seen to fluctuate considerably but the flow pat-tern in the mold stays nearly the same

Figure 9 shows the fluctuation of the downward-velocitycomponent (vz) sampled at two points which are symmet-rically located on the side-port outlet plane with a distance40 mm below the upper edge Both signals show a meanvalue of 06 ms but with significant fluctuations Thehighest frequency of the signals is around 10 Hz The veloc-ity component is mostly positive indicating that the flow ismostly downward with occasional upward excursions Short-term velocity differences are observed between the two sidesHowever averaging over a short time is seen to result inapproximately the same flow field (Figures 10 and 11)

The computed transient velocity fields in the nozzle wereaveraged over 945 seconds (37800 time instants) and are

plotted in Figure 10 The velocities of the side jets are quanti-tatively shown in Figure 11 along the nozzle-port center lineThe mean of the side jets is very symmetrical even for ashort time (945 seconds) of average Most of the fluid existthe ports from the port center region (20 to 80 mm belowthe upper edge) with some small backflow near the upperand lower edges where the fluid re-enters the nozzle

C The Thin-Slab Water Model (Case 2-W)

The transient velocities obtained from the trifurcated-nozzlesimulation were then used as inflow into the thin-slab water-model computation Before showing the computational resultsFigure 12 first presents snapshots of the dye-injection experi-ment on the water model at four instants showing the evolutionof the transient flow in the liquid pool Figure 12(a) is at 05 sec-onds after the dye exits the nozzle ports showing instantaneousjet angles of 42 deg (left-hand side) and 35 deg (right-handside) The dye flows with the jet and impinges the narrowface 07 seconds later as shown in Figure 12(b) It then splits

Fig 9mdashTime variation of downward velocity (vz) at two symmetrical points on the side nozzle ports

Fig 10mdashTime-averaged velocity fields near nozzle ports at the center planebetween narrow faces obtained from the simulation (cases 2-W and 2-S)

Fig 11mdashTime-averaged velocities along the nozzle port centerline on bothsides

09-03-364B-Eqxd 92604 841 AM Page 692

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

09-03-364B-Eqxd 92604 841 AM Page 693

694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

09-03-364B-Eqxd 92604 841 AM Page 694

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 695

696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash693

into two parts with the flow to move into the lower and upperrecirculation zones as can be seen in the plots of Figures 12 (c)and (d) The shape of the jets and the lower and upper recir-culation zones can be reconstructed from this sequence of fourplots Vortex shedding of the center jet can also be observedalthough it is obscured by the external frame of the water model

Figures 13 and 14 show a typical instantaneous velocityfield and the mean field at the center plane The mean wasobtained over a time of 485 seconds For clarity of pre-sentation the time-averaged vector plots in this work onlyshow arrows at every second grid point in each directionThe classical double-roll flow pattern can be seen in bothplots The shape of the jets and the upper and lower recir-culation zones agrees with the dye-injection observation Incontrast to the smooth time-averaged plot the instantaneousvector plot shows local turbulent structures similar to what

was seen in case 1 The oscillation of the center jet observedin the dye injection is also seen in the simulation

Figure 15 compares the computed time-averaged speed(vx

2 vz2)12 with the estimated value from the dye injections

along the jet centerline The solid line denotes the predictedvalue of the speed averaged over 485 seconds The error barshows the upper and lower bounds of the transient speed duringthe 485-second simulation indicating a large fluctuation Thedark dots are the estimated transient flow speeds obtained bymeasuring the development of the dye front on the video imagesThe predicted values reasonably agree with the measurements

D Numerical-Model Validation (Case 2-W)

Sections A and C provide model validation with experi-mental measurements comparing the predicted and measured

Fig 12mdash(a) through (d) Dye injection experiment of case 2-W at four instants

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694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

09-03-364B-Eqxd 92604 841 AM Page 694

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 695

696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

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METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

694mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 13mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-W) obtained from simulation

Fig 14mdashTime-averaged velocity vector plot at the center plane betweenwide faces (case 2-W) obtained from simulation

Fig 15mdashComparison of time-averaged speed (vx2 vz

2)12 along side jetcenterline between computation and dye injection estimate (case 2-W)

Fig 16mdashComparison of computed fluid speeds (vx2 vz

2)12 along the ver-tical line in the center plane obtained from three different grid resolutions(case 2-W)

velocities along four vertical lines (Figure 7) and along thejet (Figure 15) Further quantitative experimental validationis provided in Sections F and H for the top-surface velocity

and shape respectively Computational models also requirenumerical validation to ensure that the effects of grid reso-lution time-step size turbulence model and discretizationerrors associated with the order of the numerical scheme aresmall Further related issues are the inlet conditions and sym-metry assumption (including a half or full mold) This is acomplex subject beyond the scope of this article and is dis-cussed in detail elsewhere[3445] An example is provided herein Figure 16 which compares the computed speed (vx

2 vz2)12

along a vertical line in the caster center plane midwaybetween the SEN and the narrow face This figure comparesresults from three different computational grids a 04-million-node coarse grid the current 07-million-node grid and a14-million-node fine grid The fine grid included only one-half of the domain so its node spacings are roughly 4 and8 times finer than the other two grids respectively The dif-ferences between the right- and left-hand sides of the domainare more significant than the differences between grids Thisindicates that the mesh resolution is adequate This figure

09-03-364B-Eqxd 92604 841 AM Page 694

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 695

696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash695

also shows that the effect of adding an SGS k model[46] isvery small This indicates that either the unresolved smallturbulent eddies are not very important or that false diffu-sion from numerical discretization errors dominates over theSGS effects

E The Thin-Slab Steel Caster (Case 2-S)

The steel caster differs from the water model mainly inthe solidifying-shell boundary and the outlet In additionthe kinematic viscosity of the molten steel is 20 pct smallerthan that of water These differences might lead to a dif-ferent flow field in a real steel caster even under the sameoperating conditions as in the water model To investigatethe flow in the real steel caster a transient simulation of thethin-slab caster was performed using the same unsteady inletvelocities as the water model (case 2-W) The computedinstantaneous and time-averaged velocity fields are presentedin Figures 17 and 18 and are qualitatively similar to thoseof the water model Both of the time averages of the twosystems were taken over 50 seconds It should be notedthat in both cases the time-averaged center jet is slightlyslanted to the left-hand side indicating a long-term asym-metry Asymmetries such as these may likely be the causefor the asymmetrical defects observed in steel products

Figure 19 quantifies the development of the center jetBoth the time-averaged streamwise velocity (vz) and the rmsvalues of all three velocity components (vx vy and vz) alongthe jet centerline are shown in the figure The results revealthat the jet velocity decreases dramatically starting from thenozzle port The center jet can only penetrate to around

800 mm below the center nozzle port This is important tounderstanding inclusion transport in the liquid pool Theresults also show that the rms of the downward velocity isdominant along the center jet suggesting a strong anisotropyof this turbulent flow

In Figure 20 the downward velocity and rms values arepresented along a line 85 center-port diameters below theport in the center plane (y 0) The downward velocity atthe jet center is seen to decrease to 45 pct of the portoutlet value The high velocity near the narrow-face frontis caused by the side jets Because of the influence of side

Fig 17mdashTypical instantaneous velocity vector plot at the center planebetween wide faces (case 2-S) obtained from simulation

Fig 18mdashTime-averaged velocity vectors at the center plane between widefaces (case 2-S) obtained from simulation

Fig 19mdashTime-averaged and rms values of velocities along the center jetcenterline (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 695

696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

696mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig 20mdashTime averaged and rms values of velocities along a horizontalline 05 m below meniscus halfway between wide faces (case 2-S)

Fig 21mdashComparison of time-averaged horizontal velocity toward SENalong top surface centerline between case 2-W and case 2-S

Fig 22mdashComparison of the rms values of the velocity in Fig 21

Fig 23mdashComparison of the time-averaged downward velocity betweencase 2-W and case 2-S in the lower recirculation zones

tion between the two sides on the top surface in the watermodel which is absent in the simulation of case 2-S Thedownward velocity of the shell in the steel-caster simulationmay have stabilized the flow so that it has less oscillationOur previous studies of a 04-scale water model[2448] founda similar oscillation on the top surface with a frequencylower than 002 Hz The reason of the oscillation is stillnot clear It should be also noted that the velocity on theleft-hand side of the water model is very close to that ofthe steel caster The rms values of the velocities in Figure 21are presented in Figure 22 All the data suggest that the rmsvalues reach their maximum at 15 to 30 mm away from theSEN and then monotonically decrease toward the narrowface The predicted rms values are large usually exceeding30 pct of the local mean velocities

Figures 23 and 24 compare the mean and rms values ofthe downward velocity in both systems The data are extractedalong a horizontal line 1000 mm below the top surface and164 mm from the narrow face Figure 23 shows a bigger spa-tial variation of the downward velocity for the water model

jets as well as being confined by the shell the jet width issmaller than the self-similar free jet[47] The rms distributionagain supports the anisotropy of flow in the liquid pool

F Comparison of the Thin-Slab Steel Caster and ItsWater Model (Case 2-S vs 2-W)

Flows in the thin-slab caster and the water model havebeen investigated separately The two systems will be com-pared here All the mean values presented in this sectionwere averaged over 50 seconds Figure 21 presents themean of horizontal velocity toward the SEN along the cen-terline on the top surface The velocity estimated from thedye injection is also plotted as solid squares in Figure 21All the data show a maximum velocity in the middle betweenthe SEN and the narrow face with a value of 015 to026 ms A significant asymmetry between the left- andright-hand sides is found in the water model (case 2-W)compared to the steel caster (case 2-S) This indicates theexistence of a low-frequency (lower than 002 Hz) oscilla-

09-03-364B-Eqxd 92604 841 AM Page 696

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

It shows that the steel caster has slower downward flow nearthe walls (where the shell is found) and less upward (or reverse)flow in the central region This is likely due to the combinedeffects of tapering which restricts the flow domain the massloss from solidification which tends to even the velocity dis-tribution and the downward withdrawal of the shell whichpulls the flow downward at the casting speed An asymmetrybetween the two sides can be seen for both the water modeland the steel caster again indicating that low-frequency oscil-lations exist with a period longer than 51 seconds

G Velocity Fluctuation on Top Surface

The top-surface velocity greatly influences the harmfulentrainment of liquid slag It fluctuates with time due to itsturbulent nature Instantaneous high values of this velocity canshear off fingers of liquid slag into the steel pool[49] to formnonmetallic inclusions that cause serious defects The historyof surface-velocity variations is investigated in Figure 25(a)where the time signal of the computed horizontal velocity at

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash697

Fig 24mdashComparison of the rms values of the velocity in Fig 23

(a)

(b)

Fig 25mdashTime variations of the horizontal velocity toward SEN at the center point of the top surface (a) case 2-S and (b) a 04-scale water model[24]

09-03-364B-Eqxd 92604 841 AM Page 697

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

sudden jumps Half-pool simulations suppress these suddenjumps by imposing the symmetry boundary condition Thehalf-pool simulations may be missing other such transient phe-nomena necessitating full-pool simulations

H Liquid Level across Top Surface

The liquid level across the top surface is importantbecause it affects the ability of the liquid flux to fill theinterfacial gap between the mold and shell which is impor-tant for heat transfer and thereby for surface quality of thefinal product Figure 26 shows typical transient top-surfacelevels obtained from the simulation surface-pressure resultsfor the thin-slab caster and water model The top-surface liq-uid displacement (z) was estimated from a simple potential-energy balance

[6]z(x y) p(x y) pmean

(rl rtop)g

the top-surface center point (midway between the SEN andthe narrow face) for the thin-slab steel caster (case 2-S) is plot-ted The velocity from the narrow face to the SEN is definedwith a positive value in this plot The figure shows that theamplitude of the fluctuation is comparable to the mean value(Figure 21) We observed that the velocity occasionally has asudden ldquojumprdquo with considerable amplitude (eg the flowvelocity drops from 04 ms toward the SEN to the oppo-site direction in 02 seconds) Due to a lack of long-term mea-surements in this caster or the corresponding water model (case2-W) this behavior is compared with our previous PIV mea-surements on a 04-scale water model shown in Figure 25(b)and published elsewhere[50] The figure also shows LES pre-dictions which were obtained from two half-pool simula-tions (LES1 and LES2) on the 04-scale water model[50] It isobserved that the characteristic of large sudden ldquojumpsrdquo wasalso seen in the PIV measurements but does not exist in thehalf-pool simulations This suggests that transient interactionbetween the two halves of the caster likely causes the large

698mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

(a)

(b)

Fig 26mdashComparison of predicted and measured top surface liquid levels in (a) case 2-W and (b) Case 2-S

09-03-364B-Eqxd 92604 841 AM Page 698

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash699

Figure 26(a) shows the water-model prediction comparedwith top-surface liquid profiles measured from video imagesat three instants The predicted surface shape is in reason-able agreement with the measurements It is also consistentwith previous water-model results[51] Figure 26(b) presentsthe predicted molten-steel level at the top surface The levelis always higher near the narrow face by 2 mm in the watermodel and 4 to 6 mm in the steel caster This is becausethe upward momentum of the liquid near the narrow facelifts the liquid level there The level change is greater inthe steel system because interface movement only requiresthe displacement of some molten slag The prediction of thesteel caster top-liquid profile compares reasonably withindustry measurements (Figure 26(b))[52] Each of nine mea-surements was obtained by dipping a thin-steel sheet intothe operating steel-caster mold and recording the slag-steelinterface shape after removing it[52] Each point representsthe mean deviation of the measurements at that locationfrom the average surface level along the centerline Thisaverage level was determined to be 13 mm using Eq [6]The error bars indicate the range of the measurements ateach location Significant uncertainty in the measurementexists regarding possible rotation of the sheets The slaglayer needs to be thick enough to cover the steel in order toprovide a steady supply of molten flux into the interfacialgap to lubricate the steel maintain uniform temperature pro-files and to avoid surface defects in the solid steel productThus the height of this ldquostanding waverdquo is important to steelquality

I Asymmetries

In most Reynolds-averaged simulations symmetry isassumed between the flow in the two halves of the liquid

pool This assumption has been shown to be valid for long-term averages However transient flow in the two halves isdifferent for instance on the top surface (Figures 21 and22) and in the lower roll (Figures 23 and 24) Figure 27 fur-ther reveals the significant flow asymmetry in the lower rollThe signals present the variation of the downward velocityat two paris of monitoring points each symmetrically locatedin the thin-slab steel caster The data were sampled every0001 seconds from the simulation results Shown as solidtriangles the first pair of points is located within the sidejets midway between the SEN and narrow face 03 m belowthe top surface The other pair is located at 12 m belowthe top surface and near (35 mm from) the narrow facesto illustrate the flow in the lower recirculation zone Theplot on the top shows the velocity history at the first pairof points which shows similar variations to those in Fig-ure 9 No long-term asymmetries are observed between thesetwo points in the jets However the plot below clearly showsthat significant asymmetries can last for a relatively longtime (eg from 37 to 40 seconds) These observations sug-gest that (1) low-frequency long-term asymmetries exist inthe lower recirculation zone and (2) the asymmetries aredue to the turbulent nature of the flow in the liquid pool andnot from asymmetries imposed by the inlet jet This find-ing is important to the understanding of the behavior ofinclusion particles as particles transported to a deeper loca-tion are likely to become permanently entrapped in the steel(as discussed in detail in Part II of this article)

J Spectral Analysis

The power spectrum[38] of the turbulent velocity compo-nent (vx) was calculated at two points in the steel caster

Fig 27mdashTime variations of downward velocity at two pairs of symmetrical points showing low-frequency asymmetries in the lower recirculation zone (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 699

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

700mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

which are symmetrically located in the pool at a distanceof 156 mm from the SEN outlet and at 100 mm below thetop surface The spectral analysis was made from 137 sec-onds of simulation data sampled every 0001 seconds usingthe following equation[38]

[7a]

where

[7b]

[7c]

The result in Figure 28 shows an irregular distribution ofthe power spectrum which has high maxima at low frequen-cies (less than 1 Hz) and tends to decrease exponentially athigher frequencies as indicated with the log-scale plot Slightdifferences exist between the two points likely due to insuf-ficient sampling time A similar behavior of the power spec-trum was seen in measurements on a scaled water model byLawson and Davidson[53]

IV CONCLUSIONS

Three-dimensional unsteady turbulent flow in the moldregion of the liquid pool of standard and thin-slab casterswas computed using an in-house LES code The computedvelocity fields are compared with measurements and are seento have reasonable agreement The following observationsare concluded from this work

1 Complex turbulent structures are observed in the liquidpool Two typical transient flow structures consisting

fk k

tN1 t0 k

N

2 p

N

2 1

Ck aN1

n0Vx (tn) e

i2pfktn

P( fk) micro 1

N2 |Ck|2 k 0

N

2

1

N2 1 |Ck|2 |CNk|

2 2 k 1 p N

2 1

of simpler or complex multiple vortices are found in theupper roll The vortices break into smaller ones or emergeinto bigger ones such that the flow switches between thetwo patterns

2 Flow asymmetries are found in full-pool simulationswhich includes the short-term asymmetry (eg at thenozzle port and along the jet) and the long-term inter-mittent asymmetry (eg on the top surface and in thelower roll) The long-term asymmetry in the lower rollis due to the turbulent nature instead of asymmetries inthe inflow

3 The interaction between the two halves of the liquid poolcauses important transient flow behavior (eg suddenjumps of top-surface velocity) Imposing an asymmetryassumption suppresses sharp sudden jumps in surfacevelocities and low-frequency flow transients in the lowerrecirculation zones

4 Water models are generally representative of steel cast-ers especially in the upper region far above the water-model outlet However steel casters are likely to havesomewhat more evenly distributed downward flow in thelower-roll zone where the influence of shell thicknessbecomes significant

5 The top-surface level can be reasonably predicted from thetop-surface pressure distribution The top-surface level pro-file rises more near the narrow face in the steel caster thanin the water model which has no slag layer to displace

6 Our analysis shows anisotropy of turbulent flow in the liq-uid pool Spectral analysis suggests that most of the energyis contained in the low-frequency region (0 to 5 Hz)

The flow transients and asymmetries have important effectson many other phenomena in the liquid pool that are criti-cal to steel quality The behavior of inclusion particles willbe investigated in the second part of this article

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNo DMI-01-15486) which made this research possible Thework is also supported by the member companies of theContinuous Casting Consortium at University of Illinois atUrbanandashChampaign (UIUC) Special thanks are due to Ya Mengfor calculating the shell thickness to Ron OrsquoMalley for plantdata and insights into the fluid flow and the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

APPENDIX

The effect of the moving solidifying shell on the internalflow in the liquid pool can be represented using a velocityboundary condition which is illustrated as follows A sta-tionary control volume in the Euler frame shown in Figure A1comprises a piece of solid shell A normal velocity of themolten steel entering the control volume through the solidi-fication front (sloped edge) can be obtained from massconservation

[A1]d(rsV)

dt rsA1Vcasting rlA3vn rsA2Vcasting

Fig 28mdashPower spectrum of vx at two points in the upper mold obtainedfrom simulation data (case 2-S)

09-03-364B-Eqxd 92604 841 AM Page 700

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B AUGUST 2004mdash701

By assuming that both the shell shape and the solid densitystay constant in this Eulerian frame the normal velocity canbe expressed as

[A2]

This imposed normal velocity accounts for the mass flowcaused by continuous solidification and shell withdrawal Thenonslip condition is assumed to hold tangential to the front

[A3]

Written in terms of the x z velocity components

[A4a]

[A4b]

Eq [A4] gives the velocity boundary condition at the shellfront position The value of is evaluated at each distance zfrom the slope of the shell thickness profile given by CON1Din Figure 3 No adjustment is needed for inside and outsideradii because the top 3 m of the caster is straight

NOMENCLATURE

total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluideff effective viscosity of turbulent fluid

(

t vj

xj)

D

Dt

vz vn sin u vt cos u ars

rl sin2 u cos2 ubVcasting

vx vn cos u vt sin u ars

rl 1b sin u cos uVcasting

vt Vcasting cos u

vn rs(A2 A1)

rlA3 Vcasting ars

rl sin ubVcasting

densityp static pressurepmean average pressure over top surfacet timeg gravity acceleration (981 m2s)P power spectrum defined in Eq [7a]|Ck| modulus of Ck in Eq [7b]fk frequency defined in Eq [7c]iA areaVcasting casting speedN total sampling time steps in Eq [7]Subscriptsl fluid (liquid steel or water)s solidtop top surface material (air or slag)n t normal tangentiali j direction (x y or z) repeated indices imply summation

REFERENCES1 RC Sussman M Burns X Huang and BG Thomas 10th Process

Technol Conf Proc ISS Warrendale PA 1992 vol 10 pp 291-3042 Y Ho C Chen and W Hwang Iron Steel Inst Jpn Int 1994

vol 34 (3) pp 255-643 B Grimm P Andrzejewski K Muller and K-H Tacke Steel Res

1999 vol 70 (10)4 X Huang BG Thomas and FM Najjar Metall Trans B 1992

vol 23B pp 339-565 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25B

pp 227-336 A Theodorakakos and G Bergeles Metall Mater Trans B 1998

vol 29B pp 1321-277 J Herbertson QL He PJ Flint and RB Mahapatra Steelmaking

Conf Proc ISS Warrendale PA 1991 vol 74 pp 171-858 GD Lawson SC Sander WH Emling A Moitra and BG Thomas

Steelmaking Conf Proc ISS Warrendale PA 1994 vol 77pp 329-36

9 R Bommaraju R Glennon and M Frazee in Continuous CastingM Wolf ed ISS Warrendale PA 1997 vol 9 pp 307-18

10 BG Thomas D Lui and B Ho in Applications of Sensors in Mater-ials Processing V Viswanathan ed TMS Warrendale PA 1997pp 117-42

11 TJH Billany AS Normanton KC Mills and P Grieveson Ironmakingand Steelmaking 1991 vol 18 pp 403-10

12 WH Emling TA Waugaman SL Feldbauer and AW Cramb Steel-making Conf Proc Chicago IL Apr 13ndash16 1997 ISS WarrendalePA 1994 vol 77 pp 371-79

13 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas SteelmakingConf Proc Chicago IL Mar 20ndash23 1994 ISS Warrendale PA 1994vol 77 pp 381-88

14 J Szekely and RT Yadoya Metall Mater Trans 1973 vol 4pp 1379-88

15 SK Choudhary and D Mazumdar Iron Steel Inst Jpn Int 1994vol 34 (7) pp 584-92

16 SK Choudhary and D Mazumdar Steel Res 1995 vol 66 (5)pp 199-205

17 BG Thomas and LM Mika 2nd FIDAP Users Conf Evanstan IL1988 M Engelman ed Fluid Dynamics International Inc EvanstonIL 1988 pp 1-29

18 BG Thomas FM Najjar and LJ Mika Proc F Weinberg Int Sympon Solidification Processing 29th Canadian Institute of Mining andMetallurgy Conf Hamilton ON 1990 JE Lait and I V Samarasekeraeds Pergamon Press Inc Toronto 1990 pp 131-45

19 BG Thomas LM Mika and FM Najjar Metall Trans B 1990vol 21B pp 387-400

20 BG Thomas and FM Najjar Appl Mathematical Modeling 1991vol 15 pp 226-43

21 BG Thomas Q Yuan S Sivaramakrishnan T Shi SP Vanka andMB Assar Iron Steel Inst Jpn Int 2001 vol 41 (10) pp 1262-71

11

Fig A1mdashThe control volume for calculating boundary velocities at theshell front

09-03-364B-Eqxd 92604 841 AM Page 701

22 BG Thomas and L Zhang Iron Steel Inst Jpn Int 2001 vol 41 (10)pp 1181-93

23 X Huang and BG Thomas 35th Conf Metallurgists C Twigge-Moleceyed CIM 1996 vol 23B pp 339-56

24 S Sivaramakrishnan H Bai BG Thomas P Vanka P Dauby andM Assar Ironmaking Conf Proc ISS Warrendale PA 2000 vol 59pp 541-57

25 Q Yuan SP Vanka and BG Thomas 2nd Int Symp on Turbulent andShear Flow Phenomena Stockholm 2001 Royal Insitute of Technology(KTH) Stockholm 2001 p 6

26 K Takatani Y Tanizawa H Mizukami and K Nishimura Iron SteelInst Jpn Int 2001 vol 41 (10) pp 1252-61

27 R Sobolewski and DJ Hurtuk 2nd Process Technology Conf ProcISS Warrendale PA 1982 vol 2 pp 160-65

28 BG Thomas X Huang and RC Sussman Metall Trans B 1994vol 25B pp 527-47

29 D Gupta and AK Lahiri Metall Mater Trans B 1996 vol 27Bpp 757-64

30 D Gupta S Chakraborty and AK Lahiri Iron Steel Inst Jpn Int1997 vol 37 (7) pp 654-58

31 S Sivaramakrishnan BG Thomas and SP Vanka in MaterialsProcessing in the Computer Age V Voller and H Henein eds TMSWarrendale PA 2000 vol 3 pp 189-98

32 MB Assar PH Dauby and GD Lawson Steelmaking Conf ProcISS Warrendale PA 2000 vol 83 pp 397-411

33 J Smagorinsky Monthly Weather Rev 1963 vol 91 pp 99-16434 Q Yuan B Zhao SP Vanka and BG Thomas University of Illinois

at Urbana-Champaign Urbana IL unpublished research 200435 FH Harlow and JE Welch Phys Fluids 1965 vol 8 (112)

pp 2182-8936 J Crank and P Nicolson Proc Cambridge Phil Soc 1947 vol 43

pp 50-6737 LF Sampine and MK Gordon Computer Solution of Ordinary Dif-

ferential Equations the Initial Value Problem WH Freeman amp CoSan Francisco CA 1975

38 WH Press SA Teukolsky WT Vetterling and BP FlanneryNumerical Recipes The Art of Scientific Computing Cambridge Uni-versity Press Cambridge United Kingdom 1992 pp 490-600

39 Userrsquos Manual Hypre-High Performance Preconditioners Report NoUCRL-MA-137155 DR Center for Applied Scientific ComputingLawrence Livermore National Laboratory Livermore CA 2001

40 BG Thomas R OrsquoMalley T Shi Y Meng D Creech and D Stonein Modeling of Casting Welding and Advanced Solidification ProcessesShaker Verlag GmbH Aachen Germany 2000 vol IX 2000 pp 769-76

41 Y Meng and BG Thomas Metall Mater Trans B 2003 vol 34Bpp 685-705

42 BG Thomas RJ OrsquoMalley and DT Stone in Modeling of Cast-ing Welding and Advanced Solidification Processes San Diego CA1998 BG Thomas and C Beckermann eds TMS Warrendale PA1998 vol VIII pp 1185-99

43 FM Najjar BG Thomas and DE Hershey Metall Trans B 1995vol 26B pp 749-65

44 H Bai and BG Thomas Metall Mater Trans B 2001 pp 269-8445 B Zhao BG Thomas SP Vanka and RJ OrsquoMalley University of

Illinois at Urbana-Champaign Urbana unpublished research 200346 K Horiuti J Phys Soc Jpn 1985 vol 54 (8) pp 2855-6547 I Wygnanski and H Fiedler J Fluid Mech 1969 vol 38 pp 577-61248 S Sivaramakrishnan Masterrsquos Thesis University of Illinois Urband

IL 200049 A Cramb Y Chung J Harman A Sharan and I Jimbo Iron Steelmaker

1997 vol 24 (3) pp 77-8350 Q Yuan T Shi BG Thomas and SP Vanka Proc Computational

Modeling of Materials Minerals and Metals Processing Seattle WA2001 TMS Warrendale PA 2001 pp 491-500

51 J Anagnostopoulos and G Bergeles Metall Mater Trans B 1999vol 30B pp 1095-1105

52 RJ OrsquoMalley AK Steel Middltown OH private communication2002

53 NJ Lawson and MR Davidson J Fluids Eng 2002 vol 124 (6)pp 535-43

702mdashVOLUME 35B AUGUST 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

09-03-364B-Eqxd 92604 841 AM Page 702