in a 0.4-Scale Water Model of a Continuous Steel Caster ... · Computational and Experimental Study of Turbulent Flow in a 0.4-Scale Water Model of a Continuous Steel Caster QUAN

Computational and Experimental Study of Turbulent Flow in a 04-Scale Water Model of a Continuous Steel Caster

QUAN YUAN SIVARAJ SIVARAMAKRISHNAN SP VANKA and BG THOMAS

Single-phase turbulent flow in a 04-scale water model of a continuous steel caster is investigatedusing large eddy simulations (LES) and particle image velocimetry (PIV) The computational domainincludes the entire submerged entry nozzle (SEN) starting from the tundish exit and the completemold region The results show a large elongated recirculation zone in the SEN below the slide gateThe simulation also shows that the flow exiting the nozzle ports has a complex time-evolving pat-tern with strong cross-stream velocities which is also seen in the experiments With a few exceptionswhich are probably due to uncertainties in the measurements the computed flow field agrees withthe measurements The instantaneous jet is seen to have two typical patterns a wobbling ldquostair-steprdquodownward jet and a jet that bends upward midway between the SEN and the narrow face A 51-second time average suppressed the asymmetries between the two halves of the upper mold regionHowever the instantaneous velocity fields can be very different in the two halves Long-term flowasymmetry is observed in the lower region Interactions between the two halves cause large velocityfluctuations near the top surface The effects of simplifying the computational domain and approxi-mating the inlet conditions are presented

I INTRODUCTION

TURBULENT flow in the mold region of continuous steelcasters is associated with costly failures (eg shell-thinningbreakout) and the formation of many defects (eg slivers) byaffecting important phenomena such as top-surface-level-fluctuations and the transport of impurity particles and super-heat[1ndash4] Understanding the unsteady flow structures in thisprocess is an important step in avoiding failures and decreas-ing defects Unfortunately because of the high temperature(1800 K) of superheated steel it is difficult to conduct veloc-ity measurements directly in molten steel[5] However due tothe nearly equal kinematic viscosities of molten steel and waterwater models have been extensively used to investigate theflow in steel casters[6ndash11]

The dimensions and operating conditions of a water modelare usually chosen to have geometry and Froude number (orsometimes Reynolds number) similarities[12] with the actualsteel caster Figure 1(a) shows an example of a scaled watermodel[913] The walls of the tundish the nozzle and the moldof a water model are usually made of transparent plasticplates The mold side walls are sometimes curved to repre-sent the tapering shape of the internal liquid cavity withinthe solidifying steel shell A slide gate (Figure 1(a)) or stop-per rod is used to control the flow rate by adjusting the open-ing area in order to achieve the desired casting speed (definedas the downward withdrawal speed of the shell in an actualsteel caster) Water flows downward from the tundish passesthrough the nozzle enters the mold cavity and exits from

outlet ports near the bottom It should be mentioned that twomain differences exist between a water model and its corre-sponding steel caster First in the mold region the no-slipsolid wall of a water model does not represent the solidifi-cation occurring at the shell front Second a water modelhas a horizontal bottom plate with outlet ports while in acontinuous steel caster molten steel flows into a taperingsection resulting from the solidification Despite these dif-ferences however our recent studies have confirmed that thevelocity field in a water model generally agrees with that ina steel caster especially in the top region[14]

One of the advantages of a water model is that its trans-parent walls allow flow visualization such as dye injection[1415]

(Figure 19) and the penetration of laser light This enablesthe use of accurate and nonintrusive optical laser velocimetrytechniques[16] Two typical methods are laser-doppler veloci-metry (LDV)[1617] and particle image velocimetry (PIV)[1618]

The LDV technique measures instantaneous flow velocitiesat single or multiple points by detecting the Doppler frequencyshift of the laser light[16] while PIV is a method designed formeasuring an instantaneous planar velocity field[16] DuringPIV measurements a pulsed laser sheet is used to illuminatea desired planar section through the flow field where smallparticles (usually 1 to 20 m) are seeded into and well mixedwith the fluid A charge coupled device (CCD) camera is usedto record the images of the illuminated particles in the flowfield The time interval between two consecutive laser pulseswhich produce a pair of exposures is only a few microsecondsThe particle images are then discretized into rectangular inter-rogation areas and the particle positions are correlated toproduce a spatially averaged displacement vector By dividingthe displacement vector by the laser pulse time interval foreach interrogation area an instantaneous velocity field isobtained This procedure is repeated at 1-second time inter-vals to measure the evolution of a flow field Computers haveso improved the simplicity and speed of this method that itis now often called digital PIV (DPIV)[19] Details on PIV canbe found elsewhere[1618]

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 35B OCTOBER 2004mdash967

QUAN YUAN PhD Candidate SP VANKA Professor and BGTHOMAS W Grafton and Lillian B Wilkins Professor are with the Depart-ment of Mechanical and Industrial Engineering University of Illinois atUrbana-Champaign Urbana IL 61801 Contact e-mail bgthomasuiuceduSIVARAJ SIVARAMAKRISHNAN PhD Candidate is with the Biomed-ical Engineering Department Northwestern University Evanston IL 60208

Manuscript submitted July 23 2001

Numerical simulation is another powerful tool used to studyturbulent flow in continuous casting Models of turbulent flowcan be classified into the Reynolds-averaged approach largeeddy simulation (LES) and direct numerical simulation (DNS)[20]

Because of its low computational cost the Reynolds-averagedapproach typically with the two-equation (k-) turbulence

model has been extensively adopted in previous studies andhas produced valuable insights about the flow in continuouscasting nozzles[21ndash24] and molds[1025ndash29] However this approachlimited by its nature is not suited for studying the time evo-lution of unsteady flow structures triggered by flow instabili-ties Plant observations suggest that flow transients under thenominally steady operating conditions are very important[30]

The LES and DNS approaches are better for solving the time-dependent flow of the continuous casting process in whichthe Reynolds number is of the order of 105 Due to the pro-hibitive computational cost of DNS at high Reynolds num-bers LES is a more feasible way for solving this complexflow problem Recently a few attempts have been made toapply it to the continuous casting process[131431] The princi-pal idea of LES is that during the simulation the time evolu-tion of the large-scale (energy-containing) eddies is resolvedand the small energy-dissipative eddies are ldquofilteredrdquo The fil-tering of the small eddies generates a residual stress ten-sor[20] in the NavierndashStokes momentum transport equationwhich is included using a subgrid scale (SGS) model AlthoughLES is less expensive than DNS it still requires consider-able computational effort In this article the transient flowstructures in a 04-scale water model are investigated usingLES computations and PIV measurements

II WATER MODEL

Figure 1(a) depicts the schematics and dimensions of a 04-scale water model constructed from transparent plastic platesat the former LTV Steel Technology Center (ClevelandOH)[913] The flow rate in this water model is controlled by a

968mdashVOLUME 35B OCTOBER 2004 METALLURGICAL AND MATERIALS TRANSACTIONS B

Table I Dimensions and Operating Conditions of the 04-Scale Water Model

DimensionsConditions Value

Slide-gate orientation 90 degSlide-gate opening (area) 39 pctSEN bore diameter 32 mmSEN submergence depth 75 mmPort height width 32 31 mmPort thickness 11 mmPort angle lower edge 15 deg downPort angle upper edge 40 deg downBottom well recess depth 48 mmWater model height 950 mmWater model width 735 mm

(corresponding full scalecaster width) 1829 mm (72 in)

Water model thickness 95 mm (top) to 65 mm (bottom)

(corresponding full scalecaster thickness) 229 mm (9 in)

Outlet at the bottom of the 3 round35-mm-water model diameter holes

Inlet volumetric flow ratethrough each port 353 104 m3s

Mean velocity inside nozzle bore 0439 msCasting speed (top thickness) 102 mms

(0611 mmin)Water density 1000 kgm3

Water kinematic viscosity 10 106 m2sGas injection 0 pct

(a)

(b)

Fig 1mdashSchematics of the 04-scale water model (a) dimensions and (b) the computational domain

slide gate that moves in the mold thickness (y) direction Thebifurcated submerged entry nozzle (SEN) shown in the figurehas two downward-angled square nozzle ports with top andbottom edges angled downward at 40 and 15 deg respectivelyNo gas is injected The Reynolds number at the nozzle portbased on its hydrodynamic diameter is 12000 It is alsoshown in the figure that the mold thickness tapers from thetop (95 mm) to the bottom (65 mm) so that the mold cavityrepresents only the liquid portion in the steel caster Waterflows into the mold cavity recirculates and finally exits fromthree 35-mm outlet holes spaced 180 mm apart along the plas-tic bottom wall A photograph of flow in this water model isgiven in Figure 19 visualized using die injection Table I givesthe details of the water model geometry and its operating con-ditions Further details are available elsewhere[2324]

III COMPUTATIONAL MODEL

Flow in the 04-scale water model is solved using LESThe computational domain is shown in Figure 1(b) It startsat the tundish exit includes the upper tundish nozzle (UTN)the slide gate the SEN and the complete tapered mold cavityit ends at the mold bottom The domain is discretized usinga Cartesian grid consisting of 15 million finite volumes

A Governing Equations

In the context of LES only the large-scale flow structuresare resolved in the simulation The dissipative effect of thesmall-scale eddies which are smaller than the finite volumeand therefore filtered during the simulation is representedusing an SGS model The governing equations for the resolvedflow field account for conservation of mass and momentumas[20]

[1]

[2]

where

[3]

The symbols p and vi in Eqs [1] and [2] represent the pres-sure and filtered velocities The subscripts i and j representthe three Cartesian directions and repeated subscripts implysummation The residual stresses which arise from the unre-solved small eddies are modeled using an eddy viscosity(t) An important issue here is the selection of an appro-priate SGS model for this complex industrial flow problemIn the past a class of SGS kinetic energy (SGS k) modelshave been developed for simple problems such as flow in achannel[32ndash36] The SGS k model employed here requires solv-ing the following additional transport equation which includesadvection production dissipation and viscous diffusion[3336]

[4]

ksgs

t vi

ksgs

xi nt 0 S 20 C

k32sgs

xi av02vt

ksgs

xib

neff n0 nt

Dvi

Dt

1r

p

xi

xjneff avi

xj

vj

xib

vi

xi 0

where

[5]

[6]

where is the magnitude of the strain-rate tensor definedas

[7]

where

[8]

The parameters C and Cl can be treated as constants[33] orevaluated dynamically during the simulation by assumingsimilarity between the subgrid stress tensor and the large-scale Leonard stress tensor[36] This work adopts a static SGSk model with constant values 10 and 01 for C and Clrespectively[33]

B Boundary Conditions

The flow enters the computational domain from the topopening of the UTN which connects the tundish bottomwith the nozzle A uniform velocity of 115 ms is prescribedat the inlet opening based on the desired mass flow rate Theno-slip boundary condition is employed at the wall bound-aries The top surface of the water in the mold cavity is mod-eled as a free-slip plane (z velocity and z gradient of all othervariables set to zero) A constant pressure boundary condi-tion is used at the three outlet ports in the bottom wall wherethe gradients of all the other variables are set to zero

C Solution Procedure

The time-dependent three-dimensional filtered NavierndashStokesequations[124] are discretized using the HarlowndashWelch frac-tional step procedure[37] Second-order central differencing isused for the convection terms and the CrankndashNicolsonscheme[38] is used for the diffusion terms The AdamsndashBashforthscheme[39] is used to discretize in time with second-order accu-racy The pressure Poisson equation is solved using an alge-braic multigrid (AMG) solver[40]

D Computational Details

The computational domain depicted in Figure 1(b) is dis-cretized using 15 million Cartesian finite volumes Smallergrid spacing (08 mm) is set at the nozzle outlet port and nearthe narrow face walls The adequacy of this mesh refinementis demonstrated in separate investigations of the computationalissues in LES modeling of continuous casting[1441] The timestep (t) is set to 00003 seconds to keep the simulation stable

(CFL t max 1[42] here the CFL

number was found to be 06 so the results should beaccurate) The computational time was 24 hours for 1 secondof integration time on a Pentium IV 32 GHz PC (Linux 80Redhat Raleigh NC) Time mean and variation values werecalculated after the flow reached a statistically stationary

a vx

x

vy

y

vz

zb

Sij 1

2 avi

xj

vj

xib

0 S 0 12SijS

ij

0 S 0nt Clk32

sgs

(xyz)13


state[43] Variations are characterized by their root-mean-square (rms) values such as which is computed by

where (t2 t1) is the time

interval for the average and t is the time step size The meanand rms velocities were calculated for 51 seconds (170000time steps) and 20 seconds (70000 time steps) respectively

IV PIV MEASUREMENTS

The principle of PIV is to determine the flow velocities bymeasuring the displacement vector of illuminated particle imagesduring a known time interval as shown in Figure 2 In thiswork aluminum powder with particle diameters of approximat-ely 30 m was seeded into the fluid before the measurements[13]

A NdYAG laser was used to illuminate the flow field[13] TheCCD camera used in this work was a DANTEC-Double Image700 (DANTEC Dynamics Stovlunde Denmark) with 768 480 pixels[13] To generate enough particle images in each inter-rogation area to give an accurate average an image resolutionof 32 32 pixels per interrogation was used in this study Thisproduced a measured field of 32 19 vectors In addition toavoid problems arising from the crossover of particles near areaedges moving between adjacent areas the interrogation areaswere made to overlap each other by 25 pct

Because of our interest in the relatively large-scale flowstructures in the water model a large measurement area wasselected at the expense of the relatively low overall resolu-tion (compared to the computation) Owing to the limitednumber of camera pixels the illuminated flow domain wasdivided into three regions as shown in Figure 2 the upperregion (0 to 025 m) containing the jet and the upper tworolls the middle region (025 to 065 m) and the lower region(065 to 077 m) containing the two lower rolls Because theSEN blocks the laser the flow in each half of the upper regionwas measured separately During measurements the timeinterval between two consecutive laser pulses was set at 1 msThe number of snapshots (pairs of pulses) collected and thetime interval between them (which varies from 02 to 1 sec-ond) were determined depending on the time scales of the

a 1

(t2 t1)at2

ti t1

(v(ti) v)2tb12(viquest viquest)12

flow in the respective regions The collected data total 900snapshots of one half of the mold spaced 02 seconds apartfor the top portion 2000 snapshots of both halves spaced1 second apart 400 snapshots of one half spaced 02 secondsapart for the middle region and 200 snapshots of both halvesspaced 02 seconds apart for the bottom region

V FLOW IN THE SEN

Flow in the nozzle is important because a detrimental flowpattern may lead to problems such as clogging which bothlimits productivity and causes defects[44] In addition the SENports direct the fluid into the mold cavity which controls thejet angle the flow pattern and the corresponding steel qual-ity issues In this study flow in the UTN and SEN could notbe reliably measured using PIV due to the curvature and par-tial opacity of the nozzle wall Thus this section presentsthe computed flow field in the nozzle region and comparesit with measurements only at the port outlets

Figure 3 gives an overall view of the computed velocitiesin the UTN and SEN at the centerline slice (x 0) Theplot on the left shows a representative instantaneous veloc-ity field The time-dependent velocities in the nozzle wereaveraged over 51 seconds and are shown in the right twoclose-up plots In both the instantaneous and time-averaged


Fig 2mdashSchematics showing the PIV measurement regions Fig 3mdashComputed velocity field at the center plane x 0 of the SEN

plots the narrowed flow passage at the slide gate induceslarge downward velocities (3 ms) These velocities exceedthe mean velocity down the nozzle bore by 7 times and dimin-ish gradually with distance down the nozzle A recirculationflow is seen in the cavity of the slide gate A large elon-gated recirculation zone is also observed in the SEN beneaththe slide gate and extends almost to the nozzle ports Thisrecirculation zone is complex and actually exhibits multi-ple transient recirculation regions These recirculation flowsencourage the accumulation of impurity inclusions in themolten steel by increasing their residence time and may causeproblems such as clogging The plot on the right bottomreveals a clockwise swirl in the y-z plane near the SEN bot-tom This swirl is clearly induced by the partial opening ofthe slide gate The swirl is transported downstream with theflow to exit the nozzle ports as shown in Figure 4 whichdepicts the time-averaged velocity vectors leaving the noz-zle ports In Figure 4(a) the cross-stream velocities in theouter plane of the nozzle outlet (x 0027 m) are plotted forthe view looking into the port The single swirl persists herealso Figure 4(b) shows the velocity vectors at the center-line slice y 0 and indicates that most of the fluid exitsthe nozzle from the lower half of its ports Reverse flow is

observed in the upper portion of the port This result is con-sistent with previous work[2345] and is expected because theport-to-bore area ratio (247) greatly exceeds 1 Comparingthe velocity vectors in Figures 4(a) and (b) the cross-streamvelocity components are seen to be comparable in magni-tude to the streamwise components

Figure 5 shows the time-averaged flow speed along the nozzle port vertical centerline The PIV data shownhere were collected in the mold cavity close to the nozzleports[46] They are the average of 50 PIV snapshots spaced02 seconds apart[46] The computed speed is seen to have asimilar distribution to that obtained from PIV In both LES andPIV the ldquopeakrdquo speed occurs 3 mm above the lower edge ofthe nozzle port The computed speeds are consistently largerthan the measured values in the lower portion of the porthowever In previous work misalignment of the laser planewas suspected as the reason for this discrepancy[23] Anothersuspected reason is that the relatively large off-plane velo-city component ( 02 to 03 ms) in the lower portion ofthe port makes the tracer particles in the water model move02 to 03 mm during the 1-ms time interval between twoconsecutive laser pulses The typical thickness of the lasersheet is 1 mm Particles moving in and out of the illuminatedplane could confuse the measurement

Figure 6 presents a sequence of the computed instantaneoussnapshots of the flow at the nozzle outlet port to reveal itsevolution In Figure 6(a) a strong clockwise swirl is seen tooccupy almost the whole port area After 4 seconds the sizeof this swirl reduces to 23 of the port area with cross-streamvelocities in the other 13 portion dropping close to zero (Fig-ure 6(b)) It then breaks into many distinct small vortices1 second later as shown in Figure 6(c) and further evolves intoa nearly symmetric double swirl another 4 seconds later (Fig-ure 6(d)) The flow at the nozzle port is seen to fluctuatebetween these four representative patterns This same behav-ior was observed in visual observations of the water modelHowever the strong cross-stream flow is not seen when astopper rod is used instead of a slide gate[14]

vy

1vx2 vz

2212


(a)

(b)

Fig 4mdashComputed time-averaged velocity field exiting nozzle ports (a) view into the port and (b) slice y 0

Fig 5mdashTime-averaged fluid speed along the vertical cen-terline of the SEN nozzle ports

1vx2 vz

2212

VI FLOW IN THE MOLD CAVITY

The jet exiting the SEN feeds into the mold cavity whereit controls the flow pattern and corresponding phenomenawhich affect quality problems If insufficient superheat istransported with the jet to the top surface then the menis-cus may freeze to form subsurface hooks which may entrapinclusions and cause slivers The contour of the top surfacebeneath the flowing liquid affects flux infiltration into the gapwhich controls lubrication and surface cracks Excessive flowfluctuations can cause fluctuations in the top surface leveldisrupting meniscus solidification and causing surface defectsExcessive velocity across the top surface can shear fingersof molten mold flux into the steel leading to inclusion defectswhen the particles eventually become entrapped[47] The moldregion is the last step during which impurity particles couldbe removed without being entrapped in the solid steel slabsKnowledge of the turbulent flow in the mold region is cri-tical for understanding all of these phenomena This section

presents the details of the turbulent flow in the mold cavityof the water model

A Time-Averaged Flow Structures

After the computed flow reached a statistically stationarystate[43] the means of all variables were collected by aver-aging the instantaneous flow fields obtained at every timestep Figure 7 presents the simulated flow field at the centerplane y 0 in the mold cavity averaged over 51 secondsFor clarity velocity vectors are only shown at about everythird grid point in each direction The usual double-rollflow pattern[1025] is reproduced in each half of the moldThe two jets emerging from the nozzle ports spread andbend slightly upward as they traverse the mold region Thetwo lower rolls are slightly asymmetric even in this time-averaged plot This indicates that flow transients exist withperiods longer than the 51-second average time


(a) (b)

(c) (d)

Fig 6mdash(a) through (d) Representative instantaneous cross-stream flow patterns exiting the nozzle port obtained from LES view into the port

Figure 8 gives a closer view of the upper roll The PIVplot shown on the left is a 60-second average of 300 instan-taneous measurements The right half shows some of thecomputed velocities plotted with a resolution comparable tothat of the PIV A jet angle of approximately 29 deg isimplied by the LES results which is consistent with the flowvisualization[13] A larger jet angle of 34 to 38 deg is seenin the PIV vectors This may be due to the manually adjustedlaser sheet being off the center plane (y 0) In both LESand PIV the jet diffuses as it moves forward and becomesnearly flat 02 m away from the center The eyes of the upperrolls are seen to be nearly 02 m away from the SEN cen-ter and 01 m below the top surface The main difference

between the computed and measured velocities in the upperregion is that the computed velocities are consistently higherthan the measured values in the low-velocity regions Per-haps this is because the PIV system is tuned to accuratelymeasure velocities over a specific range (eg by adjustingthe pulse interval) which might decrease accuracy in regionswhere the velocities are either much higher or much lower

The time-averaged flow in the lower region is given in Fig-ures 9(a) and (b) Both plots are for the center plane y 0The LES data clearly show that the lower roll in the left halfis smaller and about 01 m higher than the right one This


Fig 7mdashTime-averaged velocity vector plot in the mold region obtainedfrom LES

Fig 8mdashTime-averaged velocity vectors in the upper roll sliced at y 0 obtained from PIV measurements and LES

(a)

(b)

Fig 9mdashTime-averaged velocity vectors in the lower roll region obtainedfrom (a) LES and (b) PIV measurements

confirms that a flow asymmetry exists in the lower roll regionthat persists longer than 51 seconds (the averaging time) Inthis region ten sets of PIV measurements were conductedEach set of measurements consists of 200 snapshots takenover 200 seconds Figure 9(b) presents the velocity field aver-aged from all of the measurements For all ten averages thelower roll in the right half is larger and slightly lower thanthe left one This proves that the asymmetry of the flow ispersistent over long times exceeding several minutes It wasalso observed that for all the ten sets of PIV data the down-ward velocities close to the right narrow face are always greaterthan those down the left side It is not known whether this isdue to the flow asymmetry or errors in the experiments (eglaser light diminishing as it traverses the flow field) Thislong-term flow asymmetry in the lower roll has been observedin previous work and may explain why inclusion defects mayalternately concentrate on different sides of the steel slabs[48]

B Velocities along Jets

Figure 10 compares the computed speed withPIV measured values along the jet centerline The solid linedenotes the speed obtained from the LES and averaged for51 seconds It shows that the jet exits the nozzle port at aspeed 07 ms and slows down as it advects forward It isseen that the 51-second average almost suppresses the dif-ferences between the left and right jets Except in the regionclose to the nozzle port a reasonable agreement between thecomputation and measurements is observed

C Velocities on the Top Surface

In a steel caster the flow conditions at the interface betweenthe molten steel and the liquid flux on the top surface are cru-cial for steel quality Therefore accurately predicting veloc-ities there is important for a computational model Figure 11shows the time-averaged x velocity component ( ) towardthe SEN along the top surface center line Due to a lack ofmeasurements at the current casting speed two sets of averaged

vx

1vx2 vz

2212

data from other PIV (provided by Assar)[49] are used to comparewith the LES predictions Each one of these is the averageof a group of measurements conducted on the same watermodel at a constant casting speed slightly higher (0791 mmin)or lower (0554 mmin) than that in this work It can be seenthat this velocity component increases away from the SENreaches a maximum midway between the SEN and the nar-row face and decreases as it approaches the narrow face Themaximum of 015 ms is about 13 of the mean velocity inthe nozzle bore and 15 of the maximum velocity of the jetexiting the nozzle The comparison suggests that the compu-tation agrees reasonably well with the PIV measurements

The computed rms value of this velocity component is plot-ted in Figure 12 No PIV data are available for the rms onthe top surface The figure suggests that the rms of the x veloc-ity component (( )12) decreases slightly from the SENto the narrow face The results also suggest that the rms ofthe velocity can be as high as 80 pct of the mean velocityindicating very large turbulent velocity fluctuations

Figure 13 compares the time variation of the horizontalvelocity toward the SEN near the top surface for the simulation

vxiquest vxiquest


Fig 10mdashTime-averaged fluid speed along jet centerlineobtained from LES (SGS-k model) and PIV measurements

(u 2 w2)12

Fig 11mdashTime-averaged horizontal velocity toward SEN along the top sur-face centerline

Fig 12mdashThe rms of the u-velocity component along the top surface centerline

and the measurement The data are taken at a point 20 mmbelow the top surface midway between the SEN center andthe narrow face The mean PIV signal is lower than expectedfrom the measurements in Figure 11 which shows variationsbetween the PIV measurements taken at different times Thevelocity fluctuations are seen to be large with a magnitudecomparable to the local mean velocities Both the computa-tion and measurement reveal a large fluctuating componentof the velocity with approximately the same high frequency(eg the velocity drops from 021 ms toward the SEN toa velocity in the opposite direction within 07 seconds) Thisvelocity variation is important because the liquid level fluc-tuations accompanying it are a major cause of defects in theprocess The computed signal reproduced most of the featuresseen in the measurements The signal also reveals a lower fre-quency fluctuation with a period of about 45 seconds A spec-tral analysis of the surface pressure signal near the narrowface on the top surface reveals predominant oscillations withperiods of 7 and 11 to 25 seconds which are superimposedwith a wide range of higher-frequency lower-amplitude oscil-lations Knowing that model surface pressure is proportionalto level[14] this result compares with water model measure-ments of surface level fluctuations by Lahri[50] that appearto have a period of 04 seconds and by Honeyands andHerbertson[51] of 12 seconds

D Velocities in the Lower Roll Region

Figure 14 shows the downward velocity profile across thewidth of the mold centerline in the lower roll zone (04 mbelow the top surface) As stated earlier ten sets of 200-second200-snapshot PIV measurements were conducted in bothhalves of the mold The average of all sets is shown as opensymbols The error bars denote the range of the averages ofall ten sets of measurements The solid symbols correspondto a data set with large upward velocities near the center Inall data sets the largest downward velocity occurs near thenarrow face (x 0363) The computation is seen to over-predict the upward velocity measured right below the SENThis may partially be due to the shorter averaging time (51 sec-onds) in LES compared to PIV as the PIV results indicatesignificant variations even among the ten sets of 200-secondtime averages This inference is further supported by the rmsof the same velocity component along the same line shownin Figure 15 The open symbols and error bars again repre-sent the rms velocity averaged for the ten sets of measure-

ments and the range The results indicate large fluctuationsof this vertical velocity in this region (eg near the centerthe rms value is of the same magnitude as the time-averagedvelocity) Both the time averages and rms are seen to changesignificantly across these 200-second measurements indica-ting that some of the flow structures evolve with periods muchlonger than 200 seconds Accurate statistics in the lower rolltherefore require long-term sampling This agrees with mea-surements of the flow-pattern oscillation period of 40 sec-onds (a 2- to 75-second range) conducted on a very deep watermodel[52]

Figure 16 presents the downward velocity along two linesacross the mold thickness in the center-plane midway betweenthe narrow faces (x 0) in the lower roll These results showa nearly flat profile of this velocity in the interior regionalong the thickness direction This suggests that a slight mis-alignment of the laser sheet off the center plane should notintroduce significant errors in the lower roll region


Fig 13mdashTime history of the horizontal velocity toward SEN (20 mm belowthe top surface midway between the SEN and narrow faces)

Fig 14mdashTime-averaged downward velocity component across the width(along the horizontal line 04 m below the top surface midway betweenwide faces)

Fig 15mdashThe rms of the downward velocity component along the line in Fig 7

flow patterns in the upper region Flow in this region isseen to switch randomly between the two patterns in bothLES and PIV An analysis of many frames reveals thatthe staircase pattern oscillates with a time scale of 05 to

E Instantaneous Flow Structures

The instantaneous flow pattern can be very different fromthe time-averaged one The time-dependent flow structures inthe mold cavity are presented in this subsection Figure 17(a)gives an instantaneous velocity vector plot of the flow fieldin the center plane (y 0) measured with PIV It is a com-posite of the top middle and bottom regions shown in Fig-ure 2 for each half Each of the six frames was measured ata different instant in time Figure 17(b) shows a correspond-ing typical instantaneous velocity field obtained from LESThe flow consists of a range of scales as seen by the veloc-ity variations within the flow field The jets in both halvesconsist of alternate bands of vectors with angles substantiallylower and higher than the jet angle at the nozzle port Thevelocities near the top surface and the upper roll structureare observed to be significantly different between individualtime instants and between the two halves In both halves theflow from the downward wall jet can be seen to entrain thefluid from a region below the SEN although at differentheights Thus the shape and size of the two lower rolls appearsignificantly different for both PIV and LES

Figure 18 gives a closer view of flow structures in theupper region obtained by LES and PIV The upper plot showsa computed instantaneous velocity field at the center plane y 0 The lower velocity vector plot is a composite of twoinstantaneous PIV snapshots divided by a solid line andobtained from measurements of the same flow field A ldquostair-steprdquo type of jet is observed in the left vector plot for boththe simulation and measurement This flow pattern is believedto result from the swirl in the jet (Figure 6) the swirling jetmoves up and down and in and out of the center plane as itapproaches the narrow face causing a stair-step appearancein the center plane The flow displayed in the right snapshotshows a shallower jet The jet bends upward after traveling025 m in the x direction and splits into two vortices Inthe actual steel casting process this upward-bending jet maycause excessive surface level fluctuation resulting in surfacedefects while the deeper jet shown in the left plot may carrymore inclusions into the lower roll region leading to inclu-sion defects These are the two representative instantaneous


Fig 16mdashSpatial variation of the downward velocity across the thicknessdirection (beneath SEN)

(a)

(b)

Fig 17mdashAn instantaneous snapshot showing the velocity field in the watermodel obtained from (a) PIV measurements and (b) LES

instantaneous flow also shows that the sizes of the two lowerrolls change in time causing oscillations between the two halves

15 seconds This is consistent with a spectral analysis of thevelocity signal at this location which shows strong frequencypeaks at 06 and 09 Hz and many other smaller peaks atdifferent frequencies

The LES results also suggest that the instantaneous flow inthe two halves of the mold can be very asymmetric The asym-metry does not appear to last long in the upper mold becausea 51-second average is seen to eliminate this asymmetry (Fig-ures 7 10 and 11) The instantaneous asymmetric flow inthe upper roll is also evidenced by the dye-injection photo-graph in Figure 19 This picture suggests a flow pattern similarto that shown in Figure 18(a)

Two sequences of flow structures obtained from LES andPIV respectively are compared in Figure 20 showing theevolution of the flow in the lower region In the first plot (Fig-ure 20(a)) a vortex can be seen in the left half approximately035 m below the top surface and 015 m from the center Thisvortex is seen in the next two plots to be transported down-stream by the flow In both LES and PIV the vortex is transportedabout 015 m down in the 15-second interval The computed


(a)

(b)

Fig 18mdashInstantaneous velocity vector plots in the upper region obtained from (a) LES and (b) PIV measurements

Fig 19mdashSnapshot of dye injection in the water model showing asymmetrybetween the two upper rolls

region is longer than 200 seconds The asymmetrical flowstructures shown here are likely one reason for the intermit-tent defects observed in steel slabs[53]

Asymmetric flow in the two halves is seen in both the com-putation and measurements The long-term experimental dataimplies that the period of the flow asymmetry in the lower


(a)

(b)

(c)

Fig 20mdashA sequence of instantaneous velocity vector plots in the lower roll region obtained from LES and PIV measurements showing evolution of flowstructures


Fig 21mdashSchematics showing the simplified simulations of the 04-scaledwater model

VII SIMPLFIED COMPUTATIONSIN MOLD CAVITY

Although less expensive than DNS LES still requires con-siderable computational resources for applications to industrialproblems The domain of the LES shown earlier includes thecomplete UTN the slide gate the SEN and the full moldcavity The computational cost may be lowered by reducingthe domain extent by simplifying the upstream domain thatdetermines the inlet conditions or by simulating flow in onlyhalf of the mold cavity by assuming symmetric flow in thetwo halves of the mold

This section presents results of two half-mold simulationswith simplified inlet conditions The curved tapering cavity wassimplified to be a straight domain with a constant thicknessequal to the thickness 03 m below the top surface The time-dependent inlet velocities from the nozzle port were obtainedfrom two simplified separate simulations The results are com-pared with the complete nozzle-mold simulation and PIV mea-surements presented earlier

For the first simplified simulation the unsteady velocitiesexiting the nozzle ports were obtained from a two-step simula-tion In the first step turbulent flow in a 32-mm diameter pipewith a 39 pct opening inlet (Figure 21) was computed usingLES Instantaneous velocities were collected every 001 sec-ond for 10 seconds at a location 0312 m downstream of theinlet They were then fed into a 32 32-mm rectangular duct(Figure 21) that represented the flow passage in the nozzlebottom containing the bifurcated nozzle ports Instantaneousvelocities were then collected every 001 second for 10 sec-onds a location 27 mm from the center of the duct Thesevelocities were turned by 30 deg (to match the measured jetangle) and employed as the unsteady inlet conditions for

the first mold simulation The velocities were recycled peri-odically for the duration of the mold simulation

For the second simplified simulation the inlet velocities werecomputed from a simulation of fully developed turbulent flowin a 32-mm-square duct The unsteady velocities were againcollected every 001 second for 10 seconds inclined 30 degand fed into the mold as the inlet conditions Figure 22 showsthe time-averaged cross-stream inlet velocities for these twosimulations A strong dual-swirl pattern is seen in the outletplane of the nozzle port in the first simulation (left) The cross-stream velocities for the second simulation (right) are verysmall Both of the simplified upstream simulations produce inletconditions different from that in the complete nozzle-mold sim-ulation (Figure 4) Further details on these simulations are givenelsewhere[13]

The turbulent flow in the half-mold cavity was next com-puted using the inlet velocities obtained above[13] The meanvelocity fields at the center plane (y 0) are shown in Fig-ure 23 Both of the plots reveal a double-roll flow patternsimilar to the complete nozzle-mold simulation and PIV mea-surements Comparisons of the time-averaged velocities inboth the upper and lower regions (not shown) also suggestthat these two simplified simulations roughly agree withresults of the full-mold simulations and PIV However astraight jet is observed in the second simulation which differsfrom the first simplified simulation the complete nozzle-mold simulation and the PIV The lack of cross-stream veloc-ities in the jet is believed to be the reason for the straightjet Neither of these simplified half-mold simulations cap-tured the instantaneous stair-step-shaped jet observed earlierBoth simulations missed the phenomena caused by the inter-action between the flow in the two halves which is reportedto be important to flow transients[51] Figure 24 illustrates thiswith sample velocity signals at a point 20 mm below thetop surface midway from the SEN and the narrow face com-pared with PIV It is observed that both the simplified simu-lations only capture part of the behavior of the measuredsignal The sudden jump of the instantaneous x velocity com-ponent which is reproduced by the full-mold simulation (Fig-ure 13) is missing from both half-mold simulations Thissuggests that the sharp velocity fluctuation is caused by theinteraction of the flow in the two halves The selection ofthe computational domain must be decided based on a fullconsideration of the available computational resources the inter-ested flow phenomena (eg flow asymmetry) and the desiredaccuracy

VIII CONCLUSIONS

Three-dimensional turbulent flow in a 04-scale watermodel was studied using LES The computed velocity fieldsare compared with PIV measurements The following obser-vations are made from this work

1 The partial opening of the slide gate induces a long com-plex recirculation zone in the SEN It further causes strongswirling cross-stream velocities in the jets exiting fromthe nozzle ports Complex flow structures consisting ofsingle and multiple vortices are seen to evolve in timeat the outlet plane of the nozzle port

2 A downward jet with an approximate inclination of 30 degis seen in both measurements and simulations The computed

oscillate between a large single vortex and multiple vorticesof various smaller sizes Large downward-moving vorticesare seen in the lower region

4 Significant asymmetry is seen in the instantaneous flowin the two halves of the mold cavity A 51-second aver-age reduces this difference in the upper region Howeverasymmetric flow structures are seen to persist longer than200 seconds in the lower rolls

velocities agree reasonably well with measurements in themold region The jet usually wobbles with a period of 05 to15 seconds

3 The instantaneous jets in the upper mold cavity alternatebetween two typical flow patterns a stair-step-shaped jetinduced by the cross-stream swirl in the jet and a jet thatbends upward midway between the SEN and the narrowface Furthermore the flow in the upper region is seen to


Fig 22mdashCross-stream flow patterns exiting the nozzle port in the simplified simulations

Fig 23mdashTime-averaged velocity vector plots obtained from the simplified simulations compared with full nozzle-mold simulation and PIV measurement


Fig 24mdashTime history of horizontal velocity toward SEN at points 20 mm below the top surface midway between the SEN and narrow faces

5 The instantaneous top surface velocity is found to fluctu-ate with sudden jumps from 001 to 024 ms occurringin a little as 07 second These velocity jumps are seenin both the full nozzle-mold simulations and the PIV mea-surements Level fluctuations near the narrow face occurover a wide range of frequencies with the strongest hav-ing periods of 7 and 11 to 25 seconds

6 The velocity fields obtained from half-mold simulationswith approximate inlet velocities generally agree with theresults of the full domain simulations and PIV measure-ments However they do not capture the interactionbetween flows in the two halves such as the instanta-neous sudden jumps of top surface velocity

ACKNOWLEDGMENTS

The authors thank the National Science Foundation (GrantNos DMI-98-00274 and DMI-01-15486) which made thisresearch possible The work is also supported by the membercompanies of the Continuous Casting Consortium Universityof Illinois at UrbanandashChampaign (UIUC) Special thanks areextended to Drs MB Assar and P Dauby for sharing thePIV data and helpful suggestions and to the National Centerfor Supercomputing Applications (NCSA) at UIUC for com-putational facilities

NOMENCLATURE

DDt total derivative

xi coordinate direction (x y or z)vi velocity component0 kinematic viscosity of fluidt turbulent kinematic viscosityeff effective viscosity of turbulent fluid densityp static pressuret time

a

t vj

xjb

ksgs sub-grid scale turbulent kinetic energyi grid size (in x y and z directions)

Subscripti j direction (x y z)

REFERENCES1 L Zhang and BG Thomas Iron Steel Inst Jpn Int 2003 vol 43 (3)

pp 271-912 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

3 BG Thomas in The Making Shaping and Treating of Steel 11th edCasting Volume AW Cramb ed The AISE Steel FoundationWarrendale PA 2003 p 24

4 J Herbertson QL He PJ Flint and RB Mahapatra SteelmakingConf Proc ISS Warrendale PA 1991 vol 74 pp 171-85

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

6 R Sobolewski and DJ Hurtuk 2nd Process Technology Conf ProcISS Warrendale PA 1982 vol 2 pp 160-65

7 D Gupta and AK Lahiri Steel Res 1992 vol 63 (5) pp 201-048 D Gupta and AK Lahiri Metall Mater Trans B 1996 vol 27B

pp 757-649 MB Assar PH Dauby and GD Lawson Steelmaking Conf Proc

ISS Warrendale PA 2000 vol 83 pp 397-41110 BG Thomas X Huang and RC Sussman Metall Mater Trans B

1994 vol 25B pp 527-4711 BG Thomas and L Zhang Iron Steel Inst Jpn Int 2001 vol 41 (10)

pp 1181-9312 FM White Viscous Fluid Flow 2nd ed McGraw-Hill Series in

Mechanical Engineering McGraw-Hill Boston MA 1991 p 61413 S Sivaramakrishnan Masterrsquos Thesis University of Illinois at

UrbanandashChampaign Urbana 200014 Q Yuan BG Thomas and SP Vanka Metall Mater Trans B 2004

vol 35B pp 685-70215 ERG Eckert in Fluid Mechanics Measurements RJ Goldstein ed

Taylor amp Francis Washington DC 1996 pp 65-11416 RJ Adrian in Fluid Mechanics Measurements RJ Goldstein ed

Taylor amp Francis Washington DC 1996 pp 175-29917 NJ Lawson and MR Davidson J Fluids Eng 2002 vol 124 (6)

pp 535-4318 RJ Adrian Annu Rev Fluid Mech 1991 vol 23 pp 261-30419 II Lemanowicz R Gorissen HJ Odenthal and H Pfeifer Stahl

Eisen 2000 vol 120 (9) pp 85-93


20 SB Pope Turbulent Flows Cambridge University Press CambridgeUnited Kingdom 2000 p 771

21 DE Hershey BG Thomas and FM Najjar Int J Num Meth Fluids1993 vol 17 pp 23-47

22 FM Najjar DE Hershey and BG Thomas 4th FIDAP Users ConfEvanston IL 1991 Fluid Dynamics International Inc Evanston IL1991 pp 1-55

23 H Bai and BG Thomas Metall Mater Trans B 2001 vol 32Bpp 253-67


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

26 MR Aboutalebi M Hasan and RIL Guthrie Metall Mater TransB 1995 vol 26B pp 731-44

27 XK Lan JM Khodadadi and F Shen Metall Mater Trans B 1997vol 28B pp 321-32

28 X Huang and BG Thomas Can Metall Q 1998 vol 37 (304)pp 197-212

29 BG Thomas Mathematical Models of Continuous Casting of SteelSlabs Report Continuous Casting Consortium University of Illinoisat UrbanandashChampaign Urbana IL 2000

30 J Knoepke M Hubbard J Kelly R Kittridge and J Lucas Steel-making Conf Proc Chicago IL Mar 20ndash23 1994 ISS WarrendalePA 1994 pp 381-88

31 Q Yuan SP Vanka and BG Thomas 2nd Int Symp on Turbulentand Shear Flow Phenomena Stockholm Royal Institute of Technology(KTH) Stockholm 2001 p 6

32 U Schumann J Comput Phys 1975 vol 8 pp 376-40433 K Horiuti J Phys Soc Jpn 1985 vol 54 (8) pp 2855-6534 H Schmidt and U Schumann J Fluid Mech 1989 vol 200

pp 511-6235 U Schumann Theor Comput Fluid Dyn 1991 vol 2 pp 279-9036 WW Kim and S Menon AIAA 97-0210 American Institute of Aero-

nautics and Astronautics (AIAA) New York NY 1997

37 FH Harlow and JE Welch Phys Fluids 1965 vol 8 (112)pp 2182-89

38 J Crank and P Nicolson Proc Cambridge Phil Soc 1947 vol 43pp 50-67

39 LF Sampine and MK Gordon Computer Solution of Ordinary Diff-erential Equations the Initial Value Problem WH Freeman amp CoSan Francisco CA 1975

40 Center for Applied Scientific Computing Lawrence Livermore NationalLaboratory Livermore CA [Report No UCRL-MA-137155 DR] 2001

41 Q Yuan B Zhao SP Vanka and BG Thomas unpublished research2004

42 R Madabushi and SP Vanka Phys Fluids 1991 vol 3 (11)pp 2734-745

43 H Tennekes and JL Lumley A First Course in Turbulence The MITPress Cambridge MA 1992 pp 197-222


45 FM Najjar BG Thomas and DE Hershey Metall Mater Trans B1995 vol 26B pp 749-65

46 H Bai PhD Thesis University of Illinois at UrbanandashChampaignUrbana IL 2000

47 A Cramb Y Chung J Harman A Sharan and I Jimbo Iron Steel-maker 1997 vol 24 (3) pp 77-83

48 Q Yuan BG Thomas and SP Vanka Metall Mater Trans B 2004vol 35B pp 703-14

49 S Sivaramakrishnan H Bai BG Thomas P Vanka P Dauby andM Assar Ironmaking Conf Proc ISS Warrendale PA 2000 vol 59pp 541-57

50 D Gupta and AK Lahiri Metall Mater Trans B 1994 vol 25Bpp 227-33

51 T Honeyands and J Herbertson 127th ISIJ Meeting ISIJ Tokyo 199452 D Gupta S Chakraborty and AK Lahiri Iron Steel Inst Jpn Int 1997

vol 37 (7) pp 654-5853 R Gass Inland Steel Inc East Chicago IN private communication

1992

Numerical simulation is another powerful tool used to studyturbulent flow in continuous casting Models of turbulent flowcan be classified into the Reynolds-averaged approach largeeddy simulation (LES) and direct numerical simulation (DNS)[20]

Because of its low computational cost the Reynolds-averagedapproach typically with the two-equation (k-) turbulence

model has been extensively adopted in previous studies andhas produced valuable insights about the flow in continuouscasting nozzles[21ndash24] and molds[1025ndash29] However this approachlimited by its nature is not suited for studying the time evo-lution of unsteady flow structures triggered by flow instabili-ties Plant observations suggest that flow transients under thenominally steady operating conditions are very important[30]

The LES and DNS approaches are better for solving the time-dependent flow of the continuous casting process in whichthe Reynolds number is of the order of 105 Due to the pro-hibitive computational cost of DNS at high Reynolds num-bers LES is a more feasible way for solving this complexflow problem Recently a few attempts have been made toapply it to the continuous casting process[131431] The princi-pal idea of LES is that during the simulation the time evolu-tion of the large-scale (energy-containing) eddies is resolvedand the small energy-dissipative eddies are ldquofilteredrdquo The fil-tering of the small eddies generates a residual stress ten-sor[20] in the NavierndashStokes momentum transport equationwhich is included using a subgrid scale (SGS) model AlthoughLES is less expensive than DNS it still requires consider-able computational effort In this article the transient flowstructures in a 04-scale water model are investigated usingLES computations and PIV measurements

II WATER MODEL

Figure 1(a) depicts the schematics and dimensions of a 04-scale water model constructed from transparent plastic platesat the former LTV Steel Technology Center (ClevelandOH)[913] The flow rate in this water model is controlled by a


Table I Dimensions and Operating Conditions of the 04-Scale Water Model

DimensionsConditions Value

Slide-gate orientation 90 degSlide-gate opening (area) 39 pctSEN bore diameter 32 mmSEN submergence depth 75 mmPort height width 32 31 mmPort thickness 11 mmPort angle lower edge 15 deg downPort angle upper edge 40 deg downBottom well recess depth 48 mmWater model height 950 mmWater model width 735 mm

(corresponding full scalecaster width) 1829 mm (72 in)

Water model thickness 95 mm (top) to 65 mm (bottom)

(corresponding full scalecaster thickness) 229 mm (9 in)

Outlet at the bottom of the 3 round35-mm-water model diameter holes

Inlet volumetric flow ratethrough each port 353 104 m3s

Mean velocity inside nozzle bore 0439 msCasting speed (top thickness) 102 mms

(0611 mmin)Water density 1000 kgm3

Water kinematic viscosity 10 106 m2sGas injection 0 pct

(a)

(b)

Fig 1mdashSchematics of the 04-scale water model (a) dimensions and (b) the computational domain






[1]

[2]

where

[3]


[4]

ksgs

t vi

ksgs

xi nt 0 S 20 C

k32sgs

xi av02vt

ksgs

xib

neff n0 nt

Dvi

Dt

1r

p

xi

xjneff avi

xj

vj

xib

vi

xi 0

where

[5]

[6]


[7]

where

[8]










a vx

x

vy

y

vz

zb

Sij 1

2 avi

xj

vj

xib

0 S 0 12SijS

ij

0 S 0nt Clk32

sgs

(xyz)13





IV PIV MEASUREMENTS




a 1

(t2 t1)at2

ti t1



V FLOW IN THE SEN









vy

1vx2 vz

2212


(a)

(b)



1vx2 vz

2212







(a) (b)

(c) (d)








(a)

(b)







vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992






[1]

[2]

where

[3]


[4]

ksgs

t vi

ksgs

xi nt 0 S 20 C

k32sgs

xi av02vt

ksgs

xib

neff n0 nt

Dvi

Dt

1r

p

xi

xjneff avi

xj

vj

xib

vi

xi 0

where

[5]

[6]


[7]

where

[8]










a vx

x

vy

y

vz

zb

Sij 1

2 avi

xj

vj

xib

0 S 0 12SijS

ij

0 S 0nt Clk32

sgs

(xyz)13





IV PIV MEASUREMENTS




a 1

(t2 t1)at2

ti t1



V FLOW IN THE SEN









vy

1vx2 vz

2212


(a)

(b)



1vx2 vz

2212







(a) (b)

(c) (d)








(a)

(b)







vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992




IV PIV MEASUREMENTS




a 1

(t2 t1)at2

ti t1



V FLOW IN THE SEN









vy

1vx2 vz

2212


(a)

(b)



1vx2 vz

2212







(a) (b)

(c) (d)








(a)

(b)







vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992





vy

1vx2 vz

2212


(a)

(b)



1vx2 vz

2212







(a) (b)

(c) (d)








(a)

(b)







vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992







(a) (b)

(c) (d)








(a)

(b)







vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992







(a)

(b)







vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992






vx

1vx2 vz

2212




vxiquest vxiquest



(u 2 w2)12


















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992
















(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992







(a)

(b)







(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992






(a)

(b)






(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992




(a)

(b)

(c)











VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992










VIII CONCLUSIONS















ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992












ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992





ACKNOWLEDGMENTS


NOMENCLATURE



a

t vj

xjb





















Eisen 2000 vol 120 (9) pp 85-93

































1992

































1992

Documents

in a 0.4-Scale Water Model of a Continuous Steel Caster ... · Computational and Experimental Study of Turbulent Flow in a 0.4-Scale Water Model of a Continuous Steel Caster QUAN