Impinging Jets Confined by a Conical

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FLUID M ECHANICS AND TRANSPORT PHENOMENA

Impinging Jets Confined by a Conical Wall :Laminar Flow Predictions

J oao M. Miranda and J oaoB. L. M. Campos Centro de Estudos de Fenomenos de Transporte, Dept. de Engenharia Qumica, Faculdade de Engenharia da Universidade do Porto, 4099 Porto Codex, Portugal

A laminar j et flow confined by a conical wall and an impinging plate was inesti-gated. The Naier-Stokes equatio ns were num erical ly soled by a finite difference tech-nique and the results compared with laser D oppler anemom etry data, the latter al so

coering the transition and turbulent flow regimes. Transition was found to start in theimpingement region at a jet Reynolds number of around 1,600. The inestigation con-

(centrat ed on assessin g the effects of nozzle-to-plate di stance always less than one noz-)zle diameter , jet Reynolds number, and nozzle outlet conditions. The shape of the

elocity profile at the nozzle outlet determines the entire cell flow field, whereas nozzle-to-plate distance affects the flow in the expansion region. Under certain flow and geo-metric conditi ons a recirculation zone appears in the expansion region, in theicinity ofthe plate.

Introduction

Impinging jets are encountered in many industrial applica-

tions, such as in paper and textile drying processes, in steelmills, glass tempering, turbine blades, and electronic compo-nents cooling. The jet impinges on a solid surface and spreadsout along the surface with a high radial velocity, thus induc-ing high mass- and heat-transfer rates.

Two types of liquid single-phase jets are commonly re-ferred to in the literature: submerged jets and free-surfacejets. In submerged jets the liquid jet issues into a region con-taining the same liquid at rest, whereas in free-surface jets

.the liquid jet is surrounded by ambient air by a gas . Sub-merged jets can be unconfined or confined by an upper sur-face, usually by a plate attached to the nozzle and parallel tothe impingement surface. This article describes the investiga-tion of a laminar impinging jet, confined by a conical wall

extended from the nozzle to a short distance above the im- .pingement plate 0.1 to 0.3 nozzle diameters .

.G arimella and R ice 1995 divided the flow field of an im-pinging jet into three regions: the free-jet region, the im-pingement region, and the wall region. The flow in the free-jetregion is axial and is not affected by the presence of the im-

Correspondence concerning this article should be addressed to J. B. L. M. Cam-

pos.

pingement plate; at the nozzle exit, the a xial velocity starts to

decay a nd the jet spreads to the surroundings. G ardon a nd . .Akfirat 1965 and Martin 1977 have shown that the flow of

turbulent jets starts to be affected by the impingement sur-face at approximately 1.2 nozzle diameters from this surface.In the present laminar study, the nozzle-to-plate distance islower than 1.0 nozzle diameter and, as will be shown later,the effect of the surface starts near the nozzle exit. At theimpingement region, the fluid accelerates and is forced tochange to the radial direction. At the wall region, the flow ispredominantly radial with a growing boundary layer.

In a coming article, results of experimental and numericalstudies on mass transfer from a soluble impingement platewill be presented. The emphasis in the present article is toinvestigate the flow hydrodynamics and the effects of several

geometric and dynamic para meters and to select one configu-ration for the mass-transfer experiments. The final objectiveof such studies is to simulate and investigate the flow and themass-transfer rates near a separation membrane, in particu-lar to investigate the so-called polarization effect.

The numerical methods employed to solve the Navier-Stokes equations are described in the next section. It followsthe description of the experimental setup and of the laserDoppler technique. After that, the numerical predictions areinterpreted and compared with the experimental data.

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Figure 1. The cell.

Numerical Studies

The objective of the numerical study is to predict the lami-nar flow pattern inside the conical cell. To this purpose, therewere two possibilities: either integrate numerically the properflow equations or use a commercial CFD code. The secondoption was straightforward, but the development of a new

code to study mass transfer near a separat ion membrane re-.verse osmoses and ultrafiltration processes was preferred for

this investigation. .Polat et al. 1989 reviewed the numerical literature on flowand heat-transfer characteristics of impinging jets. They re-

viewed the numerical studies on unconfined and confined by.a plane wa ll impinging jets in both the laminar and t urbulent

regimes, and summarized t he corresponding numerical tech-niques. The effects of physical and geometric parameters onflow patterns and on heat-transfer rates were also described.

Various numerical predictions of impinging jet flows havebeen carried out, most of them referring to turbulent regime.

.Miyazaki and Silberman 1972 studied the flow and heat-transfer characteristics of a two-dimensional laminar uncon-fined impinging jet. They obtained the streamwise flow veloc-ity from potential theory and solved the boundary layer and

energy equations by a finite difference method to evaluatethe flow and the heat-transfer parameters near the plate. Saad .et al. 1977 solved the Na vier-Stokes and energy equations

to predict the flow and local heat-transfer characteristics of a .confined by an upper plane wall laminar impinging jet. Mu-

. jumdar and D ouglas 1980 studied a confined by an upper.plane wall laminar jet of hot humid air impinging on an

isothermal wet surface. They solved simultaneously the flow,mass, and energy equations. No reference to numerical stud-ies about impinging jets confined by a conical wall was foundin the literature.

Brief description of the conical cell

The cell is shown in Figure 1. The liquid jet, with an aver-age velocity V, enters the cell from a circular nozzle of diam-jeter D . Within the cell the jet is confined by a conical wall,jwhich attached to the nozzle and impinges on a flat round-shaped plate of diameter D , placed perpendicularly to thepaxis of the nozzle. The nozzle-to-plate distance, H, was less

than one nozzle diameter, and the exit area height, L dis-tance between the plane containing the basis of the conical

.wall and the impingement plate , ranged from 0.1 to 0.3 noz-zle diameter.

Flow equations

The flow equations are written below, in the stream func-tion-vorticity formulation. The coordinates are normalized bythe nozzle diameter, D, the velocity components are normal-jized by the average jet velocity, V, and Re represents the jetjR eynolds number:

2 1 2rs y q 1 .

2 2

r rz r

1 2 1 2r q y s y q q . 2 .r z 2 2 2 /r z r Re r rr r z

Equation 1 is a Poisson-type of equation and Eq. 2 is theso-called vorticity transport equation. The dimensionless vor-ticity and stream function are defined as

r zs y 3 .

z r

1 1 sy ; s . 4 .z r

r r r z

These equations are written with the assumption of laminarand incompressible flow, constant fluid properties, and nobuoyancy effects.

Numerical domain and boundary conditions

The elliptical nature of the flow equations requires thespecification of the boundary conditions at all sides of thenumerical domain. The flow is axisymmetric, therefore only

.half of t he cell is taken for the numerical doma in Figure 2 .When the inlet flow is fully developed and laminar, the do-main begins at approximately one pipe diameter upstream ofthe nozzle to a ccount for the impingement effects on the flowat the nozzle exit. When the inlet flow is uniform or a devel-

oping laminar flow, the calculation domain begins at the noz-zle exit.

Boundary I: Fluid Inlet. The dimensionless velocity profileof a fully developed laminar pipe flow is

s0; sy2 1y4r2 . 5 . .r z

Figure 2. Boundaries of the numerical domain.

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For a uniform flow at the nozzle exit, the dimensionless ve-locity profile is given by

s0; sy1. 6 .r z

A dimensionless developing laminar flow profile can be wellapproximated by

nq2 n s0; sy 1y 2r , 7 . .r z n

. where n can take values from 2 laminar flow to uniform.flow .

The stream function and the vorticity function correspond- .ing to the preceding three inlet velocity profiles Eq s. 5 to 7

can be determined by application of t heir definitions Eq s. 3.and 4 using the reference, s0 for rs0 and any z.

For fully developed laminar flow:

s r2 1y2r2 ; sy16r. 8 . .

For uniform flow at the nozzle exit:

r2

s ; s0. 9 .2

For a developing laminar flow at the nozzle exit:

nnq2 1 2r .

2 n ny1s y r ; sy nq2 2 r . 10 . .n 2 nq2

Along the solid boundaries tube wall, conical wall, and im-.pingement surface the no-slip condition determines the val-

ues of the velocity components, as per the following.Boundary II: Tube Wall.

1 z s0; s0; s ; sy . 11 .z r 8 r

Boundary III: Confining Conical Wall.

1 r z s0; s0; s ; s y . 12 .r z 8 z r

Boundary I V: Im pingement Surface.

r s0; s0; s0; s . 13 .r z

z

Boundary V: Fluid Exit. The outlet is located sufficientlyfar from the zone of interest to allow the application of acondition of developed flow parallel to the impingement plate,which has no upstream influence. Thus,

s0; s0; sy . 14 .z

r r r

.Boundary VI: Cell Axis. At the axis of symmetry rs0

the magnitude of the gradients rz, rr and rz isr znull,

s0; s0. 15 .

Numerical method and grid

A finite difference technique was used to discretize the flowequations. The domain is not rectangular, and it was neces-sary to use a boundary fitted grid. The grid used is shown inFigure 3 and has three distinct zones. In zones A a nd C thegrid is orthogonal and aligned with the jet axis and the im-pingement plate, thus defining a cylindrical grid. In zone Bthe grid is still axisymmetric but nonorthogonal; the grid wasgenerated by lines aligned with the jet axis and by inclinedlines with different slopes from the conical wall to the im-pingement plate. These nonorthogonal coordinates are notcoincident with the cylindrical coordinates of the equationsto be solved, thus an adequate algebraic transformation mustbe carried out.

. The coordinates of the nonorthogonal grid are s, z Fig-.ure 4a with the direction of the unit vector schanging from

the direction of the conical wall to that of the impingementplate. The radial coordinate, r, is related with sthrough thefollowing relation

rss , 16 .

cos

. 0with ranging from 12 inclination of the conical wall to 0 .inclination of the impingement plate .

The flow equations were transformed from cylindrical intogrid coordinates by using the preceding transformation

2 2 1 2rs 1q tg q y . 2 2 / rco s sz s

tg 2 tg 2y q 17 .r z cos zs

1 r q q tg y .r z rcos s z r

2 2 21 2 tg 1 2s 1q tg q q q . 2 2 /Re cos zs rco s sz s

tg q y , 18 .

2r z r

Figure 3. Representation of the grid employed in thenumerical study.

. .In regions A and B qC, t h e gr i d had 219 7 an d 1 0053nodes in the r- and z-directions, respectively.



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Figure 4. Representation of the grid in the nonorthogonal region. . . .a Coordi n a t e s of t h e g r id an d de t a i l aroun d t h e n ode i, j ; b i l lustration of the interpolation scheme applied to the nodes connecting

zones A

B .

with

1 1 1 sy q tg ; s . 19 .z r /r cos s z r z

Equations 1719 apply everywhere in the domain, even inzones A and C where s0.

The first and second derivatives in Eqs. 1719 were dis-cretized in the internal nodes of the domain. In the Poissonequation, the derivatives were approximated by a second-order a ccurate central difference scheme. After discretiza-

tion an equation of the form

r sA q A qA q A i, j i, j 1 iy1, j 2 iq1, j 3 i, j 4 i, jy1

qA qA qA qA 5 i, jq1 6 iy1, jy1 7 iq1, jy1 8 iy1, jq1

qA 20 .9 iq1, jq1

was obtained for each node, where the coefficients A to A1 9are listed in the Appendix.

For the nodes connecting zones AB a n d B C, the sec-ond-order central-difference scheme is applicable only withthe help of auxiliary nodes located out of the grid, becausethere is a discontinuity in direction s. For the nodes connect-

ing zones AB, the discretized Poisson equation is of the fo rm

r sA q A q A qA i, j i, j 1 iy1, j 2 iq1, j 3 i, j 4 i, jy1

q A 21 .5 i, jq1

where is the value of the stream-function variable atiq1,jthe auxiliary node, represented in Figure 4b by a square sym-bol. The value of was obtained through a fourth-orderiq1,jpolynomial interpolation applied to the values in the four

. .nearest nodes to iq1, j located in the vertical line iq1of the nonorthogonal grid. The coefficients A to A are1 5listed in the Appendix. An identical procedure was applied tothe nodes connecting zones B C .

At the boundaries, the radial coordinate was transformedinto the nonorthogonal coordinate with Eq. 16, and thederivatives were approximated by forward or backward dif-ference, with an accuracy of second order, always in the in-ward cell direction. For the stream-function boundary condi-tions, an equation of the form

B q B q B s b 22 .1 iy2, j 2 iy , j 3 i, j

was obtained for each node, where the coefficients B to B1 3and b are listed in the Appendix.

The vorticity transport equation was discretized by an up-wind scheme: the derivatives of the diffusive terms were ap-proximated by central difference and the derivatives of theconvective terms by forward or backward difference; in bothapproximations the accuracy was of second order. For thenodes next to the boundaries, the first derivatives of the con-vective terms were a pproximated by fo rward or ba ckward dif-ference, with an accuracy of first order. For each node anequation of the form

C qC qC qC qC 1 iy1, j 2 iq1, j 3 i, j 4 i, jy1 5 i, jq1

qC qC qC qC 6 iy1, jy1 7 iq1, jy1 8 iy1, jq1 9 iq1, jq1

qC qC qC qC s0 23 .10 iy2, j 11 iq2, j 12 i, jy2 13 i, jq2

was obtained, where the coefficients C to C are listed in1 13the Appendix.

For t he nodes connecting zones AB and B C the interpo-lation method described for the Poisson equation was ap-plied. The vorticity transport equation was discretized in these



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nodes by an upwind scheme of first order: the derivatives ofthe diffusive terms were discretized by a second-order accu-rate central difference scheme, and the derivatives of theconvective terms by a first-order accurate forward or back-ward difference scheme.

The vorticity boundary conditions were discretized by ap-plying an identical scheme to that described earlier for thestream-function boundary conditions, and an equation of theform

A q A qA q A qA q A q A q A q A y r1 iy1, j 2 iq1, j 3 i, j 4 i, jy1 5 i, jq1 6 iy1, jy1 7 iq1, jy1 8 iy1, jq1 9 iq1, jq1 i, j i, jR s ,i, j

A 3 i, j

D q D q D s d 24 .1 iy2, j 2 iy1, j 3 i, j

was obtained for each node where the coefficients D to D1 3and d are listed in the Appendix.

After discretization, two sets of interrelated algebraicequations were obtained. One set arises from the discretiza-tion of the Poisson equation and stream-function boundary

conditions, the o ther set from the discretization of t he vortic-ity transport equation and vorticity boundary conditions. Theywere solved by an iterative procedure and in each step the

.first set was solved by the a lternating direction implicit AD Imethod and the second set by the G auss-Seidel method.

The convergence of the iterative procedure was studiedfollowing the evolution of the values of . This study re-rquires the prior identification of the region with the slowestconvergence rate, usually at the beginning of the expansionregion, fo llowed by the choice o f a representative point. Theiterative evolution of the values of in such a location wasrcarried out and is plotted in Figure 5 for Res300. The val-

.ues of converge asymptotically to a constant value .r r 1000th After the 1000th iteration the absolute difference y r r

is less than 10y3

,and the a bsolute difference between consec-utive values is always less than 10y5. Similar conclusions wereobtained regardless of the jet Reynolds number.

The convergence of the iterative procedure was also stud-ied following the evolution of the sum of the tot al norma lizedresidues. For example, for the vorticity transport equation,

Figure 5. Analysis of the convergence of the iterativeprocess.Evolution of a n d o f R along the i terations at Res 300.r s

the total normalized residue was defined by

nm i

R i, jis 1 js 1

R s , 25 .20.p

with

where p is the number of the internal nodes of the grid.The sum of the total residues, represented by R , is thes

sum of R with the residual of Eq . 23 and of all boundary20.conditions, that is,

R s R q R q R q R . 26 .s 20. 22. 23. 24.

The iterative evolution of R is also represented in Figure 5sfo r Res300. There is a decrease in the value of R along thescomputation, and after about 1000 iterations its value is 510y5, three to four orders of magnitude less than at the be-ginning of the process. Similar conclusions were obtained re-gardless of the jet Reynolds number.

The fact that, after 1,000 iterations the values of tendsrtoward a stabilized value with R still decreasing, confirmssthe convergence of the iterative method to correct values.

A criterion of convergence wa s established: the iterativeprocess was completed when during one hundred iterationsthe absolute difference between consecutive values of wa srless than 10y5 and the value of R was also less than 10y5.s

The accuracy of the numerical method was determinedfrom solutions on successively refined grids. The rms error

.defined by Fletcher 1988 , and ba sed on the normalized ve-

locity components, was used for that purpose:

1r2nm i2rms s y rp , 27 . . . r r ri,j i,j /

is 1 js 1

where starred and non-starred values represent quantitiescalculated with grids having p and pnumber of nodes, re-spectively.

Three tests were performed to study the solution accuracy.The grids used, and the values of rms found are shown inTable 1. The values of an d taken for reference werer zthose obtained with the finest grid. The schemes used in the

Table1. Resultsof theAccuracyTestsatRes685

rmsG rid Nodes .Z one A Z one B qC E q . 27

r z r z r z6 25 25 14 0.15 0.04411 49 50 27 0.034 0.01121 97 100 53 0.0059 0.002641 193 200 105 R eference R eference



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Figure 6. Acrylic test section employed in the laser ex-periments.

discretization of the equations were of second order in al-most all the domain and so, when the grid spacing is halved,the new value of rms is expected to be about 4 times lowerthan the previous one. The ratios between consecutive rmsvalues in the table are of this order of magnitude, and so it isreasonable to infer that the solution of the algebraic equa-tions is converging to the exact solution. The grid chosen tosolve the equations was that in the third line of the table andthe respective solutions at Res685 have an rms error lowerthan 0.01. This value increases with jet Reynolds number, andis about 0.02 at Res1,685.

Experimental Studies

Test secti onFigure 6 shows the acrylic test section. The enclosing box

had flat external and internal walls to attenuate the refrac-tion of the laser beams. The internal diameter of the ap-proaching pipe was 11.7 mm and its length was sufficient toguarantee a well-developed velocity profile at the nozzle exit.The pipe and conical wall could be displaced to change the

.exit area height L in the ra nge 1.1 to 3.3 mm. The nozzle-to-plate distance, H, ranged from 7.6 mm to 9.6 mm and thediameter of the impingement plate was D s77.1 mm.p

The fluids employed were ta p wa ter a nd a queous solutionsof glycerol, and they circulated in a closed circuit at a con-trolled temperature of 25C. The density and dynamic viscos-ity of the water and glycerol solutions at this temperature

were measured. The flow rate was a djusted by a control valveand measured by a rotameter.

L DA measur ements

The velocity field inside the conical wall was not measuredbecause the refraction of the laser beams did not allow acontrol volume to be formed. To overcome this difficulty, aliquid with a refractive index matched to that of the acrylicmust be used, but for the present purpose it was sufficient to

measure the mean radial velocity and the radial turbulenceintensity in the cell aperture L , with no curved surface to becrossed by the laser beams.

A 100-mW argon-ion laser was used and the LDA oper-ated in the dual-beam forward-scatter mode. The flow direc-tion was discriminated by an optical frequency shifting of 0.5or 0.7 MHz from a Bragg cell . The front lens had a focallength of 160 mm, and the half-angle between the two fo-cused beams was 6.45. The measuring volume dimensions

.were 0.86 mm in the longitudinal direction of the bea ms Xand of 0.10 mm in the two transverse directions, Y and Z .see Figure 7 for the definition of the coordinate system .

The scattering particles were natural contaminants of thefluid and the scattered light was collected by a 300-mm lensand focused on to the pinhole of a photomultiplier beforewhich was placed an interference filter of 514.5 nm. The pho-tomultiplier signal was band-pass filtered to remove high- andlow-frequency noise, and the resulting signal was processedby a TSI 1990C counter o perating in the single-measure-ment-per-burst mode, with a frequency validation setting at1% with 10r16 cycle comparison. The counter was interfacedby a 1400 Dostek card with a 80486-based computer, whichprovided all the statistical quantities through a purpose-built

software.The test section was placed on an orthogonal coordinate

table that allowed displacements in all directions. The dis-placement uncertainties were 0.2 mm in the X- and Y-di-rections and of 0.01 mm in the Z-direction.

The test section was aligned with the laser beams so that .1 the plane containing the laser beams was parallel to the

.impingement plate; 2 the bisectrix of the angle defined bythe laser beams was perpendicular to a lateral wall of the test

.section; 3 the direction of the bisectrix was coincident with .one of the orthogonal coordinates X in Figure 7 . With this

alignment, the velocity component in the direction perpen- .dicular to the bisectrix Y in Figure 7 was measured, that is,

the radial velocity component according to Figure 7.

Seven transverse profiles of the radial-velocity componentwere measured to investigate the flow development. Fo r eachprofile, the radial velocity was measured at several pointsalong the cell a perture L , the minimum distance between twoconsecutive measurements being 0.05 mm.

The displacements along Ywere done, taking point B as a .reference Figure 7 . This point was visually defined a s the

Figure 7. Laser system.



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interception of the laser beams with the periphery of theplate; the positioning uncertainty was of the order of magni-tude of the displacement error of the test section along Y.For the displacements along Z the reference was the platesurface; the positioning uncertainty was of the order of mag-nitude of the dimension of the control volume along Z.

Velocity profiles were also measured along the radius OA .Figure 7 to confirm the flow symmetry. The relative devia-tions between the radial velocity values in axisymmetrical po-

sitions were less than 3%.

Flow Patterns

Flow patterns obtained by numeri cal simul ation

The flow inside the cell can be analyzed in three distinctregions: the impingement region, where the flow changes fromaxial to radial due to the presence of the plate; the wall re-gion, where the radial flow is confined to the downside by theplate and to the upside by the recirculating fluid; and theexpansion region, where the fluid is confined by the plateand conical wa ll. The extension of ea ch region and the corre-

sponding flow patterns depends on geometric HrD, LrD ,j j. and HrD and hydrodynamic parameters Re and velocityp

.profile at the nozzle exit . The free jet region upstream of theimpingement is not co nsidered here beca use, as will be shown,the small separation between the plate and the nozzle exitaffects the flow, that is, the impingement region starts at thenozzle exit.

Im pingement Region. The impingement plate imposes ashift in the jet flow direction. The fluid decelerates in theaxial direction, losing kinetic energy that is converted intopressure energy. The deceleration starts a t the nozzle exitand intensifies on approaching the plate along the axis. Theincreased pressure is then transformed into radial momen-tum of the fluid.

The jet Reynolds number determines the pressure andvelocity fields in the impingement region, and can be

interpreted as the ratio between the axial inertial flux of

momentum and the radial viscous flux of momentum. Thisinterpretation is analogous to that usually found in energyand mass transport, and is adequate to explain the develop-ment of the flow. Thus, the kinematic viscosity, , becomesthe momentum diffusivity by analogy. The entire flow field isthree-dimensional and complex; however some details of itsdevelopment can be emphasized:

1. As soon as the f luid exits the inlet pipe to enter the cell,it ceases to be submitted to the pipe-wall shearing stresses.

At low jet Reynolds numbers, the momentum diffusivity en-ables a fast radial momentum transfer from the center to theperiphery of the jet, thus changing the shape of the axial ve-locity profile and inducing some radial velocity. This transferprocess is masked by impingement effects at the nozzle exitregion.

2. The internal shearing stresses, which are proportionalto the momentum diffusivity, play a shock-absorber role dur-ing the axial deceleration of the fluid. At low jet Reynoldsnumbers, this da mpening eff ect elongates the decelerationzone axially toward the nozzle.

3. In the immediat e vicinity of the cell axis and o f the plate,the fluid turns radially. This stagnation zone has the highestpressures, a nd consequently the flow in the vicinity becomes

radial with the highest momentum. At low jet R eynolds num-bers, the momentum diffusivity q uickly spreads this radialmomentum axially, and the fluid placed at the periphery ofthe jet and far from the plate also shifts its flow direction.

In summary, the radial and axial extension of the deceler-ated zone is largely dependent on the jet Reynolds numbervalue. This is shown well in the streamline plots of Figure 8aat two different values of Re. The lower the jet Reynoldsnumber, the larger the zone of influence of the plate, that is,fluid further away from the wall changes in direction.

The cell region bounded by the conical wall and by theradial flowing fluid is occupied by a recirculating region.

The flow in the impingement region can be modified bygeometric or hydrodynamic parameters. In this respect, the

numerical simulations showed tha t:

Figure 8. Streamlines representation forRes100 and for Res685 at LrrrrrDs0.1 and differentinlet conditions.j . .a Pa rabolic velocity profile at the nozzle exit ; b uniform velocity profile at the nozzle exit .



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Figure 9. Radial velocity profiles and streamline bor-dering the first recirculation zone for Res685. .a Pa rabolic velocity profile at the nozzle exit for LrD s 0.1j

.a n d f o r LrD s 0.3; b parabolic and uniform velocity pro-jfiles at the nozzle exit and LrD s 0.1.j

1. For any jet Reynolds number, a change of the dimen-sionless distance LrD , within the range 0.1 to 0.3, does notjaffect the flow field; the streamlines for LrDs0.1 are iden-jtical to tho se pertaining to LrDs0.3. The corresponding ra-jdial velocity profiles at the exit of the impingement regionare compared in Figure 9a for Res685, and they are similar.

2. The velocity profile at the nozzle exit determines theflow pattern in the impingement region. The decelera-tionracceleration rates of the fluid depend on the initial ra-dial momentum distribution o f the jet. I f the momentum pro-

file at the nozzle exit is uniform instead of para bolic for the.same bulk velocity , the zone of increased pressure is larger

and the fluid in the periphery of the jet shifts direction far-ther from the plate. In Figure 8b a spread flow and a small

recirculation zone a re observed.Wall Region. The predominantly radial flow in the wall

region is confined by the impingement plate and the recircu-lation region. The recirculation length increases radially withthe jet R eynolds number, a s can be assessed f rom the stream-line plots of Figure 10a corresponding to LrDs0.1.j

At low jet Reynolds numbers, the low radial velocities atthe exit of the impingement region, together with a high-momentum diffusivity, provide conditions fo r a fast expan-sion of the fluid to the entire cell cross-section area. In spite

of the adverse pressure gradient in the expansion region, noseparation was observed for the fluid flowing next to the plate.

With increasing jet Reynolds numbers, the high radial ve-locity in the thin layer next to the impingement plate and thelow local momentum diffusivity delay the expansion of thefluid.

When LrD increases, the thickness of this fluid layer re-jmains practically unchanged, and there is a small extensionof the recirculation zone in the radial direction. The radial

velocity profiles at the inlet, middle, and outlet of the wallregion and the streamline bordering the recirculation regionsare represented in Figure 9a for Res685 and two values of

.LrD 0.1 and 0.3 . At the inlet and middle of the wall region,jthe radial velocity profiles coincide, but they are not compa-rable at the exit due to the difference in cell geometry. Theinlet velocity profile has a sharp shape, with the highest ve-locity near the plate. Along the wall region, the boundarylayer develops and the maximum radial velocity moves awayfrom the plate.

The velocity profile at the nozzle exit has a large influenceon the flow in the impingement region and consequently alongthe wall region. Figure 9b compares the radial velocity pro-files at the inlet, middle, and outlet of the wall region and

the shape of the recirculation zones for the uniform andparabolic profiles at the nozzle exit at Res685. For a uni-

Figure 10. Streamlines bordering the recirculationzones for parabolic velocity profile at thenozzle exit. .a First recirculation zone for different Rea t LrD s 0.1;j .b second recirculation zone for different Rea t LrD s 0.3.j



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form-velocity inlet, the recirculation zone is shorter, thechannel along the wall region is thicker, and the fluid has alower velocity than for the parabolic inlet.

Expansion Region. The expanding flow downstream of therecirculation depends on t he jet R eynolds number as follows:

1. For low jet Reynolds numbers, if the radial-flow cross-section a rea increases with r for the present case, the maxi-

mum area normal to the flow is reached at rs1.7 for LrDsj.0.1 , the fluid flows against an adverse pressure field with

loss of momentum, but no fluid separation from the plateoccurs.

2. When increasing jet R eynolds numbers, a second recir-culation zone develops close to the impingement plate. Fluidflows radially along the wall region against an adverse pres-sure gradient, and hence with a constant loss of momentum.By the time the cross-section area increases at the end of thefirst recirculation zone, the remaining momentum is notenough to prevent fluid separation from the plate. The exten-sion and location of this second recirculation zone dependson Re, LrD , and the velocity profile at the nozzle exit.j

As stated earlier, for increasing values of LrD , there is ajslight extension of the upper recirculation zone, as shown inFigure 9a. In spite of this extension, the cross-section areaincreases and the fluid expansion occurs in a more adversepressure field. This results in an enlargement of the secondrecirculation zone; the corresponding limiting streamlines forLrDs0.3 and several values of Re are represented in Fig-jure 10b. For Res1000, the recirculation zone is located closeto the exit of the cell, and for increasing values, the recircula-tion zone exits the cell, originating an inflow of liquid andpromoting instability.

For any jet Reynolds number and in the investigated rangeof LrD , when the profile at the nozzle exit is uniform, thejsecond recirculation zone is not observed. The momentumlost by the flow along the wall region is not enough to induceseparation during fluid expansion.

For low jet Reynolds numbers, regardless of what the ve-locity profiles at the nozzle exit are, the radial velocity pro-

.files coincide after the expansion Figure 9b .

Experimental data

Velocity Profiles. The experimental work had two ma in ob- .jectives: 1 to study the conditions for laminar-to-turbulent

.transition; and 2 the validation of the simulation data.These were carried out by measuring the radial velocity

profiles of the flow in the aperture L near the impingementsurface, where the laser beams did not cross any curved sur-face. In Figure 11a, the experimental velocity profiles a recompared with the numerical predictions for two values of

Reand t he agreement is quite good, except for profiles in thevicinity of the impingement region. Some numerical simula-tions were carried out to assess the effect of the shape of theinlet velocity profile, a nd in particular for profiles pertaining

.to non f ully developed flow different values of nin Eq . 7 ; itwas observed that a small deviation from a fully developedprofile was enough to justify that disagreement. This is clearlyshown in Figure 11b, which compares experimental data withpredictions, taking ns2.5 for Res685 and ns3.5 for Res1,685; the agreement along the whole plate is now quite good.

Figure 11. Transverse profiles of the radial velocitycomponent.Comparison between experimental dat a and numerical

.predictions, considering a a fully developed laminar pro- .f i le a t t h e n ozzle e xit ; b a de vel opin g l amin ar prof i le a t

the nozzle exit, ns 2.5 for Res 685 and ns 3.5 for Res .1685 n is the exponent of Eq. 7 .

Since the length of the inlet pipe is theoretically enough toestablish a fully developed profile, the presence of small per-turbations to the flow in the pipe may be a possible reasonfor having an undeveloped velocity profile at the inlet.

The profiles of Figure 11a are representative of the numer-

ous profiles obtained for different Rea nd LrD.jL aminar-to-Turbulent Transition. In their review article,

.Polat et al. 1989 observe that for short nozzle-to-impinge-ment plate distances, the flow field is considered laminar upto Res2,500, but they also point out that this value has notbeen experimentally confirmed.

In order to investigate the conditions for the onset of tran-sition, the mean and fluctuating radial velocity componentswere measured by LDA, the latter quantified by the statisticrms:



10/13

Figure 12. Representation of v and of rmsrrrrrV vs.Re at LrrrrrDs0.1.r j j . .The symbols represent laser data and the solid l ines numerical laminar predictions: a rs 0.94 an d zs 0.034; b rs 1.79 a nd zs 0.034;

. .c rs 2.22 a nd zs 0.034; d rs 2.65 a nd zs 0.034.

1r22N y . .r ri

rms s , 28 .Ny1

is 1

.where is the ith-instantaneous radial-velocity reading; r i ris the mean radial-velocity; and N is the number of data

.points in the sample N 4,000 .For each jet Reynolds number in the range 100 Re

30,000, the measurements were performed in eight radial po-sitions along the flow for the same axial coordinate.

The experimental values of the radial velocity were com-pared with the corresponding predictions in laminar flow. Theagreement was good for the laminar regime, but otherwise apersistent deviation showed the onset of transition. The on-set of transition is also marked by a sudden increase in therms values. This comparison is carried out in Figures 12a 12d,which are representative of the LrDs0.1 case. In each fig-jure part, and rmsrV are plotted against Refor each radialr jposition in a semilog scale.

Fo r Reabove 200, the data represented in Figures 12a and

12b were obtained in the wall region, in positions below therecirculation zone. A good agreement is observed at low jetReynolds numbers, but above a critical value there is a per-sistent deviation when comparing experimental and numeri-cal mean-velocity values. This critical jet Reynolds number isapproximately the same at which rmsrVstarts to increase. InjFigure 12a such critical Re is around 1,400, while in Figure12b it is around 1,600.

Fo r Reabove 600, the data represented in Figure 12c wereobtained below the recirculation zone. The mean a nd the

fluctuating velocities vary in the same way as previously, withthe experimental values of beginning to deviate from therpredictions when the rmsrVvalues start to increase at an Rejaround 1,600.

The data of Figure 12d were measured in the region wherethe second recirculation zone develops. The experimentalvalues of

begin to deviate from the predictions at an Re

raround 1,600, whereas the rmsrV starts to increase earlier,jat an Rearound 1,400.

In conclusion, the majority of the experimental data sug-gest that the laminarturbulent transition occurs at aroundRes1,600. Similar values of critical Reobserved along dif-ferent ra dial positions suggest that transition probably beginsat the impingement region.

An identical study was carried out for LrDs0.3 and thejsame conclusions were drawn.

Flow confined by: conical walls. parallel plate

The flow patterns inside a cell with a plate attached to thenozzle and parallel to the impingement surface were ob-

tained by numerical simulation. That flow pattern is pre-sented in Figure 13 and compared with that obtained in aconical cell, for Res400 a nd with the same no zzle-to-im-pingement-plate distance. The flow patterns are similar, ex-cept that with the parallel-plate confinement both recircula-tion zones are longer. When the confinement plates are short,the recirculating zones burst through the outflow boundary,leading to an inflow that promotes instability inside the cell.

.For the geometry studied DrDs6.6, HrDs0.65 this re-p j jcirculation zone bursts at an Ref400.



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Figure 13. Flow pattern developed in a conical cell vs. a cell with parallel plates atRes400 and HrrrrrD s0.65.j

Conclusions

The laminar flow of a jet confined by a conical wa ll extend-ing from the nozzle to a short distance above an impinge-ment plate can be analyzed separately in three regions: theimpingement region, the wall region, and the expansion re-gion. The jet Reynolds number and the inlet velocity profilehave a strong influence on the whole flow, while the nozzle-to-plate distance is influential only in the expansion region.At low jet Reynolds numbers, the fluid far from the plateacquires radial velocity and a short recirculation zone close

to the conical wall is observed. At high jet Reynolds num-bers, this recirculating zone enlarges, the fluid flows radiallyin a thin channel attached to the impingement plate, and asecond recirculation zone develops in the expansion region,close to the plate. If the nozzle-to-plate distance increases,the second recirculation zone enlarges. For the uniform ve-locity profile at the nozzle exit, instead of pa rabolic, the fluidflows in a thick channel with a low radial velocity, and thesecond recirculation zone does not develop. Transition fromlaminar to turbulent flow probably begins in the impinge-ment region at an Re around 1600. The effect of the conicalconfinement on the flow is felt in the size of the recirculationzones: for the same jet Reynolds number the recirculationzones are longer when the confinement is accomplished by

parallel plates. Then, for small impinging plates, the conicalwall has a stabilizing effect on the flow, preventing the recir-culation zones from bursting through the exit of the cell.

Acknowledgments

The authors acknowledge the financial support given by JNICT,p r o j e c t P B I C rC rC E Nr1337r9 2, a n d b y F . C . T. , P r o je c tP R AX I SrCrE Q Ur12141r1998.

Notation

msnumber of nodes of the grid along boundary IVnsnumber of nodes along the ith verical line of the gridi

rsdimensionless radial coordinate, RrDjRsradial coordinate, m

sdimensionless radial velocity, VrVr r j sdimensionless axial velocity, VrVz z jVsradial velocity, m sy1rVsaxial velocity, m sy1z

zsdimensionless axial coordinate, ZrDjZsaxial coordinate, m

Literature Cited

Fletcher, C. A. J., Computational Techniques for Fl uid Dynamics I,Springer Series in Com putational Physics, Springer-Verlag, Berlin .1988 .

G ardon, R ., and J. C. Akfirat, The R ole of Turbulence in Deter-mining the Hea t Transfer Chara cteristics of Impinging J ets,Int. J.

.H eat M ass Transfer, 8, 1261 1965 .G arimella, S. V., a nd R . A. Rice, Confined and Submerged Liquid

Jet Impingement H eat Transfer, ASM E J. H eat Transfer, 117, 871 .1995 .

Martin, H., Heat and Mass Transfer Between Impinging Gas Jets .and Solid Surfaces, Ad. Heat Transfer, 13, 1 1977 .

Miyazaki, H., and E. Silberman, Flow and Heat Transfer on a FlatPlate Normal to a Two-Dimensional Laminar Jet Issuing from a

.Nozzle of Finite H eight, Int. J. H eat M ass Transfer,15, 2097 1972 .Mujumdar, A. S., Y. K. Li, and W. J. M. Douglas, Evaporation

U nder an Impinging Jet: A Numerical Mod el, Can. J. Chem. Eng., .58, 448 1980 .P ola t , S ., B . H u a ng , A. S . Mu ju md a r, a nd W. J . M. Dou glas,

Numerical Flow and Heat Transfer Under Impinging Jets: A Re- .view, Ann. Re. Numer. Fluid Mech. Heat Transfer, 2, 157 1989 .

Saad, N. R., W. J. M. Douglas, and A. S. Mujumdar, Prediction ofHeat Transfer Under an Axi-Symmetric Laminar Impinging Jet,

.Ind. Eng. Chem., Fundam., 16, 148 1977 .

Appendix

The coefficients of the discretized flow equations, pre- .sented in this Appendix, are expressed for the node i, j

represented in Figure 4a.The coefficients of the discretized Poisson equation are

computed from the following expressions:

A sA q A q A A s A y 2A y 2A1 1,1 1,2 1,3 2 j 1,1 j 1,2 j 1,3

A sy A qA q A q A q A y 2A .3 1 2 4,1 4,2 4,1 4,2

A sA qA yA yA4 4,1 4,2 6 7

A s A y 2A yA yA5 4,1 4,2 8 9

2A z y z r y r . .0 i, jq1 i, j iq1, jy1 i, jy1

A s6r y r r y r . .i, jy1 iy1, jy1 iq1, jy1 iy1, jy1 jy1

A sy2 A7 jy1 6

2A z

yz r

yr . .

0 i, j i, jy1 iq1, jq1 i, jq1A sy8r y r r y r . .i, jq1 iy1, jq1 iq1, jq1 iy1, jq1 jq1

A sy2 A ,9 jq1 8

where

2 j jA s0

z y z z y z z y z . . .i, j i, jy1 i, jq1 i, j i, jq1 i, jy1



12/13

2 1q 2 .jA s1,1 2r y r r y r . .i, j iy1, j iq1, j iy1, j j

2 2A z y z y z y z r y r . . .0 i, j i, jy1 i, jq1 i, j iq1, j i, j

A s1,2r y r r y r . .i, j iy1, j iq1, j iy1, j j

r y riq1, j i, jA s1,3

r r y r r y r . .i, j i, j iy1, j iq1, j iy1, j

2 1q 2 .jA s4,1

z y z z y z . .i, j i, jy1 i, jq1 i, jy1

z y z .j i, jq1 i, jA s4,2

r z y z z y z . .i, j i, j i, jy1 i, jq1 i, jy1

and

z y ziy1, j iq1, j sj

r y riq1, j iy1, j

2 2z y z q r y r . .

' i

y1, j i

q1, j i

q1, j i

y1, j

sjr y riq1, j iy1, j

r y r z y zi, j iy1, j i, j i, jy1s s .j

r y r z y ziq1, j i, j i, jq1 i, j

The coefficients of the discretized vorticity transport equa-tion a re computed from t he f ollowing expressions:

C sC qC qC qC1 1,1 1,2 1,3 1,4

C sy2C q C y 2C qC2 j 1,2 j 1,3 j 1,4 2,1

C sy C qC qC qC qC y 2C q C3 1 2 4,2 4,3 5,1 4,2 4, 31 ri,j

qC qC qC qC q y.10 11 12 13 2 rRe r i, ji, j

C sC qC qC yC yC4 4,1 4,2 4,3 6 7

C sy 2C q C qC yC yC5 4,2 4,3 5 ,1 8 9

2C z y z r y r . .0 i, jq1 i, j iq1, jy1 i, jy1

C s6r y r r y r . .i, jy1 iy1, jy1 iq1, jy1 iy1, jy1 jy1

C sy2 C7 jy1 6

2C z y z r y r . .0 i, j i, jy1 iq1, jq1 i, jq1

C sy8 r y r r y r . .i, jq1 iy1, jq1 iq1, jq1 iy1, jq1 jq1

C sy 2 C9 jq1 8

2 2C r y r C r y r . .1,1 i, j iy1, j 2,1 iq1, j i, j

C sy C sy10 112 2r y r r y r . .i, j iy2, j iq2, j i, j

2 2C z y z C z y z . .4,1 i, j i, jy1 5,1 i, jq1 i, j

C sy C sy ,12 132 2z y z z y z . .i, j i, jy2 i, jq2 i, j

where

1C sy A0 0

Re

r y r . r i, j iy2, ji, j 0 yri,j~ r y r r y r . .C s i, j iy1, j iy1, j iy2, j1,1

0 0ri,j

r y riq1, j i, jC s1,2

r r y r r y r Re . .i, j i, j iy1, j iq1, j iy1, j j

2 1q 2 .jC sy1,3 2r y r r y r Re . .i, j iy1, j iq1, j iy1, j j

2 2C r y r z y z y z y z . . .0 iq1, j i, j i, j i, jy1 i, jq1 i, j

C s1,4r y r r y r . .i, j iy1, j iq1, j iy1, j j

0 0 ri,j

~ r y r .C

s r iq2, j i, ji,j2,1

0ri,j r y r r y r . .iq1, j i, j iq2, j iq1, jz y z q . . i, j i, jy2 z j ri,j i,j

q 0 yz j ri,j i,j~ z y z z y z . .C s i, j i, jy1 i, jy1 i, jy24,1 q 0 0z j ri,j i,j z y z .j i, jq1 i, j

C s4,2r z y z z y z Re . .i, j i, j i, jy1 i, jq1 i, jy1

2 1q 2 .jC sy4,3

z y z z y z Re . .i, j i, jy1 i, jq1 i, jy1

q 0 0 z j ri,j i,j~ z y z q . .C s i, jq2 i, j z j r i,j i,j5,1

q 0z j ri,j i,j z y z z y z . .i, jq1 i,j i, jq2 i, jq1

with A , , and previously defined.0 j jFor the nodes located next to the boundaries, the first

derivatives of the convective terms in the vorticity transportequation were approximated by a first-order scheme, and sosome coefficients are expressed in a different way

C sC sC sC s0.10 11 12 13

ri,j 0 yri,j r y r~ i, j iy1, jC s1,1 0 0ri,j 0 0 ri,j

~ rC s i,j2,1 0ri,j r y riq1, j i, jNovember 1999 Vol. 45, No. 11 AIChE J ournal2284


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q z r ji,j i,j q 0 yz r ji,j i,j z y z~ i, j i, jy1C s4,1 q 0 0z r ji,j i,j q 0 0 z r ji,j i,j

q ~ z r jC s .i,j i,j5,1 q 0z r ji,j i,j z y zi, jq1 i, j The coefficients of the boundary equations a re com-

puted from the following expressions:

Boundary I

n2 rnq2 1 .i, j2B s B s0; B s1; bs r y1 2 3 i, j

n 2 nq2

D s D s0; D s1; dsy nq2 2 nrny1. .1 2 3 i, j

Boundary II

1B s B s0; B s1; bs1 2 3 8

4 y z ziy1, j iy2, jD s D s0; D s1; ds ,1 2 3 2 rj

with rs r y r s r s r .j i , j iy1,j iy1,j iy2,jBoundary III

1B s B s0; B s1; bs1 2 3 8

D s D s0; D s1;1 2 3

4 y 4 y z y zr r z z i, j iq1, ji,jy1 i,jy2 i,jy1 i,jy2dsy q ,

2 z r y r 2 zi iq1, j i, j i

with zs z y z s z y z .i i , j i , jy1 i,jy1 i,jy2Boundary IV

B s B s0; B s1; bs 01 2 3

4 y r ri,jq1 i,jq2D s D s0; D s1; ds1 2 3 2 zi

with zs z y z s z y z .i i,jq1 i,j i,jq2 i,jq1Boundary V

B s1; B sy4; B s3; bs01 2 3

2 rjD s1; D sy4; D s3q ; ds0,1 2 3

ri, j

with rs r y r s r y r .j i , j iy1,j iy1,j iy2,jBoundary VI

B s B s0; B s1; bs 01 2 3

D s D s 0; D s1; ds0.1 2 3

M anuscript receied Jan. 28, 1999, and reision receied July 27, 1999.


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Impinging Jets Confined by a Conical