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Ali M. JawarnehAssistant Professor

Department of Mechanical Engineering,Hashemite University,Zarqa 13115, Jordan

e-mail: jawarneh@hu.edu.jo

Georgios H. VatistasProfessor

Department of Mechanical and IndustrialEngineering,

Concordia University 1455 DeMaisonneuve Blvd.West,

Montreal, H3G 1M8 Canadae-mail: vatistas@me.concordia.ca

Reynolds Stress Model in thePrediction of Confined TurbulentSwirling FlowsStrongly swirling vortex chamber flows are examined experimentally and numericallyusing the Reynolds stress model (RSM). The predictions are compared against the ex-perimental data in terms of the pressure drop across the chamber, the axial and tangen-tial velocity components, and the radial pressure profiles. The overall agreement betweenthe measurements and the predictions is reasonable. The predictions provided by thenumerical model show clearly the forced and free vortex modes of the tangential velocityprofile. The reverse flow (or back flow) inside the core and near the outlet, known fromexperiments, is captured by the numerical simulations. The swirl number has been foundto have a measurable impact on the flow features. The vortex core size is shown tocontract with the swirl number which leads to higher pressure drop, higher peak tangen-tial velocity, and deeper radial pressure profiles near the axis of rotation. The adequateagreement between the experimental data and the simulations using RSM turbulencemodel provides a valid tool to study further these industrially important swirlingflows. �DOI: 10.1115/1.2354530�

IntroductionSwirling flow occurs in many engineering applications, such as

ortex separators, pumps, gas turbine combustors, furnaces, sprayryer, the vortex valve, the vortex combustor, and gas-coreuclear rocket. In modern combustors, swirl is used to produceood mixing and to improve the flame stability. In all confinedortex applications, it is important to understand adequately theverall flow field evolution as a function of both the geometricalnd flow parameters. A good knowledge of these flows will im-rove the design and performance of a variety of vortex devices.

It is well known that the tangential velocity of the confineduid changes from free to forced vortex as the flow approaches

he axis of rotation. The static pressure in an attempt to balancehe centrifugal force will reduce from a maximum value near theylindrical wall of the chamber to a minimum on the axis ofotation. Depending on the inlet swirl intensity, the pressure insidehe core might drop below the outside ambient, thus inducing aeverse flow. Escudier et al. �1� demonstrated experimentally thexial and swirl velocities distributions using Laser Doppler An-momerty �LDA� measurements. The experiments were per-ormed with water for a range of exit diameters. The observationevealed a remarkable change in the vortex structure as the exitiameter is reduced, where the vortex core size changes from ahick core to a thin core. In addition, the axial velocity was foundble to develop profiles ranging from jetlike to wakelike shapes,hus revealing the evolution of the reverse flow. Vatistas �2� re-orted a model for single- or double-celled intense vortices, de-ending on the values of scaling constants. It was shown that thexial velocity component may attain profiles ranging from jetlikeo wakelike. The last was an attempt to mathematically simulatehe reverse flow conditions. Sullivan’s �3� two-celled vortex

odel can also approximately simulate the direction reversal ofhe radial and axial velocity components near the axis of rotation.

The major obstacle in numerical modeling of complex turbulentwirling flows is the selection of appropriate turbulence closureodels. In simple flow cases, the k−� model performs well. How-

ver, for strongly swirling flows that involves severe streamline

Contributed by the Fluids Engineering Division of ASME for publication in theOURNAL OF FLUIDS ENGINEERING. Manuscript received February 9, 2005; final manu-

cript received March 22, 2006. Assoc. Editor: Ugo Piomelli.

ournal of Fluids Engineering Copyright © 20

bending it fails. The last conclusion is clearly evident in a varietyof studies; see for example the work of Nallasamy �4�, Nejad et al.�5�, and Weber �6�. A review of second-moment computations forengineering flows has been provided by Launder �7�, Leschziner�8�, and Ferziger and Peric �9�. The results of these computationsdemonstrate the superiority of RSM over eddy-viscosity modelsfor curved flows, swirling flows and recirculating flows. Jones etal. �10� have studied the performance of second moment closureturbulence models for swirling flow in a cylindrical combustionchamber. The models are found to predict mean and turbulent flowquantities well. German and Mahmud �11� have shown that theoverall agreement between the measurements and the predictionsobtained with both the k−� and Reynolds-stress turbulence mod-els are reasonably good. However, some features of the isothermaland combusting flow fields are better predicted by the Reynolds-stress model. Jakirlic et al. �12� have shown numerically usingthree versions of the second-momentum closure and two eddy-viscosity models that the second-momentum models are superior.However, difficulties in predicting accurately the transformationfrom free- to forced-vortex modes or the determination of thenormal stress components inside the core still remain. Vortexchamber flows at low Reynolds number via direct numericalsimulations were investigated by Orland and Fatica �13�. Jonesand Pascau �14� and Hoekstra et al. �15� used the k−� turbulentmodel and a Reynolds stress transport equation model of a strongconfined swirling flow. Once more, comparisons of the resultswith measurements show the superiority of the transport equationmodel, where k−� gave large discrepancies between the measuredand predicted velocity fields.

Since a Reynolds stress model �RSM� takes into considerationthe effects of severe streamline bending due to swirl in a moreappropriate way than the one- and two-equation models, it is bestsuited for the present study. The aim of this paper is to study theflow features in a vortex chamber experimentally and numericallyusing FLUENT �Fluent Inc.�, and to compare the results obtainedusing a Reynolds stress model to available experimental data ofvortex chambers operating under different swirl numbers. Perfor-mance assessment of the RSM in the predicting turbulent, strongly

swirling vortex chamber flows will also be one of the objectives.

NOVEMBER 2006, Vol. 128 / 137706 by ASME

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Experimental SetupThe experiments have been conducted using a jet-driven vortex

hamber similar to the one utilized by Vatistas et al. �16�. Theain difference between the two is that in the latest version,

hown schematically in Fig. 1. It has a cylindrical configurationith constant cross-sectional area �Ro=7 cm� and a central axisutlet and circumferential inlets. Swirl is imparted to the fluid viahe vortex generator shown in Figs. 1 and 2. It has four perpen-icular air inlets where the compressed air is induced. The re-uired set of inlet conditions is obtained by the insertion of theppropriate vortex generator blocks �swirler� into the vortex gen-rator assembly a long the periphery of the vortex generator. Aumber of openings of a circular cross section �din� are drilled atspecified angle �=30 deg. When the air flow passes through the

wirlers, it is guided to enter the vortex chamber in the radial andangential directions so that swirl is formed inside the vortexhamber. The swirler has 16 holes with diameter din=1.267 cmnd inlet area �Ain=20.177 cm2�. Chamber diameter ratio ���,hich is defined as the ratio of the diameter of a vortex chamber

Do� to the diameter of the exit hole �De�, was varied from �2.5, 3.33, 3.67, 4.0, 5.01, 5.29, 5.80, 6.47, 7.08 to 7.45. Chamberspect ratio ���, which is defined as the ratio of the chamberength L to the diameter of a vortex chamber �Do� was fixed at=3.00. Area ratio ���, which is defined as the ratio of the total

nlet area �Ain� to the cross-sectional area of the vortex chamber

o was fixed at �=0.131.The measurements were made at inlet air flow rate Qin

0.0187 m3/s, which is corresponding to Reynolds number Reo11,592, which is defined based on the average velocity as

Reo =4Qin

��Do

he static pressure is measured by a series of taps located aheadf the tangential ports and is averaged by connecting in parallel allhe pressure pickup tubes into a common tube. The measurementsf the mean gage pressure ��p= pin− pa� were obtained using a

Fig. 1 Schematic of the vortex chamber

Fig. 2 Inlet flow boundary condition

378 / Vol. 128, NOVEMBER 2006

U-tube filled with Meriam oil, having a specific gravity equal to1.00. The estimated uncertainty is less than ±8% for the pressuredrop measurements. A rotameter was used to measure the volu-metric flow rate of the inlet air. This was carefully calibrated instandard conditions �1 atm and 20±0.5 % °C�. For the flow rateused, the uncertainty was estimated to be ±2%.

3 Computational Details

Governing Equations. In Reynolds averaging, the solutionvariables in the instantaneous Navier-Stokes equations are decom-posed into the mean and fluctuating components. For the velocitycomponents: ui= ui+ui� where ui and ui� are the mean and fluctu-ating velocity components. Likewise, for pressure and other scalarquantities: = +� where denotes a scalar such as pressure.Substituting expressions of this form for the flow variables intothe instantaneous continuity and momentum equations and drop-ping the over-bar on the mean velocity u yields the momentumequations. They can be written in Cartesian tensor form as

�

�xi�ui� = 0 �1�

�

�xj�uiuj� = −

�P

�xi+

�

�xj��� �ui

�xj+

�uj

�xi−

2

3�ij

�ul

�xl��

+�

�xj�− ui�uj�� �2�

Equations �1� and �2� are called Reynolds-averaged Navier-Stokes�RANS� equations. Additional terms now appear that represent theeffects of turbulence. These Reynolds stresses, −ui�uj�, must bemodeled in order to close Eq. �2�.

Reynolds Stress Transport Equations. The Reynolds stressmodel �17� involves calculation of the individual Reynoldsstresses, ui�uj�, using differential transport equations. The indi-vidual Reynolds stresses are then used to obtain closure of theReynolds-averaged momentum �Eq. �2��. The transport equationsfor the transport of the Reynolds stresses, −ui�uj�, can be writtenas follows:

�

�xk�ukui�uj��

Cij

=�

�xk��t

k

�ui�uj�

�xk� +

�

�xk��

�

�xk�ui�uj���

− �ui�uk��uj

�xk+ uj�uk�

�ui

�xk�

Pij

+ �ij −2

3�ij� �3�

The term on the left-hand side of Eq. �3� represents the convec-tion, the terms on the right-hand side represent the turbulent dif-fusion as proposed by Lien and Leschziner �18�, molecular diffu-sion, stress production, pressure strain, and the dissipation,respectively. The pressure strain term �ij is simplified accordingto the proposal by Gibson and Launder �19�:

�ij = �ij,1 + �ij,2 + �ij,w �4�

�ij,1 = − C1�

k�ui�uj� −

2

3�ijk� �5�

�ij,2 = − C2��Pij − Cij� −1

�ij�Pkk − Ckk�� �6�

3

Transactions of the ASME

wuC�t

wLnt

T

weK

eurllfl

TsAt

Tb

wt

Tg

4

stsi

J

�ij,w = C1��

k�uk�um� nknm�ij −

3

2ui�uk�njnk −

3

2ui�uk�nink� k3/2

Cl�d

+ C2��km,2nknm�ij −3

2ik,2njnk −

3

2 jk,2njnk� k3/2

Cl�d�7�

here C1=1.8, C1�=0.5, C2�=0.3, nk is the xk component of thenit normal to the wall, d is the normal distance to the wall, and

l=C�3/4 /�, where C�=0.09 and � is the von Kármán constant

=0.4187�. The scalar dissipation rate � is computed with a modelransport equation similar to that used in the standard k−� model

�

�xi��ui� =

�

�xj��� +

�t

�� ��

�xj�C�1

1

2Pii

�

k− C�2

�2

k�8�

here k=1.0, C�1=1.44, and C�2=1.92 are constants taken fromaunder and Spalding �20�. When the turbulence kinetic energy iseeded for modeling a specific term, it is obtained by taking therace of the Reynolds stress tensor

k =1

2ui�ui� �9�

he turbulent viscosity �t is computed similarly to the k−� model

�t = C�

k2

��10�

here C�=0.09. Because of severe pressure gradients, the non-quilibrium wall functions were used near wall as proposed byim and Choudhury �21�.

Inlet Conditions for the Reynolds Stresses. Whenever flownters the domain, the values for individual Reynolds stresses,

i�uj�, and for the turbulence dissipation rate ��� can be input di-ectly or derived from the turbulence intensity and characteristicength. The turbulence intensity I can be estimated from the fol-owing formula derived from an empirical correlation for pipeows

I =u�

Vavg= 0.16�Reo�−1/8 �11�

he turbulence length scale l is a physical quantity related to theize of the large eddies that contain the energy in turbulent flows.n approximate relationship between l and the physical size of

he vortex chamber diameter Do is

l = 0.07Do �12�

he relationship between the turbulent kinetic energy k and tur-ulence intensity I is

k =3

2�VavgI�2 �13�

here Vavg is the average axial velocity. The turbulence dissipa-ion rate � can be determined as

� = C�3/4k3/2

l�14�

he values of the Reynolds stresses explicitly at the inlet areiven by

ui�uj� = 0 and ui�2 =

2

3k �15�

Turbulence Modeling in Swirling FlowsThe problem is considered to be an incompressible, steady, axi-

ymmetric, and turbulent swirling flow. In this case, we can modelhe flow in two-dimensional �2D� �i.e., solve the axisymmetricwirl problem� and incorporate the prediction of the swirl veloc-

ty; see Fig. 3.

ournal of Fluids Engineering

The difficulties associated with the solution of strongly swirlingflows can be attributed to high degree of coupling in the momen-tum equations. High fluid rotation gives rise to large radial pres-sure gradient, which drives the flow in the meridional plane. This,in turn, determines the distribution of the swirl in the field. Nu-merical instabilities that are attributed to momentum coupling re-quire special solution techniques in order to obtain a convergedsolution. Hence, a segregated, implicit solver, which is wellsuitedfor the sharp pressure, and velocity gradients are more appropriatefor the flow under consideration. The mesh is sufficiently must bealso sufficiently refined in order to resolve the expected large flowparameter gradients. The under-relaxation parameters on the ve-locities were selected 0.3–0.5 for the radial and axial, and 0.9 forthe azimuthal velocity components.

There is a significant amount of swirl in the chamber. The ap-propriate choice depends on the strength of the swirl, which canbe gaged by the swirl number. To characterize the degree of swirl-ing flow in a vortex chamber, a swirl number S is introduced.Based on Gupta et al. �22� definition,

S =G– �

G– zRe�16�

where G−� is the axial flux of swirl momentum,

G– � =0

Ro

VzV�r2dr �17�

G– z is the axial flux of axial momentum,

G– z =0

Ro

Vz2rdr �18�

To simplify the calculation of swirl number, the free-swirl velocityprofile and the average axial velocity are assumed inside the vor-tex chamber.

V� =V�inRo

r, Vz = Vavg �19�

The ultimate form of the swirl number �S� can be determined as

S =V�in

Vavg� �20�

Then, the simulations were performed for different swirl numbersvaried from S=12.5–20.

Grid Generation. A major challenge in calculating the flowinside the vortex chamber is providing an adequate description ofthe geometry. Because of the complex geometry of the vortexgenerator, control over the grid is limited, making it difficult toreduce the size of mesh without losing accuracy in the results.Also, the grid size is limited by the computer memory available.This leads to use axisymmetric problem, the formulation of 2Dgrid generation are shown in Fig. 4. Triangular mesh elements andan unstructured grid were used. A grid independent solution studywas made by performing the simulations for three different gridsconsisting of 30,000, 43,000, and 50,000 nodes. The mean swirl

Fig. 3 Computational domain

velocity for the three different grid sizes is shown in the Fig. 5.

NOVEMBER 2006, Vol. 128 / 1379

fiiitb

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1

Boundary Conditions. Boundary conditions have to be speci-ed in order to solve the governing equations; see Fig. 3. At the

nlet, the values can be calculated from the given conditions at thenlet, boundary; see Fig. 2. The total inlet velocity vector �Vin� haswo components Vr,in and V�,in and they are related to each othery:

V�,in = Vin cos �, Vr,in = Vin sin �, Vin =Qin

Ain�21�

owever, at the outlet boundary there is no information about theariables and some assumptions have to be made. The diffusionuxes in the direction normal to the exit plane are assumed to beero. The pressure at the outlet boundary is calculated from the

Fig. 4 Computational grid near the exit

Fig. 5 Grid independent solution study

380 / Vol. 128, NOVEMBER 2006

assumption that radial velocity at the exit is neglected since itdoes not have the space to develop, so that the pressure gradientfrom r momentum is given by

�p

�r=

V�2

r�22�

At the solids walls, the no-slip condition was applied where thevelocities at the walls were specified to be zero. The centerlineboundary was considered axis of symmetry.

Discretization Scheme. The pressure-velocity coupling ishandled by using the SIMPLE-algorithm, the pressure staggeringoption scheme was used for the pressure interpolation, the firstorder upwind schemes were used for momentum, swirl velocity,turbulence kinetic energy, turbulence dissipation rate, and Rey-nolds stresses. Convergence was assumed when the residual of theequations dropped more than 3 orders of magnitude.

5 Results and Discussion

Pressure Drop Coefficient. Pressure drop or loss can be re-garded as energy loss from the point of view of energy conserva-tion. In the vortex chamber, pressure drop occurs mainly throughthe dissipation of the swirl velocity as proposed by Jawarneh et al.�23�. The pressure drop coefficient is defined as

Cp =2�p

Vin2

The estimated uncertainty for the pressure drop coefficient �Cp�has appeared at the maximum of ±9%. Figure 6 compares thepresent experimental data to the RSM prediction of the pressuredrop coefficient �Cp� for aspect ratio �=3.0 and inlet angle �=30 deg. It is clear that as the diameter ratio ��� increases, thepressure coefficient �Cp� increases. Stronger vortices will be pro-duced by increasing the diameter ratio, resulting in a higher tan-gential velocity and hence a higher pressure drop. It can be seenthat the Reynolds stress model gives good agreement with theexperimental data and the percentage difference error between thepredicted and experiments is �10%.

Mean Swirl Velocities Profiles. The predicted and measured

Fig. 6 Pressure drop coefficient

radial profiles of mean tangential, axial velocities and radial pres-

Transactions of the ASME

s1fnc�ag�cstta

J

ure for the chamber at station H=0.8L are shown in Fig. 7, 8, and0, respectively. Figure 7 shows the mean swirl velocities profilesor the configuration ��=30 deg, �=3.0, �=0.131� at Reynoldsumber �Reo=11,592� and the predicted swirl velocity results areompared with available experiments data �LDA� from Yan et al.24� at three diameter ratios ��=2.5,3.33,4.0�. It is shown thebility of RSM to capture the free-vortex and forced-vortex re-ions. Because of the intense swirl by increasing the swirl numberS�, a high level of swirl momentum is transported around theentreline and the consequence is the formation of the intensewirling vortex along the center-line. In the latter figure, the peakangential velocity increases with increasing the diameter ratio orhe swirl number, and the location where the tangential velocity ismaximum moves towards the vortex chamber center.

Fig. 7 Mean swirl velocity

Fig. 8 Dimensionless mean axial velocity

ournal of Fluids Engineering

Mean Axial Velocity Profile. The mean axial velocity compo-nent is able to develop profile ranging from jetlike to wakelikeshape as shown in Fig. 8. The predicted axial velocity results arecompared with available experiments data �LDA� from Escudieret al. �1�. The reverse flow �backflow� in the vortex core is due tothe reduction of static pressure to values that are below the ambi-ent and the stagnation point is appeared clearly in Fig. 8. Figure 9shows the predicted axial velocities vectors near the chamber exit,a flow-reversal region is found in the vortex core.

Mean Radial Pressure Profiles. The following analysis illus-trates the mean pressure distribution profiles ���, and the pre-dicted radial pressure will be compared to available experimentaldata. The mean pressure distribution profiles ��� is defined ac-cording to the following equation:

��r� =2�p�r� − p�r = 1��

Vin2 , where r =

r

Ro

Figure 10 compares the predicted radial pressure coefficient ���to the experimental results �25,26� at three diameter ratios ��=2.5,3.33,4.0�. A good agreement with the experiments is ob-served especially with high-diameter ratios �i.e., �=4.0�, wherethe flow field is under strong swirling condition. Increasing theswirl number �S� leads to deeper pressure profiles.

6 ConclusionsConfined vortex flow inside the vortex chamber was investi-

gated both experimentally and computationally at different swirlratio. The RSM model is able to predict the flow features, such asthe press drop, tangential, axial velocities, and the radial pressureprofiles. The prediction shows the behavior of the mean tangentialvelocity distribution where a forced-vortex inside the core and afree-vortex outside the core are existed and agree with the experi-mental data. The reverse-flow, which is associated with the axialvelocity profile, is captured inside the core region and close to thechamber exit. The swirl number has sufficient impact of the flowfeatures, the vortex core size contracts with increasing the swirlnumber leads to more pressure drop �energy loss�, the peak tan-gential velocity grows up, and deeper radial pressure profiles. Acomparison of the results with measurement shows clearly thesuperiority of the Reynolds-stress turbulence model in capturing

Fig. 9 Predicted axial velocity vectors near the exit

the major features of a confined, strongly swirling flow.

NOVEMBER 2006, Vol. 128 / 1381

N

G

1

omenclatureAo � cross-sectional area of the vortex chamberAin � total inlet areaCp � pressure coefficient �2�p /Vin

2 �De � diameter of the exit port �2Re�Din � diameter of the inlet portDo � chamber diameter �2Ro�

k � turbulent kinetic energyL � chamber lengthp � static pressure

pa � ambient static pressurepin � static pressure at the inletQin � inlet volumetric flow rate

r ,� ,z � radial, tangential and axial coordinaterespectively

r � normalized radius �r /Ro�Re � radius of exit port

Reo � Reynolds number �Reo=4Qin /��Do�Ro � radius of the chamberS � swirl number

ui ,uj ,uk � velocity components in Cartesian coordinatesV� ,Vz � mean tangential and axial velocity components

Vin � total velocity vector at the inletVavg � average axial velocityV�,in � inlet tangential velocity componentVr,in � inlet radial velocity component

reek Symbols� � area ratio �Ain /Ao�

�p � static pressure difference �Pin− Pa�� � radial pressure �2�p�r�− p�r=1�� /Vin

2 ��ij � Kronecker delta

� � turbulence dissipation rate� � kinematics viscosity

Fig. 10 Mean radial pressure

382 / Vol. 128, NOVEMBER 2006

� � dynamic viscosity�t � eddy or turbulent viscosity � density of the fluid� � inlet angle� � aspect ratio �L /Do�� � diameter ratio �Do /De�

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LDA Measurements of Confined Turbulent Vortex Flow,” J. Fluid Mech., 98,pp. 49–63.

�2� Vatistas, G. H., 1998, “New Model for Intense Self-Similar Vortices,” J. Pro-pul. Power, 14�4�, pp. 462–469.

�3� Sullivan, R. D., 1959, “A Two-Cell Vortex Solution of the Navier-StokesEquations,” J. Aerosp. Sci., 26�11�, pp. 767–768.

�4� Nallasamy, M., 1987, “Turbulence Models and Their Applications to the Pre-diction of Internal Flows,” Comput. Fluids, 15�2�, pp. 151–194.

�5� Nejad, A. S., Vanka, S. P., Favaloro, S. C., Samimy, M., and Langenfeld, C.,1989, “Application of Laser Velocimetry for Characterization of ConfinedSwirling Flow,” Trans. ASME: J. Eng. Gas Turbines Power, 111, pp. 36–45.

�6� Weber, R., 1990, “Assessment of Turbulence Modeling for Engineering Pre-diction of Swirling Vortices in the Near Burner Zone,” Int. J. Heat Fluid Flow,11�3�, pp. 225–235.

�7� Launder, B. E., 1989, “Second-Moment Closure and Its Use in ModellingTurbulent Industrial Flows,” Int. J. Numer. Methods Fluids, 9�8�, pp. 963–985.

�8� Leschziner, M. A., 1990, “Modelling Engineering Flows With Reynolds StressTurbulence Closure,” J. Wind. Eng. Ind. Aerodyn., 35�1�, pp. 21–47.

�9� Ferziger, J. H., and Peric, M., 1999, Computational Methods for Fluid Dynam-ics, 2nd ed., Springer-Verlag, Berlin.

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�11� German, A. E., and Mahmud, T., 2005, “Modelling of Non-Premixed SwirlBurner Flows Using a Reynolds-Stress Turbulence Closure,” Fuel, 84�5�, pp.583–594.

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�13� Orland, P., and Fatica, M., 1997, “Direct Simulation of Turbulent Flow in aPipe Rotating About Its Axis,” J. Fluid Mech., 343, pp. 43–72.

�14� Jones, W. P., and Pascau, A., 1989, “Calculation of Confined Swirling FlowsWith a Second Momentum Closure,” ASME Trans. J. Fluids Eng., 111, pp.248–255.

�15� Hoekstra, A. J., Derksen, H. E. A., and Akker, V. D., 1999, “An Experimentaland Numerical Study of Turbulent Swirling Flow in Gas Cyclones,” Chem.Eng. Sci., 54, pp. 2055–2065.

�16� Vatistas, G. H., Lam, C., and Lin, S., 1989, “Similarity Relationship for theCore Radius and the Pressure Drop in Vortex Chambers,” Can. J. Chem. Eng.,67, pp. 540–544.

�17� Launder, B. E., Reece, G. J., and Rodi, W., 1975, “Progress in the Develop-ment of a Reynolds-Stress Turbulence Closure,” J. Fluid Mech., 68�3�, pp.537–566.

�18� Lien, F. S., and Leschziner, M. A., 1994, “Assessment of Turbulent TransportModels Including Nonlinear RNG Eddy-Viscosity Formulation and Second-Moment Closure,” Comput. Fluids, 23�8�, pp. 983–1004.

�19� Gibson, M., and Launder, B. E., 1978, “Ground Effects on Pressure Fluctua-tions in the Atmospheric Boundary Layer,” J. Fluid Mech., 86, pp. 491–511.

�20� Launder, B. E., and Spalding, D. B., 1974, “Numerical Computation of Tur-bulent Flows,” Comput. Methods Appl. Mech. Eng., 3�2�, pp. 269–289.

�21� Kim, S. E., and Choudhury, D., 1995, “A Near-Wall Treatment Using WallFunctions Sensitized to Pressure Gradient,” ASME FED, 217, Separated andComplex Flows, pp. 273–280.

�22� Gupta, A. K., Lilly, D. G., and Syred, N., 1984, Swirl Flows, Abacus, Tun-bridge Wells, England, UK.

�23� Jawarneh, M. A., Vatistas, G. H., and Hong, H., 2005, “On the Flow Devel-opment in Jet-Driven Vortex Chambers,” J. Propul. Power, 21�3�, pp. 564–570.

�24� Yan, L., Vatistas, G. H., and Lin, S., 2000, “Experimental Studies on Turbu-lence Kinetic Energy in Confined Vortex Flows,” J. Therm. Sci., 9�1�, pp.10–22.

�25� Lam, H. C., 1993, “An Experimental Investigation and Dimensional Analysisof Confined Vortex Flows,” Ph.D. thesis, Department of Mechanical and in-dustrial Engineering, Concordia University, Montreal.

�26� Alekseenko, S. V., Kuibin, P. A., Okulov, V. L., and Shtork, S. I., 1999,“Helical Vortices in Swirl Flow,” J. Fluid Mech., 383, pp. 195–243.

Transactions of the ASME