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AIAA-2000-0134Advanced Turbulence Model forTransitional and Rotational Flows inTurbomachineryHsin-Hua TsueiConcepts, ETI, Inc.White River Junction, VT

andJ. Blair PerotDept. of Mechanical and Industrial EngineeringUniversity of MassachusettsAmherst, MA

For permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics,1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.

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AIAA-2000-0134ADVANCED TURBULENCE MODEL FOR TRANSITIONAL AND ROTATIONALFLOWS IN TURBOMACHINERY

Hsin-Hua Tsuei*Concepts ETI, Inc.White River Junction, VTandJ. Blair Perot+Dept. of Mechanical and Industrial EngineeringUniversity of MassachusettsAmherst, MA

INTRODUCTION

Turbulence Modeling for CFDComputational Fluid Dynamics (CFD) for aircraftengine and airframe design is now commonplace.Practical CFD calculations require models forapproximating the effects of turbulent flow structure.Many turbulence models have been developed, but theneed for more robust, computationally efficient, andeasy-to-use turbulence models applicable to wideclasses of aerodynamic and aerothermodynamic flowproblems has been well established. Turbomachineryflows include a variety of complex flow phenomena,including laminar-to-turbulent transition andrelaminarization, adverse pressure gradients, flowseparation, secondary flow mixing, rotor-statorinteraction, tip clearance flows, combustion, and heattransfer. Each of these flow characteristics must betaken into account in turbomachinery engineering. Acommon thread among all these phenomena isturbulence. Turbulence is taken into account in todaysengineering calculations through the use of turbulencemodels. However, current commonly used turbulencemodels perform relatively poorly in predicting certaintypes of turbomachinery flows, especially in flowsexperiencing transition and rotation. For example, thewidely used k-E turbulence model always underpredictsthe transition. Todays turbulence models can beadjusted to obtain a fairly accurate solution for certaintypes of flows, though success typically depends on the

* Senior CFD/Project Engineer. AIAA member. Assistant Professor. AIAA member.

skill of the fluid dynamics expert who must tweak themodeling using ad hoc damping functions ormanipulations of the turbulence modeling parameters.It is highly desirable to have a powerful, generalizedturbulence model which accurately models the fullrange of turbulence effects across a wide r ange ofcommon turbomachinery flows without substantial userinteraction.Turbulence modeling is an ongoing area of research andthere have been many claims of improved turbulencemodels which have later disappointed users. Pastefforts at improved turbulence modeling havehistorically failed because they improved accuracy onlyfor a specific flow condition, such as shear, vortex flow,etc. These new turbulence models were not generalizedto work under a wide variety of flow conditions.Without detailed knowledge on the part of the user ofthe models formulation, plus additional tweaking foreach specific application, substantial improvements inaccuracy usually could not be achieved. In addition,many of these more complex models incurred a majorpenalty in computer memory use and computation timeas a result of going to a two- or three-equationformulation, while not providing sufficientimprovement in accuracy over the less demandingmodels to justify this increased computation time. Inaddition, most researchers, despite the fairly substantialnature of some of the past efforts, have failed to confirmtheir models with basic testing and first-principlestechniques.

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Despite attempting to capture complex flow fieldphysics, such as recirculating regions, shock-waveboundary l ayer interactions, horseshoe vortices, 3Dboundary layers, secondary flow effects, tip clearanceflows, and rotation and curvature effects, all the existingturbulence models fail to predict transition occurring inturbomachinery flows (Hirschltl). Therefore, ensuringthe reliability of numerical predictions is crucial toturbomachinery engineering design. There are a varietyof turbulence models available, ranging from algebraicmodels to Reynolds Stress Transport Equation (RSTE)models. Within each family, based on differentassumptions or on different empirical data, even largernumbers of variants have been derived. The propersetup and calibration of turbulence models can be acritical role in todays practical CFD applications sincenone of the available models offer universal validity.On the contrary, all of them are based, to a greater orlesser degree, on empirical techniques. More advancedturbulence models, like the RSTE, or the Large EddySimulation (LES), are capable of capturing complex 3Dturbulent flow physics but at an unacceptable cost incomputer memory requirements and computational cost.The algebraic models, such as Baldwin-Lomax, andone- or two-equation models, such as k-E are based onthe eddy viscosity hypothesis, which assumes that theReynolds stress is a function of the mean flow strainrate, with isotropic turbulence exchanges similar to themolecular viscosity. Algebraic models are widely usedin CFD codes because of their simplicity and lowcomputational cost. No additional transport equationsare required to be coupled -with the Navier-Stokesequations. The version developed by Baldwin-Lomaxt21is the most popular of the algebraic turbulence models.The Baldwin-Lomax model assumes that the turbulentlayer is formed by two regions, an inner and an outerregion, with different expressions for t he eddy viscositycoefficient under the assumption of isotropicturbulence. However, because the Baldwin-Lomaxmodel is not able to take into account the transport anddiffusion of turbulence, difficulty arises in the vicinityof separation or reattachment points. However, it isgenerally considered that algebraic models provideacceptable results for attached boundary layers,especiaily for external applications.Applications of the Baldwin-Lomax model toturbomachinery flow problems were reported andsummarized by Hirscht) and DawesR, and thediscrepancies between the measured data and thepredicted results were documented. An attempt wasreported by Nakahashi t41 o apply the Baldwin-Lomaxmodel to predict transition in a turbine cascade by

introducing a transition criterion based on the maximumeddy viscosity exceeding a preset value. At a highincidence angle and a high Mach number, the transitionwas underpredicted, leading to an inaccurate globalflow field when compared to Schlieren pictures,primarily because of the presence of a separated flowregion induced by shock-wave boundary layerinteractions.A variety of two-equation models has been developedover the years, the most popular being the k-c model.The k-E was developed by Launder and Spaldingt51, andsolves the transport equations of turbulent kineticenergy and dissipation rate. These two equations haveto be computed simultaneously with the 3D Navier-Stokes equations. Various forms of the k-c model areavailable, with the differences between the variousversions being the treatment of near-wall low Reynoldsnumber turbulence. The first attempt to apply the two-equation k-E model to turbomachinery flows waspioneered by Haht61, and an acceptable compromisebetween the range of validity and the requiredcomputation time was made. Prediction of 3D non-isotropic turbulence remains one of the modelsdeficiencies. Application of the k-E model to variousturbomachinery flows, such as a subsonic turbine bladeand a transonic compressor, were performed byMatsuot71, and Jennions and TurnerIs]. Transition wasalso modeled using the k-E model as noted by Savilllgl.The application of the k-E model to transition requiredthat a high free steam turbulence be imposed, which ledto calculations predicting transitions prematurely. Itwas found that imposing a high upstream turbulencelevel to trigger transition led to the turbulence kineticenergy dissipating into the boundary layer more quicklythan occurs in real flows. Another extreme case inwhich the k-c model failed dramatically was theprediction of an impinging jet on a plate as described inLeschziner and LaunderlrOl. It appeared that the lineareddy viscosity stress-strain relation caused excessivelevels of turbulent kinetic energy and eddy viscositycompared to the measured data. This result issignificant for turbomachinery flows since a stagnationpoint always exists near the leading edge of a blade.Heat transfer at the leading edge is also poorlypredicted today.The more advanced RSTE models take into account theeffects of anisotropy, rotation, and curvature (Mansour,Kim, and Moinlt 1). They are able to capture 3Dturbulent fl ow physics much better than the algebraic orthe two-equation models. The drawback is that thesemodels require a total of seven transport equations to becalculated for the Reynolds stress tensor and Navier-

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Stokes equations. In addition, some implementations ofthese models have proven numerically stiff and,consequently, computationally unstable. These modelsneed to be further developed before they can be appliedto practical modeling of turbulence in turbomachinery.In summary, although substantial work has beenperformed in the area of turbulence modeling, thesearch for a reliable and economical turbulence modelfor turbomachinery continues. The current turbulencemodel is expected to capture the full details of 3D non-isotropic turbulence and has the potential to moreaccurately model transition in a wide variety of flows.This greater accuracy is expected to be achieved with acomputational cost similar to an enhanced k-E modelwhile providing detailed Reynolds stress informationfor the mean flow.Types of Turbulence ModelsThere are a number of different types of turbulencemodels already available, including algebraic models,one- or two-equation models, Reynolds Stress TransportEquation (RSTE) models, and the Large EddySimulation (LES). Each of these models predictssatisfactorily accurate results for certain cases.However, a turbulence model havi ng universal validityis still lacking. In todays competitive environment,turbomachinery designers must work effectively withineven shorter product design cycles, requiring fast runtimes and efficient design tools. As a result, thealgebraic models (such as the popular Baldwin-Lomaxmodel) and the two-equation turbulence models(particularly the k-E model) have been widelyincorporated into CFD tools for both government andindustry. These two models generally require lessdemanding computational resources and are able toprovide reasonably accurate results in a reasonablytimely fashion. However, the Baldwin-Lomax algebraicturbulence model often leads to unrealistic predictionsfor some types of flows commonly found inturbomachinery, such as flow in recirculating regions,secondary flows, shock-wave boundary layerinteractions, etc. The two-equation k-E model generallyprovides a better solution than the Baldwin-Lomaxmodel, but it still lacks accuracy for transition androtating flows. More advanced turbulence models, suchas the RSTE or the LES, although capable of modelingmore complicated turbulent flows (particularly for non-isotropic turbulence), often require prohibitivecomputation time. These advanced turbulence modelsare still in the academic development stage and requireadditional refinement before they can be beneficial forpractical product design cycles. A reliable and accurateturbulence model is desired which combines the

advantages of the fast run time of algebraic or two-equation models and the non-isotropic, complexturbulent flow prediction capability of the moreadvanced turbulence models.The present advanced turbulence model is based on aset of partial differential equations with theoreticalprocedures and boundary conditions formulated for anew set of two turbulence variables. This model wasdeveloped by Perot1121*1131. The current techniquemodels the scalar and vector potentials of the turbulentbody force, rather than the Reynolds stress tensor oreddy viscosity. This approach has an advantage overeddy viscosity models (such as Baldwin-Lomax, k-E, q-o, etc.) in that no constitutive relationship between theReynolds stress tensor and the mean velocity gradientsneeds to be hypothesized. (This advantage allows thepossibility of a strong disequilibrium between theturbulence and the mean flow, such as has been shownto occur in rotating flows.) The present turbulencemodel also has an advantage over the RST equationmodels in that the same turbulent flow physics can becaptured with a significant reduction in model closurecomplexity and associated computational cost. Due toits relative computational simplicity, the present modelwill be more robust than RSTE models and better ableto account for wall and surface effects, yet has acomputational cost similar to that of many enhancedtwo-equation models, which often add one, and moreoften two, additional partial differential equations.THEORETICAL FORMULATION AND MODELIMPLEMENTATION FOR COMPRESSIBLE

TURBOMACHINERY FLOWSFor the sake of simplicity, incompressible, isothermalflow, the IJeynolds Averaged Navier-Stokes (RANS)equations can be written as:

~+v.(UU)=-Vp+v.vU-v.R (14v.u=o (lb)

where u is the mean velocity vector, p i s pressure, v isthe kinetic vi scosity, and R = uu is the Reynoldsstress tensor. The Reynolds tensor represents thecorrelation of the fluctuating velocity components withfour components for two-dimensional flows and ninecomponents for 3D flows. The fundamental goal ofRANS models is to hypothesize a relationship betweenthis tensor and the mean flow variables so thatEquations (la) and (1 b) can be solved. The sameprinciples exist for the general (compressible) RANSequations.

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The key to developing a model which avoids the use ofa constitutive assumption, and yet does not involve thecomplexity of the full Reynolds stress transport closureused in the existing RANS models, is to focus on thefact that the Reynolds stresses actually contain moreinformation than is required to calculate the mean flow.Only t he divergence of the Reynolds stress tensor(a body force vector) is needed to evaluate the meanflow, so the remaining (non-equilibri um) terms can beneglected. A new set of turbulent variables wasdeveloped to model the scalar and vector potentials ofthe turbulent body force, rather than the Reynolds stresstensor or an eddy viscosity. The turbulent potentials(4 and ~JJ) re defined mathematically by the followingequations.

(24v,yl = 0 (2b)

The second equation is a constraint on the vectorpotential. These equations can be rewritten to expressthe turbulent potentials explicitly.V2$ = V. ( V. R )-V=v=Vx(VR)

(34(3b)

The boundary conditions on these equations areconstructed intuitively. Both potentials are required togo to zero at infinity, at a wall, or at a free surface.Note that by definition (Equation 2a) the scalarpotentials are those portions of the turbulence thatcontribute to the mean pressure but do not affect themean vorticity, or vorticity of the mean flow,(o= V x u). Only the vector potential has the ability toaffect the mean vorticity, and it only transportsvorticity; it does not create or destroy vorticity.In flows with a single inhomogeneous direction (saythe y-direction), Equations 3a and 3b simplify toCp Rz2, w1 = -R23, w2 = 0, ~3 = R,z. For this reason, itis also reasonable to view the vector potentials as aconceptual generalization of the shear stress (uv) toarbitrary geometries and three dimensions. Theserelations will be used in evaluating model predictionsagainst experimental data and against Reynolds stressdata from published Direct Numerical Solution (DNS)data.In two-dimensional (2D) flow, the vector potential isaligned perpendicular to the flow and has only a singlenonzero component (~3). In full 3D flow, the vectorpotential must obey the divergence free constraint. Thisimplies that the vector potential can be computed, even

in 3D flows, with a cost roughly equivalent to the scalarpotential. Therefore, the overall complexity andassociated computational cost of solving a model for theproposed turbulent potentials should be roughly lessthan half of that of a full Reynolds stress closure.The transport equations for the turbulent potentials werederived by Perot (1995,1997):

~=V(v+vr/cr,r,)V~ + P, + D,Dt (44

3 =V~(v+vr/c~~)V~ + P, + D,Dt (4b)where P+ P, represent the production terms for theturbulent potentials, and D,, D, are the dissipationterms, respectively.

The general modeling philosophy is to use existingReynolds stress transport models, and the assumption ofa single inhomogeneous direction, to guide theconstruction of the turbulent potential source terms.Other guiding factors have been the construction of amodel that naturally obtains the correct asymptoticbehavior near walls and free surfaces, and a modelwhich corresponds correctly to shearing, both initiallyand in the long time limit. Obtaining the correct rapidresponse for shear flows is important because thissituation is closely related to many engineeringapplications. Examples are the freestream turbulencewhich suddenly encounters an obstacle, such as a bladein turbomachinery, and the initial development oftripped boundary layers and free shear layers.The turbulent potential transport equations aresupplemented by transport equations for the turbulentkinetic energy and the dissipation. These equationswere chosen because they are widely used forengineering solutions, and because data for both k and Eare widely avail able. These equations enable thesystem to behave correctly in time developing orconvection dominated situations, and help to make theequations system more robust in situations where themean flow gradients are small. Other two-equationmodels, such as Ww and k/L models could easily besubstituted if they were preferred.The turbulent potential model was integrated into theBTOB3D turbomachinery CFD code. The BTOB3Dcode was developed by Prof. W. Dawesti41, ofCambridge University, for general 3D centrifugal andaxial turbomachinery flows. The turbulence modelimplemented in the BTOB3D code is the algebraicBaldwin-Lomax model. This particular CFD code is


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widely used in the turbomachinery industry around theworld and is popular for its fast solution times, whichhave helped make CFD a practical tool for daily designoptimization.FUNDAMENTAL VALIDATIONS

A series of fundamental benchmark turbulent flow casesfor model validation is finished. This model hassuccessfully demonstrated the ability to predict aturbulent channel flow at different Reynolds numbers,transition and relaminarization of a channel flow, arotating channel Row, a turbulent mixing layer, and aturbulent boundary layer with adverse pressure gradient.With these promising features, which are specificallyapplicable for turbomachinery flows, it is proposed tofurther develop this turbulence model to extend itsapplicability to all types of turbomachinery, includingcompressors, turbines, and pumps, to provide accurateengineering solutions. Several representativebenchmark test results, of particular interest forturbomachinery flow predictions, are presented belowusing the turbulent potential model. It should be notedthat turbomachinery flows are some of the mostcomplex flows to model. Therefore, it is reasonable toassume that a turbulence model developed and proveneffective for turbomachinery applications will be widelyapplicable to other types of flow as well.Rotation PredictionRotation is an integral part of turbomachinery andcannot be ignored when developing a generalturbulence model. Rotation produces a Coriolis forcethat can be coupled with the turbulence and affectsmachine performance. At a first level of approximation,a turbulent eddy acts like a gyroscope with individualangular momentum. When rotation is imposed,turbulent eddies try to maintain their angularmomentum to balance the rotation. This process alterseddy orientation, and hence, influences the conventionalturbulence structure, leading to a reduction of turbulentdissipation rate and the creation of non-isotropic flows.The turbulent potential model shows correct qualitativepredictions due to rotation. Figure 1 shows the modelprediction results for rotating channel flow. TheReynolds number is 7,900. The results are shownacross the entire channel since the flow is no longersymmetric about the centerline. The proposed modelcorrectly predicts the qualitative effects of stabilizationon the upper wall and enhanced turbulence productionon the lower wall.

Turbulent Kinetic Energy

Normal Stress (R.22)

-1 ; --_.._.- ----- Shear Stress\___ - (R12) --2 I I I I I I I-1 -.75 -.5 ~25 0 0.25 0.5 0.75 1

Btm Yh TOP

Figure 1. Predictions, using the turbulent potentialmodel, of streamwise variation of flow parametersfor rotating channel flow, Re = 7,900.Prediction of a Boundary Layer with an AdversePressure GradientAdverse pressure gradient boundary layers represent asituation where the classic turbulence modelingassumptions are not well approximated. In particular,the turbulence is not in equilibrium with the mean flow,which violates the eddy viscosity hypothesis. However,such flows of adverse pressure (diffusing flows) are atthe heart of turbomachinery. Two-equation models tendto have some problems predicting adverse pressureboundary layers. However, models which predict shearstress directly (Johnson and Kinglt51, and Bradshaw,Ferris and Atwell1161, and full Reynolds stress closure)generally show considerable more success with thesetypes of flows. Since the proposed model also directlypredicts a quantity similar to shear stress, it is expectedto perform well in these situations.Figure 2 shows the computed streamlines for theadverse pressure gradient experiment of Samual andJoubertli71. The vertical direction has been exaggeratedby a factor of ten to highlight the rapid growth of theboundary layer under the influence of an adversepressure gradient. The predicted friction coefficient asa function of the downstream distance is shown inFigure 3. Circles indicate the experimental data and thesolid line represents the numerical prediction. It isapparent that the agreement between the experimentaldata and the predicted results is very good. Most two-equation models tend to underpredict the frictioncoefficient as separation is approached, leading tocalculations which predict separation prematurely.

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Begina&enepressure gradient Staiion Stat&12

Figure 2. Computed streamlines for an adversepressure gradient boundary layer. Note that thevertical coordinate has been expanded by a factorof ten.O.CO35

0.003

0.0025

I

.

c,o,

o.al150.001

O.OCO5L0~0 1.2 1.8 2.4

oomreem Distance, x-x0

A 3

Figure 3. Measured and predicted frictioncoefficient versus downstream distance for anadverse pressure gradient boundary layer.ADVANCED MODEL VALIDATIONS AGAINSTDNS AND LASER VELOCIMETRY DATA

Prediction of Fullv Developed Turbulent ChannelFlowThe turbulent potential model was further validatedagainst a published DNS data for a fully developedturbulent channel flow. The DNS computation wasconducted by Moser, Kim and Mansour1181. TheReynolds number of the fully developed channel flowwas 8,000, based on the half channel width. Thenumber of grid points used in the DNS simulation wason the order of 10,000,000, a very large number of gridsuch that the computation can only be carried out on asupercomputer. By computing the fully developedchannel flow and compared with DNS data, the resultsimplied the accuracy of the turbulent potential models

for predicting turbulent production, dissipation rate, andother turbulent transport quantities like cross product ofthe Reynolds stresses.The current computation using the turbulent potentialmodel was performed on a 59 x 97 mesh system, asignificant reduction in grid size compared to the DNScalculation. The mesh is stretched toward the wall andrepresents a grid system of the first grid point well intothe viscous layer region, y+ < 1 O. Figure 4 showed thenon-dimensionalized fully developed velocity profilesfor the turbulent potential model and the DNS data.The radial direction was also non-dimensionalized bythe half channel width. Excellent agreement wasobserved between the turbulent potential model resultsand the DNS data. These velocity profiles literallycoincided with each other. Detailed turbulent transportproperties will be discussed in the followingparagraphs.

Figure 4. Velocity profile compared to DNS data.Figure 5 shows the non-dimensionalized turbulentkinetic energy profiles. As can be seen from this figure,the turbulent potential model compared very well nearthe channel centerline to the log layer, almost agreedwith the DNS data exactly. The location of the peak ofTKE agreed well with the DNS data (y+ - 15),however, the magnitude was overpredicted about 2% bythe turbulent potential model. The TKE decayinglinearly to zero in the viscous sublayer region wasclearly resolved by the present model. Figure 6 showsthe non-dimensionalized turbulent dissipation rate. Thedissipation remained small in the core region near thechannel centerline, then increased monotonically towardthe wall. On the edge of viscous sublayer layer,y+ - 10, the DNS data showed that the dissipation ratejumped to a slightly higher l evel, instead of increasingcontinuously. This jump was captured by the turbulent


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potential model, however, the location of the jump wasoverpredicted by about 1%. Both DNS data and thecurrent model results indicated the dissipation rateincreased linearly in the viscous sublayer toward thewall, which is a source of generating turbulentdissipation. On the wall, the turbulence modeloverpredicted the dissipation by about 15%. Thediscrepancy could be a result of the wall boundarycondition used for the epsilon equation, which isarguably the most difficult part to model. At the presentmoment, there is no good experimental data, numericalmodel or estimate for the turbulent di ssipation at a wall.

1 2 , 1 5 6WMX

Figure 5. Turbulent kinetic energy profilecompared to DNS data.

Figure 6. Turbulent dissipation rate profile com-pared to DNS data.The next two figures demonstrate the predictivecapability of the turbulent potential model for secondorder closure turbulent transport quantities. As stated inthe earlier section, the turbulent potential model was

derived from the full Reynolds stress model, therefore itpossesses the functionality to predict Reynolds stresscomponents, yet avoids the numerical stiffness problemthat is always associated with the Reynolds stressmodel.Figure 7 represents the non-dimensionalized Reynoldsstress component vv compared with the DNS data. Inthe viscous sublayer, the results compared well withDNS data, showing a near linear increase in this region.In the log layer, the peak location and magnitude werebeing predicted almost precisely. The present modelresults deviated from the DNS data only near thechannel centerline, but the overall agreement is good.Figure 8 shows the comparison for another non-dimensionalized Reynolds stress, uv. Among all thecomparisons made with DNS data, this is the onlyturbulent transport quantity that was not compared wellwith the DNS data. The turbulent potential modelconsistently overpredicted, by about 5%, from channelcenterline to close to the edge viscous sublayer. Thepeak location was captured; however, the magnitudewas larger than the DNS data.Overall, the computed fully developed turbulentchannel flow results using the turbulent potential modelcompare very well with the DNS data, which providedfundamental understanding of the model behavior andperformance. Furthermore, detailed examinations ofeach individual turbulent transport quantities revealedthe capability of the turbulent potential model inpredicting not only the mean flow but also the local andnear wall turbulent flows.

I I I I I !I/

Figure 7. Reynolds stress vv profile compared toDNS data.


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Figure 8. Reynolds stress uv profile compared toDNS data.

Validation Against Annular Diffuser LaserVelocimetrv DataAnnular Diffuser Rig Setup and Test Results

This axisymmetric rig is attractive for a proof-of-concept effort because it provides a flow field whichexhibits some key turbulent characteristics whileremaining simple enough to be reasonably assumed 2D.A cross section of the annular diffuser rig used is shownin Figure 9. This rig consists of a bellmouth-type inletleading to a simple annular diffuser that dumps into anexhaust plenum. This existing rig was designed atConcepts, ETI, Inc. (CETI) with full instrumentationcapabilities, including windows for laser velocimetry(LV) measurements. Figures 10 and 11 show theoverall view and the close-up view of the annulardiffuser test rig.

L---II-d-

Figure 9. Cross-sectional view of annular diffuser rig.

Figure 10. The overall view of the annular diffusertest rig.

Figure 11. The close-up view of the annular diffuser.The flow enters the diffuser from right to left.With the rig as configured, it will be possible to takecomplete LV traverse measurements across the passageat several stations. The purpose of this test is to use LVtechniques to resolve a series of fundamentalturbomachinery flow field phenomena, where the flowis under the influence of curvature (rotation), adversepressure gradient (diffusion), and laminar-to-turbulenttransition. This sophisticated, yet straightforward testaccomplishes all of these objectives and will providekey validation data for the 2D compressible model.Note that CFD codes and the associated turbulencemodels must be able to calculate flows in this simplegeometry if flows in moie compleq turbomachinerygeometries are to be predicted.


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The tests were run at two different flow rates: a highflow rate of 1.26 kg/s and a low flow rate of 1.01 kg/s.The tests were conducted at room temperature andatmospheric pressure. The L2F traverse data, in thespanwise direction for total velocity at two stations, asindicated in Figure 9, were collected. The spanwisedata traverses were plotted to show the velocity profilesat these two measurement stations in Figures 12 and 13for the high and low flow rate cases, respectively.

0 20 a m +

Figure 12. Measured L2F velocity profile for thehigh flow rate case.s .__-.............- ._- ..-......-1 I I I I 1mx Ki_,

Figure 13. Measured L2F velocity profile for thelow flow rate case.Computational Results

The computational results using the algebraic Baldwin-Lomax turbulence model, the two equation k-c modeland the turbulent potential model are presented in thissection. The computational mesh was 101 x 101 for the

annular diffuser, as shown in Figure 14. The mesh isstretched on both the hub and shroud side of the annulardiffuser to provide grid resolution in the viscoussublayer for the near wall turbulence behavior. Thisenables the accurate prediction of turbulent productionand dissipation rate, and therefore, leads to a moreaccurate Reynolds stress calculation for higher orderclosures. The mesh was uniformly distributed in thethroughflow direction. The flow entered in the annulardiffuser from left to right, then turned by the 45 bend,then moved up in the radial direction. As radiusincreases, the cross-sectional area increases. Thisprovides an adverse pressure gradient environment forthe flow. In addition, the turning serves the purpose ofproviding the mechanism of rotation. The inletboundary conditions for the computation were assumedto be the room temperature and atmospheric pressure,without swirl. The freestream turbulence intensity levelis set to be 1%.

0.140.13

Eo.120.11

0.10.090.060.070.06

0.02 0.04 0.06 0.06 0 1 0.122

Figure 14. The computational mesh for theannular diffuser. The mesh size is 101 x 101.The representative computational results for the highflow rate case using the turbulent potential model arepresented in the following. The Reynolds numberbased on the channel height is about 250,000.Figure 15 shows the total velocity contours. For thehigh flow case, the flow speed at the inlet is about220 m/s, accelerates around the corner to 240 m/s,enters the diffusing section of the channel and slowsdown gradually, then exits the annular diffuser at about100 m/s. The low momentum region (with lowervelocity) near the shroud line, after the turning of thediffuser, has been observed. This phenomenon,associated with the adverse pressure gradient, resemblesthe general flow field in a centrifugal compressor andaccelerates around the bend to cause local pressure dropto a minimum value of 60 KPa in this region. The exitgives a very good indication for the validation effort.Figure 16 represents the static pressure contours in the


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annular diffuser. This figure clearly shows the adversepressure gradient throughout the diffuser, withexception of the region around the turning corner. Thelocal flow field static pressure is about 91 KPa for thehigh flow rate case. These two figures represent theglobal flow field phenomenon.

Figure 15. Total velocity contours in the annulardiffuser for the high flow rate case.

0.170.160.150.140.13

=0.120.11

0.10.090.06

I9000066000660006400062000

0.060.02 0.04 0.06 0.06 0.1 0.12-PL

Figure 16. Static pressure contours in theannular diffuser for the high flow rate case.Figure 17 shows the turbulent kinetic energy contours,while Figure 18 depicts the turbulent dissipation ratecontours. In Figure 17, turbulent kinetic energy wasgenerated close to both the hub and the shroud of theannular diffuser due to the specific velocity gradientnear the wall. Because of the turning and the

subsequent low momentum and its associated strongervelocity gradient along the shroud line, the TKEmagnitude is obviously seen larger than at the hubregion. The TKE penetration into the core region canalso be observed. In the core region, the TKE remainsat a low level until the flow exits the diffuser. A similartrend can be found in the turbulent dissipation ratecontours, as shown in Figure 18.

0.16 TKE-0.150.140.13

=0.120.11

0.10.090.060.070.06

0.02 0.04 0.06 0.06 0.1 0.122Figure 17. Turbulent kinetic energy contours in theannular diffuser for the high flow rate case.

0.140.13

=0.120.11

0.10.090.060.070.06

0.02 0.04 0.06 0.08 0.1 0.12z

Figure 18. Turbulent dissipation rate contours inthe annular diffuser for the high flow rate case.Two Reynolds stress components, vv and uv arepresented in the next two figures. For a two-dimensional flow, these components represent thedominant second order quantities. Figure 19 shows thevv contours. This figure is similar to the TKEcontours, since the production of this Reynolds stresscomponent is proportional to turbulent kinetic energy.Large concentrations of vv near the turning region and


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the shroud line can clearly be observed. In Figure 20,the contours of uv is shown. Note that the productionof this Reynolds stress component is related to the meanvorticity of the flow field. For a channel flow, thevorticity goes from negative in the bottom half of thechannel due to negative velocity gradients, to zero at thecenterline, to positive in the upper half of the channelbecause of positive velocity gradients. Therefore, theuv contours shows exactly the same phenomenon asthe vorticity for a channel flow. In Figure 20, at thediffuser inlet, it is seen that the zero contour l ine dividesthe channel in half. Then the zero contour l ine starts todeviate from the centerline of the diffuser channel,depending upon the local vorticity field of the flow.Despite its sign, the magnitude of the uv contoursremains low in the core region, while being strong nearthe wall boundaries. The diffusion of these Reynoldsstress components into the core region, as well asdownstream locations, were observed in both Figures19 and 20. This agrees with the other computedturbulent quantity contours and the global diffuser flowphenomenon.Detailed examination of the computational results forthe low flow rate case reveals a similar turbulent flowpattern for both the global flow field and the detailedturbulent transport quantities. These contours for thelow flow rate case will be included here for simplicity.The computed results, however, will certainly bepresented in the following section to compare with theL2F test data.

0.16 vv contarn0.150.140.13

Eo.120.11

0.10.090.060.070.06

0.02 0.04 0.06 0.06 0.1 0.12Z

Figure 19. Reynolds stress component vvcontours in the annular diffuser for the high flowrate case.

0.170.16 "V coMou*0.150.140.13

=0.120.11

0.10.090.060.070.06

0.02 0.04 0.05 0.06 0.1 0.12ZFigure 20. Reynolds stress component uvcontours in the annular diffuser for the high flowrate case.Note that the Reynolds numbers for both the high flowrate case (250,000) and the low flow rate case(175,000) are large enough to impose numericalstiffness and model infidelity for a number ofturbulence models. Very often a number of modelingconstants in these turbulence models, for example thetwo equation k-E model, need to be tweaked to produceresults for a particular case. The tried-and-errorprocedure was a standard practice for most of theexisting turbulence models. During the course of usingthe turbulent potential model, all the modeling constantsremained unchanged, i.e., without being tweaked toachieve the computational results. Furthermore, theBTOB3D turbomachinery CFD code converged withoutencountering numerical stiffness or instability, whenrunning the turbulent potential model.

ComDarisons with L2F DataAs indicated in Figure 10, the L2F data traverses weretaken at two stations. The first data station is locatedimmediately downstream of the 45 bend, when boththe hub and shroud curvatures return back to zero. Thesecond data traverse is about halfway through thediffusing section of the diffuser channel path. The datataken at these two stations are far enough away from thediffuser exit, that downstream exhaust distributionshould have no effect on the measured quantities. Thecomputational results at these two locations arecompared with the L2F data for both the high and lowflow rate cases. We will begin comparisons anddiscussions with the low flow rate case, which has amoderately high Reynolds number of 175,000.There are three turbulence models used in thisvalidation study: the algebraic Baldwin-Lomax model,

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the two-equatfon k-E model, and the current turbulentpotential model. Note that all three models areimplemented into the BTOB3D turbomachinery CFDcode, sharing the same numerical algorithm and thesame numerical dissipation characteristics. Therefore,there is no bias in this study towards a particularturbulence model, nor as to whether to use oneturbulence model from one CFD code, and anothermodel from another CFD code. In this manner, thecomparisons made with data using the three turbulencemodels become meaningful.Figure 21 shows the comparisons with measuredvelocity profile at the first data traverse station, whileFigure 22 represents the comparisons at the second datalocation. As can be seen in Figure 21, generallyspeaking, all three turbulence models used predictedvelocity profiles that are relatively close to themeasured data at this location. The Baldwin-Lomaxmodel underpredicted the velocity profile consistentlyby less than 5% throughout the span. The k-E modelagreed with the data well from hub to about 30% span,then deviated from data slightly, overpredicted by lessthan 3%. The turbulent potential model showed aslightly better result than the k-c model, by agreeingwith the data from hub to about 60% span location, thenoverpredicted the near shroud flow by less than 2%.

Figure 21. Comparison with L2F data at the firstdata traverse location for the low flow rate case.In Figure 22, at the second data station, the Baldwin-Lomax model underpredicted the velocity profilefurther, the velocity magnitude being 10% lower thanthe measured data. This is understandable, because at afurther downstream location, more turbulent transportprocesses (production, dissipation, etc.) should have

happened. The isotropic turbulence, inner-outer l ayermatching assumption, without the consideration ofadverse pressure gradient, curvature (rotation), and thediffusion effects, the algebraic Baldwin-Lomax modelsimply misrepresented the complicated turbulenttransport phenomena in the annular diffuser channel.Both the k-e and the turbulent potential modelspredicted nearly the same results from hub to about60% span location, however, both undepredicted byabout 5% compared to data. From 60% to about 80%span, the turbulent potential model outperformed k-Emodel by almost agreeing with the measured data, whilethe k-E model underpredicted by 10%. Finally, all threemodels overpredicted the data point at 88% spanlocation by at least 25%. The measured velocityindicated a more significant diffusion process occurredat this span location and all the turbulence models didnot come close to what was measured. Thismisprediction might suggest that, in reality, t heturbulent fl ow might be more strongly coupled with theadverse pressure gradient and curvature than thepresently available models can handle.

Figure 22. Comparison with L2F data at thesecond traverse location for the low flow rate case.Comparisons with the measured data for the high flowrate case are presented in Figures 23 and 24. At thefirst data traverse location, which is shown in Figure 23,the Baldwin-Lomax model overpredicted the velocityprofile by about 10% consistently for most of the span.Near the shroud line, the 88% span data point showedthat Baldwin-Lomax model overpredicted close to 20%.On the contrary, the k-c model consistentlyunderpredicted the velocity profile by 7% compared todata. The 88% span data point was again greatlymissed by the k-c model. However, the two equation

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k-E model results did indicate a better agreement thanthe Baldwin-Lomax model. The turbulent potentialmodel showed amazingly good agreement with the L2Fdata, from hub to about 80% span location, with severaldata points falling on the predicted profile.Furthermore, the turbulent potential model predictedexactly the 88% data point; however, the predictedresults showed a stronger velocity gradient from 80% to88% span compared to the measured data. At this datalocation, it is clear that the turbulent potential modelcaptured the correct flow characteristics, whileBaldwin-Lomax and k-E models did not agree well withdata, by mispredicting both the core flow and boundarylayer velocity gradients.

5Figure 23 Comparison with L2F data at the firstdata traverse location for the high flow rate case.

At the second data station, as indicated in Figure 24, theL2F data represented a thick boundary layer, possiblydue to curvature and adverse pressure effects from the60% span location to the shroud line. In this figure, bothBaldwin-Lomax and k-E models underpredicted themeasured velocity profile. Combining with the firstdata location prediction, the Baldwin-Lomax model didnot provide a consistent predictive trend (overpredictingat location 1, while underpredicting at location 2). Forcomplicated turbulent flow physics like the annulardiffuser, therefore, the Baldwin-Lomax model cannot becounted on to provide a reliable solution. The k-Emodel consistently underpredicts the velocity profile inthe core region at both locations. This s eemed like astep increase compared to the Baldwin-Lomax model.Judging from the k-E results, the signature strongdissipation characteristics prevailed again, to lead tofast penetration of TKE from the boundary layer to thecore region. As a result, the model likely underperdictsthe mean flow in the core region, while overpredicts the

flow in the near wall region. Figure 24 clearly showsthis trend of the k-E model. On the other hand, thepresent turbulent potential model predicted an excellentvelocity profile compared to the data, from hub to about50% span location. In this region, the computationalresults agreed with the data very closely. In addition, asshown in the previous figure, the turbulent potentialmodel predicted the velocity at 88% span locationalmost precisely. However, a stronger velocity gradientin the near wall region was still observed compared tothe measured data. The turbulent potential model hasshown a promising potential to predict turbulent fl ow inturbomachinery, yet can be improved in the near wallmodeling portion of the model.

Figure 24. Comparison with L2F data at thesecond traverse location for the high flow ratecase.Combined with the finding in the previous figure, theresult has shown that the turbulent potential model hassuccessfully captured the complicated turbulenttransport phenomenon for this diffuser flow field. Inother words, the turbulent potential model predicted arealistic transport of TKE, dissipation, and other secondorder turbulent closure quantities, from their source ofgeneration (the near wall region) to the mean flow area(core region). The two most popular turbulencemodels, the algebraic Baldwin-Lomax and the twoequation k-E models, did not reach close agreement withthe L2F data for the annular diffuser flow field, whichcan be considered to be a representative turbomachineryflow field. Turbulence models, need to show goodagreement with data for this simplified annular diffuserflow field before they can be applied to accurate andrealistic turbomachinery flow predictions. Afterconducting this validation study, it is clear the turbulentpotential model outperformed both the Baldwin-Lomaxand the k-E models by predicting more accurate results


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when compared to the L2F data, for both the low andhigh flow rate cases, without tweaking any modelingconstants for these two high Reynolds number flows.This achievement is a step increase in turbulencemodeling capability for complicated turbomachineryflow fields.SUMMARY

The development and validation of the turbulentpotential model has demonstrated a clear start towardsdeveloping a comprehensive turbulence model, not onlyfor turbomachinery, but for all aspects of turbulent fl owfield applications. Progress has been made ondeveloping the 2D compressible flow model,particularly for turbomachinery flows, integration intothe BTOB3D turbomachinery CFD solver supported byan annular diffuser rig test with L2F test data forvalidation. A series of supplementary validation studiesusing the turbulent potential model against classicalturbulent flow bench mark cases, including DNS data,was also given. The validation results clearly indicatethat the turbulent potential model surpasses two widelyused turbulence models, the Baldwin-Lomax model andthe two equation k-E model. CETI will continue thiswork for further advancement of this much neededtechnology.ACKNOWLEDGMENT

The research was funded by the National ScienceFoundation. The financial support is highlyacknowledged. The L2F test was conducted byD. Hinch, D. Karon, and R. Burrell at CETI. Theireffort is highly appreciated. Thanks also goes to Mrs.Brenda LeDrew for her devoted assistance on creatinggraphics of this paper.

REFERENCES1. Hirsch, C., CFD Methodology and Validation forTurbomachinery Flows, AGARD Lecture Serieson Turbomachinery Design Using CFD, 1994.2. Baldwin, B., and Lomax, H., Thin LayerApproximation and Algebraic Model for SeparatedTurbulent Flow, AIAA 78-0257, 1978.3. Dawes, W. N., The Devel opment of a SolutionAdaptive 3D Navier-Stokes Solver forTurbomachinery, AIAA 91-2469, 1991.4. Nakahashi, K., Nozaki, O., Kikuchi, K., andTamura, A., Navier-Stokes Computations of Two-and Three-Dimensional Cascade Flow Fields,AIAA 87-1315, 1987.5. Launder, B. E., and Spalding, B., MathematicalModels of Turbulence, Academic Press, 1972.

6. Hah, C., Numerical Solution of Three-Dimensional Flows for Modern Gas TurbineComponents, ASME 87-GT-84, 1987.7. Matuso, Y., Computations of Three-DimensionalViscous Flows in Turbomachinery Cascades,AIAA 91-2237, 1991.8. Jennions, I. K., and Turner, M. J., Three-Dimensional Navier-Stokes Computations ofTransonic Fan Flow Using an Explicit Flow Solverand an Implicit k-E Solver, ASME 92-GT-309,1992.9. Savill, A. M., Evaluating Turbulence Predictionsof Transition - an ERCOFTAC SIG Project,Advances in Turbulence, Nieuwstadt Ed. pp. 555.Kluwer Academic Publishers, 1992.10. Leschziner, M. A., and Launder, B. E., Eds.,Proceedings of the 2nd ERCOFTAC-IAHRWorkshop on Refined Flow Modeling, Univ. ofManchester, UK, 1993.11. Mansour, N. N., Kim, J., & Moin, P., Reynolds-Stress and Dissipation-Rate Budgets in a TurbulentChannel Flow, J. Fluid Mech. Vol. 194, pp.15-44,1988.12. Perot, J. B., AND Moin, P., Shear-free turbulentboundary layers. Part 2. New concepts forReynolds stress transport equation modeling ofinhomogeneous flows. J. Fluid Mech., Vol. 295,pp. 229-245, 1995.13. Perot, J.B., A New Approach to TurbulenceModeling, SBIR Phase I Final Report,Aquasions, Inc., 1997.14. Dawes, W. N., A Computer Program for theAnalysis of Three-Dimensional ViscousCompressible Flow in Turbomachinery BladeRows, Whittle Laboratory, Cambridge, UK, 1988.15. Johnson, D. A., and King, L. S., A MathematicallySimple Turbulence Closure Model for Attachedand Separated Turbulent Boundary Layers, AIAAJ. vol. 23, pp. 1684-1692, 1985.16. Bradshaw, P., Ferris, D. H., and Atwell N. P.,Calculations of Boundary-Layer DevelopmentUsing the Turbulent Energy Equation, J. FluidMech., vol. 28, pp. 593-616, 1967.17. Samual, A. E., and Joubert, P. N., A BoundaryLayer Developing in an Increasingly AdversePressure Gradient, J. Fluid Mech. vol. 66, pp.481-505, 1974.18. Moser, R. D., Kim, J., and Mansour, N. N., DirectNumerical Simulation of Turbulent Channel Flowup to Re = 590, Physics of Fluids, Vol. 11, 1999.


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