Application of Modern Turbulence Models to VorticalFlow Around a 6:1 Prolate Spheroid at Incidence

S.-E. Kim and S. H. Rhee,Fluent Inc., Lebanon, NH 03766, USA

Davor CokljatFluent Europe Ltd., Sheffield, United Kingdom

This paper is concerned with prediction of turbulent flow past a 6:1 prolate spheroid at a range of incidenceangles. A finite-volume based Reynolds-Aaveraged Navier-Stokes (RANS) solver is employed for the computationin conjunction with several engineering turbulence models in popular use today. Attempts are made to improveperformance of the second-moment closure models by modifying length-scale equations. The numerical resultsare compared with the experimental data in terms of crossflow separation pattern, static pressure, skin-friction,and wall-shear angles on the body surface, and variation of lift and pitching moment with incidence angle. Theprediction accuracy varies widely depending on the turbulence model employed. The fidelity of the predictionsshown by some turbulence models such as Wilcox’ k-ω model and the Reynolds-stress transport models withmodified length-scale equations are highly commendable.

IntroductionDespite its simple geometry, flow around a prolate spheroid in

maneuvering carries a rich gallery exhibiting a variety of com-plex three-dimensional turbulent shear flows, featuring stagnationflow, highly three-dimensional boundary layer under the influenceof strong pressure gradients and streamline curvature, cross-flowseparation, and formation and evolution of free-vortex sheet andensuing stream-wise vortices (See Figure 1 for an illustration). Allthese features of spheroid flows are the archetypes of flows aroundairborne and underwater vehicles at incidence or in maneuvering,warranting an in-depth study.

The subject flow has been studied by many others experimen-tally and numerically. Among the most relevant to the presentstudy are the early works of Meier et al.1, 2 and the more re-cent works of the group at Virginia Polytechnic Institute (VPI).3–6

The series of experimental studies conducted by these two groupsprovide most comprehensive experimental data, revealing salientphysics of the flow and offering an invaluable dataset useful tovalidate computational fluid dynamics (CFD) codes. On the nu-merical side, there have been quite a few studies using reducedNavier-Stokes equations in earlier days and, more recently, usingfull Naiver-Stokes or Reynolds-averaged Navier-Stokes (RANS)equations.7–12

Given the flow features of both fundamental and practical in-terest as alluded to in the beginning, the present work is gearedtoward evaluating how well one can predict the subject flow nu-merically. For the spheroid to be studied, we chose the one whichhas been extensively measured by the group at VPI. The com-putational study presented in this paper covers the entire rangeof incidence angle experimentally investigated (α 10

30).

As with other complex three-dimensional turbulent shear flows,turbulence modeling plays a significant role for the present flow,affecting the prediction accuracy, especially in light of the chal-lenging features of the subject flow noted earlier. In pursuit of the

Z

Y

X

Fig. 1 Cross-flow separation and streamwise vortices on a 6: 1 prolate spheroid at α 30

: The pathlines are computed

using the present CFD solution.

best possible predictions, several popular engineering turbulencemodels with good track-records for similar flows were employed,including eddy-viscosity transport model, a family of k-ω models,and second-moment closure (SMC) models. In addition to assess-ing the fidelity of these models for the present flow, we will alsoexplore some avenues to improving the performance of some ofthe selected models. All the computations were carried out usinga finite-volume based a RANS solver.13, 14

The paper is organized as follows. We start by looking at someof the salient features of the flow and pondering upon their impli-cations to turbulence modeling of the subject flow. To that end, thenumerical results are utilized to illustrate some of the significantcharacteristics of the flow. This is followed by a brief overviewof the numerical method and turbulence models employed in this

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41st Aerospace Sciences Meeting and Exhibit6-9 January 2003, Reno, Nevada

AIAA 2003-429

Copyright © 2003 by Fluent Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

work. Finally the computational results will be presented alongwith the discussion.

Preview of Main Flow FeaturesHow can we characterize the present flow? A visual impression

of the mean flow in question is aptly portrayed by Figure 1, whichwas generated using the numerical solution for the α 30

case.

The figure serves nicely to highlight the most prominent featuresof the flow, i.e, crossflow in the boundary layer, free vortex sheet,and stream-wise vortices. The whole phenomena depicted here arean embodiment of the so-called “crossflow” or “open” separation,the significance of which can be recognized from the fact that thestructure of separation and its change with incidence angle greatlyaffect maneuvering characteristics of the body such as forces andmoments acting on it. One motivation to characterize the meanflow is that a good understanding of it, if qualitative, often enablesone to surmise whether or not certain turbulence models will beadequate for the flow at hand. In this vein, one of the useful ques-tions to ask for the present flow is: how the mean flow deforms?

Much insight to the mean flow along this line can be gainedfrom contours of a normalized invariant of deformation tensor15

defined as:

D Si jSi j

Ωi jΩi j

Si jSi j Ωi jΩi j

(1)

where Si j and Ωi j are strain and rotation tensors defined bySi j ∂Ui ∂x j

∂U j ∂xi 2 and Ωi j ∂Ui ∂x j ∂U j ∂xi 2,

respectively.This invariant, which ranges between -1 and 1, is a convenient

measure of the relative importance of strain, shear, and rotation.Note that D 1 for pure strain, D 0 for pure shear, and D 1for pure rotation. Figure 2 shows contours of this invariant inthe crossflow plane at x l 0 772 and the incidence angle ofα 20

. According to the figure, the subject flow is largely shear-

dominated (D 0) in the leeward boundary layer. However, theflow becomes predominantly rotational (D 1) near the core ofthe stream-wise vortices. Evidently, the mean flow portrayed hereis highly three-dimensional and carries significant extra rates ofstrain and/or rotation, providing an acid test for turbulence mod-els. As regards the effects of rotation on turbulence at the mostfundamental level, say, homogeneous isotropic turbulence, it isa well-established fact that rotation inhibits energy transfer fromlarger to smaller eddies, decreasing the decay rate of TKE.16 Itis also well known that, when a mean shear exists, rotation caneither delay or accelerate the energy transfer depending on the rel-ative orientations of the mean shear and the rotation, attenuatingor accentuating turbulence accordingly. The well-known effectsof streamline curvature, either convex or concave, which is alsorelevant to the subject flow, can be explained in the same way.

How rapid the mean flow is strained, sheared or rotating is alsoof interest from a turbulence modeling standpoint. Time-scaleof mean flow (1 S or 1 Ω) normalized by turbulence time-scale(k ε), i. e., Sk ε (S 2Si jSi j) and Ωk ε (Ω 2Ωi jΩi j), aregood measures for that. Contours of these “relative” strain androtation in the crossflow plane at x L 0 772 are shown in Fig-ure 3 and Figure 4. In spite of the fact that these quantities werecomputed using numerical solutions, the level of these two quan-tities (Sk ε Ωk ε 1) as shown in the figures suffices to indicatethat the subject flow is distorted quite rapidly. This has negative

Section at x/L = 0.772Contours of deformation-ratioProlate Spheroid at alpha = 20 deg.

1.00e+00

8.24e-01

6.47e-01

4.71e-01

2.95e-01

1.19e-01

-5.76e-02

-2.34e-01

-4.10e-01

-5.86e-01

-7.63e-01Z

Y

X

Fig. 2 Contours of the deformation invariant (see Equa-tion (1) for its definition) in the crossflow plane at x L 0 772and α 20

computed using the RSTM-2 result

x/L = 0.772Contours of relative-strain6:1 Prolate Spheroid at alpha = 20 deg.

3.24e+01

2.91e+01

2.59e+01

2.27e+01

1.94e+01

1.62e+01

1.30e+01

9.72e+00

6.48e+00

3.24e+00

3.96e-04Z

Y

X

Fig. 3 Contours of relative strain (Sk ε) at x L 0 772 (α 20) computed using the RSTM-2 result

x/L = 0.772Contours of relative-rotation6:1 Prolate Spheroid at alpha = 20 deg.

1.23e+01

1.11e+01

9.83e+00

8.60e+00

7.37e+00

6.14e+00

4.91e+00

3.68e+00

2.46e+00

1.23e+00

2.01e-04Z

Y

X

Fig. 4 Contours of relative rotation (Ωk ε) at x L 0 772(α 20

) computed using the RSTM-2 result

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implications for almost all aspects of turbulence modeling. Botheddy-viscosity model (EVM) and SMC suffer when the mean flowunder consideration is rapidly sheared, strained, or rotating.17

All these considerations on the mean flow set the stage for thefollowing question: how far the turbulence in a given flow is fromequilibrium? It is an important question to ask, inasmuch as al-most all the phenomenological turbulence models in use today,including most sophisticated SMC-based models, rely upon localequilibrium assumption in one way or another. Equilibrium refersto a state where production of turbulent kinetic energy (TKE),which is an event involving larger eddies, is balanced by its dissi-pation rate occurring at small scales. In highly complex turbulentflows like the present one, however, equilibrium state is less thanlikely to be attained. Several mechanisms are responsible for caus-ing nonequilibrium of turbulence. In the present flow, the salientfeatures of the present flow discussed so far act to make the turbu-lence to depart from equilibrium. It is surmised that rapid strainand rotation, streamline curvature, and adverse pressure gradientin the boundary layer play significant roles. Thus, turbulencemodels that are better capable of representing these physics andtheir effects on turbulence are likely to be more successful for thepresent flow than others which do not.

The experimental studies cited in the beginning extensively dis-cuss and provide valuable insight into the salient features of thepresent flow. Among many others, we find particularly notewor-thy the discussion by Chesnakas and Simpson3 regarding turbu-lence anisotropy based on their measurements of the individualReynolds stresses and the mean velocity field. By analysing thestresses and mean rates-of-strain, they assessed the validity ofisotropic eddy-viscosity assumption for the present flow. Theyfound that the Reynolds stresses are largely aligned with the strainrates inside the boundary layer at a low incidence angle (α 10

).

However, they become grossly mis-aligned almost everywhereelse, especially along the free vortex sheet and near the vorticeson the leeward side of the body and at high incidence angle. Asthey concluded, this suggests that turbulence models based onisotropic eddy-viscosity are likely to perform poorly, warrantinguse of SMC.

In summary, there is much evidence indicating that the presentflow is a complex, highly three-dimensional flow with rapid strainand rotation, and with highly nonequilibrium, anisotropic turbu-lence. All these observations and thoughts influenced our overallturbulence modeling strategy adopted in this study; the model se-lection and the rationales behind our attempts to modify someturbulence models.

Numerical MethodA steady form of Reynolds-averaged Navier-Stokes (RANS)

equations are used as the governing equations, since the salientfeatures of the flow we are mainly interested in can be found in av-eraged or mean flow. A cell-centered finite-volume method is em-ployed along with a linear reconstruction scheme that allows useof computational elements (cells) with arbitrary polyhedral topol-ogy, including quadrilateral, hexahedral, triangular, tetrahedral,pyramidal, prismatic, and hybrid meshes. The velocity-pressurecoupling and overall solution procedure are based on SIMPLE-type segregated algorithm adapted to unstructured mesh. The con-vection terms are discretized using a third-order upwind scheme,and the diffusion terms using central differencing scheme. The

high-order terms are treated using a deferred correction approach.The discretized algebraic equations are solved using a pointwiseGauss-Seidel iterative algorithm. An algebraic multigrid methodis employed to accelerate solution convergence. The details of thenumerical method are described by Mathur and Murthy,13 Kim etal.,14 and Kim.26

Turbulence ModelsAmong many choices, we screened the turbulence models

based on their relevance and track-records for aerodynamics ap-plications. So the k-ε models are omitted in the paper. Table 1summarizes the models chosen for the paper. The list includesfour EVM and three SMC models. The details of the models canbe found in the cited papers.

Abbreviation DescriptionSA Eddy-viscosity transport model18, 19

SST Blended k-ω /k-ε 20

KO-1 High-Re k-ω 22

KO-2 Low-Re k-ω 22

RSTM-1 Second-moment closure26

RSTM-2 Second-moment closure (present)RSTM-3 Second-moment closure (present)

Table 1 Turbulence models used in this study

Eddy-Viscosity Transport Model of Spalart and Allmaras

In the Spalart-Allmaras (SA) model,18 one directly solves thetransport equation for an effective viscosity, ν. The SA modelhas become rapidly popular especially in the aerospace commu-nity due to its commendable performance for boundary layer flowssubjected to adverse pressure gradient. The SA model used in thisstudy is identical to the original model except one thing. In the SAmodel, the production of ν is computed by:

Gν ρCb1 S ν (2)

where Cb1 is a model constant, and ν the effective viscosity. In theoriginal model, S in Equation (2) is computed from a modulus ofrotation rate tensor as:

S 2Ωi jΩi j Ω (3)

For thin boundary layer flows, it would be of no consequencewhether the modulus of rotation-rate tensor or that of strain-ratetensor is used, since they are comparable in magnitude. How-ever, adopting vorticity magnitude has an unwanted consequencefor swirling or vortical flows like the present flow where rotationdominates over strain in the vicinity of vortex core, leading tospuriously large production of ν. The original S-A model indeedperformed very poorly for the present flow. To avoid the problem,we adopted what Dacles-Mariani et al.19 proposed, i. e.,

S Ω Cν min 0 S Ω (4)

where S 2Si jSi j and Ω 2Ωi jΩi j are moduli of strain-rate and rotation-rate tensors, respectively, with Cν being an ad-justable constant of an order of 1. We simply took the valueCν 2 0 from the cited reference.

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k-ω Models

We adopted here three variants of k-ω models in popular usetoday. The first two k-ω models, denoted as KO-1 and KO-2in Table 1, are based on the recently revised version of Wilcox’k-ω models.22 The difference between the Wilcox’ originalmodel21 and the revised one lies in the “shear-correction” and“vortex-stretching” terms which were newly added in the revisedmodel to improve the performance mainly for free shear flowssuch as far-wakes, mixing layers, and jets. The new model also hasa re-calibrated low-Reynolds number model designed to accountfor transitional effects. KO-1 and KO-2 thus refer to the revisedWilcox’ models without and with the low-Reynolds number mod-ification, respectively. Another variant of k-ω model used in thisstudy is Menter’s k-ω model20 often referred to as shear-stresstransport (SST) k-ω model in the literature. The SST model is es-sentially a “two-zone” model that blends a variant of k-ω model inthe inner layer and a k-ω model obtained essentially by transform-ing the traditional k-ε model in the outer layer. In addition, theSST model clips turbulent viscosity based on the argument thatthe structural similarity between k and uv (k uv a1

0 3)should be preserved, which can be used to set an upper limit onturbulent viscosity as:

νt min

kω a1 k

Ω (5)

It should be noted that, in the SST model, this limit is appliedwithin boundary layer only. As for the present flow, the leewardside of the flow will be mostly unaffected by the clipping, exceptthe near-wall region. And the model essentially reduces to a tradi-tional k-ε model outside the boundary layer.

Second-moment closure models

The baseline model of the Reynolds-stress transport models(RSTM) used in this study consists of Rotta’s model23 for slowredistribution term, isotropization of production (IP) model ofFu et al.,24 and wall-reflection model of Gibson and Launder.25

The baseline model, which will be called RSTM-1 hereafter, wasimplemented in unstructured mesh based finite-volume RANSsolver, and has been been validated for a number of complex three-dimensional internal and external flows.26, 27 The unique featuresof the implementation include: isotropic turbulent diffusion mod-els for Reynolds-stress and dissipation equations, high-order dis-sipation term designed to prevent decoupling of Reynolds stressesand mean velocity field arising from co-located, cell-centered fi-nite volume discretization scheme. The wall-reflection effects inthe pressure-strain correlation were included with the aid of a wall-proximity function that allows wall distance and wall normals tobe computed for arbitrary wall configurations.

In the RSTM-1 model, turbulence length-scale is obtained bysolving the transport equation for ε given by:

ρDεDt

∂∂x j

µ µt

σε ∂ε∂x j Cε1

12

Piiεk Cε2ρ

ε2

k(6)

where Pi j is the production of Reynolds stresses, σε 1 3, Cε1

1 44, Cε2 1 83.

Despite more-than-modest improvements over lineark-ε models, the performance of the RSTM-1 model fell short ofour expectation in comparison to the remarkable performance

of the same model for a similar flow.27 Its rather disappointingperformance begs a question as to what the main culprits could be.Among many leads alluded to earlier, including the shortcomingsof the SMC itself for non-equilibrium flows like the subjectflow, we decided to track down the length-scale model equation,namely, ε-equation. As discussed earlier, the boundary layerand free vortex sheet with large shear/strain and strong rotationmake conventional length-scale equations, i.e., the ε-equation inEquation (6), less than adequate for the present flow. To inves-tigate the impact of the length-scale equation on the prediction,we adopted here two alternative ε model equations. One of themis the ε-equation proposed by the turbulence modeling group atNASA Glenn (Shih et al.28), which was developed starting froman exact equation for mean-square vorticity fluctuation (ωkωk).This enstrophy-based ε-equation has been used mostly in thecontext of two-equation k-ε models. Luo and Lashiminarayana29

adopted it in conjunction with a RSTM to study duct flows. Theyreported that this new ε model equation was capable of predictingthe turbulence enhancement observed in boundary layer nearconcave walls significantly better than the traditional model.The new ε-equation was also claimed to better describe theprocess of vortex stretching and spectral energy transfer, and hasactually been found to perform significantly better than traditionalk-ε models for boundary layers involving rapid strain and severeadverse pressure gradient. All these benefits seem relevant to thepresent flow. The new ε-equation reads:

ρDεDt

∂∂x j

µ µt

σε ∂ε∂x j ρC1Sε ρC2

ε2

k µε ρ (7)

where S 2Si jSi j), and

C1 max 0 43 η

η 5 η Sk ε

C2 1 9 σk

1 0 σε 1 2

In the RSTM-2 model, the standard ε-equation in the baselinemodel was replaced by the new ε-equation in Equation (7).

The strong non-equilibrium turbulence in the present flowprompted us to look for models known to better deal with that.And we employed the modification proposed by Durbin30 andChen and Kim31 as the second alternative. The modified ε-equation is given by:

ρDεDt

∂∂x j

µ µt

σε ∂ε∂x j C ε1Pk

εk Cε2ρ

ε2

k(8)

where σε 1 3, Cε2

1 83, and C ε1 is computed from:

C ε1 Cε1 1 aPk ε (9)

where Cε1 1 4 and a 0 05. As can be noted, the model

parameter, C ε1 in the “production-of-dissipation” term is a func-tion of Pk ε. This term adds more dissipation as production ofTKE becomes larger than dissipation rate, suppressing spuriouslylarge TKE frequently encountered in complex flows. The baselinemodel with the standard ε-equation replaced by Equation (8) andEquation (9) is denoted as RSTM-3 in this paper.

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Near-wall treatment

The choice of near-wall treatment depends on the resolution ofthe near-wall mesh in use. For the present study, we employedboth fine and coarse meshes. The fine meshes are such that theentire near-wall region is resolved down to wall including viscoussublayer. SA, SST, KO-1, and KO-2 models, all of which can benaturally integrated to wall, were run on the fine meshes. Whenfine meshes are used, the wall boundary conditions for the meanvelocity and turbulent quantities essentially exploit no-slip condi-tion at walls. For ω, we “fix” the asymptotic value of ω as y 0at wall-adjacent cells, using:

ωP 6ν

βyP2 (10)

where yP is the distance from the wall to the cell center, and β 0 075.

The coarse meshes were designed to skip the viscosity-affectedregion by placing the first grid (cell-center) points in fully tur-bulent region. We then can employ wall functions, namely, thelaw-of-the-wall and related hypotheses, to derive the wall bound-ary conditions for the mean velocity and turbulence quantities suchas Reynolds stresses and ω.14, 26 All RSTM computations weremade on the coarse mesh only. Among the EVMs, SST and KO-1models were run on the coarse mesh using the wall functions.

Computational DetailsBody geometry and computational conditions

The experiments conducted at the Virginia Polytechnic Institutewith 1 37m-long model of a 6:1 prolate spheroid are simulatedin the computations. The Reynolds number (ReL) based on thefreestream velocity (U0) and the body-length (L) is 4 2 106.In the experiments, the body was supported by a sting mountedat the rear-end of the body. The body was also mounted a tripat x L 0 2 to help trigger laminar-to-turbulent transition. Thesting-mount were not modeled in the computations. However, weattempted to mimic the effects of the trip using a numerical trig-ger. More on this will be discussed shortly in the beginning of the”Results” section.

The coordinate system adopted in this study is such that the pos-itive x-axis is in the streamwise direction, y points to the upwardvertical direction, and the x y plane makes the vertical symm-metry plane. The origin of the coordinate system is placed at thefore-end of the spheroid.

Solution domain, mesh, and boundary conditions

The solution domain covers 2 75

x L 3 425 and 2 75

y L z L 2 75, in the streamwise and lateral direction, respec-tively. Single-block hexahedral meshes are used. Four differentsizes of mesh were used to see the effects of different near-wallmodeling strategies (wall function vis-a-vis near-wall-resolvingapproach) and also to ensure mesh-independency of the numeri-cal solutions. Two of them (540,000 cells, 1,380,200 cells) weredesigned to penetrate the viscous-sublayer. Two other meshes(345,000 cells, 1,120,000 cells) were made for the wall functioncalculations. For each pair of meshes, mesh-dependency of thenumerical solutions were found to be insignificant. Therefore,we present here the results for the respective coarser meshes :345,000-cell mesh for wall function calculations, and 540,000-cell

mesh for the near-wall model calculations. The y values at thewall-adjacent cells of the fine and the coarse meshes were neary 1 0 and y 30, respectively, over most of the body sur-face.

The symmetry of the geometry and the flow allowed us tomodel only a half of the domain. Thus, the domain boundary con-sists of the body surface, upstream/far-field inlet, vertical plane ofsymmetry, and exit boundary. The wall boundary conditions wereapplied as described earlier. On the inlet boundary, freestreamconditions were specified. On the exit boundary, the solutionsvariables were extrapolated.

The steady computations were carried out for three incidenceangles, namely, α 10

20 30

. The numerical solutions were

deemed converged when scaled residuals for all solution variablesdrop by four orders of magnitude. The lift and pitching momentwere also monitored to ensure full convergence of the solutions.

ResultsEffects of boundary layer tripping

To mimic the effects of the physical trip mounted at x L 0 2,we employed a numerical trigger by which the flow in the up-stream of the trigger was treated as being laminar, whereas theturbulence models were enforced to become fully effective rightat the trigger. Several runs were made with the k-ω models withthe numerical trigger. However. the numerical results did notshow any meaningful differences from the fully-turbulent resultsin terms of the global features of the flow such as the surfacequantities and the lift and moment, except a very small region inthe laminar region. Thus, we made all the subsequent computa-tions assuming that the flow is fully turbulent over the entire body,which are presented in the paper.

Crossflow separation pattern

Line of crossflow separation is usually defined as a (hypotheti-cal) line in a given wall-shear vector field onto which neighboringwall limiting streamlines converge. The separation line can be vi-sualized by surface flow visualizations, as long as convergence ofthe wall limiting streamlines is strong enough. The VPI grouplooked at several other indicators3–5 and studied the correlationsbetween those and the separation location determined from theflow visualization. One of the proposed indicators was the loca-tion of minimum wall-shear (or skin-friction). The locations ofskin-friction minima were found near the separation yet consis-tently on the further leeward side of the separation line.

For comparison with the experimental data available at x L 0 729, two circumferential angles were read off from the numer-ical results. One is the angle at which the skin-friction becomesa minimum, which enabled us to make direct comparison withthe measurement. Figure 5(a) shows the results. It should bementioned that all the EVM results shown here were obtained onthe fine mesh, whereas the RSTM results are based on the coarsemesh. Quite a large scatter is observed among the results, withSA, KO-2 and RSTM-2 deviating farthest from the experimentaldata. In the experiment, the location where the circumferential ve-locity changes its sign (direction) was found closer to the actualseparation location. Figure 5(b) shows the angles thus-determinedfrom the numerical results, along with the separation location de-termined from the oil flow visualization. In comparison to theminimum C f angles in Figure 5(a), the angles determined this way

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shift leeward by a substantial amount. The overall trends shownby the different turbulence models remain largely unchanged fromwhat were seen earlier. Particularly noteworthy is that KO-2 andRSTM-2 models better predict the actual separation locations thanthe locations of minimum C f . It is also seen that the SA and SSTmodels continue to predict a delayed separation.

The experiments also reveal that, at higher incidence angle(e.g., α 20

), another separation line emerges. The numerical

results from some of the turbulence models also showed a clear ev-idence for the secondary separation. Figure 6 depicts the limitingwall streamlines numerically visualized using the result of KO-2model. Note that the body in the figure was rotated around its axisto provide a better view of the surface flow pattern. Clearly visibletoward the leeward symmetry plane is a line starting from a littledownstream of x L 0 6, onto which the neighboring streamlinesconverge. By x L 0 772, the secondary separation line becomesfull-fledged, as Chesnakas and Simpson3 described in their paper.In addition, the diverging streamlines between the two separationlines imply that the flow re-attaches to the body surface along aline (denoted as reattachment line in the figure). The reattachmentline is located much closer to the secondary separation line. Allthese features predicted by the KO-2 and RSTM models are ingood accordance with the experimental observations.3

Surface quantities

The behaviors of the surface quantities like pressure (Cp p p0 0 5ρU2

0 ), skin friction (C f τw 0 5ρU20 ), and wall-shear

angle (βw arctan WU ) carry rich information not only on the sur-

face flow itself but off-the-wall structures like the free vortex layerand streamwise vortices. At α 10

, the lowest incidence angle

we computed in this study, the crossflow and resulting separationare rather weak. And the differences among the turbulence modelswere found to be also small, although the differences become morediscernible as one goes to the rear and leeward side of the body.Although not shown here, SA model grossly under-predicted thecircumferential variations of all surface quantities, while the KO-2model tends to overpredict them. Overall, the results of the SSTand KO-1 models were about right. The behaviors of the pre-dicted surface quantities appeared to be largely consistent with theexperimental observations with regard to the incipient crossflow atx L 0 6 and the fully-developed separation line at x L 0 772.

The surface quantities exhibit much richer features at α 20.

And the differences among the models also become far more no-ticeable. Figure 7, Figure 8, and Figure 9 depict the circumfer-ential distributions of Cp, C f , and βw at α 20

, respectively. It

should be noted that the results shown in these figures were ob-tained using the fine (near-wall-resolving) mesh. The experimen-tal data3, 4 show that the primary vortex is located near φ 158

in the crossflow planes at both x L 0 6 and x L 0 772. Thisis manifested by the conspicuous dips in the measured Cp datashown in Figure 7. The predictions widely vary among the models.The SA model barely captures the feature, giving the shallow-est dip and yielding the minimum pressure location more leewardthan all the other models and the data. The KO-2 model seems toclosely capture both the locations and magnitudes of the minimumCp. KO-2 also best reproduces the Cp-plateau seen right after theprimary separation which was found in the experiments to occurnear φ 123

and φ 112

at x L 0 6 and x L 0 772, re-

spectively. Furthermore, at x L 0 772, the KO-2 model appears

5 10 15 20 25 30 35Angle of attack (deg.)

90

120

150

180

φ (d

egre

e)

Measured (min. Cf)

SASSTKO-1KO-2RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(a) Circumferential angle of minimum Cf

5 10 15 20 25 30 35Angle of attack (deg.)

90

120

150

180φ

(deg

ree)

Oil flowSASSTKO-1KO-2RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(b) Circumferential angle of circumferential vel. sign change

Fig. 5 Change in circumferential location of crossflow separa-tion with incidence at x L 0 729 determined by (a) minimumC f (b) circumferential velocity

Frame 003 26 Mar 2002 No Data Set

Frame 004 26 Mar 2002 No Data Set

Frame 005 26 Mar 2002 No Data Set

primary separation line

secondary separation lineleeward

windward

x/L = 0.600 x/L = 0.772

reattachment line

Fig. 6 Wall limiting streamlines showing the pattern of thecrossflow separation at α 20

- based on the KO-2 prediction

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0 30 60 90 120 150 180φ(degree)

-0.4

-0.3

-0.2

-0.1

0

Cp

MeasuredSASSTKO-1KO-2

(a) x/L = 0.600

0 30 60 90 120 150 180φ(degree)

-0.4

-0.3

-0.2

-0.1

0

Cp

MeasuredSASSTKO-1KO-2

(b) x/L = 0.772

Fig. 7 Surface pressure (Cp) distributions at α 20

pre-dicted with the fine mesh

to capture, far closer than other models, the small kink caused bya secondary vortex above the surface, which was found to occur atφ 140

in the experiment. However, KO-2 model deviates from

the data and other models’ results on the windward side of the pri-mary separation. It is interesting to see large differences amongthe three k-ω models. Evidently, the low-Reynolds number mod-ification in the KO-2 model seems to play a significant role inmaking a large difference between the KO-1 and KO-2 model re-sults. The SST model gives only marginally better results than theSA model, falling behind the KO-1 and KO-2 models.

As discussed in the foregoing, the circumferential locations ofminimum C f and the separation locations are correlated to eachother. The experiments indicate that the minimum Cf locations arelocated consistently on the leeward side of the actual separationline. This can be seen in Figure 8. Note that the primary separa-tion lines were observed at φ 123

and φ 112

in the crossflow

planes at x L 0 6 and x L 0 772, respectively. Again, the pre-dictions widely vary in terms of the minimum Cf locations, withthe maximum difference of approximately 15

between the SA

and the KO-2 models. The predicted locations of the Cf min-ima are seen to shift windward gradually as one goes from SA,SST, KO-1, KO-2 models. It is also observed that KO-1 and KO-2models predict the C f minima (and separation) a little too early incomparison to the data, which is largely consistent with the resultsshown in Figure 5. The distributions of predicted wall-shear angle

0 30 60 90 120 150 180φ(degree)

0

0.002

0.004

0.006

0.008

Cf

MeasuredSASSTKO-1KO-2

(a) x/L = 0.600

0 30 60 90 120 150 180φ(degree)

0

0.002

0.004

0.006

0.008

Cf

MeasuredSASSTKO-1KO-2

(b) x/L = 0.772

Fig. 8 Skin-friction (C f ) distributions at α 20

predictedwith the fine mesh

shown in Figure 9 give us a measure of the strength (magnitude)of the crossflow and the rate of convergence (or divergence) ofthe wall limiting streamlines near the separation or reattachmentlines. The separation and reattachment lines are likely to manifestthemselves as local peaks and zero-crossings in these plots. Thecomparisons among the models are largely in line with what Cp

and C f distributions show. The crossflow becomes progressivelystronger in the order of SA, SST, KO-1, and KO-2 models.

Thus far, we have looked at the surface quantities predicted us-ing the fine mesh which resolves down to wall including viscoussublayer. Now we will present the results obtained on the coarsemesh using the wall function approach. Figure 10, Figure 11, andFigure 12 depict the distributions of Cp, C f , and βw at α 20

,

respectively, predicted using the coarse (wall function) mesh. Theuse of coarse mesh and wall functions precluded the KO-2 model(k-ω model with a low-Reynolds number modification) from thecomparisons shown here. First of all, the results from the SSTand KO-1 models show only marginal differences between thecoarse (wall function) mesh results and the corresponding finemesh results. Practically, this finding is quite significant, inas-much as it gives a credential to the wall function approach andthe results based thereupon. The RSTM-1 model, the baselineSMC, performs clearly better than the SST model in terms of cap-turing the local variations of the surface quantities. Although notshown here, it should be mentioned that the RSTM-1 model yieldsmeasurable improvements over linear k-ε models which failed to

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS

0 30 60 90 120 150 180φ(degree)

-40

-20

0

20

40

β w (

deg.

)

MeasuredSASSTKO-1KO-2

(a) x/L = 0.600

0 30 60 90 120 150 180φ(degree)

-40

-20

0

20

40

β w (

deg.

)

MeasuredSASSTKO-1KO-2

(b) x/L = 0.772

Fig. 9 Wall-shear angle (βw) distributions at α 20

pre-dicted with the fine mesh

capture the prominent features of the flow. However, the KO-1model seems to outperform the RSTM-1 model. The RSTM-2 andRSTM-3 models are seen to bring remarkable improvements overthe baseline RSTM model, outperforming the k-ω models. It isquite interesting that seemingly minor changes in the ε-equationlead to such disproportionately large differences in the results.These improvements in the prediction of the surface quantities aredirectly carried over to the prediction of force and moment, as willbe shown later.

Comparisons of the coarse mesh results shown here and withthe fine mesh results presented earlier indicate that, insofar ashigh-Reynolds number flows are concerned, CFD predictionsbased on wall functions are perhaps better than has been com-monly believed. A similar conclusion was made by Kim27 basedon the numerical study of flow around a ship hull which sharessome of the salient features with the present flow.

Lift and pitching moment predictions

The lift acting on slender bodies like prolate spheroids is char-acterized by a nonlinear increase of lift with incidence angle. Thenonlinear lift is often called “vortex” lift because the augmentedlift is due to the low pressure near the core of the vortices which isimpressed upon the nearby body surface. Accurate prediction ofthe location and strength of the vortices is therefore prerequisite to

0 30 60 90 120 150 180φ(degree)

-0.4

-0.3

-0.2

-0.1

0

Cp

MeasuredSST (wall fn.)KO-1 (wall fn.)RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(a) x/L = 0.600

0 30 60 90 120 150 180φ(degree)

-0.4

-0.3

-0.2

-0.1

0

Cp

MeasuredSST (wall fn.)KO-1 (wall fn.)RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(b) x/L = 0.772

Fig. 10 Surface pressure (Cp) distributions at α 20

pre-dicted using wall functions

successful prediction of forces and moments.

The predicted lift (CL FN 0 5ρU20 L2) and pitching moment

coefficients (CM Mz 0 5ρU20 L3) for the full range of incidence

are plotted in Figure 13 along with the measured ones. The resultsshown in Figure 13 were obtained on the fine mesh. Like the re-sults presented so far, the predictions vary in a wide range. Thedifferences among the models increase with the incidence. Allmodels underpredict the lift. The SA and the KO-2 model yieldthe lowest and the highest lift, respectively. The lift predictionsare largely consistent with the behaviors of the surface quantitiesdiscussed before. Interestingly, the predicted pitching moment co-efficients shows an opposite trend. The SA model matches closestwith the measurement, while the KO-2 model deviates most fromthe data.

Figure 14 depicts the result based on the coarse (wall func-tion) mesh. It is noteworthy that the lift predictions by SST andKO-1 models based on wall functions are slightly higher than thecorresponding results based on the fine mesh. However, the dif-ferences are deemed insignificant, insofar as we are comparingthe results based on two drastically different near-wall modelingstrategies. The RSTM-1 prediction is largely comparable to theSST and KO-1 model results. The RSTM-2 and RSTM-3 resultsare in commendable agreements with the data.

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0 30 60 90 120 150 180φ(degree)

0

0.002

0.004

0.006

0.008

Cf

MeasuredSST (wall fn.)KO-1 (wall fn.)RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(a) x/L = 0.600

0 30 60 90 120 150 180φ(degree)

0

0.002

0.004

0.006

0.008

Cf

MeasuredSST (wall fn.)KO-1 (wall fn.)RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(b) x/L = 0.772

Fig. 11 Skin-friction (C f ) distributions at α 20

predictedusing wall functions

Summary and ConclusionsThe turbulent shear flow past a 6:1 prolate spheroid at the

Reynolds number of 4 106 was studied numerically using theReynolds-averaged Navier-Stokes equations. The study coveredthe steady flow for a full range of incidence angle (α 10

30).

Both fine meshes and coarse meshes were used to investigatethe effects of near-wall treatment. Several turbulence models,including Spalart-Allmaras’ one-equation model, Menter’s SSTk-ω model, Wilcox’ k-ω model, and three variants of Reynolds-stress transport model, were employed to evaluate the perfor-mances of the models for the present flow. The numerical resultswere compared with the experimental data in terms of several keyaspects of the flow such as pattern of the crossflow separation, dis-tributions of surface pressure, skin-friction and wall-shear angle,and finally variation of lift and pitching moment with incidenceangle.

The fidelity of the predictions varied in a wide range amongthe turbulence models. Some turbulence models employed herewere found to be reasonably successful in predicting the salientfeatures of the flow such as the separation patterns, wall pressuredistribution, skin-friction distribution, lift force, and pitching mo-ment reasonable well.

Among our significant findings are:

Mimicking transition numerically - treating the boundarylayer on the forebody as a laminar region and trigerring

0 30 60 90 120 150 180φ(degree)

-40

-20

0

20

40

β w (

deg.

)

MeasuredSST (wall fn.)KO-1 (wall fn.)RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(a) x/L = 0.600

0 30 60 90 120 150 180φ(degree)

-40

-20

0

20

40

β w (

deg.

)

MeasuredSST (wall fn.)KO-1 (wall fn.)RSTM-1 (wall fn.)RSTM-2 (wall fn.)RSTM-3 (wall fn.)

(b) x/L = 0.772

Fig. 12 Wall-shear angle (βw) distributions at α 20

pre-dicted using wall functions

turbulence downstream of the trip (x L 0 2) - affects thepredictions very little.

Some of the eddy-viscosity based models such as Spalart-Allmaras’ model and k-ω models that have shown consid-erable promise in complex two-dimensional or quasi-two-dimensional flows were found to perform rather poorly forthe present flow.

Overall, the revised k-ω model of Wilcox (1998) seems towork better than other EVMs employed in this study. In par-ticular, the low-Reynolds number version (KO-2) leads to ameasurable improvement in all important aspects of the flow.

The baseline Reynolds-stress transport model (RSTM-1),which performed admirably for similar three-dimensionalboundary layers yet with weaker crossflow and rotation, wasless successful for the present flow.

Minor modifications in the length-scale equation in theReynolds-stress transport models result in significant im-provements of the predictions.

Resolving the viscous sublayer brings only a marginal differ-ence. The wall function based results are largely comparableto the results based on the near-wall resolving meshes.

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0 5 10 15 20 25 30Angle of attack (deg.)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

CL

Measured SASSTKO-1KO-2

(a) Steady lift force

0 5 10 15 20 25 30Angle of attack (deg.)

-0.008

-0.006

-0.004

-0.002

0.000

CM

Measured SASSTKO-1KO-2

(b) Steady pitching moment

Fig. 13 Force and moment predictions with the fine mesh (a)lift (CL) (b) pitching moment (CM)

AcknowledgementThe authors acknowledge use of Fluent Inc.’s software and

thank the members of the product development group at FluentInc.

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