Full text of DOI:10.2514/2.7291

AIAA JOURNAL
Vol. 41, No. 9, September 2003
Turbulence Modeling Applied to Flow over a Sphere
¤
†
‡
George Constantinescu, Matthieu Chapelet, and Kyle Squires
Arizona State University, Tempe, Arizona 85287-6106
Numerical simulations of the subcritical  ow over a sphere are presented. The primary aim is to compare
prediction of some of the main physics and  ow parameters from solutions of the unsteady Reynolds-averaged
Navier–Stokes (URANS) equations, large-eddy simulation (LES), and detached-eddy simulation (DES). URANS
2
predictions are obtained using two-layer k–", k–!, v –f, and the Spalart–Allmaras model. The dynamic eddy
viscosity model is used in the LES. DES is a hybrid technique in which the closure is a modiŽ cation to the Spalart–
Allmaras model, reducing to RANS near solid boundaries and LES in the wake. The techniques are assessed
by evaluating simulation results against experimental measurements, as well as through their ability to resolve
time-dependent features of the  ow related to vortex shedding. Simulation are performed at a Reynolds number
4
of 10 , where laminar boundary-layerseparation occurs at approximately83 deg. With the exception of two-layer
k–", RANS predictions of the streamwise drag are in reasonable agreement with measurements. The pressure
and skin-friction coefŽ cients along the sphere are adequately predicted by the RANS models, with DES and LES
results in better agreement with measurements in the aft region. ProŽ les of the mean velocity and turbulent kinetic
energy in the near wake from all of the techniques are similar. The nearly axisymmetric URANS solutions predict
the value of the main shedding frequency, albeit at substantially reduced amplitude compared to the LES and
DES. DES compares favorably with LES in that both techniques resolve eddies down to the grid scale in the wake
and are better able to capture unsteady phenomena, including formation of Kelvin–Helmholtz instabilities in the
detached shear layers and the associated high frequency in the  ow.
I. Introduction
a laminar shear layer by contact with a strongly turbulentlayer, and
models should be relatively insensitive to freestream values of the
turbulencevariables.
A. Background
N important class of separated ows are those in which the lo-
cation of  ow detachmentis not Ž xed by the geometryand are
Satisfaction of the Ž rst requirement described, adequate predic-
tion of boundary-layer growth and separation, generally requires
closure models that resolve the near-wall  ow, rather than ap-
proaches that employ wall functions. Previous work indicates, for
example, the importance of resolving the viscous wall layer to ob-
A
not subjectto externaleffectsthat are used to force unsteadiness,as
occurs for many bluff body  ows. The main interest of the present
study is prediction of the  ow around a sphere in the subcritical
regime, a complex case characterized by large-scale vortex shed-
ding, transitional shear layers, and a turbulent wake with random
and periodic Reynolds stresses of comparablemagnitudes. Simula-
tion of the  ow around a sphere is then an important benchmark for
techniquesused to predict massively separated  ows.
2
;3
tain sheddingsolutionsin RANS calculations. Each of the RANS
approachesconsideredin this work resolvethe near-wallregionand
can be integrated all of the way to the wall: two-layer k–" (Ref. 4)
2
low Reynolds number k–! (Ref. 5), v – f (Ref. 6), and Spalart–
7
Allmaras (S-A). These models invoke the Boussinesq approxima-
Most engineering predictions of separated  ows are obtained
from solutions of the Reynolds-averaged Navier–Stokes (RANS)
equations, with an increasing interest in the use of large-eddy sim-
tion, by the use of one or more equationsto determine the isotropic
eddy viscosity,and, therefore,provide a crude representationof the
anisotropy of the Reynolds stresses. Although often sufŽ cient in
attached regions not far from the thin shear layer approximation,
it is difŽ cult to satisfy the second criterion outlined, accurate cap-
ture of the Reynolds stresses after separation, by the use of RANS
approaches that employ the Boussinesq approximation. The util-
ity of more complex RANS models, for example, Reynolds-stress
closures, for satisfaction of this requirement remains a subject of
1
ulation (LES). As discussed by Spalart, there are important cri-
teria that RANS models should satisfy to yield accurate predic-
tions of complex  ows: Models should correctlypredict the growth
and separation of the boundary layers and accurately capture the
Reynolds stresses after separation. For the sphere considered in
this work, boundary layers are laminar at separation (subcritical
regime), with transitionoccurringin the separatedshear layers. Ac-
curate prediction of separation in this case then requires that in the
laminar boundary layers (and, in general, in the laminar regions
of the  ow), turbulence models should accept vanishing solutions
for the Reynolds stresses, or at least small values without in uence
on the turbulent layers. In addition, the behavior of the models at
turbulent/nonturbulentinterfacesshouldallow the contaminationof
1
debate.
Previous applicationsof these models to separated ows include
predictionof the  ow aroundtriangularcylindersusing k–" (Ref. 8)
2
and v – f (Ref. 3), rectangularcylindersusing k–" (Refs. 9 and 10),
1
1;12
circularcylindersusingS-A,
and two-dimensionalairfoilsusing
1
3
S-A. Averaging of the shedding solution in a RANS calculation
typically yields a signiŽ cant fraction of the Reynolds stress, in turn
lessening the burden on the turbulence model. These studies show
that it is possible to resolve time-periodic shedding solutions with
a wake frequency generally comparable to the lowest frequency
measured in experiments. The amplitude modulation of the actual
shedding is difŽ cult to represent, however, and is one cause of the
Received 12 October 1999;revision received 19 February 2003;accepted
for publication 12 March 2003. Copyright °c 2003 by the American In-
stitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of
this paper may be made for personal or internal use, on condition that the
copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,
1
2
$
22 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/03
10.00 in correspondence with the CCC.
Research Engineer, Mechanical and Aerospace Engineering Depart-
ment; currently Research Engineer, Iowa Institute Hydraulics Research—
Hydroscience and Engineering,University of Iowa, 100 C. Maxwell Stanley
Hydraulics Laboratory, Iowa City, IA 52242-1585.
sometimes poor predictionsof integral quantitiessuch as the drag.
There are relatively few RANS or unsteady RANS (URANS)
¤
14
predictionsof spheres in the turbulentregime. Drikakis computed
the  ow past a sphere using an artiŽ cial compressibility method
and grid sizes up to about 58,000 grid volumes. Steady-state, that
is, time-independent,subcritical,and supercritical ows were com-
†
Research Assistant, Mechanical and Aerospace Engineering
1
5
k–" model. Predictionsof
Department.
puted with the Baldwin–Lomax and the
‡Professor, Mechanical and Aerospace Engineering Department.
thepressurecoefŽcientC
forthesubcriticalcasewereinreasonable
p
1
733

1
734
CONSTANTINESCU, CHAPELET, AND SQUIRES
agreementwithmeasurementsbeforeseparation,thoughpredictions
of the pressure coefŽ cient in the wake were rather poor. For the su-
percritical  ow, the pressure coefŽ cient was inaccurate over most
of the sphere, which results in large errors in the drag.
shedding solutions are obtained.12 As discussed in Sec. III, the dif-
ferent character of the URANS solutions of the sphere compared
to other cases such as the  ow around a circular cylinder might be
attributed in part to the reduced coherence of the shed structures in
the wake.
Techniques that resolve more  ow details, such as LES, are at-
tractive for massively separated ows becausea signiŽ cant fraction
of the turbulent motions are not modeled, instead being directly
resolved on the grid. LES then provides a powerful approach for
capturing the Reynolds stresses in separated regions, where the
stresses are dominated by the geometry-dependent large eddies.
Tomboulideset al.16 used LES (in addition to direct numerical sim-
Unlike RANS approaches, the role of the turbulence model is
less dominant in LES and DES where the large scales, which con-
tribute strongly to unsteady effects, are resolved. Consequently, a
reŽ ned evaluation of the URANS is possible by bringing out as-
pects of these calculationsrelative to the LES and DES predictions.
Note that because boundary-layer separation is laminar, the DES
predictions constitute an LES away from the boundary performed
in this investigation with a one-equation transport model for the
subgrid-scale eddy viscosity. An evaluation of the DES is, there-
fore, possible in the subcritical regime via comparison to the LES
predictions obtained with a more established approach. For super-
critical ows that experienceturbulentboundary-layerseparation,a
similarevaluationof DES predictionsis somewhatmore ambiguous
due to the need to model the wall layer in LES.
In the following section, the numerical method, boundarycondi-
tions,and turbulencemodelsare summarized,along with discussion
of aspects concerningvalidation of the simulations. URANS, LES,
and DES resultsare thenpresentedalongwithcomparisonstoexper-
imental measurements. The main focus in Sec. III is on prediction
by these techniques of the overall parameters of engineering inter-
est, as well as some unsteadyfeatures of the  ow. Conclusionsand
recommendationsare presented in Sec. IV.
1
7
ulations at lower Reynolds numbers to predict the wake behind a
sphere using a subgrid model based on renormalization group the-
4
ory. Predictions were obtained at a Reynolds number of 2 £ 10 ,
where there is a laminar boundary-layer separation. Tomboulides
1
6
et al. resolved the higher frequencies present in time histories of
the streamwise drag caused by instabilities in the detached shear
layers. These investigatorsalso obtainedpredictionsof the drag that
were in good agreement with measurements.Although not exhaus-
tive, this study demonstrates that resolution of features related to
vortex sheddingand three-dimensionalityof the  ow are important
to the capture of unsteadyeffects. Such effects are more difŽ cult to
resolve by the use of closure models, which parameterizeall scales
of motion as in Reynolds-averagedmethods.
LES, despiterequiringlessempiricismthanRANS, carriesa high
computational cost when used to resolve thin features of fully tur-
bulent  ows.1
8;19
To combine the most favorable aspects of RANS
and LES, Spalart et al.19 proposed a hybrid technique known as
detached-eddy simulation (DES), essentially reducing to RANS
near solid boundaries and LES away from the wall. The closure
is based on a simple modiŽ cation to the S–A model and takes ad-
vantage of the usually adequate performance of RANS models in
the thin shear layers where these models are calibrated.In separated
regions where the  ow is far from the thin shear layer approxima-
tion, an LES treatment is used to resolve time-dependentstructures
thatare not treatedas well with RANS. Most of the stress in the LES
region is resolved, rather than modeled, and stress anisotropiesare
calculateddirectly.DES and LES predictionsare used in the present
investigationsto provide a basis for evaluation of unsteady effects
predicted by RANS models.
II. Simulation Overview
A. Numerical Approach
In the present study, the incompressible  ow around a sphere
is computed with a fractional step method. The governing equa-
tions are transformed to generalized curvilinear coordinates with
the primitive velocitiesand pressureretained as the dependentvari-
ables. The base numerical method was previously employed for
21;22
computation of steady  ows.
Extension to time-accurate cal-
culations was performed with a double-time-steppingalgorithm as
23
described by Johnson and is summarized hereafter.
Withina physicaltime step, the momentumand turbulencemodel
equations are integrated in pseudotime using a fully implicit algo-
rithm. In the Ž rst step of the fractionalstep method, an intermediate
velocity Ž eld is obtained by advancingthe convectionand diffusion
termsusingan alternatedirectionimplicitapproximatefactorization
scheme. The intermediateŽ eld is obtained with the currentpressure
Ž eld and does not satisfy the continuity equation. A Poisson equa-
tionis then solvedfor the pressure,and the resultingsolutionis used
to update the intermediate velocities so that continuity is satisŽ ed.
Advancement in pseudotime is continued until a converged solu-
tion of the equations is obtained. Local time-stepping techniques
are used to accelerate the convergence of the resulting system of
equations. Source terms in the turbulence-modelequations are also
treated implicitly.
B. Objectives and Approach
The main goal of this paper is to evaluate the performance of
some of the leading RANS models by the use of predictions of
2
the subcritical  ow over a sphere: two-layer k–", k–!, v – f , and
S–A. Predictions are evaluated against the experimental measure-
ments of Achenbach,20 as well as results from DES and LES. The
present contributionfocuses on a Reynolds number in the subcriti-
4
cal regime, Re D 10 , where there is a laminar boundary-layersep-
aration from the sphere surface. The laminar separation relaxes
resolution requirements near the surface, in turn permitting per-
formance of whole-domain LES using a grid of approximately
5
5
£ 10 points.
All terms in the pressure Poisson equation are discretized with
second-orderaccurate central differences.Odd–even decouplingin
the pressureŽ eld, associatedwith the discretizationof the equations
on nonstaggered grids, is circumvented through use of the opera-
RANS predictionsof the unsteady ow are obtainedfrom numer-
ical solution of the time-dependent Navier–Stokes equations. One
of the challenges for the RANS models is the application outside
their calibration range, that is, in a massively separated  ow that is
far from the thinshearlayerswherethemodelconstantsin the turbu-
lenttransportequationsare calibrated.A priori,it might be expected
that this would result in poor predictions.However, URANS results
discussed in Sec. III show that predictions of quantities such as the
pressure coefŽ cient and skin friction along the sphere are mostly
adequate.The main deŽ ciencyassociatedwith the URANS simula-
tions is relatedto the more generalproblemthatReynoldsaveraging
suppressesmuch useful informationfrom the solution. For the sub-
critical ow around the sphere consideredin this study, frequencies
associatedwith the main instabilitymodesand the energycontentof
these frequencies are not well resolved, and the URANS solutions
become essentially axisymmetric and nearly steady. This behavior
differs from related applications of URANS methods to massively
separated ows such as the circularcylinder,in which time-periodic
2
4
tor developedby Sotiropoulosand Abdallah. The momentum and
turbulence transport equations are discretized using either second-
or Ž fth-order-accurateupwind differencesfor the convectiveterms,
whereas all other operators are calculated using second-order cen-
tral differences. The overall discretization scheme is second-order
accurate in space, including at the boundaries. Effects of the or-
der of upwinding and the dissipation associated with the odd–even
decoupling algorithm are discussed in Sec. II.D.
Solutions are obtained on a domain that extends from the
sphere surface (r D 0:5D where D is the sphere diameter) to
15 diameters in the radial direction. The governing equations are
solved on an O–O grid with .r; Á; µ/ the radial, polar, and az-
imuthal directions, respectively. The conditions at the upstream
boundary (r D 15D, 0< Á < 0:55¼) consist of a uniform veloc-
ity. Turbulence variables at the in ow boundary were set to their

CONSTANTINESCU, CHAPELET, AND SQUIRES
1735
2
¡8
3
¡6
¡1
,
thresholdvalues [k=U D 10 , "=.U =D/ D 10 , !D=U D 10
implementation of boundary conditions for the turbulent quantities
at the wall.
¡
8
and º
t
=.U D/ D 10 ]. The velocity components and turbulence
variables at the downstream boundary (r D 15D, 0:55¼ < Á < ¼)
are obtained with second-order extrapolation from the interior of
the domain. No-slip conditions on the sphere surface are imposed.
The pressure boundaryconditionon the sphere and at the upstream
and downstream boundaries are obtained from the surface-normal
momentum equation. Periodic boundaryconditionsare imposed on
allvariablesinthe azimuthaldirection.On the polaraxes,(Á D 0, ¼),
the variables (pressure, velocity, and turbulence quantities) are ob-
tained by averaging over the azimuth a second-order accurate ex-
trapolation of these variables.
4) The fourth model investigated is the S–A model. The S–A
7
model computes the eddy viscosityby the use of a single transport
equation.Thelow-Reynolds-numberversionof the modelpossesses
favorablenumerical characteristicsin terms of near-wall resolution
and stiffness properties. SpeciŽ cation of the boundary condition at
the wall for the modiŽ ed eddy viscosity is trivial.
Each of the models just summarized are linear eddy-viscosity
closures and do not account for effects such as the misalignment
between the turbulent stress and strain rate, presume equilibrium,
etc. In addition, the models are calibrated in  ows that are close
to the thin shear layer approximation.Applicationof these models,
all of which resolve the wall layers, to the time-dependent predic-
tion of a complex three-dimensional ow far from their calibration
range, is important to assessingmethods used to predict other  ows
encounteredin applications.
The calculationswere carriedout on a mesh of about450,000grid
nodes (101£ 42 £ 101 points in the r, µ, and Á directions)with the
C
Ž
rst point off the wall situated at about r D 0:2, and about eight
C
points within r < 10 (based on an estimate of the friction velocity
of 0.04). The maximum grid spacing near the sphere surface is
roughly nine wall units in the polar direction and 42 wall units in
the azimuth(at Á D 0:5¼). The effectof grid reŽ nement is discussed
in Sec. II.D. For the k–" model, turbulentboundary-layerproŽ les of
k and " were initiallyprescribednearthe spheresurface.The steady-
state k–" solutions were subsequently used as the initial Ž elds for
calculationsperformed with other turbulence models.
C. Turbulence Models: LES and DES
In LES, the subgrid-scale (SGS) stress is closed by the use of
28
the dynamic eddy viscosity model. In the dynamic approach, the
resolved scales are explicitly Ž ltered and the Germano identity is
used to compute the subgrideddyviscosity.The dynamic procedure
possesses the correct asymptotic behavior near solid surfaces and
differentiates between laminar and turbulent regions of the  ow,
without damping or intermittency functions.
The equations were advanced with a physical time step of
:02D=U, where U is the freestream velocity. This insured ap-
0
proximately 220 time steps per cycle, corresponding to the main
shedding frequency. The convergence criterion at every time step
was that the maximum value of the velocity and pressure residuals
should be smaller than 10 . This insured a reduction of more than
two orders of magnitudefor the norm of the velocitycomponentsto
reach incompressibility at each physical time step, and yielded an
adequate damping of errors throughout the entire domain. An av-
erage of about 60 subiterations were used to converge the solution
per physical time step.
In the dynamic procedure,a “test Ž lter” is applied at coarser res-
olution to obtain the “test-Ž eld” stress. By the use of the Germano
identity, an expression may be developed for calculation of model
coefŽcients. When the  owŽ eld is sampled, a mechanism is pro-
vided for dynamic models to respond to changes in the turbulence
due to perturbationsin the  ow. As a consequence,some nonequi-
librium effects are accounted for, despite the use of an isotropic
eddy viscosity model. The Smagorinsky model is used as the base
in this work, and the dynamic procedure yields a coefŽ cient that
varies in space and time. Test Ž ltering is applied in the statistically
homogeneousazimuthal directionwith the Ž lter width equal to two
times the local grid spacing. Although it would also be possible to
Ž lter over the other dimensions locally, test Ž ltering only over ho-
mogeneous directions offers theoretical advantages in the dynamic
¡
4
B. Turbulence Models: RANS
URANS solutionsareobtainedby solvingthe Reynolds-averaged
and turbulence model equations in a time-accurate fashion. As the
grid is reŽ ned, the turbulence model will continue to dominate the
ow, and, consequently, the quality of the RANS solution will re-
main strongly dependenton the turbulence model. Four turbulence
models are investigatedin the URANS:
) TheŽ rstmodelinvestigatedisthetwo-layerk–". Thetwo-layer
k–" model of Chen and Patel is straightforwardin implementation

2
9
formulation. To avoid numericalinstability,the SGS viscosityob-
tained after the model coefŽcient is averaged is also constrainedto
be positive through truncation to zero of any negative values.
The DES formulation used in this study is based on a modiŽ ca-
tion to the S–A model such that the closure reduces to RANS in
1
4
2
5;26
andhas beenappliedto predictionofa wide rangeof  ows.
Two-
1
9
layerk–" reducesin the inner layer to a simple one-equationmodel,
requiring an additional parameter, the matching distance between
the two-layers,to which the solutionfor  ows at low Reynoldsnum-
bers can be sensitive.22 The length scale applied in the inner layer
to determine the dissipation rate and eddy viscosity is a function of
the turbulence Reynolds number, k y=º and is inappropriate for
the prediction of laminar separation. It should be expected, there-
fore, that the two-layer model will be inadequate for the subcritical
regime of the sphere, as will be illustratedin Sec. III. The principal
motivation for application of the model in the present investigation
is for generationof reasonableinitial Ž elds for solutionsperformed
with other turbulencemodels.
boundarylayers and to LES away from the wall. The modiŽ cation
is a redeŽ nition of the length scale of the destruction term in the
eddy viscositytransport equation, written hereafterwithout the trip
terms that were not used in the present simulations:
µ ¶
2
1
=2
Q
£
¤
Q
Dº
1
º
Q
2
D c SºQ C r ¢..º CºQ/rºQ/Cc .rºQ/ ¡ cw1 fw
(1)
b1
b2
Dt
¾
d
whereeº is the working variable,
2
2
e
S ´ S C .eº=· d / fv2
;
f
v2 D 1 ¡ Â=.1 C Â fv1
/
(2)
and S is the magnitude of the vorticity. The eddy viscosity º
is
t
2
) The next model investigated is the k–!. The low-Reynolds-
5
obtained from
number version of the k–! model of Wilcox typically predicts
adverse-pressure-gradient boundary layers more accuratelythan its
counterpart k–" formulation at no additional computing cost.5;27
RANS predictionsof external ows using k–! are known,however,
to be sensitiveto freestreamvaluesof the turbulencevariables,espe-
cially in the near wake. The implementationof boundaryconditions
for k and ! at solid surfaces is trivial.
) The third model investigated is the v – f . The v – f model
of Durbin supplementsk–" with one additionaltransportequation
forv ) and an ellipticrelaxationequation.The model has been suc-
cessfully applied to prediction of equilibrium and nonequilibrium
¯
¡
¢
3
3
3
v1
º
t
Deº fv1
;
f
v1 D Â
 C c
;
 ´eº=º
(3)
where º is the molecular viscosity. The function f is given by
w
µ
¶
1
6
6
1
6
C c
eº
w3
6
fw D g
;
g D r C cw2.r ¡ r/;
r ´
6
g C c
e
S· d
2
2
2
2
3
w3
6
(4)
2
(
The wall boundary condition is eº D 0. The constants are cb1
D
3
2
3
2

ows. An issue is the computational cost of the model (»20%
0:1355, ¾ D , cb2 D 0:622, · D 0:41, cw1 D cb1=· C .1 C cb2/=¾,
increase in CPU time compared to two-equation models) and the
c
w2 D 0:3, cw3 D 2, and cv1 D 7:1. The DES formulation is obtained

1
736
CONSTANTINESCU, CHAPELET, AND SQUIRES
e
e
when the distance to the nearest wall d, is replaced by d, where d
is deŽ ned as
e
d ´ min.d; CDES1/
with
1 ´ max.1x; 1y; 1z/ (5)
Near walls, 1 is larger than d, and the standard S–A model is ob-
tained.Consequently,predictionof boundary-layerseparationis de-
termined in the RANS mode of DES. Away from solid boundaries,
the closureis a one-equationmodel for the SGS eddy viscosity.The
additional model constant CDES D 0:65 was calibrated with simula-
1
3
tions of homogeneous turbulence and is used without modiŽ ca-
tion in this study. In both the LES and DES predictions presented
here, the convectiveterms were discretizedby the use of Ž fth-order,
upwind-biasedŽ nite differences.
a)
c)
e)
D. Simulation Validation
Variants of the  ow solver used in this study employing several
RANS closureshavebeentestedextensivelyfor predictionof steady

ows and accurately predicted complex three-dimensionallaminar
2
1;22;25;30;31
and turbulent  ows.
Preliminary validation of the time-
accuratesolverwas performedthroughcalculationof the  ow pasta
sphere in the laminar, unsteady regime (Re D 300) and comparison
of drag predictions (both streamwise and lateral) against published
results.
At Re D 300, vortices are shed regularly from the sphere. By the
use of the solver employed in the current study, predictions of the
drag coefŽ cient C
respectively. Johnson and Patel obtained 0.656 and 0.0035, and
and its rms amplitude are 0.6556 and 0.0032,
d
3
2
16
Tomboulidesetal. computed0.671and 0.0028,respectively.Roos
and Willmarth33 measuredC
D 0:629. The mean value and the am-
b
)
d)
f)
d
plitudeofthe resultantlateral-forcecoefŽ cientwere 0.065and0.017
with the present method, and 0.069 and 0.016 in Ref. 32. Predic-
tions of the sheddingStrouhal number are 0.136, which agreeswell
Fig. 1 Mean streamwise velocity (upper Ž gures) and resolved-scale
turbulence kinetic energy (lower Ž gures) in the near wake: ——, DES-
CG; – – –, DES-FG; a) and d) x/D = 0.6, b) and e) x/D = 1.2, and c) and
f) x/D = 2.0.
3
2
with the correspondingvalues obtained by Johnson and Patel and
1
6
Tomboulideset al. of 0.137and 0.136,respectively.Sakamoto and
3
4
Haniu measured a value of the Strouhal number in the range of
0
.15–0.16 at Re D 300. Simulation results showed that the second-
order extrapolationused at the downstream boundarydid not nega-
tively affect the unsteady  ow, that is, no spurious oscillations nor
odd–even decoupling were detected near the boundary or within
the computational domain. Further validation of the time-accurate
solver for turbulent  ow simulations is provided in the results sec-
tion, where comparison with experimental data is provided for the
main parameters describing the unsteady shedding.
Grid sensitivity was assessed for the RANS calculations when
the mesh was reŽ ned in the polar direction, with a grid of
1
51£ 42 £ 101 points. Calculationswere carriedout using the k–!
and S–A models. Solutions on the Ž ner grid agreed well with those
on the coarsermesh, the pressure coefŽ cient and skin friction along
the sphere differing by less than 5% on the coarse and Ž ne grids.
Grid dependencefor the eddy-resolvingcalculationswas evaluated
by the use of DES predictions on two grids in which the azimuthal
resolutionwas increasedby a factor of two: DES-coarse grid (DES-
e
Fig. 2 Pressure coefŽ cient (averaged over the azimuthal coordinate):
–
– –, k–!; –¢¢–, k–"; ¢ ¢ ¢ ¢, v2–f; –¢–, S–A; - - - -, LES; ¤, DES-CG; ——,
CG) DES-Ž ner grid (DES-FG). The mean streamwise velocity V
1
20
DES-FG; and , Achenbach.
¨
and resolved-scaleturbulencekineticenergy K in the nearwake are
shown in Fig. 1. The mean velocity exhibits little variation on the
two grids, with a slightly higher centerline velocity on the coarse
grid at x=D D 1:2. Predictionsof K for these cases are also similar
to turbulencekineticenergylevels slightlylarger close to the sphere
studyshowthat,althoughthe employmentsecond-orderupwinddis-
cretizations for the convective terms was sufŽ cient for the RANS
solutions, use of higher-order discretizations (Ž fth-order upwind
biased differences) was important for resolving the spectral distri-
bution of energy and shedding parameters at higher wavelengths
and frequencies. Investigations into the role of the model and the
in uence of numerical dissipation were also conducted when cal-
culationsthat did not explicitly include the subgrid model were in-
cluded.Solutionsbecamenumericallyunstablein suchcalculations,
in turn providing evidence of the importanceof the subgrid models
to obtainingconvergent,and accurate,predictions.In general, these
featuresconcerningthe orderof the upwindscheme, role of the sub-
gridmodel, etc., are consistentwith relatedLES studies. Although
the nondissipativenatureof centeredmethods are preferablefor tur-
bulence simulations, low-order schemes for the convective opera-
tors can suffer from dispersive errors, for example, as can occur on
stretched grids. This complicates application of these methods for
(
x=D D 0:6) on the Ž ne grid, consistentwith the Ž ner mesh support-
ing a largerrangeof scales.The mean pressureand skin-frictiondis-
tributions over the sphere, shown in Figs. 2 and 3 for the two grids,
are also in good agreement. Correspondingly,the predictionsof the
streamwisedrag are similar, differingin C
d
by 1.5% in between the
two computations. The Strouhal numbers associated with the main
shedding for DES-FG and DES-CG were 0.190 and 0.200, respec-
tively. The Strouhal number ranges associated with the shear-layer
instabilitiesshowed more variation,rangingbetween 1.6 and 2.4 for
the Ž ne grid,and between1.9 and 2.1 for the coarsegrid.In addition,
the higher-frequencyrange on the Ž ne grid was more energetic.
The effectof numericaldissipationon the DES and LES solutions
was investigated by Constantinescu and Squires.35 Results in that
3
6

CONSTANTINESCU, CHAPELET, AND SQUIRES
1737
use in turbulence-resolvingsimulations at high Reynolds numbers
and in complex geometries.
In spite of the limitations, higher-order upwind-biased methods
III. Results
In this section, the techniques URANS, LES, and DES are
assessed through comparison against measurements, as well as
throughintercomparisonoftheresultsobtainedwitheachtechnique.
The comparisons between the four URANS models and LES and
DES predictions serve to evaluate the capability of these models to
resolve unsteady features of the  ow accurately.
In an attempt to minimize numerical uncertainties and allow a
focus on the differences between the various techniques used to
predict the sphere, results presented in this section were obtained
with the same  ow solver, grid, and boundary conditions. The mo-
mentum and turbulence-transport equations are integrated all of
the way to the wall, and the viscous wall layers are adequately
resolved. In turn, it should be possible to assess the predictive
capabilities of the techniques independently of the effects arising
from numerical resolution, because insufŽ cient resolution of the
wall layer can adversely affect the capability of a model to cap-
have yielded reasonablepredictionsof the mean  ow and low-order
1
3;36;37
statistics in previous LES studies.
Previous work by Johnson
3
2
and Patel using the numerical method employed in this study has
shown a negligible effect of the artiŽ cial dissipation in the Pois-
son equation on the formal order of accuracy of the method. In the
present work, the spectral content at high frequenciesis not overly
damped.In addition,Ž ner-scalefeatures,suchas the developmentof
the Kelvin–Helmholtz instabilitiesin the detached shear layers, are
resolved only via application of the Ž fth-order upwind discretiza-
tion. Though not shown here, these features are not resolved using
second-orderupwind biased differences of the convective terms.
2
;3
ture unsteadyfeatures such as vortexshedding. Also note that the
computationalcost of the various models will vary, for example, in-
creasesin computationalrequirementsfor RANS closuresthatcarry
more transportequations.Simulations performed with k–! (and k–
"
) required about 15% more CPU time than calculationsperformed
2
with S–A. The computational cost of v – f as measured in terms
of CPU usage was approximately40% greater than for simulations
with S–A.
Contours of the out-of-plane vorticity component from the DES
and LES in an r–Á plane are shown in Fig. 4. Figure 4 shows an
array of vortices of varying scale in the detached shear layers. The
shear layers are laminar at separation, and transition takes place
over a certain distance with Kelvin–Helmholtz instabilities in the
shear layers developing into vortex tubes shortly downstream of
the sphere. The process is especially clear in the LES, but DES
predictions also show the shear-layer vortices. (In Fig. 4 the same
contour interval is used to allow direct comparison between the
plots.) Vortex development is also apparent in the pressure Ž eld
(not shown), where alternatingregionsof low and high pressure are
Fig. 3 Skin-friction coefŽ cient (averaged over the azimuthal coordi-
nate): – – –, k–!; –¢¢–, k–"; ¢ ¢ ¢ ¢, v f; ¢ , S A; - - - -, LES; ¤, DES-CG;
2
–
– –
–
2
0
—
—, DES-FG; and ¨, Achenbach.
Fig. 4 Out-of-plane vorticity contours in an azimuthal plane: 24 increments in the contour levels from ¡10 to 10 U/D.

1
738
CONSTANTINESCU, CHAPELET, AND SQUIRES
apparentinvisualizations.Intherecirculationregion,a largenumber
of smaller-scaleeddies,nonexistentin the URANS calculations,are
observedin the vorticityŽ eld from the DES and LES, which are able
to resolve eddies down to the grid scale.
the transition region at the sphere surface is well deŽ ned with the
S–A model (and, therefore, also DES). For the other closures, an
approximate method was used. DES and S–A predictions of Á are
t
practically the same and are also close to that obtained with k–!.
2
The RANS models,on the otherhand,suppressthe smallereddies
and, in general, reduce the three-dimensionalityof the  ow. With
the exception of the k–" model, the shear layers computed by the
use of the different models are somewhat similar to the LES/DES,
but with a smoother distributionof the vorticity in the recirculation
region. As will be apparent later, the URANS predictionsevolve to
nearly steady and axisymmetric solutions. The Kelvin–Helmholtz
instabilitiesthat develop into vortex rings in the near wake must be
modeled by the RANS closures.In general,the scaleof the detached
shear layers and range of vorticity values recorded in these regions
are comparablein the URANS, LES, and DES calculations.Results
LES and v – f seem to predict an earlier transition, immediately
following separation at 87 deg. Finally, for the k–" model, transi-
tion begins at approximatelythe same angle as separation. Though
no experimental data are available for the location of the transition
region, the models (except k–") correctly predict a subcritical  ow
4
at Re D 10 as Á
s t
< Á .
Figures 2 and 3 show the distribution of the mean pressure co-
efŽ cient C and skin-friction coefŽ cient C
over the sphere. The
p
f
results are averaged in time and spatially over the azimuthal di-
rection. In Figs. 2 and 3, the symbols represent the experimental
5
measurements for the subcritical ow at Re D 1.62£ 10 (Ref. 20).
2
obtained with v – f predict the most elongated shear layers and are
Because the mean drag and  ow physics are somewhat similar over
the range of Reynolds numbers in the simulations and measure-
ments, important differences between the present calculations at
closest to the LES and DES visualizations.
URANS predictions of the mean streamwise drag coefŽcient C
and rms value, back pressure coefŽcient Cp;µ D 180, main shedding
frequency Strouhal number, the separation angle Á , and the tran-
sition angle Á , are summarized in Table 1. The transition angle as
d
4
Re D 10 and the availableset of experimentaldataare not expected.
s
The agreement between simulation and experimentis quite good in
the acceleration and part of the deceleration region for all simula-
tions, with the exception of the k–" results. The relative increase
in the pressure coefŽ cient near the aft region of the sphere is most
accurately captured in the DES and LES. The k–! distribution is
also close to the measurements, except near the downstream stag-
t
predictedby variousmodelsis calculatedwith a “turbulenceindex,”
7
a measure of when the closure model becomes active. For the S–A
model, the turbulenceindex is deŽ ned as @eº=@r=.·u
/ where u
is
¿
¿
the friction velocity. For the other models, the eddy viscosity was
used in the evaluation, and, though a more coarse approximation,
the aim was only to use the evaluation to deduce the range of rapid
increase indicating the activation of the model.
nation point where the relative increase in C
is clearly higher than
p
the one occurringin the experimentalmeasurementsand DES/LES
results. The same trend, more pronounced, is apparent in the S–A
2
As shown in Table 1, most of the RANS models are in reasonable
prediction. Results obtained with v – f provide an improved pre-
2
0
agreement with the measurements of Achenbach and Sakamoto
and Haniu,34 as well as the LES and DES predictions. The two-
layer k–" predictions are poor, an expected result given that the
length scale prescription in the near-wall region is appropriate for
fully turbulentboundarylayers. Applicationof the two-layermodel
to the laminar sphere boundarylayers severelydamagespredictions
by this model. Table 1 also shows that the pressure recovery in
the aft region is overpredicted by the RANS models, whereas the
unsteadiness of the  ow is better represented in the LES/DES and
diction in the aft region compared to the other URANS models.
The k–" predictions more closely resemble a supercritical solu-
tion, though the drag prediction for the supercritical  ow at higher
Reynolds numbers is around 0.2, lower than the k–" result of 0.291
(cf., Table 1).
2
ThedistributionsoftheskinfrictionfromtheLES, DES, andv
resultsare in good agreementwith measurements(Fig. 3). Note that
– f
0
:5
the Re scaling of the skin friction in Fig. 3 is appropriate for
the subcriticalregime with laminar boundary-layerseparationfrom
re ected in the improved prediction of Cp;µ D 180
.
The reasonable predictions of C
summarized in Table 1 with
d
LES, DES, and the k–! model, when compared to experimental
data, might indicate the superiority of these models. However, un-
certainitiesin the experimentalmeasurementsare not available,and
it is important to remark that a different set of experimental mea-
surements might favor one of the other closures. As will also be
subsequently shown, k–! calculations evolve to essentially steady

ows without vortex shedding. It is not clear whether this result is
speciŽ c to the moderate Reynolds number considered in this study,
or whethersheddingsolutionsmight be apparentat higherReynolds
numbers where there is a larger-scale separation between the shed
vortices and Ž ner-scale turbulence.
With the exception of k–" results, boundary-layer separation in
all simulations occurs at around 85:5 § 1:5 deg, slightly later than
2
0
the value of 82:50 deg measured by Achenbach. The 2–3-deg
azimuthal variation of Á
sequence of the shedding. Achenbach did not report variations
in Á in time or along the separation line and, consequently, it is
s
in the LES and DES predictions is a con-
2
0
s
not possible to report the rms variation in Table 1. The turbulence
index used to calculate the polar angle interval corresponding to
Fig. 5 Temporal variation of streamwise drag coefŽ cient: – – –, k–";
–¢¢–, k–!; –¢–, v2 f; - - - -, S A; ¢ ¢ ¢ ¢, LES; and ——, DES-CG.
–
–
Table 1 Effect of turbulence model on  ow parameters
Model
Cd
Cp;µ D 180
Sr
Ás , deg
Át -deg
5
4
a
k–"
0:291§ 7:70 £ 10¡
0.096
¡0:132
¡0:207
¡0:165
¡0:229
¡0:277
¡0:28
0.180
0.210
0.220
0.220
0.195
0.200
——
103.0
86:5 § 0:5
86.0
101¡109
¡
a
k–!
0:400§ 1:56 £ 10
92¡110
a
2
¡3
v – f
0:370§ 2:08 £ 10
0:435§ 2:12 £ 10
0:393§ 1:43 £ 10
0:397§ 7:00 £ 10
0:40 § 0:01
87¡88
¡
¡
¡
5
2
3
S–A
LES
DES
Experiment
87.0
93¡108
a
85:0 § 1:0
85:5 § 1:5
0.195
86¡88
93¡108
82:5
a
Calculation of Át using an approximate method.

CONSTANTINESCU, CHAPELET, AND SQUIRES
1739
the sphere. Based on the measurements, the skin friction is mildly
overpredicted in these particular simulations for 90 < Á < 140 deg
and mildly underpredictedfor 140< Á < 180 deg, although, as re-
marked earlier, the accuracyof the experimentalmeasurementswas
not reported. These differences are accentuated in the k–! results,
as well as the prediction obtained with S–A, which shows a deeper
minimum around Á D 160 deg. Overall, all models (except k–")
adequately predict the growth and separation of the boundary lay-
ers, with the turbulence models remaining dormant in the sphere
boundarylayers. Transition to turbulencein the presentsimulations
occursvia the triplessmode.11 Nonzero valuesof the eddy viscosity
are convected from downstream and ignite the turbulence model in
the separating shear layers.
The temporal variation of the drag coefŽcient in the k–", k–!,
and S–A results in Fig. 5 show virtually no departure from the av-
2
erage value, with a correspondingrms C
of nearly zero. The v – f
d
prediction, which shows a more visible temporal variation and for
which the rms C
is 0.0021, is neverthelessnearly an order of mag-
nitude lower than that obtained with LES or DES. Based on this
d
4
2
result, for the  ow around the sphere at Re D 10 , the v – f predic-
tions of time-dependent features of chaotic unsteady  ows appear
superiorto the other RANS models.Compared to the RANS results,
the rms C
d
is substantially greater in LES and DES. Resolution of
the large eddies and alternate vortex shedding in these calculations
are responsiblefor the larger deviations from the average drag (cf.,
Table 1 and Fig. 5). The effects of vortex shedding and its resolu-
tionby the simulationtechniquesare alsoillustratedin Fig. 6, where
the time history of lateral-force coefŽ cients is shown for DES and
k–!. There are important differences in the magnitude and nature
of the lateral forces between the DES and the RANS prediction.
(
The LES results are similar and not shown.) Lateral forces in the
RANS are small, whereas DES predicts a more signiŽcant varia-
tion around the mean with no regular pattern, consistent with the
random nature of the shedding,and in agreement with observations
from experiments. The nearly zero side forces predicted from the
RANS are also consistent with the nearly axisymmetric solutions
to which these calculations evolve. Figure 6 also shows that the
frequency associatedwith the lateral-forcevariation is roughly two
times lower than the sheddingfrequencyin the time variationof the
drag. This effect is also apparent in the URANS, though the rms
values of the lateral force coefŽ cients are close to zero. Note also
that there are no peaks at higher frequencies in the DES histories
Fig. 6 Temporal variation of lateral force coefŽ cients: ¢ ¢ ¢ ¢, y com-
ponent DES; – – –, z component DES; –¢¢–, y component k–!; –¢–, z
component k–!; and ——, Cd for DES-CG (with vertical offset of 0.35).
Fig. 7 Power spectra of the streamwise drag coefŽ cient.

1
740
CONSTANTINESCU, CHAPELET, AND SQUIRES
(
nor in the LES historiesnot shown in Fig. 6). The higherfrequency
than in the DES and LES. Other maximums are also apparent near
3
component evident in the C
histories arises from the formation
the main peak in Fig. 7, similar to the DNS results at Re D 10 of
d
1
6
of Kelvin–Helmholtz instabilities in the detached shear layers (see
Refs. 34 and 38). In the  ow past a sphere, the vortices that form in
these shear layers are vortex rings, approximatelyperpendicularto
the main  ow direction. The induced force and perturbationsin the
shed structures will then be primairly in the streamwise direction
with the higher-frequencyvariationsconsequentlymore apparentin
the streamwise force.
Tomboulides et al. The LES and DES time histories of C
d
and
of their power spectra are also qualitatively similar in the sense
that a second high-frequency band is captured between Sr D 1:30
and 1.85 in the LES, and between Sr D 1:90 and 2.10 in the DES.
3
4
Sakamotoand Haniu indicateda Strouhalnumberbetween1.8and
4
2.5 for the high-frequency mode at Re D 10 . The high-frequency
band predicted in the LES shows a possibly better capture than
in the DES of the development of Kelvin–Helmholtz instabilities
in the detached shear layers. The high-frequencycomponent is not
presentin anyoftheRANS results,indicativeofthesemotionsbeing
parameterizedby the turbulence model. Though the Strouhal num-
ber of the high-frequency component is somewhat underpredicted
compared to measurements, the present results would indicate that
LES/DES adequately predicts the dynamics of the vortex shedding
process.
Predictions of the shedding frequencies (Table 1) are within
3
4
1
0% of the experimental measurements of Sakamoto and Haniu,
Sr D 0:195§ 0:005, for all simulations, with LES and DES yield-
ing the best agreement. Wake frequencies were obtained by cal-
culation of the power spectrum of the streamwise drag varia-
tion. Figure 7 shows that there is a clear maximum around the
aforementioned value of the Strouhal number in each simulation.
The vertical scale shows the energy content in the DES and LES
predictions is characterized by a considerably larger amplitude
Contours of the turbulence kinetic energy K are shown in Fig. 8.
The distributions shown are obtained when the times are averaged
over at least six shedding cycles and spatially averaged over the
azimuthal direction.(When the DES solution was averagedover 12
shedding cycles, a negligible change in the statistics was yielded.)
Only the resolved turbulent kinetic energy is shown in Fig. 8 for
the LES and DES predictions. Figure 8 shows that all calculations
capture the pocket of high K behind the sphere in the recirculation
region, where the level of the turbulent shear stresses (not shown)
are also large (due to the distortion of the mean  ow and associ-
ated increase in turbulence production). DES and LES predictions
agree reasonablywell with one another,bothqualitativelyas well as
quantitatively.The k–! predictions are roughly similar to the LES
and DES results, though there Fig. 8 shows that the region of high-
est kinetic energy levels is closer to the sphere, at around x=D D 1,
2
compared to the URANS results, with the exception of v – f ,
where the energy content is about an order of magnitude smaller
2
rather than x=D D 2 as in the LES and DES. The v – f predictions
compare favorablywith LES and DES in that the region of elevated
K are closer than for k–!, though the actual values in the region of
high kineticenergyare about30% lower. In addition,RANS predic-
tions of kinetic energy levels are also higher in the detached shear
layers compared to LES/DES.
Eddy viscosities from the RANS and subgrid viscosities for the
LES and DES are shown in Fig. 9. The lower levels in the wake for
the LES and DES, as compared to the RANS results, are a conse-
quence of most of the turbulent stress being resolved (about 90%
of the total stress) in these simulations, whereas for the RANS, all
of the stresses are modeled. For similar mean  ows, the level of the
Fig. 8 Contours of turbulence kinetic energy: 25 increments in the
contour levels from 0 to 0.08 K/U .
2
Fig. 9 RANS eddy-viscosity and LES/DES subgrid viscosity in the sphere wake: 25 increments in the contour levels from 0 to 25 ºt/º.

CONSTANTINESCU, CHAPELET, AND SQUIRES
1741
turbulentstressescan be roughlydeducedfrom the magnitudeof º
t
.
viscosityand do not fully test the capabilitiesof the method. Super-
critical  ows with turbulent boundary-layer separation introduce
more empiricism and strongly challenge feature-resolving tech-
niques such as LES and DES. At high Reynolds numbers, LES pre-
dictions will require a differenttreatment of the wall layer, whereas
DES predictions in  ows with turbulent boundary-layerseparation
will be more sensitive to RANS modeling approximations in pre-
dicting boundary-layergrowth and separation.
Though not shown here, the distributionsof the averaged turbulent
stresses are similar for all of the techniques and, consequently, the
URANS eddy viscosityfor the present simulations should be about
one order of magnitude higher than the SGS viscosity in the LES
or DES, in agreement with Fig. 9. The qualitativedifferencesin the
instantaneous distributions between LES/DES and URANS calcu-
lations is again a consequenceof the large eddies being resolved in
the LES/DES and the SGS viscositylevelsin theseturbulentregions
being higher than in the surrounding ow. In the URANS cases, the
Acknowledgments
This work is supported by the U.S. OfŽ ce of Naval Research
t
distributionofº re ectsthetime-averagedvaluesin thewakewhere
(
1
Grants N00014-96-1-1251, N00014-97-1-0238, and N00014-99-
-0922, Program OfŽ cers L. P. Purtell and C. Wark). The authors
the shedding is not well represented. The URANS results also ex-
hibit the regionof highesteddy viscosityin the recirculationregion,
whileDES and LES resultsshowthe correlationwith shedstructures
in the wake. In the detached shear layers, RANS eddy viscosities
are predictablyhigh, while the structuresare mostly resolved in the
LES and DES.
gratefully acknowledge valuable discussionswith P. R. Spalart.
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1
Spalart, P. R., “Strategies for Turbulence Modeling and Simulations,”
International Journal of Heat and Fluid Flow, Vol. 21, 2000, pp. 252–263.
2
Rodi, W., “On the Simulation of Turbulent Flow past Bluff Bodies,”
IV. Summary
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pp. 3–19.
4
sphere at Re D 10 . One-, two-, and four-equation isotropic eddy-
3
2
Durbin, P. A., “Separated Flow ComputationsUsing the k–"–v Model,”
viscosityturbulencemodels were employed, with results compared
to experimentalmeasurementsand predictionsfrom LES and DES.
The  ow experiencedlaminar boundary-layerseparationwith tran-
sition to turbulence occurring via the development and breakdown
of Kelvin–Helmholtz instabilitiesin the detached shear layers.
With the exception of the two-layer k–" model, URANS predic-
tions of the pressure coefŽcient, skin friction, and (by association)
the streamwise drag, were in reasonable agreement with measure-
ments. Predictionsof turbulencekinetic energy and the shear stress
were similar to LES and DES results. The relative accuracy of the
URANS in predicting this  ow was somewhat unexpectedbecause
AIAA Journal, Vol. 33, No. 4, 1995, pp. 659–664.
4
Chen, H. C., and Patel, V. C., “Near-Wall Turbulence Models for Com-
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Wilcox, D. C., “Reassessment of the Scale-Determining Equa-
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Johansson, S., Davidson, L., and Olsson, E., “Numerical Simulation of
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two-dimensionalURANS typically results in large overpredictions
of the drag.11 URANS solutions did not, however, adequately re-
solve shedding mechanisms, leading to a poor prediction of wake
frequenciesand especiallytheenergyassociatedwiththemain insta-
bilitymodes.This behaviorisunlikeURANS of thecircularcylinder
where the main sheddingis more prominentlyrepresented.This dif-
ferencein the characterof RANS solutionsfor the sphereand cylin-
der might be explainedin part by the wavelengthsmost responsible
for the main shedding and the ability of a model to capture them.
The k–! predictionsof the pressureand skin-frictioncoefŽ cients,
the mean drag, and the position of the laminar separation location
were in closest agreement with LES and DES, but rms variationsin
the dragand lateralforceswere essentiallyzero.Resultsobtainedby
9
Franke, R., and Rodi, W., “Calculation of Vortex Shedding past a Square
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0
Bosch, G., and Rodi, W., “Simulation of Vortex Sheddingpast a Square
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Shur, M., Spalart, P., Strelets, M., and Travin, A., “Navier–Stokes Sim-
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12Travin, A., Shur, M., Strelets, M., and Spalart, P. R., “Detached-Eddy
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the use of v – f for these quantities were also satisfactory. Predic-
tions of the unsteady features of the  ow with v – f were superior
2
compared to the other models, for example, the amplitude of the
drag coefŽcient variations were the highest, though still sensibly
lower than those in the LES and DES. The S–A model predicted a
somewhat higher mean drag coefŽ cient, whereas the rms value of
the drag was the smallestof all of the RANS models. These Ž ndings
should be consideredalso in the context of computationalcost, that
is, the S–A runs were the most efŽ cient in view of CPU considera-
tions and yield acceptable predictions of integral quantities such as
C , C , and C .
The improved comparison against experimental measurements
from the LES and DES relative to the RANS results is related in
large part to the vortex shedding process being mostly resolved,
rather than modeled. Details of the sheddingthatmust be accounted
for in a RANS model are difŽ cult to parameterize accurately. For
applications in which time-dependent information is crucial, tech-
niquessuchas LES and DES are advantageous.LES predictionsthat
use the dynamic eddy-viscositymodel exhibited less SGS dissipa-
tion and consequently yielded more energetic turbulent solutions.
Therelativelybetterperformanceofthe LES comparedto theDES is
consistentwith the DES solutionsbeing obtainedwith a suboptimal
subgrid model, that is, one that is constrained by the RANS cali-
bration inherentto S–A. In addition,for the laminar boundary-layer
separation considered in this study, DES predictions constitute an
LES performed with a one-equationtransportmodel for the subgrid
13Shur,M. L.,Spalart,P. R., Strelets, M. K., and Travin, A. K., “Detached-
Eddy Simulation of an Airfoil at High Angle of Attack,” Proceedings of the
4
th International Symposium on Engineering Turbulence Modelling and
Measurements, edited by W. Rodi and D. Laurence, Elsevier, Amsterdam,
1
999, pp. 669–678.
14Drikakis, D., “Development and Implementation of Parallel High Res-
olution Schemes in 3D Flows over Bluff Bodies,” Proceedings of Parallel
CFD Conference—Implementation and Results Using Parallel Computers,
Elsevier Science, New York, 1995, pp. 191–198.
p
f
d
1
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Baldwin,B.,and Lomax,H., “ThinLayer ApproximationandAlgebraic
Model for Separated Turbulent Flows,” AIAA Paper 78-257, Jan. 1978.
1
6
Tomboulides, A. G., Orszag, S. A., and Karniakidis, G. E., “Direct and
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