NASA Technical Reports Server (NTRS) 20020012430: The Lag Model, a Turbulence Model for Wall Bounded Flows Including Separation

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by  NASA Technical Reports Server (NTRS)

AIAA 2001-2564
The Lag Model, a Turbulence Model
for Wall Bounded Flows Including
Separation
M. E. Olsen
T. J. Coakley
NASA Ames Research Center
Moffett Field, CA 94035
t-
15th AIAA Fluid Dynamics Conference
June 11 -June 14, 2001 /Anaheim, CA
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AIAA 2001-2564
The Lag Model, a Turbulence Model for Wall Bounded Flows
Including Separation
M. E. Olsen *
T. J. Coakley t
NASA Ames Research Center
Moffett Field, CA 94035
A new class of turbulence model is described for wall bounded, high Reynolds number
flows. A specific turbulence model is demonstrated, with results for favorable and adverse
pressure gradient fiowflelds. Separation predictions are as good or better than either
Spalart Almaras or SST models, do not require specification of wall distance, and have
similar or reduced computational effort compared with these models.
Introduction without invoking the full formalism of the Reynolds
stress models. The basic idea is to take a baseline
One difficulty with current one and two-equation '
turbulence models is the inability to account di- two-equation model and to couple it with a third (lag)
rectly for non-equilibrium effects such as those encoun- equation to model the non-equilibrium effects for the
tered in large pressure gradients involving separation eddy viscosity. The third equation is designed to
and shockwaves. Current turbulence models such as predict the equilibrium eddy viscosity in equilibrium
Spalart's one-equation model, 5 the classic k- e and flows.
Wilcox's k - w z two-equation models have been de- We show results obtained with a lag model based on
signed and tuned to accurately predict equilibrium the Wilcox k-w model. Applications to four flows are
flows such as zero-pressure gradient boundary- layers given including an essentially incompressible flat plate
and free shear layers. Application in more complex flows, an essentially incompressible adverse pressure
flows can be problematical at best. Although there gradient flow with separat(on, 2a transonic bump flow 3
have been many attempts to modify or correct basic with a shock wave and separation, a three dimensional
one- and two-equation models, most of these attempts transonic wing flow. 4 Results using the new model are
have been only marginally successful in predicting compared with results obtained with Spaiart's model 5
and Menter's k-co SST model. 6 Results obtained with
complex flows.
More complex models such as Reynolds stress mod- the new model show encouraging improvements over
results obtained with the other models.
els have been investigated extensively, primarily for
relatively simple flows but also for complex flows. In
The Lag Model
most cases these models give somewhat better predic-
tions than the simpler one and two equation mou_,s,-'-' The differential equations defining the lag model are
but for complex flows they do not perform much bet-
ter than the simpler models. One theoretical advan- Dk - - ek+ + (1)
tage of Reynolds stress models is that they directly P Dt
account for non-equilibrium effects in the sense that Dw = - + (2)
the Reynolds stresses do not respond instantaneously P-_
to changes to the strain rate but more realistically D_/t
= (I(RT) pw (VtE --_Vt) (3)
lag them in time and/or space. Unfortunately, The P'-_--
Reynolds stress models are usually considerably more where:
complicated and numerically stiff than the one- and
two- equation models, and this has prevented their k/to Rt = pk/Ftw
_¢tE =
wide application for complex flows.
ports 2 _ow _- {xpS2
In this paper we introduce a new three equation Pk =
_*pkw e,, = t3pw2
model designed to account for non-equilibrium effects ek =
*Research Scientist, NASA Ames Research Center, Associate
s=
Fellow AIAA
tResearch Scientist, NASA Ames Research Center, Associate
Fellow AIAA
Thin plY.per il a work of the U.S, Government lind im not subject to ©opy- sii
rlsh_ protection in the United Stlatee,2001
American Institute of Aeronautics and Astronautics
(c)2001AmericanInmtRuioofAmronmutie&sAatronm_k:orP_d)fls_wdithPermissioonfAuilwr(mr)and/oAru_or(s)_'sodng _mniz4dlon.
100 '' '_''J ..... "_ ..... "_' ..... "_ ...... "_ " ''""J
[ (RT "t"RT,) l (4)
a(RT)= L(RT+,T.).I
5O
with parameters
(_t
O_ : 5/9
p : 0.075 20
ao R 0.35
R_ = 1 p" : 0.09
RTm m _| Ok R 0.5 I0
0". = 0.5
5
The model equations are compoasd of an underlying
model, (k- w) with the lag equation augmenting
the system. The k - w model, given by the first two
equations is unchanged from the standard model, 2
except that _t is now given by a field PDE, Eq.(3),
instead of k/w.
The boundary conditions are those of a convected I(_-_ .....1_-3.......0.011 ...... 0.1"....... 1' ....... 10'1.......100
scalar: specified on Inflow boundaries and extrapo- Rt
lated on outflow boundaries. At inflow boundaries, k,
w and _t are set to constants _h_ refen-ed to as Fig. 1 Lag Source RT Factor
)coo,Woo, and _t_ _=k_/woo). In this paper, koo/U_
model at low values of RT so that it will have less lag,
was chosen to be 0.0001, corresponding to • turbulent
and will act more like the unmodified model in these
Intensity of 0.8%. The value of k external to the wall
conditions. In effect, this term "turns on" the lag only
bounded flows is, of course, substantially lower than
in turbulent regions (RT >> 1).
this, since k decays in the absence of mean strains.
The model requires the storage of one additional
woo is chosen to yield a low value of _tE (here chosen
field variable over the SST model, but does not require
as one tenth the molecular _), and _t is set to _t_. The
the wall distance information used in both the SST
model's predictions are insensitive to the values chosen
for these constants as is shown in the results. At the and SA mod_els, thus freeing up the storage required
solid walls a "no lag" boundary condition is enforced: for that variable. As the model_, is computatioually
the eddy viscosity is set to its equilibrium value: zero simple, not requiring the Vit" V_t or Vk. Vw terms
for hydrauiiciy smooth walls, finite for rough walls. present in the other models, it actually requires less
The cases discussed in this paper all use the smooth CPU time per iteration than either SST or SA models,
wall conditions. T Rough wall boundary conditions are and similar or fewer iterations for convergence.
also possible in the manner of the k - w model, 2 but
Numerical Method
are not discussed in this paper.
The turbulent eddy viscosity is governed by Eq.(3). The solutions obtained in this paper were done with
This equation simply says that the eddy viscosity goes • modified version of the OVERFLOW s.s code. For
to its equilibrium value (_tt) along • streamline like the mean flow equations, the existing 3rd order upwind
a first order dynamical system with a time constant scheme or the central/matrix dissipation z° method
given by l/(aw). The stability of the turbulence was used. The full Navier Stokes equations were
model as • dynamical system is ensured by the form solved, as opposed to utilizing the thin _ approxi-
of this source term, given a stable underlying model. mation. Converged solutions were indistinguishable in
There is no diffusion term in this equation, and evo- terms of skin friction and velocity profiles. Matrix dis-
lution of the eddy viscosity is dependent only on its sipation was used with 2nd and 4th order smoothing
upstream history and the underlying equilibrium eddy coefficients of 2 and 0.1, respectively. The eigenvalue
viscosity at that point. limiters were set to zero, and Roe averaging was used
The a term of Eq.3 of the source for the "YTEq.(4) to form the matrix. Muitlgrld was employed, both as
governs the amount of lag present in the model. This grid sequencing for startup, and during the relaxation
term is made up of three factors. The leading con- process. 3levels of muitigrid were used for all solutions
stant, a0, controls the amount of lag in the model. presented in _his paper.
The higher the value of a0, the less lag(shorter time The turbulence model equations were spatially dis-
constant) in the system, and the closer will follow the eretized using a 2nd order upwind method with a
underlying turbulence model. The second factor in sin-sod limiter. This is adeparture from the 2 equa-
Eq.4 goes monotonically from 100 to one as RT goes tion models implemented in OVERFLOW, which use
from 0 to infinity(Fig. 1). This in effect stiffens the Ist order upwind. 1st order upwind was the initial
American Institute of Aeronautics and Astronautics
presented in this paper. Subsonic Flat Plate
The turbulence model equations was spatially dis- The fine grid for this case is
cretized using a 2nd order upwind method with a 101(streamwise)xl01(wall normal). Nearly iden-
min-mod limiter. This is a departure from the 2 equa- tical results were obtained on a 51x51 grid. The wall
tion models implemented in OVERFLOW, which use normal grid stretching for these cases was 1.1 and 1.2
1st order upwind. 1st order upwind was the initial respectively, with initial _+ spacings of 0.1 and 0.2.
implementation, but this proved too dissipative, and The initial 4 wall normal points were equispaced.
led to excessive grid density requirements for grid in-
5x10-3 .... i .... i
dependent solutions.
_D
The reason for this can be seen in the 3rd equa- Cf
tion, which implements the history effects (lag) of the " M=O.2 Flot Plate
model. Using a 1st order upwind on this equation is '.'_ _, a Kormon-Schoener
"_ , -- Lag
analogous to using a 1st order time integrator to in-
tegrate an ODE, with grid spacing analogous to the 4x10_ 3 ,_o _A T
ODE time step size. When the 1st order upwinding
was replaced with a 2nd upwinding, the gridding re-
quirements dropped back to what would normally be
required for grid independent solutions with other tur-
3×1o-3 __..
bulence models. The additional work required to solve
_. 1_1,.
the third equation is offset by the relative simplicity
of the underlying model. Convergence is rapid and
robust, as implemented in OVERFLOW.
Results
2x10 -3 5/00 10i00 200....0.... 5000 1'04 Re2x'041
Dissipation of Isotropic Turbulence
Isotropic turbulence has no mean strain, so that the
Fig. 2 Subsonic Flat Plate Skin Friction
decay of k and w follow those of the underlying model,
here the Wilcox k- tv model. The _t equation uncou-
5O
ples from the other two equations, and the equation /
t
governing the evolution of the eddy viscosity becomes: _J÷
Law of the Wall
25 7:
_'V t
- a0(k- act)
0t • Re_- 2055
,o
2O
This decoupling is aesthetically beautiful, from the
viewpoint of _t as a ratio of turbulent stess to mean
15 /il
strain. When the mean strain vanishes the turbulent "l
str_ses are not effected by the value of the eddy vis-
cosity. Similarly, the lag equation does not affect the
decay of turbulent kinetic energy built into the under-
lying model under conditions of zero mean strain.
Equilibrium Channel Flows
For fully developed channel shear flows, such as Cou- 0 ...... i ........ t .... ,..j ........ i .i
ette, or fully developed pipe flows, the model again 1 10 100 1000 104
÷
Y
decouples and reproduces the results of the underly-
ing model. The differential equation becomes: Fig. 3 Law of the Wall Velocity Profiles
For flat plates, the model roughly reproduces the
_t
u-_-x = a0(k- ayvt) original model performance, since the flow is slowly
varying in the streamwise direction, and the model
and as _ = 0 for fully developed Couette or chan- forces wt to its equilibrium values. Comparison for
nel flows,8xthis simply enforces vt = k/v0. In the same low speed flows (actually Moo = 0.2, Fig. ) shows a
manner as in the decay of isotropic turbulence, if the good comparison with Karman-Schoener correlation.
This was expected from the underlying k - ¢vmodel's
underlying model does a good job under these con-
ditions (which k - ¢v does) then the Lag model will predictions for smooth flat plates. The law of the wall
is also well reproduced (Fig. 3).
alSO.
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(c)2001American InstituteofAeroneutl¢/& Altroneuflce orPub,shedwithPermbsionofAuthor(s}and/orAuthor(m)S'pormoflng C_enizatlon.
5x10-3 .... j ........ ,
\ c,
Cr
0.6
M..,O.2. k,.,- 10"IUZ,,
_ Korrnon-Schoenherr
,_'_._ .... R,=IO -I
,,,,o-" -..
0.4
0.2
3x10 -3 . ':__
',q'dt
0
2x10 -3 , •
500 1000 2000 "50C)0" " 104 2x 104
Re
-0.5
Fil. 4 Woo Dependence
Fig. 6 Driver CS0 Surface Prusuru
wall distance information, and are under consideration
for use as future baseline models. The V+ require- 4x I0"3 .... ' , , , ' • •
ments were also investigated, and similar behaviour to
the underlying k - w model were found. T Improve-
men_ sad extensions of the which include roughness
are available for the k-w model, zs and axe also under
consideration for future model improvement.
Driver CS0 Separated Case
2x10"4.
This is a low speed separated case. s The axisym-
metric geometry (Fig. 5) is defined by an external
streamline determined from experimental data, and
wall pressures axe available in addition to velocity pro- 104.
files and skin friction.
Y[]
Fill. 6 Driver CSO Flowfleld
Upstream boundary conditionswere constanttotal Fig. ? Driver CSO Skin l_'iction
pressureand temperature,withstaticpressureallowed
to varyand velocitydirectionalignedwith thecylin- 100 x 80 grid and demonstrated grid independence in
der axis. The .outerstreamline was treated as an
the same manner as Bardina et. al.T Surface pressures,
inviscldwall.The viscouswall(thesurfaceofa cylin- skin friction and velocity profiles agreed with the Rue
der) isa no-slip,adiabaticwall. The downstream grid results. This case was one which demonstrated
staticpressurewas adjusted tomatch theexperlmen- the need for handling the turbulence model convection
telstaticpressure(Fig.6)upstream oftheinteraction operator with 2nd order upwind (minmod limiter), as
region(x_ -.438m). Upstream lengthof the cylln- the solutions on the two grids'are effectively identical
dricalbody was adjusted so that the computed the when the"2nd order operator IS employed, but show
boundary layerthicknessat x -- -0.438m matched
slight differences when computed with afirst order up-
theexperiment. This "entrylength"was thesame for wind operator for the turbulence model.
allthreemodels.
The pressurevariationpredictedby the Lag model
The standard OVERFLOW low Mach number pre- iscloserto the experimental data than either the
conditioningwas employed. The gridused inthesecal- Spalart-Alrnarasor SST models, although both give
culationswas 200(axial) by 160(radial), an extremely reasonableagreement. This improvement inpressure
finegrid. The calculationswere repeated with a
variation prediction could be important for internal
American Institute of Aeronautics md Astronautics
, , , , I i • , , I .... I,,,,1,,,,1 .... IL,
I:I]
] 1:3Exp q] LJ .1_ I_
0.01
0 , • , , , , , , , , , , r, i , , • i , .... i,,,,i,,,,l,i' .... I ' ' ' ' I .... I .... I''''1''''1'
0.8 1 0.6 0.8 1 0.4 0.6 0.8 1 0.4 0.6 0.8 0.2 0.4 0.6 0.8
0.06
0.04
0.02
0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6
Fig. 8 Driver CS0 Velocity Profiles
i 1. i i i i , i i I i i i i l i i I i i i I / i i | i i , . I • i I i i i i / i
, , , = _0.46_1.T1 [_..i tiltl _. 0..3,30m Ep _" _ '_--" =-- .
1 [] Expe riment[-]
0.02 -_ ..... :4J ['1
040i,
2x'10a 0 ' '---'2x'10 ........ ' .... 2 '
,,,I,,,,1,
0.1 = -0.0127 _"= +0.058 _ = +0.101 _=+0.152 = .
Y
0,08
0.06
0.04
0.02
0
-2x10 -'_ 0 -2×10 .3 0 -2x10 -3 0 ' -2×10- 0
u'v'/U. 2
Fig. 9 Driver CSO Stress Profiles
5
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(c)200iAmerl_nInstituteofAerocmutt_• _flronautlcsm'PublishedwithPermissioonfAuthor(sa)nd/orAuthor(s)S'ponsorinOgrgonlzation.
low speed flows with small separations. The model's grid spacing for both normal and fine grids had • g+
ability to more accurately predict the pressure varin- less than 0.17 upstream of the shock.
tlons will be repeated in later test eases.
The skin friction prediction of all four models ts
shown in Fig. T. All four predict the skin friction rea- g kJu. 2 R,.
*sonably well, with the SA model predicting a reattach- o o 10-3 10-1
ment point slightly downstream of the experimental
data and the k- w model falling to predict separa-
o lO-e I0-I
tion. All models predict cf too low Just upstream of
2.5xI0-3
separation, from x - -.1 to x ffi0.
The velodt_ profiles (Fig. 8) also show good agree.
ment. The differences between _e underlying k- w
model and the lag model are relatively small upstream
of separation, but the lag model predicts the seps-
0
rated profiles more accurately in the separated region e
0
Cx>0). 0
0
The shear stress profiles are similarly well predicted 0 00
o :ff:::l'''*
(Fig. 9), for both the evolution of the maximum shear ; : : : Ov: .:-':,,,/
stresss and its location. The lag ofthe model isevident
x/c
most clearly in this figure. Upstream of the separa-
tion, the shear stress predicted by the lag model is
0.5 1 1.5 2
true to it's name, and lags the underlying k-w mode]
especially evidently at the _. = -0.076 station. By Fig. 10 Moo = 0.875 Skin Friction Insensitivity to
the/. -- 0.101 station, all of the models are predicting kee and Rtoo values
roughly the same shear stress, even though the velocity
profile predictions (Fig. 8) show the greatest scatter at The insensitivity of the solution to freestream
this location. One of the model's aught imperfections choices of Rtoo and koo is illustrated in Fig. 10. Here,
can be seen in Fig.8 and Fig.g, as the outer edge of the predicted x component of akin friction is shown
the boundary layer has a kink not seen in the other for • range of choices of these parameters. The koo
models. range corresponds to an initial freestream turbulence
intensities from 0.08% to 2.5%. The Rt_ range corre.
Bachalo-Johnson Bump sponds to initial eddy viscosities of from 0.1 to I0-3
This test case features the transonic interaction of molecular visc_ity. The surface pressure prediction
a fully developed turbulent boundary layer with the variations are just as insensitive to these variations in
pressure field crested by a circular arc bump on the freestream turbulence values.
surface of a cylinder. The surface pressure distrlbu- The Lag model reproduces the experimental pres-
tions for various freestream Mach numbers are avail- sure distributions (Fig. II) as well as either the
able, and velocity and Reynolds stress profiles are Spalart-Alimaras or Menter SST models, a distinct
available for the Moo = 0.875 case. improvement over the underlying k- w mode], which
Upstream boundary conditions were constant total consistently misses the shock location and underpre-
pressure and temperature, with static pressure allowed dicts the extent of the flow separation.
to vary and velocity direction aJigned with the cylin- The velocity profiles show the progression of the
der axis. The outer edge of the flowfleld was treated flowfleld through the separatlon(Fig. 12). The Lag
by extending the grid 8 bump chords away from the model predicts a separation point intermediate be-
wall, and utilizing characteristic(no reflection) bound- tween the predictions of the SST and SA models, and
ary conditions. The viscous wall(the surface of a has a flow recovery better than the SST, though it still
cylinder) isa no-slip, adiabatic wall. The downstream does not recover as rapidly as expeflment. In these ve-
static pressure was held at poo Upstream length ofthe locity profiles, there is no obvious kink at the edge of
cylinder was adjusted to match computed boundary the boundary layer in contrast to the CS0 flowfleld.
layer thickness at x ffi-0.25m to experiment as done The shear stress profiles(Fig. 13) show the Lag
in the Driver case, and again this "entry length" was model's increased prediction of cm,_ downstream of
the same for all three models. the separation point, though it is not as large as mea-
The grid for this case is 181(streamwise)x78(wall sured in the experiment. Note that this plot has the
normal). Fine grid solutions with a 358 x 161 grid wall distance logarithmic, expanding the inner region
were indistinguishable to plotting accuracy, in terms of the boundary layer. There is a consistent under-
of both surface pressures and velocity profiles at both prediction of the stresses in the inner layer by all of
Moo = 0.875 and Moo = 0.925 cases. The wall normal the models in the separated region, and all the models
6
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0.3
• i
P/Pt
#
M=,=0.900 M=,=0.925
0.4
0.5
0.6
x/c
x/c
0.7
0.5 1 1.5 0.5 1 1.5 0.5 1 1.5
Fig. 11 Moo = 0.875, 0.900,0.925Bachalo-Johnson Bump Pressure Distributions
i , , i=ii ioi I 5, i[ 6i _i ,iAi i i i i I i i i [ I i i I ,
0.09 = 0.625 I I:;:1._ _ = 0.688
0.08 = -0'25 i
0.07
0.06 Lag rl
0.05 -4 -!
0.04
0.03 -!
0.02 :1
: =
0.01 rl .... ,.... #,
0
0 0.5 t 0 0.5 1 1.50 0.5 1 1.5 0 0.5 1 1.5
l i I i i i I 1 J i l i I i L , , ,, l,, , _ | _,, , _,, ,
0.09 $ = 0.875 'la
0,08 t=o.8_3 if -i-4 Jn
0.07
0.06
0.05
0.04 /
0.03
--.-i i/ /
0.02
......t / / / I
0,01 ., .
0
' 'q ' ' ' ' 1 ' ' ' ' 1 ' " ' _1 ' ' ' ' I ' ' ' ' I ' '
0 0.5 1 1.5 0 0.5 1 0 0.5 1 0 0.5 1
, , I , , , , I , . , , l . , i i i i [ i i , ,. , , .
0,09
0,08
0.07
0.06
0.05
0,04
0.03 i _ = 1"315,,_
0.02 i
0.01
0 g, " _1 ,'-/_" / , _ , ._
' ' I ' ' ' ' 1 ' ' ' ' 1 ' ' " '
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
Fig. 12 Bump Velocity Profile Comparisons, Moo = 0.875
7
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I i I | i i i i I I .... I
0.1
" " $A,
O.Ol -- k-_
Q
10.3
1
0.1
0.01
10-3
0 -0.01 0 -0.01 0
1
-- 1.0
0.1
[]
[]
0.01
10.3
-0.01 0 -0.02 -0.01 0 -0.02 -0.01 0
I
0.1
- H- I
10.3
-0.02 -0.01 0 -0.01 0 0
FI&. 13 Moo ==0.875Bump Shear Stress Profiles
8
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tween the predictions of the SST and SA models, and The main feature of this new class of models is to
has a flow recovery better than the SST, though it still introduce a lag into the response of the eddy viscosity
does not recover as rapidly as experiment. In these ve- to rapid changes in the mean flowfields so as to era:
locity profiles, there is no obvious kink at the edge of ulate the responses seen experimentally. Virtually all
the boundary layer in contrast to the CS0 flowfield. turbulence models generate Reynolds stresses that re-
The shear stress profiles(Fig. 13) show the Lag spond too rapidly to changes in mean flow conditions.
model's increased prediction of _m_, downstream of Even the Reynolds stress models predict overly rapid
the separation point, though it is not as large as mea- response of the Reynolds stresses to changes in mean
sured in the experiment. Note that this plot has the flow conditions. This is in large part due to the mod-
wall distance logarithmic, expanding the inner region els' need to accurately reproduce equilibrium flows.
of the boundary layer. There is a consistent under- The lag equation gives the existing models an addi-
prediction of the stresses in the inner layer by all of tional degree of freedom, without tampering with their
the models in the separated region, and all the models typically good ability to predict equilibrium flows.
seriously underpredict the measured maximum shear
stress. Summary
The Lag model gives good results for mild to mod-
ONERA M6 Wing
erate 2D separations, and agrees well with 3D cases
The ONERA M64 is a venerable 3D test case. In the
tested• It works well for skin friction prediction at in-
conditions from _ = 3° to _ = 5° range from a nearly compressible and supersonic Mach numbers, and pre-
attached flowfield to a relatively extensive separation, •dicts separation well for incompressible and transonic
as shown in Fig. 14. test cases. The model does not require wall distance,
The grid used in these computations in contrast to both SST and SA models, and its sim-
was the one shipped as a test case with plicity is such that the computational effort is roughly
OVERFLOW, which has grid dimensions of equivalent to the simpler 1 and 2 equation models•
269(streamwise) x35(spanwise) x67(wall normal), Future work will include efforts to remove freestream
with 201 points streamwise along the wing surface. effects, and will look at free shear flows, along with
The t3+ of the first point off the surface was below other experimental test cases.
1.25 over the entire wing surface. No grid resolution
study was performed. References
The Cp predictions of the model are compared with [l]Wilcox, David C.. "Turbulence Modeling for CFD'.
experiment in Fig. 15. The Lag model produces a good DCW Industries, Inc, 1993. "
prediction over this entire range of conditions. The [2]D.M. Driver. "Reynolds Shear Stress Measurements in a
Separated Boundary Layer Flow _. AIAA Paper 91-1787, 1991.
largest discrepancies in this range are at the wing root,
[3]Bachalo, W.D. and D.A. Johnson. "Transonic Turbu-
and are more likely due to the tunnel wall interference, lent Boundary Layer Separation Generated onan Axisymmetric
as the flow is attached in this region. As can be seen in Flow Model". AIAA Journal, 24:437-443, 1986.
the "oil flow" pictures of Fig. 14, all these cases have [4]Schmitt V. and Charpin F. "Pressure Distributions
some separation, from a "incipient separation" at 3° on the ONERA-M6 Wing at Transonic Mach Numbers". In
AGARD Report, number 138in AR, pages BI-I - BI-44, 1979.
to the rather extensively separated 5° case.
[5]Spalart, P.R. and S.R. Allmaras. "A one equation Tur-
All four models _ve good predictions at o¢= 3° and bulence Model forAerodynamic Flows". La Reche_he Aerospa-
0¢= 4° conditions, but at ot = 5° the separation there lisle, 1:5-21, 1994.
is an appreciable difference in the predictions provided [6]Menter, F.R. "Two Equation Eddy Viscosity Model for
by the various models. The wing tip separation pro- Engineering Applications". AIAA Journal, 32:1299-1310, 1994.
[7]Bardina, Jorge E., Huang, Peter G., and Thomas J.
gression in particular appears to be well captured by
Coakley. "Turbulence Modeling Validation, Testing, and De-
both the Spalart-Almaras and the Lag model. The velopment". NASA T/VI110446, April 1997.
SST model has more extensive separation than exper- [8]Buning, Pieter. Get al. Overflow user's manual. Version
iment, and the separation predicted by the k-to model 1.8, NASA Ames Research Center, February 1998.
is less extensive than experiment. [9]Jespersen, D, Pulliam, T. H., and P.G. Buning. Recent
enhancements to overflow. AIAA Paper 97-0664, January 1997.
Discussion [10]Swanson, R. C. and Eli Turkel. "OnCentral-Difference
and Upwind Schemes". Journal ol Computational Physics,
The lag equation could be coupled to virtually any 101:292-306, 1992.
model. We have chosen to couple it to the k - to IlllF.R. Morner. "Influence of l_reestream Values on k-
model for this example. Another possible implementa- to Turbulence Model Predictions". AIAA Journal, 30(6):1657-
1659, 1992.
tion would be to lag the Reynold's stresses, as opposed
[12]Thomas J. Coakley. "Development ofTurbulence Models
to the eddy viscosity, via an equation of the form
for Aerodynamic Applications". AIAA Paper 97-2009, 1997.
[13]Hellsten, Antti and Seppo Laine. "Extension of the k-to-
D't'ti_ = a'(Rx) to (2p.tt sii- "rii) SST Turbulence Model for Flows Over Rough Surfaces". AIAA
Dt
Paper 97-3577-CP, AIAA AFM Conference, August 11-13 1997,
to account for anisotropic effects seen in 3D flows. New Orleans, LA, 1997.
American Institute of Aeronautics and Astronautics