Implementation of a Two-Equation k-co Turbulence ... - NTRS - NASA

NASA-TM- 107080

NASA Technical Memorandum 107080 AIAA-96-0383

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Implementation of a Two-Equation k-co Turbulence Model in NPARC

Dennis A. Yoder and Nicholas J. Georgiadis Lewis Research Center Cleveland, Ohio Paul D. Orkwis University of Cincinnati Cincinnati, Ohio

Prepared for the 34th Aerospace Sciences Meeting and Exhibit sponsored by the American Institute of Aeronautics and Astronautics Reno, Nevada, January 15-18, 1996

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IMPLEMENTATION OF A TWO-EQUATION k-o) TURBULENCE MODEL IN NPARC Dennis A. Yoder,* Nicholas J. Georgiadis* National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio and Paul D. Orkwis* University of Cincinnati Abstract

Pt

stagnation pressure

The implementation of a two-equation k-o) turbulence modelintotheNPARC flowsolverisdescribed.Motivation for the selection of this model is given, major code modifications are outlined, new inputs to the code are described, and results are presented forseveral validation cases: an incompressible flow over a smooth fiat plate, a subsonic diffuser flow, and a shock-induced separated flow. Comparison of results with the k-Emodel indicate that the k-o) model predicts simple flows equally well whereas, for adverse pressure gradient flows, the k-o) modeloutperformstheotherturbulencemodelsinNPARC.

R

ratio ofturbulent kinetic energy production to dissipation

Rk, Rs,Rto

k-_ turbulence model constants

Ret

Reynolds number based on turbulent quantities

Rex

Reynolds number based on axial position

Re0

Reynolds number based on momentum thickness

SR

surface roughness coefficient

_Symbols a

speed of sound

t

time

Cf

skin friction coefficient along flat plate

Uref(k-_)

reference velocity for turbulent kinetic energy limiter

Cab C_2

k-_ turbulence model terms u

velocity

Ucl

centerline velocity for Fraser diffuser

u¢

friction velocity

uo,

free-stream velocity

u+

normalized velocity

-uv

Reynolds stress

x,y

Cartesian coordinates

y+

distance from wall normalized by shear length scale

C_t

k-_ turbulence model constant

fix,fl, t"2

k-e turbulence model terms

H

throat height of Sajben diffuser

I

turbulence intensity

k

turbulent kinetic energy

kR

average sand-grain surface roughness

Mt

turbulent Mach number

P

static pressure

°

*AIAA member

1 American Institute of Aeronauticsand Astronautics

(x,ot0,a*

k-to turbulence model constants

_,13"

k-to turbulence model constants

generally involves solving additional transport equations. Thus, as the model complexity increases, so does the effort required to implement the model. Higher level turbulence models are also more computationally expensive because they require more computer memory and cpu time. Two-equation models are commonly used because they are considered to provide relatively good results for the demand they place on computer resources. The most widely used two-equation turbulence model, the k-E, solves two transport equations for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energye. Important groundwork onthis model was performed by Jones and Launder1 and Launder and Spalding.2The Jones and Launder model is often referred to as the standard k-¢ model. The k-tomodel is similar to the k-_model except that the second turbulent quantity to is the specific dissipation rate (dissipation rate per unit turbulent kinetic energy). Kolmogorov3was the first to propose this type of model. Since then, othershave made modifications to theoriginal formulation. Among these are Menter 4 and Wilcox5. Wilcox6presents an interesting historical overview of the k-to model. Recent efforts at NASA Lewis Research Center to model the flow field inside subsonic diffusers have demonstrated the inability of current models to accurately predict flows with adverse pressure gradients. The k-to model has been shown to predict separated flows better than the standard k-E model, especially in flows with adverse pressure gradients.7'8As a result, the k-to model of Wilcox5 was recently installed in both the two- and three-dimensional versions of NPARC. Because the k-_ model isatwo-equationmodel,itdoesnotrequire additional cpu time relative to the k-Emodel, and since it is similar in form to the k-e model, a similar algorithm can be used. Furthermore, this model has the benefit of improved transition simulation and is capable of more realistic rough-wall treatments. This report provides an overview of some of the turbulence models already available in NPARC, a discussion of the implementation of the k-to model in the code, and results obtained using the new model for several flow fields.

boundary layer thickness rate of turbulent kinetic energy dissipation l,t

dynamic viscosity

IJ.t

turbulent viscosity

/at,max

maximum turbulent viscosity limiter

v

kinematic viscosity

FI

production term in k-_ model

p

density

Ok,_,Oco

turbulent Prandtl numbers

to

specificturbulent kineticenergy dissipation rate

_k, _co

k-to turbulence model terms

Subscripts: i,j

computational corrdinates

max

maximum

min

minimum

Introduction In NASA's High Speed Research (HSR) program, computationalfluid dynamics (CFD)is increasingly being usedtodesignandevaluateinletandnozzleconfigurations for High Speed Civil Transport (HSCT) applications, Accuratelypredictingflowseparationsand mixingbetween primary and secondary flows in highly turbulent mixerejector nozzles as well as pressure losses and bleed flows in supersonic mixed-compression inlets is a major challenge. The turbulence model employed is often the limiting factor in these types of simulations and, as a result, is often blamed for providing poor agreement with experimental data. Thus, the search for better models is ongoing, Turbulencemodelsvaryfromrelativelysimplealgebraic models to one-equation, two-equation, and full Reynolds stress models. Moving to the next level of complexity

NPARC Code The NPARC code, previously known as the PARC codegandoriginallydevelopedattheArnoldEngineering Development Center (AEDC), uses the Beam and Warming10approximate factorization algorithm to solve theReynolds-averagedNavier-Stokesequationsat discrete grid points. Generalized boundary conditions allow the user to specify any portion of a grid as a boundary. Noncontiguous multiple block interfacing can also be 2

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usedtosimplifythegridgenerationofcomplexgeometries, Cooper 11discusses recent features of the code. The NPARC code contains three algebraic turbulence models, the first of which is the Thomas model.12 It computes turbulent viscosity for wall-bounded and free shearlayerflowsandhasbeenoptimizedforthelatter.The second, the Baldwin-Lomax model,13 only computes turbulent viscosity in wall-bounded regions. An algebraic RNG (renormalized group theory) model 14 is also available. NPARC also contains the Baldwin-Barth one-equation model 15and a two-equation turbulence model based on the Chien low-Reynolds-number k-_ model,16which is presented below in detail to underscore its similarity to the k-o) model:

and Cg=0.09, CEI= 1.35,C_2= 1.80, Ok= 1.0, Ce= 1.3. The turbulent Mach number is defined as Mt2 = 2k/a2, where a is the reference speed of sound, and is used in Sarkar'scompressibilitycorrection.]7Thiscompressibility correction has been incorporated into the NPARC code to compensate for the apparent increase in turbulent dissipation at higher Mach numbers. The z-equation (3) has been relatively insensitive to compressibility effects and thereforerequiresnomodification. The termsonthe right-hand sideof equations (2) and (3) correspond to the diffusion, production, dissipation, and near-wall damping terms, respectively. One of the difficulties with the k-_ model is that there is no natural boundary condition for e near a solid surface. Since the turbulent kinetic energy and the turbulent viscosity are both zeroalong a wall, the dissipation rate is set to zero as

k2 l.l.t= Cgfgp--_-

well. It is the near-wall damping terms of equations (2) and (3) that allow all the turbulent quantities to be set to zero near a viscous surface. Another problem with the Chien model is that it requires that the distance from the wall be calculated. In complicated geometries, and especially in three-dimensionalcases, this can be difficult tOcompute. Lang and Shihl8 also point out that near-wall damping terms whichuse y+arenot desirable,particularly near separation regions.

_(pk) +_=_ _(Puik) Ot _X i

(1)

_ r( I.t+_gt ") _k ] _Xi[ _ Ok) _XiJ

+17-pE(1 + Mt2)-2g.k2 -

_(pE) Ot

O(PuiE)

_

OXi

_x i

[(

(2)

shortcomings. TheWilcoxk-t,omodeldoes The value of (onothave near a viscous theaforementioned wall can be related to thesurface roughness of thewall. This condition can be enforced at the boundary and does not require the use of near-wall terms or y+. The Wilcox k-co model is formulated as follows:

"__ ] Oe OX i _tt

. _2 £ (-0.5y) + C_fl I'I z-.- c_2f2P-z--- 2!.t__- e r, r, y

_1.t = _

(3)

(9)

_

co

where

O(Pk) I-O(Puik------_) = _--_-[(IJ.+ I'tt 1 3k ] _t _xi _xi

Lk°)axiJ

17=

au----k (au--k _Xi + 3Xj u__-ti ] J

(4)

0X i

flt = 1.0 - e(-0"0115y+) •

, pk

+rl-I * po k[1 + kF(Mt)] (10) (5)

fl = 1.0

(6)

f2 = 1.0- 0.22e -`Ret,6)2

(7)

Ret = Pk----_2 pz

_

_

_xi

=_x i

Oo)_xi. ]

+,.

(8)

3 American Institute of Aeronautics and Astronautics

(11)

where

I-I=gt p

_+

_Xi [ _Xi

o_*=

5 °c=9

[3*=

ui] _Xj)

_"Rk ) I + (Retl _,Rk)

a0

+ Ret (_-)

(12)

(13)

(oc*)-1

(14)

(Ret) 1+ _,Ro )

__ 5 + (Retl 4 9 18 _.Rs ) 100

below, comparisons are made between the models using thisrelation. However,since [3*is afunction ofRet, which in turndepends uponco,theconversion of_ to omakes the assumption that [3*= 0.09.

(15)

Becauseofthesimilarityin thek-_ andk-o formulations, the current implementation makes use of the existing Chien k-_ algorithm available in NPARC. The algorithm for solving the k-_ equations dates back to the work of Nichols19 with recent modifications by Georgiadis, Chitsomboon, and Zhu.20The most significant changes occurred in thecalculation of the source terms involved in the k- and o-equations. The source terms are those on the right-hand side of equations (10) and (11) excluding the diffusion terms. The k-to model is selected by setting IMUTUR (or IMUTR2) equal to 7. Since it is similar in form to the k-_ model and uses the same solution algorithm, users Usersare advisedto useappropriate near-wall gridspacing shouldexpect stabilityandconvergence to be comparable. alongviscoussurfaces.Thishelpstoresolvetheboundary layersurfaces.and the large o-gradients which occur near these

//fRet"_4

1+_ Rs) CompressibilityCorrections [3=-_3 40

(16)

Ret = p...k_k go

(17)

and c k=2 ¢so=2

o_o=[3/3 ct0=1/10

Rs=8

A compressibility correction has been added to the model equations to enhance predictions at higher Mach numbers. The corrections described by Sarkar 17 or Wilcox21may beselectedbysettingthe variablesCOMPK, COMPW, and CMT0 according to the following table:

COMPK

COMPW

CMTO

Correction type

1.0

0.0

0.00

Sarkar (k-eq.)a

1.0

1.0

0.00

Sarkar (k+o eqs.)

1.5

1.5

0.25

Wilcox

0.0

0.0

0.00

None

Ro=2.7

R k=6

The compressibility correction is given by

aDefanlt.

F(M t)= Max(Mt2- Mt20,0.0) where the turbulent Mach number is the same as before, and the constants default to _k = 1.0, _o = 0.0, and Mt0 = 0.00. By comparing the k-o_and the k-_ models, a relation between _ and _ can be found. Wilcox5 defines this relation as 0 = v../([3*k).In the results to be presented

These variablescorrespondto _k,_¢o,andMto,respectively, in equations (10) and (11) and have been included in the TURBINnamelistblock. Asdiscussed in reference 21, the Sarkar correction only affects the k-equation in the k-_ model. When used with the k-comodel, however, corrections should be added to both the k- and o-equations. Computations for theSajben 22diffuser strong-shock case

4 AmericanInstituteofAeronauticsand Astronautics

,

(discussed in the Results section) indicate that adding the Sarkar correction to only the k-equation provides the best results. The Wilcox correction was of little use since the turbulent kinetic energy led to an Mt value thatwas below the cutoff turbulent Mach number CMT0 = 0.25. As a result, the Sarkar correction to the k-equation was chosen as the default setting, but the other options have been included because of the lack of extensive testing of highspeed flows. These options should provide the greatest amount of flexibility and limit future code modifications,

The NPARC code computes the k_ value specified for the individualboundarythroughthe auxiliarypressurevariable and compares it with the smooth-wall value given by SRDEF. It then uses the larger of these values to compute to atthe boundary.Failure to specify the auxiliary pressure willresult in akRvalue ofzero (default), and computations will be performed using the default surface roughness givenby SRDEF.Because boundary conditions 62 and 68 use the auxiliary pressure variable for other purposes, the surfaceroughnessgivenbySRDEFis used.No calculations have been performed using any surface roughness value other than the default value (kl_= 1.0), and users are advised to do the same until such time as this option can be more fully investigated. All other boundary conditions remain unchanged. As withthek-e model,theturbulentquantitiesareextrapolated from the interior of the flow field along slip walls and at

Boundary_Conditions The boundary value for coalonga viscous wallis related to the surface roughness of the wall. This relation is given by WilcoxS: u2 to = -_ SR v

at y = 0

free boundaries. Values of turbulence intensity I and turbulent viscosity I.ttmay be specified for free inflow boundaries as described by Georgiadis, Chitsomboon, and Zhu.2° Using a fixed inflow will cause the turbulent quantities alongthat boundary to remain unchanged from those in the restart file.

(18)

where

1,0? k_ )

k_ < 25

SR =

Stability Considerations (19)

The limiters used by NPARC to increase convergence andstabilitybycappingthevaluesoftheturbulentquantities

100 k_

k_ > 25

at both the high low extremes were modified to accommodate the and k-tomodel. These limiters are used on the interior of the flow field, not along boundaries. The current implementation is similar to that used by the k-e model, but because to approaches infinity near a smooth viscous boundary,numerical stability problems may arise ifit is limitedin this region.In regions where the turbulent kinetic energy is found to be very small, k is set to a minimum value, but to is not changed. If to becomes too small,both kand to are setto minimum values. Should the turbulentkinetic energyexceed themaximum value given by kmax= 0.10xU2f(k-_),it is set to the maximum value and to is computed using

.4-

andis valid forkR= uxkR/Vvalues up to 400.The NPARC code has been modified to allow the user to input the wall surface roughness for no-slip wails. This value is read in through the auxiliary pressure variable. The variable SRDEF has been added to the TURBIN namelist block and is used to specify the default smooth-wall surface roughness. When this variable is used, the surface roughness along all the viscous boundaries can be adjusted at once. The hydraulically smooth k_ value of 1.0 is the default value. In specifying the surface roughness for

•

either values than zero as kl_and valuesinput, less than zerogreater are treated askRaretreated (averagesand-grain surface roughness nondimensionalized by the reference length). According to reference 23, the effective sandgrain surface roughness is classified as follows:

k_ < 5

/ co= Maximum(to_- Calculated; (x*pk max/l.tt.max

wherel.tt,maxisspecifiedthroughtheuserinputTMUMAX. No upper limit check is made onco.Users who experience convergence difficultiesor erroneous resultsshould check to make sure the upper limits for the turbulent kinetic energy (adjusted through the variable UREFKE) and the turbulent viscosity (adjusted through TMUMAX) are set sufficiently high. The output file (26) lists the iteration

Hydraulically smooth

+

c_

,•\ 0.00 (b) 0.000

Baldwin-Lomax -- -- k.-_

_-"2.00 _

I 0.002

! 0.004

o • "k_ _ -- -- J- 0.006 0.008

I 0.010

_ 0.25 (b) 0

_ 50

Turbulent kineticenergy,k/(u=)2

O3

"" ""..... I 150 200

I 250

I 300

2.50

_.

?= =0.75

100

.....

Turbulentviscosity,Ilt/I_

1.25 -

1.00

.,,

_ .......

_. -

Model k-_ k-_

_ _ 2.00 6

',\_,%

o

= (/} 1.50 •_

0.50 -

>o 1.00

, \

'_ 0.25 -i:5 0.00 (c) 0.000

_ Baldwin-Lomax ---- k-E ....... k.-oo

_ 0.50 i5

• \ "•\

I

0.002

........

"1,__ -- -_

0.004

1 0.006

] 0.008

0.25

" _' 0

100

Turbulent kinetic energy, k/(u=)2 Figure 2._Turbulent kinetic energy profiles along flat plate for Reynolds number based on plate location Rex. (a) Rex = 1xl 06. (b) Rex = 4xl 06. (c)Rex = I xl 07.

200

300

400

..-"

!

500

Turbulent viscosity, I_t/l_ Figure 3._Turbulent viscosity profiles along flat plate for Reynolds number based on plate location Rex. (a) Rex = lxl 06. (b) Rex = 4xl 06. (c) Rex = 1xl 07.

ll AmericanInstituteof Aeronautics and Astronautics

I

600

1.50 --

30 -

[ _._1.25 _-\ £_,,

_ Baldwin-Lomax -- -- k-_

+ 25 -

....... k-(_

=

_0.75

"'',

"_ _,

0.25

_ ___ Baldwin-Lomaxk_E _.j.-""

O 10 E z

"" ",,_

0.00 0.0( 00

5

I I i ,I -'_; I 0.0003 0.0006 0.0009 0.0012 0.0015 Reynoldsstress,-uv/(u=)2

0/I 1

1.50 -

, W,I,I,I 10

(b) , , i ,i,1,I 1000

f , i ,_,_,I 100 Y+

30 Baldwin-Lomax --k-_ ....... k-o)

1.25 _ "_

o + 25 [] = _

_"

{n

Reichardtdata Klebanoff-Diehl data

"o15

0.50 --

=

o []

o

Reichardtdata Klebanoff-Diehldata Baldwin-Lomax

....... k,-_

"_

._

.

_0.75 -_

-_ 15 N

8 0.50-

.q_fq';'_

"

>

8

" ..-_._,

0.25 -

5

"-

(c) 0.00 0.0000

! 0.0003

I 0.0006

! 0.0009

0.0012

0.0015

0/I

_

Reynoldsstress,-uv/(u=) 2 Figure4._Turbulent shear stressalongflat platefor Reynolds number based on plate locationRex. (a) Rex = lxl 06. (b) Rex = 4xl 06. (c) Rex = I xl 07.

,I,1,1 10

!

, 1,,,i,I

100

i

, i,,,I,I

1000

Y+ Figure5._Velocity profilesalongflat plate for Reynolds numberbased on plate locationRex. (a) Rex = lxl 06. (b)Rex = 4xl 06. (c) Rex = I xl 07.

12 AmericanInstituteof AeronauticsandAstronautics

0.008 -

"

Two-dimensional _t __ 0.006

....

_t.,

Three-dimensional O

30.004

Wieghardtdata

O O 0

0.002

I

0

2000

I

0

0

0

0

0

I

0

0

!

0

0

I

I

4000 6000 8000 10 000 Reynoldsnumberbased on momentumthickness,Re0

12 000

Figure6.---Comparisonof two- and three-dimensionalk-o}skinfrictionresultsfor flat plate.

,-- Reducingsection /

/4

/1 //

Straightpipe --, \

lI

5° half-anglediffuser --, \\

// I

\

II

\\

\

x =0

x = 0.642 m

(a)

(exit plane)

.

Straightpipe

Conical diffuser Straightpipe

(b)

\

I

I

x =0

Diffuser exit

Figure7.--Fraser diffuser.(a) Geometry.(b) Computationalmesh.

13 AmericanInstituteofAeronauticsand Astronautics

Outflow

15 -O

Fraserdata Baldwin-Lomax Baldwin-Barth k-=;

u ....

E E10

_

- ......

k-_(two-dimensional) k-ca imrp__.-nJm_nslona.

..

°

I

o e-

/2

0 (a) -0.2

I 0.0

l I 0.2 0.4 0.6 Normalized velocity, U/Ucl

0.8

I 1.0

60 O

Fraser data Baldwin-Lomax Baldwin-Barth

50

E Ei 40

....

k-_

....... _ --

k-e_(two-dimensional) k-oJ(three-dimensional)

|

-.,1

-_ O J_ €=

30

t

€-.

_ 20 ffl

Q

lO f f

o

-0.2

0.0

I

I

0.2 0.4 0.6 Normalizedvelocity,U/Ucl

I

0.8

Figure8.--Fraser diffuserprofilesat two axial locationsx. (a) x = 117 mm. (b)x = 642 mm.

14 American Institute ofAcronantics andAstronautics

I

1.0

0.0150 -O

....

Fraser data Baldwin-Lomax Baldwin-Barth k-_

....... _ --

k-_o(two-dimensional) k

_ Baldwin-Lomax --k-¢ ....... k-_o

0.8

_ Baldwin-Lomax -- -- k-E -----k-oo

0.2 (b)

o.o -50

,,- %''"

i

!

I

0

50

100 150 200 Velocity, u, m/s

I

I

250

300

1.0-

_- 0.5

0.80.4 0"35

"-..

0

5

_£ 0.6 .o •"_ o _ 0.4 -

10

Axial position,x/H Figure14.DPressure distributionsfor Sajbendiffuser (strong shock). (a)Top surface. (b) Bottomsurface.

o Sajben data _ Baldwin-Lomax -- -- k-_

"'_'o"_"-._.. _, I o ",..\/ o)i/

>

0.20.0 -50

_20 0

50 100 150 Velocity, u, m/s

200

250

Figure15._Sajben diffuservelocityprofiles(strongshock) at three axial positionsx]H. (a) x/H = 4.6. (b)x/H = 6.3. (c)x/H = 7.5.

19 AmericanInstituteofAeronauticsandAstronautics

10-

°Oo

0.8 --

,I-

O "_0.6

--

O

Sajbendata Three-dimensional Two-dimensional

o

O

O

-- 0.4 ->

0.2 --

o.o

I

0

I

50

o,

o

100

!

150

I

200

250

Velocity,u, m/s Figure16.---Comparisonof k-toresultsfor Sajbendiffuser(strongshock)at axial positionx/H = 7.5.

1.0 --

_0"0_-_-_

0.8 --

_

- 0.6 --

_o.

_

O

Sajbendata

--

Sarkar k-eq.

0 O _._._

Sarkar k- + _o-eqs.

_ 0.4 ->

_/

.... w,cox

ji/

. ......

/i F

None

0.2--

0.0

-50

I

0

I

50

I

100 Velocity,u, m/s

n

.

150

°l

200

Figure17.--Comparison of compressibilitycorrectionsfor Sajbendiffuser(strongshock)at axial positionx/H = 6.3.

2O AmericanInstituteofAeronauticsand Astronautics

I

250

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January1996

TechnicalMemorandum

4. TITLE AND SUBTITLE

5. FUNDING NUMBERS

Implementationof a Two-Equationk-o)TurbulenceModelin NPARC WU-537--02-23

i6. AUTHOR(S)

DennisA. Yoder,NicholasJ. Georgiadis,and Paul D. Orkwis 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

8. PERFORMING ORGANIZATION

REPORTNUMBER

NationalAeronauticsand SpaceAdministration LewisResearchCenter Cleveland,Ohio 44135-3191 9. SPONSORING/MONITORING

E-9955

AGENCY NAME(S) AND ADDRESS(ES)

10. SPONSORING/MONITORING AGENCY REPORT NUMBER

NationalAeronauticsand SpaceAdministration Washington,D.C. 20546-0001

NASATM- 107080 AIAA-96-0383

11. SUPPLEMENTARYNOTES

Preparedfor the 34thAerospaceSciencesMeetingandExhibitsponsoredby theAmericanInstituteofAeronauticsand Astronautics,Reno,Nevada,January 15-18, 1996.DennisA. Yoderand NicholasJ. Georgiadis,NASALewis Research Center;,PaulD. Orkwis,Universityof Cincinnati,Cincinnati,Ohio45221.Responsibleperson,NicholasJ. Georgiadis, organizationcode 2740, (216)433-3958. 12a. DISTRIBUTION/AVAILABILITYSTATEMENT

12b. DISTRIBUTIONCODE

Unclassified-Unlimited SubjectCategory02 This publicationis available fromthe NASACenterfor AerospaceInformation,(301)621--0390. 13. ABSTRACT (Maximurn 200 words)

The implementationof a two-equationk-o)turbulencemodelintotheNPARCflowsolver is described.Motivationfor the selectionof this model is given,majorcode modificationsareoutlined,new inputsto the codeare described,and results are presentedfor severalvalidationcases: an incompressibleflowovera smoothflatplate,a subsonicdiffuserflow,and a shock-inducedseparatedflow.Comparisonof resultswiththe k-emodelindicatethatthe k-o)modelpredictssimpleflows equallywellwhereas, for adversepressuregradientflows,thek-o) modeloutperformsthe other turbulencemodelsin NPARC.

14. SUBJECT

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