AIM 89-0566 Separated Flow Predictions Using a Hybrid k-UBackfIow Model U. Goldberg and S. Chakravarthy, Rockwell International, Thousand Oaks, CA

27th Aerospace Sciences Meeting January 9-12, 19891Ren0, Nevada

For pemdssion to copy or republish, contact the American Institute of Aeronautics and Astronautics 0 370 L’Enfant Promenade, S.W., Washington, D.C. 20024

SEPARATED FLOW PREDICTIONS AIM-89-0566 USING A HYBRID k-L/BACKFLOW MODEL

Uric1 C. Goldhrrg* and Snkuninr R. Chakravaithy*' ILockw,-ll lntcrnationai Scir:~c~: C'cnt,rr

Thonsaiid Oaks. Califnmia 91360

The k - and I, distributions are comhincd to form an eddy viscosity ( l i t ) ficld which is added to the mol<.rular viscosity i n thc diffusion terms of the Reynolds-Averaged Navicr-Stokes (RANS) equations. The high Reg k-eqiin- tion, in 2D Cartesian consrrvation form, reads as follows:

A ncw hybrid one equation k L/backflow modcl has bccn uscrl to romputr sevaral turbulent flow prohlcnis in- volving dctiic!;ccJ flm: rcgions. Tlir model incorporates al- gchrnir ncnr- wall trmt.incnts for thr kinetic encrgy of t in-

( p , ) . Tlir iiear~~~wall formulation dcpcnds on whother the flow is at!iiclierl or dctnclied. This approach obviates the nced to usc wall functions. Aareemcnt hetween nrrdictions

a 3 a . - ( p k ) + -(pvk) + -(pvk) at a x ay

bolancc (k), thc length scale ( L ) , and thc rddy viscosity

- and cxpcriniciitd data is gcnrrally very good tliroughout thr Mach mimhcr range.

Introduction where thc production term is given by ___ Rcccntly, a hyhrid Baldwin--Lomax/hackRom model

has hrrn applied srwessfiilly to predict a variety of sep- ~~

irrateti t:rhiilent, f l ~ w s ( ' - ~ l . Since Ihc model is an &ehr& m e , it wns neccssary to invoke an ad hoc method i n or- der to prodiuce a tiirhulence history effect of the detached

i t rcgion exists). By introducing a transport equation for the thv kinetic eiiergy of turbulence, these history effects are

the dissipation t,erm is given by

p i 2 D = pCn-Rr (Cd % 0.S9) ,

flow region on the flow downstream of reattachment (if such

automatically takcn into account. = C, ,~&LR~: (c,, = 0.09)

L viscosity is dctermined by

In thc prcsent paper, a one-equation k-L tixhulcncc model i, clescrihed in some detail and results are shown

and the comhined viscosity is

for several flow prohlems involving detached fl, : rcxions at Pk = !A + IL<r/Uk ( U k = 1.0) - transonic, supersonic, and hypersonic flow regimes.

In Eq. (1) iA::3 = 1 for axisymmetric flows and zero for 2D flows. Re is the refciencc Rcynolds number. The I c ~ ~ L Turbulence Model

The k - L turhulencc model comhines a PDE for the kinetic encrgy of turbnlencc (k) with an algcbraic cxpres- Jim for the dis!,ribution of the length sczlc (I,). Thus, the velocity scale (IistriRution within the flow field, A, is de- termined from a well established and reliable cquation (the k- equation) while retaining the degrees of frecdom afforded through the algebraic approach to the determination of the length scale. For exaniple, length scale distrilmtions can he tailor-made for wa:l/wnkr shear layers, jets, mixing layers, etc. Thc more problematic r-equation, which is used to de- termine the length scale in the k-e model, is thus avoided.

In the present approach, the high turbulonce Reynolds number (Re, = d h / u >> i) form of the &equation is solved together with a near-wall treatment which depends on whether the flow is attached or detached. For attached flows, Gor~ki's[~l irlgchraic viscous aiihlayer treatment is used, while for detached flows Goldberg'slz] algebraic back- floiv turbulence model is invoked.

v * Memher Technical Staff, Member AIAA ** Managcr, CFD Department, Member AIAA

Copyright 0 American Inxitute of AerOnaUlics and Astronautic$, h e . . 1989. All rights reserved. 1

The length scalc is composed of two parts (see Fig. 1): an inner hranch and an outer branch. The outer hranch consists of two linear sections given by

ivllrre S is a measure of the local shear layer thickness, determined as the distancc from the wall (or wake core) to the location where the vorticity diminishes to below a small fraction of its local maximum value, A is the distance from the core of the make (if one exists) to the location of local maximum vorticity (see Fig. 1) and K = 0.4 is the von Kirmin Constant. Note that the choice of A enables smooth transition from wall-hounded flows to fully turhu- lent wakes. Thus, the outer scale grows linearly from zero (for solid surfaces) or CLA (for wakes) to C d , its maxi- mum value, which is reached when y/6 = C,/K % 0.2. The value of CL depends on whether the flow is attached or de- tached. The fraction of turbulence energy that contributes to turbulent inertia (and, therefore, to / i t ) depends on the local flow conditions thro>igh the interaction between the

."

1. k - - L model length IC& pmfilr and wmenclature.

turbulence and the mean How. In detached How regions, the state of the turbulence is highly anisotropic and the energy of turbulence is distributed almost equally among the three Reynolds stresses u ' v ' , u", and 7 (for nominally 2D flows). Since the eddy viscosity { L , accounts for the shear stress u l d only, its magnitude within detached flow regions must be reduccd coinpared to that in the attached portions of the flow, w1it.r~ almost ail the turbulent iner- tia is contained in m. Under hypersonic flow conditions an additional mcclianisni exists which further reduces the turbulent inertia (and thus 1 1 0 , namely, the large increase i n heat transfer rate within separated flow regions. This causes a corresponding increase in turbulent dissipation, leaving a lesser amount of energy available for turbulent inertia. These effects are~~reflected in the following table of recommended values for Cr,:

_ -

-

attached detached CL Hows Hows and wakes

non-hypersonic h: (0.40) 0.30

hypersonic ti 0.23

For solid surfaces, the inner !ength scale is determined by the hehaviol of the kinetic energy of turbulence and its

dissipation within the viscous sublayrr (see next scct,iw on near-wal~ t,rratrricnt). In gcncra~, L - k 3 1 2 / e and ;iStcr SUI>- stituting for k and 6 their s o l h y c r fr,nnnlation, t,lic: resiilt.

wlwrr suljscripts I I axid tu stand for cuhlirycr cdgc a d wall, rcqxctivcly. Tlios, for solid surfxcs , thc length sci~le grows as y3 from zero at tlic wall until it reaches ;L magnitude corresponding to L,,,,. This occiirs in thc viciiiiiy OS the viscous sublayer edgc, at which point. I, is dotcrinincd by the outer tiranch. For wall h d c d i:ows, tlic lcngtli scale is, therefore, givcn as (SCC Fig. 1)

L = inin {Li, , Z,,,,)

Pcar--Wall Treatment

(I) Attached Flow

Following Gorskil'], k and pI within the viscous sub- layer are determined a!gebraically from thc following Sor- mulae:

where the viscous sublayer edge, yL,, is Iocatctl at

and the value of k at tlii siiblnycr cdge is: \i

P w I; ,

The vahir of the dissipation, e , within the sub1ayc.r is pivw

" - fi Pl.

by

~

Finally, ihe liricar interpolawr, C,, is Bivcn by

21, C" = C,' + (C*" - C,) - YS

wl,e*e csu = 2 & JE .

~~ ~

(2) Det;schcd Flow

For detached !low, k and p, withhi the hackflow region are determined algebraically using Goldberg's:') backflow turbulence model. For details, see the Appendix.

Initial and Boundary Conditions for Eq. (1)

As initial conditions, a small nonzero value is pre- scribed for k in the eiitire~computational domain. Typi- cally, k = is used.

~~ ~~

At a solid surface, k is set to zero. This is done even though Eq. (1) does not apply in the near vicinity of walls.

--

w...j, Thc justific;ition is that ,k is replaced i u the near wall rcgion as dcscrihcd ;\hove. If the shcar layer is in a wake region or if a slip wall Irmnidary condition applies, the normal dcrivntivc of k is sct to zero: (&/ay), = 0.

Mm.08,5 SHOCK

RICI I IC"L*I ,OI & ]o,,s2.m, $ 0.5

MODI1S h,l ocm, r ' m ' r m l U

T h c hybrid I: L/i,ackflow modrl has been incorpomted 0 3

into a f i n i t r : volurnc nnificd time/spacc marching zonal schcnrc RANS and co&.'l, TVD featuring iliscnAization an for implicit high acc~iracy. upwiiidMiased Liiie 02Fd 0.55

Gauss Seidcl relaxation method was used for space-

ploycd for tinro mnrrhing. 0 0 p (1.46

z marching whilr approximatt factorization method was eni- ~, , , .--,

0 as As n first validntiom m s r , turhiilcnt flow ovri ii flat ~ * * platc was compatcd using a 100 x 35 grid and was conrp;wcrl 0 6 0 9 1.3 - 0

wit11 tlic exprrimcntal data of Wieghardt['I at ~ n c h 0.1. ./r "8"-

3. Mach 0.9 axisymmctric bump flow problem: wall pres- Fignrc 2 shows skin friction and vclocity profile mmparisons for two loc;itions. The agrccmeiit is excellent. sure, skin friction, and velocity profiles comparisons.

As a further validation case! a Mach 2.55 flow cwcr a 24" ramp has been computed and compared with data by Settles, et al.~11s121 and by Dolling and Murphyl"I. Figirrc 4 shows wall pressnre and &in f4ction distributions rrs,dting f i r m a calculation donr on a coarse (60 x 40) grid. Good

SKIN FRICTION PROFILE 1 .o

0.8

N- 0.6 ngrecmcnt is obtaincd. z j 0.4 - IC16904

0.2

5.00

0 0 1 2 3 4 5

X. METER 3.75

0.5 8

0.4 P

a '* 2.50

- .. 9 0.3 1.25

LL Y .to DATA. SETTLES Li 0 DATA, DOLLING a 0.2 r'

.. - k - L t BACKFLOW MODELS -

0.0 0.250 rq 0.1

0 0.125 0 0.203 0.405 0.608 0.810 1.01

UIU,

.\\ 0 N

2. Turbtilent flow over a flat plate: comparison between

Next, the axisymmetric hump flow experiment at M, = 0.875 by Bachalo and John~on[~ l was computed us- ing a 75 x 36 grid and was compared with data['] as well as with calculations using the Johnson-King (J-K) turbulence cmdelr"1. Figure 3 shows wall pressur-. skin friction a i d -3.0 -0.5 2.0 4.5 1.0

two velocity profiles in the detached flow region. Agree- rfd,

ment with data is very good in all cases and, in general, the current turbulence model produces results very similar to those obtained with the J-K model.

0 0.0

predictions and data. z -0.125

-0.250

-' *. Mach 24' comPression Corner 'Ow Problem: pressure and skin friction comparisons.

3

The ncxt two cases are ill tlic hypersnnic flow rrgimc. Experimental data wcre obtnincd by Horct,tu;m, et t t l . l in] for a Mach 7.2 flow over an axisyminctric ogive cylindrr configuration with a shock gcnerator ring. Figwc 5 s1~oivs

surface prrssure, heat transfer, and skin frirtion compir- isons hctwcen the current model, cqx!rinmitaI data for a 15" shock generator angle, and calcnlirtions by Vuong a d C~akloyl '~] using the q-~w and q-w c tnrbulcrrcc inod- CIS (the latter being a y-w model corrartrd for hypcrsnnic cffccts). The computatious wrre doric on similar grids. 111

tlrr present work, the shock generator was trcatcd as itn iuliabatic surface over which a laminar hrnmdary layer (IC- velopcd. At the trailing edge of the shock grncrntor, locatad a t 5 = 154.48 cm (measured from the nose of thc ogive), an cxpansion took place. This expansion intcrxtcd with thc shock reflecting from the cylinder stirfacc. Uotli the q~ w c and the I ; ~ ~ L models predict tlic upsttlrwn influence Iathcr wcll; the k I, model, howcvcr, p d i c t s liirgc post rrattachinent overshoots as does the miginol q ~ ~ u ~ n i ~ l c l .

II.'*a, I

5. Mach 7 ogive-cylindcr slrock~boriuclary layer interac- tion flow problcm: wall presmirc, I rmt transfer and skin friction comparison..

r krn, I / r m ,

0. Mach 9 38" compression corner flow problem: wall pressure and heat transfer comparisons and skin fric- tion prediction.

The last comparison is for a Mach 0.2 flow (wrr a 38" compression ramp, for which expcrimcntnl data i i iv x v i d ihlc by Coleman and Stollery['61. Figurc F shows stirfacc ~ T C S S I I T ( I and licat transfcr comparisons bet (using a 100 x 50 grid) arid data, as well as skin friction prr- diction. Thr ngrccment is very gnod. Upstrcnni iriilizviirc is prcdictcd very well and prcssurc and 1ir:at transfer k v v k are calciilatcd correctly, both in the lnrgc lmckfimv wgim i ~ n d i n tlic post rcattadiinent zone. A crhilntion i i s i i ig thr q-w c turbulence modell'51 is also shown. Thc k ~ I , ~ i m l e l predicts better the post rcattaclnnciit Ijcliavior, in p;trt,icii- lnr the peaks in both press~m: and I x n t trilnsfrr.

V

- Coiicli~sions

A 11cw hybrid k-Llhnckflow onc~~equation turln~lr!~w model has Ijccn introduced and dcmonst,rated fnr s c ~ c r n l flow problems across the Mach number range, includii:g <lctnched flaw regions. The special featiircs of t h i s ~norlrl are thc near-wall trcatnicnts for k , L, and / i t nndrr Ix>th attached and detaclied flow conditions, obviating thc nrcd to usc wall fimctions whose validity, cspccially for dctnclird flnws, is highly qucstionablc. Tlir good agrccmmt hct computations and experimental data suggest,s t,liat tliis model is a promising one. Fiirthcr rrsrilts using it will l i p

rqmrtcd in future work.

APPENDIX

Tlir: following algrhraic distribution of eddy visrnsity

is prcscrihed throughout the backflow region ( s w Fig. 7 fnr

dcfinitirms of the nomenclature):

7. Schematic of detached flow bubble and nomenclature

v

4

with

whrrc

cI =n.353 , Cz -U.i86

Referairrs

1. U.C. Goldberg and S.R. Chakravarthy, “Prediction of Sepaiatcrl Flows with a New Turbulence Model,” pro- cerdiiigs of the 5th Intcrnational Conference on NII- rneiical Mctirods in Laminar and Turbulent Flow, Pt. 1, Taylor, Habashi, Hafez, Eds., pp. 560-571, Mom treal, July 1967.

2. U.C. Goldberg and S.R. Chakravarthy, “Prediction of Sepn:-atetl Flows with a New Backflow Turbulence Modcl,” A I A A Journal, Vol. 26, No. 4, April 1986,

405~~40s.

3. U.C. Gol~lbcrg, “Sq)aratcd Flow Predictions Using a New Turlmlencr Model,” proceedings of the 6th Syin- posinm on Turbiile~~t Shear Flows, pp. 15-5-1-15-5-6, Tonlousc, Franc?, Srptemhcr 1987.

4. J . J . Gorski, “A New Near~~Wall Forniddtionfor the k-- L Equations of Turbiilcnce,” AIAA Paper No. 86-0556, .January 19SF.

5. U.C. Goldherg, “Scpnrated Flow Treatment with a New Turbulcncr Model,” AIAA Journal, Vol. 24, Oc- tober 10SF, pp. 1711-1713.

F. S.R. Cliakravarthy, I<.-Y. Saema, U.C. Goldberg, J.J. Gorski, and S. Osher, “Application of a New Class of High Accuracy TVD Schemes to the Navier-Stokes Equations,” AIAA Paper No. 85~~0165, January 1985.

7. S.R. Chakravarthy, “The Versatility and Reliability of Euler Solvers Based on High-Accuracy TVD Formula- tions,” AIAA Paper No. 86-.-0243, January 1986.

8. I<. W&$inrdt, “Flat Plate Flow,” proceedings of the AFOSR-IFP-Stanford Conference on Cornputation of Turbulent Bonndary Layers, D.E. Coles and E.A. Hirst, E&., Vol. 2, 1968, pp. 98-123.

9. W.D. Birclialo and D.A. Johnson, “An Investigation of Transonic Turhulcnt Boundary Layer Separation Gen- erated 011 an Arisymmetric Flow Model,” AIAA Paper No. 7% 1 4 3 , 19i9.

10. D.A. Jolmson and L.S. King, “A Mathematically Sim- ple Turlidencr Closiire Model for Attached and Sep- aratcd Tiirlntlrnt Bcnindary Layers,” AIAA Journal, Vol. 23, No. 11, Koreinher 1985, pp. 1684~~1692.

11. G.S. Settlcs, I.E. Vas, and S.M. Bogdonoff, “Details of a Slmck-Sepnrntrd Turbulent Boundary Layer at a Compsecsioii Conrrr,” AIAA Journal, Vol. 14, No. 12, Dcc.rinl~n. 19X, pp. 1700~~1715.

12. G.S. Settles, T.J. Fitzpatrick, and S.M. Bogdonoff, “Detailed Study of Attached and Separated Compres- sion Corner Flow Fields in High Reynolds Numher Su- pcrsonic Flow,” AIAA Journal, Vol. 17, No. 6, June 1979, pp. 579-585.

13. D.S. Dol1ir.g and M.T. Murphy, “Unsteadiness of thc Separation Shock Wave Structure in a Supersonic Compression Ramp Flow Field,” AIAA Journal, Vol. 21, Dcccrnlx~ 1983, pp. 1628-1634.

14. C.C. Horstnian, M.I. Kussoy, T.J. Coaklry, M.W. Ro- l)esin, and J.G. Marvin, “Shock-Wave Induced Turbu- lent Boundary-Layer Separation at Hypersonic Speeds,” AIAA Paper 75-4, 1975.

15. S.T. Vuong and T.J. Coakley, “Modeling of Turbulence for Hypersonic Flows with and without Separalion,” AiAA Paper 87- , 1987.

16. G.T. Coleman and J.L. Stollery, “Heat nansfer in Hy- personic Turbulent Separated Flow,” Imperial College of Science and Technology, Department of Aeronautics, Report #72-05, March 1972.

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