Workshopon Engineering Turbulence Modeling

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NASA Conference Publication 10088 ICOMP-92-02; CMOTT-92-02

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Proceedings of the Workshop on Engineering Turbulence Modeling sponsored by the Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and Transition NASA Lewis Research Center Cleveland, Ohio August 21-22, 1991

II II I

N92-2451 4 ENGI ,',IE_ RI NG (,'_ASA-CP-IOOBR) WORKSHOP ON -- THRU-- 510 P CSCL 20D I IUP,_UL E _ICE ML]r_E L [NG (NASA) N92-2453I Uncl as G313:, O086B,Bl

NASA Conference Publication 10088 1COMP-92-02; CMOTT-92-02 Workshopon Engineering Turbulence Modeling

Proceedings of the Workshop on Engim'ermg Turbulence Modeling sponsored by the lnstitute fi_r Computational Mechanics in Propulsion and Cemer fi_r Modelillg of Turbu h'nce and Transition NASA Lewis Research Center Cleveland, Ohio August 21-22, 1991

fiJ/_A National Aeronautics and Space Administration

Office of Management

Scientific and Technical Information Program

_992

/_r/ICOMP }} 2 ORGANIZING COMMITTEE

T.-H. Shih

J. L. Lumley

P. Moin

M. Goldstein

L. A. Povinelli

E. Reshotko

J. M. Barton

The proceedings are edited by

L. A. Povinelli

W. W. Liou

A. Shabbir

T.-H. Shih

PRECEDING PAGE BLANK NGT F_LMED ! !

Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

Table of Contents

Workshop Program ...... 1

Session I Turbulence Modeling in CFD and Algebraic Closure Models

The current status of turbulence modeling in CFD and its future prospects B. E. Launder ...... 7

Comments D. M. I3ushrlcll ...... 17

Discussion ...... 37

The present state and the future direction of eddy viscosity models D. C. Wilcox ...... 39

Comments P. R. Spa]art ...... 71

Comments T. J. Coakley ...... 79

Comments N. J. Laaag and T. Chitsomboon ...... 87

Discussion ...... 95

The present state and the future direction of algebraic Reynolds stress models D. B. Taulbee ...... 101

Comments A. O. Demuren ...... 145

Discussion ...... 157

PRECI'D|_JG PAGE BLA_'_g ,NOT F._LMEDv Cct_tcr for Modcli71 9 of Turbulence and Transition V_:_rk_1,,ot, on Engin_.cria 9 "l_urb.lc_c_ Modf_li.fl- 1991

Session II Second Order Closure and PDF Method

The present state and the future direction of second order closure models for incompressible flows T.-H. Shih ...... 163

Comments J. R. Ristorcelli, Jr ...... 219

Comments C. G. Speziale ...... 229

Discussion ...... 243

The present state and the future direction of second order closure models for compressible flows T. B. Gatski, S. Sarkar and C. G. Speziale ...... 249

Comments J. R. Viegas aad P. G. Huaag ...... 277

Comments W. W. Liou ...... 297

Discussion ...... 299

The present state and the future direction of pdf methods S. Pope ...... 301

Comments E. E. O'Brien ...... 327

Comments J. Y. Chen ...... 337

Comments M. M. Sinder ...... 351

Discussion ...... 357

vi C_nter for Modeling of Turb_tlcncc and Transition |_/orksl,.ol_ o1_. Ez_.!linccring Turbult zsc_' _4odcling - 1991

Session III Unconventional Turbulence Modeling

The present state of DIA models and their impact on one point closures A. Yoshizawa ...... 361

The present state of two-point closure schemes and their impact on one point closures J. Weinstock ...... 391

The present state of RNG and their impact on one point closures S. Orszag ...... 409

The present state of application of RDT to unsteady turbulent flows R. R. Mankbadi ...... 435

The role of experiments in turbulence modeling W. K. George ...... 463

Numerical Simulation: Its contributions to turbulence modeling J. H. Ferziger ...... 487

Discussion ...... 513

vii _S r WORKSHOP ON ENGINEERING TURBULENCE MODELING

Center for Modeling of Turbulence and Transition (CMOTT) ICOMP, NASA Lewis Research Center Room 175, Sverdrup Building, Cleveland, Ohio

August 21-22, 1991

1. OBJECTIVES

The focus of this meeting will be one-poiut closure models (i.e. from algebraic models to higher order moment closure schemes and pdf methods) which can be applied to CFD computations. However, other schemes helpful in developing one-point closure models, such as RNG, DIA, LES and DNS, will be also discussed to some extent.

2. FORMAT

This meeting will consist of three sessions and will last about one and half days.

Each session will have three or five position presentations. In the first two sessions each position presentation (40 minutes) will be followed by a comment presentation (10 minutes) and a discussion. In session III, there are five position presentations (30 minutes) and one discussion. The presentations will be made by invited speakers and the discussions will be led by the session chairman (see outline of the workshop for details).

The viewgraphs of the presentations will be collected to be distributed later.

3. ORGANIZING COMMITTEE

T.-H. Shih (Chairman) J. L. Lumley (Honorary Chairman) P. Moin M. Goldstein L. A. Povinelli E. Reshotko J. M. Barton 4. OUTLINE OF THE WORKSHOP

August 21, 1991 (Wednesday)

08:00-08:15 am Registration

08:15-08:30 am Welcome by L. Povinelli

Session I: Turbulence Modeling in CFD and Algebraic Closure models. Chairman: E. Reshotko

08:30-09:10 am B.E. Launder, "The current status of turbulence modeling in CFD and its future prospects." 09:10-09:20 am D.M. Bushnell, "Comment paper." 09:20-09:40 am Discussion

09:40-10:20 am D. Wilcox, "The present state and the future direction of eddy viscosity models." 10:20-10:30 am P. Spalart, "Comment paper." 10:30-10:40 am T. Coakley, "Comment paper." 10:40-11:00 am Discussion

11:00-11:40 am D. Taulbee, "The present state and future direction of algebraic Reynolds stress models." 11:40-11:50 am A.O. Demuren, "Comment paper." 11:50-12:00 am Discussion

12:00-01:30 pm Lunch Break

Session II: Second Order Closure and PDF Method. Chairman: J.L. Lumley

1:30-2:10 pm T.-H. Shih, "The present state and the future direction of second order closure models for incompressit)le flows?'- _ 2:10-2:20 pm J.R. Ristorcelli, Jr., "Comment paper." 2:20-2.30 pm C.G. Speziale, "Comment paper." 2:30-2:50 pm Discussion

2:50-3:30 pm T.B. Gatski, "The present state and the future direction of second order closure models for compressible flows." 3:30-3:40 pm J. Viegas, "Comment paper." 3:40-3:50 pm G. Huang, "Comment paper." 3:50-4:10 pm Discussion

4:10-4:50 pm S. Pope, "The present state and the future direction of pdf methods." 4:50-5:00 pm E.E. O'Brien, "Comment paper." 5:00-5:10 pm J.Y. Chen, "Comment paper." 5:10-5:20 pm Discussion

6:30-9:00 pm Banquet (Pierre Radisson Inn, Great Northern Blvd.)

2 Cenler for Modeling of Turbulence and Transilion IVorkshop on Engineering Turbulence Modeling- 1991

Session I

Turbulence Modeling in CFD and Algebraic Closure Models

August 22, 1991 (Thursday)

Session III: Unconventional Turbulence Modeling. Chairman: J.H. Ferziger

08:30-09:00 am A. Yoshizawa, "The present state of DI A models and their impact on one point closures." 09:00-09:30 am J. Weinstock, "The present state of two-point closure schemes and their impact on one point closures." 09:30-10:00 am S. Orszag, "The present state of RNG and its impact on one point closure." 10:00-10:30 am R.R. Mankbadi, "The present state of application of RDT to unsteady turbulent flows."

10:30-11:20 am W.K. George, J.H. Ferziger "The role of experiments and DNS & LES in supporting turbulence modeling efforts." 11:20-11:40 am Discussion

11:40-12:00 Concluding Remarks.

5. INVITED PARTICIPANTS

University •

Names Phone No. Affiliation J.L. Lumley (607) 255-4050 Cornell University S. Pope (607) 255-6981 Cornell University J.R. Ristorcelli, Jr. (607) 255-3420 Cornell University J.H. Ferziger (415) 723-3840 Stanford University S. Orszag (609) 258-6206 Princeton University E. Reshotko (216) 368-3812 C.W.R. University I. Greber (216) 368-6451 C.W.R. University E.E. O'Brien (516) 632-8310 SUNY, Stony Brook D. Taulbee (716) 636-2334 SUNY, Buffalo W.K. George (716) 636-2593 SUNY, Buffalo C.P. Chen (205) 895-6154 U. of Alabama, Hunstville R. So (602) 965-4119 University of Arizona S. Elghobashi (714) 856-6131 University of California, Irvine A.O. Demuren (804) 683-3727 Old Dominion University M.-C. Lai (313) 577-3893 Wayne State University B.E. Launder (061) 236-3311 UMIST, U.K. A. Yoshizawa (03) 402-6231 University of Tokyo, Japan

Government •

Names Phone No. Affiliation J. Bardina (415) 604-6154 NASA Ames Research Center T. Coakley (415) 604-6451 NASA Ames Research Center G. Huang (415)604-6156 NASA Ames Research Center J. Viegas (415)604-5950 NASA Ames Research Center D.M. Bushnell (804)864-5705 NASA Langley Research Center T.B. Gatski (804) 864:5552 NASA-Langley=Research Center J. Morrison (804) 864-2294 NASA Langley Research Center K. Gross (205) 544-2262 NASA Marshall Research Center C. Schafer (205) 544-1642 NASA Marshall Research Center J. Weinstock (303) 497-3924 NOAA

Industry and Other Institutes :

Names Phone No. Affiliation P. Spalart (206) 865-5587 Boeing Company A.K. Singhal (205) 536-6576 CFD R.C. D. Wilcox (818) 790-3844 DCW Industries, Inc. M.S. Anand (317) 230-2828 GMC S. Sarkar (804) 864-2194 ICASE C. Speziale (804) 864-2193 ICASE O.P. Sharma (203) 565-2373 P. & W. M. Sindir (818) 7i8-4623 Rocketdyne J.Y. Chen (415) 294-3424 Sandia Lab.

NASA Lewis Research Center: (216) 433-4000

A. Ameri D. Ashpis J. Adamczyk J.M. Barton T. Bui C. Chang S. Chen T. Chitsomboon R. Claus R. DeAnna R.G. Deissler B. Duncan P. Giel M. Goldstein G. Harloff A. Hsu K.H. Kao S. Kim S.W. Kim T. Le M.F. Liou M.S. Liou W.W. Liou C.J. Marek R. Mankbadi E.J. Mularz L.A. Povinelli D.R. Reddy L. Reid D. Rigby B. Rubinstein J. Schwab J. Scott A. Shabbir T.H. Shih J.S. Shuen F. Simon C. Steffen W.M. To C. Towne T. VanOverbeke C. Wang Z. Yang S.T. Yu N92-245i5

LeRC Title

S 9 2.----

Turbulence modelling in CFD: /(5 Present status, future prospects

Brian Launder UMIST, Manchester, UK

• To provide a (personal) ,flew of the status of turbulence modelling for use with the fully averaged equations of motion, energy, etc.

• To give concrete examples of what types of problem can/should be tackled with different levels of closure model

• Particular emphasis on applications in turbomachiner 7 and near-wall treatments

• Models considered: 4,

(3sotropic) Eddy V'Lccosity Models ('EVM) Algebraic Second-Moment Closures (ASM) Differential Second-Moment Closures (DSM)

No space to consider numerical strategies needed for non-EVM treatments

LcRC [8/91[. Eddy viscosity models - I: Physical Basis I 3

• Can be interpreted as an implication of the turbulent kinetic energy (k) equation for a simple shear flow {Tj¢(x=_ U z , U a : O) when production and dissipation of k are effectively in balance (Pk = e)

Main industrial interest is in applying turbulence models in conditions where these conditions are not satisfied!

Seem to perform worst in 2D curved flows and where body forces act in direction of primary velocity gradient

Compressibility effects on turbu]encc not adequately accounted for with an eddy-v_scositY stress-strain relation

Nevertheless models of this type are relatively easy to use and will be replaced only where demonstrably superior alternatives are available

PREGEDIN( PAGE BLANK NOT FILMED Eddy viscosity models - 17: Choice V • Available in versions requiring solution of 0-4 turbulent transport equations, of which one is (usually) that for k

• No extensive testing beyond 2--equation level

• Generalized statement of 2-equation model

"t - ctzk{Q } Dk _"- dk + Pk - _

! ka_b _t - dz + Cz' .._ _ ez2 zk,t + sz J z Most popular strategy takes _ - k al_lfi as second dependent variable mainly because S_ can be taken as zero in many simple flows

Need for no.___n-zero S_ becomes evident in separated and impinging flows to prevent excessive near-wall length scales developing

18,911 dyNear-Wall os,oStrategy e

EVM's rarely give satisfactory levels of uiu j away from wall vicinity: if Reynolds stresses are important there, second-moment closure is needed

• More difficult to devise suitable ASM's/DSM's for near-wall sublayer

• Hence most current research on EVM's concerned with treatment of this "low-Reynolds-number" region

• Log laws are generally inadequate .....

• ..... even a mixing-length scheme is better

One-equation models for sublayer currently seem a good compromise, especially if used with a "floating" r (the length-scale gradient)

18,1 yLow-Re osi,mok-_ models o I 6 • Devised by reference to 2--dimensional flows _ to lap_JL_ walls

Fairly satisfactory in predicting laminarization and diffusion--controlled transition

Return results of uncertain accuracy when used in 3D. separated or impinging flows or on curved walls

Need for about 40 nodes across sublayer means that computations at this level for 3D flows are only just feasible

• Further development work still required, guided by DNS data banks (Rodi, Mansour)

At present it is often better to use a one--equation EVM across sublayer blended to a two-equation model in fully turbulent region

8 Eddy viscosity models - V: Application

Problem: Flow through square duct rotating in orthogonaI mode

Relevant to: Internal cooling of turbine blades

Importance of both Coriolis and | , . buoyancy forces

111 Experimental data of Wagner et al (1989); computations 13o, Iacovides, Launder C1991)

3D parabolic code with 35 x 67 x 200 grid covering half cross section and 20 hydraulic diameters

• Standard k-, model in core matched to one-equation low-Re treatment across sublayer; a 0 = 0.9 in both regions

Nil 1.5 Nu o i.o f |,a "...... pressure.

suct Ion LS.

Lo tL4, $ Io ,$ Jo _ M

Satisfactory results for this very complex flow due to weak influence of force fields in turbulence equations and to predominant importance of sublayer region

Standard two-equation low-Re model gives far worse results than one-equation model

LeRC I Second-Moment Closure (SMC) .I 9

Modelling level based on approximated set of rate equations for Reynolds stres.ses and any other influential second moments (e.g. heat fluxes)

Convective transport of u i uj together with stress generation due to shear, buoyancy, CorioLis fortes etc. all handled exactly at this level

A modelling level intrinsically better able to cope with complex flo_s than EVM's

• Approximations needed for

Pressure-strain correlation, Dissipation, 'ij (and hence ._ Diffusion, dij LeRC Statusof SMC - ] The Basic Model

• A simple closure based on:

Rotta's linear return-to-isotropy concept for non-linear part of

Esotropization-of-production concept for linear ("rapid _) pans of _ij, Daly-Harlow generalized gradient diffusion hypothesis for dij.

Local isotropy lot" t_ equation used in model

has been extensively applied in 2-D and 3-D subsonic flows

Performance nearly always superior to k-c EVM - often markedly

Scheme now becoming available in many commercial codes (FLUENT, FLOW3D, PHOENICS, etc.)

Le.RC [ 8/91[ Status of SMC- I/ The Basic Model (cont'd)

Empirical extension of model to low-Re sublayer has been extensively applied by Shima (1989) and colleagues to laminarizing flows

This model apparently does not do well in high M boundary layers, however (Huang, personal communication)

• "Waft echo" part of _j performs quite incorrectly in impinging flows

• Performs rather poorly in free flo_ (round/plane jet "anomaly"; strong/weak shear flow *paradox")

Application of Basic Model - I The Faired LeRC _ Annular Diffmer (J°nes & Mariners, 1989)

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10 I 911 o, I 13 plane channel flow. Laun.der et al (1987"1 • Because secondary flow,s are ab_nL Coriotls effects on stress _Lo components is only agency provoking asymmetric flow • , , .+-, ,,

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LeRC 181911 Applieadon of Basic Model- It1 Axisyrnmetric Impinging Jet, Craft & Launder (1991)

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LeRC Application of Basic Model - IV 3D 2et Impingement. Ince & Le._hziner (1990) , 15

-II -41 -4 It I, II "_ = =12 i ------: - _llll _+_""_:i...... +.-.____i_--i _-i i _- .... <_ll;t ,......

--._+---- ,lil - -_+I__L-_BI I+_++._+_.-"++-:+ = ,

, _ _ _+ml -,-_,,,.+, • -+ -- + - ---._

11 LeRC I8/91 I Algebraic Stress Transport Hypothesis (ASTH) V

• _uitably approximating transport (convection and diffusion) of u i uj in terms of k transport, the closure becomes one where:

• _ equations are solved for u iuj's " a differential equation is solved for k

• This is what we mean by an ASM closure

• Technique is most powerful where transport terms are small ... i,e. in wall flows

• Most widely used ASTH's not coordinate frame invariant

• Properly invariant versions have been proposed (Ahmadi, ICASE) but do not so far seem to have been extensively tested in crucial flows

LeRC S-bend, Abou Haidar et al (1991) 7

L | -- A _L I_TTLLL_ I _ x

When transport is small, useful reduction in computational effort achieved while retaining virtually same results as DSM

Nature of ASTH is then unimportant

System of equations is stiffer than when DSM is used: convergence is often more difficult

ASM's are on their way out; not worth developing new software for this level of model

12 LeRC 1.,911 on,Me. I

• .Approximation of !Dij (and other processes) designed to comp]LE.___ cxtre.me stales of turbulence: e.g. isotropic turbulence, 2--component turbulence, ....

• Proper frame indifference

• Extensive use made of stress anisotropy invariants:

A 2 = aijaij ; A 3 - aikakjaji

aij - (u i uj - _ij UkUk)/k

• Extensive use made of results of direct numerical simulations

• Algebraically far more complex than basic model but greater numerical stability can lead to a reductio...___.._n in overall computer time

• Exlensive testing in homogeneous flows 0CASE, Stanford, Cornell, LeRC, UMIST)

• Moderate testing in 2D free shear flows

• UMIST model tested in 2D recirculating and colliding flows

• Far greater width of applicability demonstrated than with basic model e

• Only known test for swirling recireuladng flows not fully successful

• Dimensionless rates of spread of equilibrium free shear flows

plow Expt Modell_slc NewI_ode] [Plane jet 0.105-0.110 0.100 0,110 [Round Jet 0.095 0.105 0.098 IPlarue wake 0.098 0.078 0.098 [Plane plume 0.12 0.078 0.118 0.16-0.20 0.176 Mixlng layer 0.16

0.6 .i--._ -.-. Basic Model • Colliding Round Jets "0. S / "4 UMIST model ! ! _-, ooo Data Witze (1974)

0.0

--_'_ _ 0.1 0 I 2 3 t 5 6 ? e, 9

13 i+,,11 I 22 • More intricate interconnection among stress and strain components in mean--_train (rapid) part of _j

• Weakening of return-to-isotropy coefficient as stress field becomes more isotropic and also as it approaches 2--component limit

• Diminished effect of mean strain on evolution of (

Dt d_ + {._I- + _te aU. 2 _2

where c_t - 0.7 (versus 1.44)

c_2 - 1.92/(I + 1.65A_A) (versus 1.92)

A • 1 - _ (^2 - A3)

DNS data bank suggest _ij far less isotropic than usually °4._.__ , presumed; behaviour of e t 2 e_pecially strange 3

3 Peak level of c at wall

I Bradshaw et al (1987) have shown inhomogeneous effects on _ij very Important in buffer layer

• "Wail-reflection" models of ._ij. need to consider different constraints imposed by parallel and impinging wall flows and free-surface flows

near-wall turbulence • cl8,eij modelled ! to satisfy exact component ratios at wall l I,,

• Inhomogeneous effects on mean---strain part of _j accommodated through use of _ velocity gradient

2x_m + c I I_n _ _ Launder & Tseleptdakls (1991) OXkOXn ] t

New wall-reflection model designed to handle impinging and parallel shear flows (but, alas, no....!t free surface effects), Launder and Craft 0991)

Best way of placing _ maximum at wall is to solve transport equation for

f3k _l _ _" = _ - 2v I.x_J Kawamura (1991)

14 LeRC Application to plane channel flow Launder & Tselepidaki.'s (1991) I 25

Z@. c I - 0.3; 'wall-echo'effect

_2"/ u" Z 0 retained _tu, L6

I.l

O.@

2. e

0 U_?'u 1. @ % u_U 1.6

t.2

c I - 0.4; 'wall-echo' 0.8 effect dropped

x I

Application to turbulent impinging jet I._RC 26 Craft & Launder (1991)

• "_/Pm.., _0 -tO. 2. 50

0.0

°0.

-O.k 0.2 • .... w (-_ 'l=lU° .,76

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tl 8

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LeRC J8/911 .....Present status of new generation DSM's gl 27

• Substantial advances demonstrated over earller models for free shear flows

Significant unresolved (or incompletely resolved) problems remain in modelling near-wall turbulence that limit range of applicability of available models

Much remains to be done in high-speed flows: present suggestions for modelling extra terms and/or physical processes seem generally to be "quick futes"

• Many improvements foreseen over next 3-5 years

New-generation closures now being incorporated into general purpose 3D solvers

15 18,911 I 28 • EVM's, ASM's and DSM's will remain in use though with steady decline in importance of EVM's and ASM's in favour of DSM's

Improved versions of low-Re two-equation EVM's should lead to more reliable predictions of separated flows than at present

New-generation DSM closures will soon (2-3 years) replace basic model even in commercial codes

Further refinement of sub-models in second moment closures can be expected throughout this decade

Increasing attention to interfacing SMC with higher order approaches such as LES

Increasing use of two-time-scale schemes providing distinct time scales for large and (fairly) small eddies

Extensive collaborative testing/assessment of turbulence models currently underway - coordinated/organized by Professor P. Bradshaw (Stanford) (with a little help from .ILL and BEL)

Outcome of that exercise will offer a more complete and objective view of state of turbulence modelling than is currently available

If I

16 N 9 2 - 2 _5_6 '

Center for Modelino of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

Comment on: The current status of turbulence modeling in CFD and its future prospects

D. M. Bushnell

NASA Langley Research Center

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DISCUSSION

J. Bardina (to B.E. Launder)

I just have a question for Prof. Launder. You dismissed the algebraic Reynolds stress models real fast. To me a natural way to go from two equation models to higher models is to go through algebraic relations first. Do you think that the poor predictions are only due to the poor models and not related to the numerical instability issues.

B.E. Launder (reply)

It's a matter of taste and depends on the problem you are looking at. It's true that if one is thinking of it in that relation, the idea of using just the same two equation k - ¢ or k - w or more complicated stress-strain relation got some appeal. But nobody would suggest ASM is an improvement of physics over k - ¢. The problem you encounter in stiffer equations make it an unattractive level to fall to. It's beginning to pass. It could be that for a particular discrete set of problems it would make a lot of sense. On the overall if you can not use an eddy viscosity model, you should just bite the bullet and use the Reynolds stress model.

A.K. Singhal

I would like to make a comment about the use of nomenclature. You can notice that even the names used for the Reynolds stress models by the invited speakers were different. It would be very helpful for industry if modelers could use the same nomenclature.

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68 S.O -

4.5

4.0 JZ__ rO 0 -

3.5

3.0 8 / "_ 2.5 -

._ EXPERIMENT

'_o0 ,_'6o _ .o _sETT_E.s_ ( "_._0- _ DOLLING _ COMPUTATIONS, k - ( MODEL 1.5- ,_: v:___13

220 .I INTEGRATION TO WALL .5 1.0 j (a) 0 , [ I I [ I [ I ] I

3.0

p 2.5 /

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(b) _ t -1 1 I 1, l| J I 1 I I I I -12 -9 -6 -3 0 3 6 9 12 15 18

Fig. 17 Comparison of computations based on wall functions ulth the k-c model of turbulence, of a calculation based on the 3ones-Launder near-ua11 terms In this k-£ model, and of measurements for the two-dlJnenslonal, supersonic, 24 ° compression corner experiment: H = 2.84. (a) Surface-pressure distribution; (b) Skin-friction _oefficient distribution.

67 20 ° Plan_r Corner-

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N92-2451t_

Comment on : The Present State and Future Direction of

Eddy Viscosity Models

T. J. Coakley NASA-Ames Research Center Moffett Field_ CA 94035

Workshop on Engineering Turbulence Modeling August 21-227 1991

PRECEDiI'JG PInkiE bLANK NOT FfCJ_EG

79 Summary and Comments

Wall bounded flow._: The k -w model is probably best

Adverse pressure 5radient flows

Compressible fiat plate flows_ esp. with heat transfer

Roughness - blowing- transition

Simplicity - no damping functions

Numerical stability - eg leading edge start

But: Separation and reattaehment still are problems (all models)

Free Shear Flows: The k - e is probably best

k - w solutions depend on free stream to

k - e more corrections and improvements available

This presentation: Compressible-hypersonic flows (NASP)

General discussion of k- to model with corrections

Comparison of model prediction for flat plate flows k-to, q-to)

Comparison of model predictions for a separated ramp flow

8O Generalizations of the k-w Model

P-d-[= ptS- pkD - fl°pwk + [(p + akGT)k,i]j

,t,, p-g = _(,,s- C,pkD) Zp_ + [(_, + ¢_,_)_,j],_

2 #T = a*pk/w, S = (u_,j + ui, _- _&iu_,k)u_,j, D = uk,_

Baseline Model

/9"= 9/100, fl = 3/40, Q_* --19

O'L -- I t O'k = _7_ = ½_

Low Re Model (Transition)

oc, _" =fns(_), o'L =5/18

Compressible Dissip. (Compressible free shear layers) _, _"= f,_(V-_l_)

Compression Mod. (Compressible sepaxatio_t) v_ - 2.4, (dp_/dt= 0, _ = _/_)

Algebraic length scale (Reattachment heat transfer)

Vorti¢.ity length scale (Incompressible separation)

81 "--, ...."....

;' '--i"' ' " I "_' ' ' I ' ' '-r117 ' ' _ .... i .... i '"' ' ' L.... '_..... '. /t_ °, :, ." = !!i:

o o o o q

l0 i0 I ,,,....'"' .... "' ' : .

",,, I ,.

. _ _ o _ _ i0 i0 82 83 Hypersonic Cylinder-Flare 35 ° flare angle M = 7.05

, a ,", I _ _ _ I I J 'IV ' I ' ' ' '" - I I - I['l ,,

3O _ 0 experiments "",__ o - ; J-Lk-_,model /;_ _ __._...._ - " baseline !_f : - -- -- L-rood II_ ------L-rood + C-rood il i tl ...... 2 " L-=o,_+S-mo_ 117 10 - - - _9__' 0 -20 -10 0 10 20

s (cm)

3O o experiments J'L k-c model tI baseline L-rood x _" 20 .... L-rood + C-rood L-rood + S-rood I

0 10 ! surface heat transfer

0 -20 -10 0 !0 20

s (cm) 219--13

84 Hypersonic Cylinder-Flare 35 ° flare angle M = 7.05

3O o experiments k-e model baseline L-rnod .... L-rood + C-rood 20 L-rood + S-mod

10 surface pressure

0 -20 -10 0 10 2O

s (cm)

3O o experiments k_ model baseline L-rood £- 20 .... L-rood + O-rood ...... L-mod + S-mod

0 -20 -10 10 2O s (cm) 21e--14

85 Hypersonic Cylinder-Flare 35 ° flare angle M - 7.05

J I I I I j 4O

o experiments q-m model baseline L-mod L-rood + C.,-mod L-rood + S-mod

10 surface pressure

0 -20 -10 0 10 20

s (cm)

i j- --i- t---r-- I _, r- i , 0 , r.

30 - o experiments q-_ model baseline m \ I-_, el, L-rood x m P_o \, L-rood + C-mod m L-rood + S-mod ! P I/ "_ 0 10 - surface heat transfer

/.ll r""! i i i I. i ! ! i 0 __1 ! -20 .-10 0 10 20

s (cm) 210-15

86 N92-24520

_2 © 0 0

O" o_ !

! 0

,,2°o ,--4 Z _0 I 0 O_

87 Q

, ! ' i ' 0

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0 \ \ \ q 0

q o , I 0 0 0 0 0 d 0 0 d I'O +n

88 o o 0 It') I"--

_=_ II

0 d 0 LO

+ < Z

m 0 0 d I.D eq

o d o 0 B d -- 0

° !

1-1

rl o

I I

I I + .<

Z / • I m 1 I © I I tO C_

0 0 0 0 0 0 0 0

90 q o . I

o %

-F

o. 0

Z I ! I I I

• i I l I, i j ] n i J I I I , I l

q o o 0 o 0 0 c_ d c_ c_ C_l +fl

91 O d O

LO O (5 O II

Z ! ! I I I

• I I ' ' I I I , O 0 m I I I d + I O >_ I , I (N

Z Z .< O O (2 O

O I I iP O 0 ,i

92 0 0 l ' I I ' 0 I

! I

I ..e=- k_ _ ® S"! I 0 I 0 I 0 II I'O I t-- I

I I

I I © 0 / 0 + 0 I eq I

/ ! I

0 d © o

o d 0 0 0 0 O. O. _d -_ r4 N ,-- o +_

98 ,_ O_

2_ .<

_o_ __ _ <_ < <_

94 Center for Modelin 9 of Turbulence and Transition Workshop on Engineering Turbulence Modeling - 1991

DISCUSSION

D.M. Bushnell

I would like to make a comment pertaining to the problem of numerical resolution and numerical fidelity. People are showing all kinds of results with various models without keeping track of how well they are doing numerically. There should be, at some stage, in turbulence modeling community some agreement of some ? calculation with some standard code and then stick with that kind of quality and fidelity the whole way through. Without this I am not sure what I am looking at quitehonestly.

R. Mankabadi (to D. Wilcox)

I noticed Wilcox in his talk gave k -e an incomplete grade for the case of unsteady boundary layers. You may like to know that Howell (1980) and Ramaprian (1983) used k- e model to calculate unsteady boundary layer and found that k - e model could not predict this if the amplitude of oscillation or frequency is high.

D. Wilcox (to T.Chitsomboom)

I have a comment about the T. Chitsomboom's slide we just saw for Re = 1410. I have grid independent solutions for that so I'll be very suspect of these results.

T. Chitsomboom (reply)

I didn't mention, and you didn't either, that k - w model is quite sensitive to w boundary condition. Difficulty we encounter is how to specify w at the wall. Solutions we have shown are about in the middle range; we can get some better results than this and also some worse.

95 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modelin 9 - 1991

D. Wilcox (to T. Chitsomboom)

You have to be careful with the w boundary condition because if this is messed up then the boundary layer is all whacked up and I am sure that that is the case in this computation.

M.S. Anand (to D. Wilcox)

I like to preface my question by mentioning that we do pdf methods and for this purpose we need the time scale information. For this purpose we solve w equation (either a mean w equation or a stochastic w equation). In the limited calculations I have done, I have not noticed the sensitivity to free stream w you talked about. I have done calculations of single axial jets with or without co-flow; non-turbulent and very low turbulence co-flow. If there is sensitivity could you clarify what the sensitivity is duc to and shouldn't there be sensitivity to k - e models too ?

D. Wilcox (reply)

No sensitivity in k - e because it's just not there. You can vary freestrealn dissipation all over the place and get the same answer. In k -w model I am cheating a little in one regard. I am always doing the similarity solutions and not marching. There are only two values of w which satisfy this. I vary these a little bit in calculations. Neither of these values gives good spreading rate but these more or less bracket it. You are not seeing the sensitivity because you are calculating the free stream w. If you change the initial value of w to start your calculation you'll see it.

M.S. Anand (to D. Wilcox)

That's what I am doing.

96 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling - 1991

D. Wilcox (reply)

I'll look into numerics. I believe my similarity solutions are really good.

W.K. George

I would like to focus our attention away from the nifty gritty of solutions and focus on when should we expect eddy viscosity models to work. Tennekes and Lumely remind you that it was included in their book to show why eddy viscosity has problems. The fact that you have a local model, it can not handle separation or flow pass separation. Although Reynolds stress models are a straightforward increase in complexity but it's a quantum leap in adding physics.

B.E. Launder (to D. Wilcox and T. Gatski)

I would like to clarify a thing on Dave's and Tom's talk. Last time I saw the k -w model it seemed to me that the second equation had in it a supplementary source term. I see Dave nodding and Tom saying no.

D. Wilcox (to B.E. Launder)

There ain't none.

B.E. Launder (to D. Wilcox)

So it's cleaned up in the current marketed version. It is very interesting step. As I indicated in my talk if for separated flows one wants accurate results we need to use an extra source term in dissipation equation which will remove difficulty with adverse pressure gradients. It's not a problem of high Mach number. It is intrinsically a problem of separated flow; we get too big of a length scale near the wall and too high of heat transfer coefficient. And

97 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

if separation is provoked by a shock wave, the separated flpweould be zero Mach number or a seven Mach number. Survey shows when one shifted attention from adverse to favorable pressure gradients k- w did better in the adverse pressure gradient, and k - e predicted much better transition and re-laminarization_ Tom Coakley said it_you want to predict transition using k -0., 2 model you have to put Re effects which are absent.

D. Wilcox (to Spalart)

Several points were made that I should answer to. First, a complete model refers to terminology used by ?? a few years ago. It simply means you can use it with out knowing anything whatsoever about the flow, like an appropriate mixing length. Thus a two equation model is about as simple of a model as you can get,

P. Spalart

Why then isn't a one equation model complete?

D. Wilcox (to Spalart)

Because you still need to specify a length scale.

You also seemed alarmed that I was using Clauser's data to tune the k - w model. The perturbation method solution yields a similarity solution that demands that fit be constant. You must compare with data where /_T is constant. Whether or not this is a limited data set, it is the only way that is formally consistent with the perturbation solution. When I go to a non- similar solution, the perturbation solution results are certainly bourn out. That is why i use Clauser's data. It's dictated by a mathematical necessity.

98 Center .for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling - 1991

P. Spalart

How many of the moments of the turbulence satisfy this right kind of scaling? Scaling which is U,- equals a constant?

D. Wilcox

Well, I don't know but this shouldn't invalidate the analysis.

You were worried about whether the velocity profiles and the shear stress data came from the same calculation-absolutely! You're only talking about one data point out near the edge.

C. Speziale (to Spalart)

Are you saying that there is no destruction term if there's no wall? Essen- tially the destruction term disappears?

P. Spalart (reply)

Yes.

C. Speziale (to Spalart)

How then would you do in a non-equilibrium shear flow? According to this, the only thing that the eddy viscosity term can do is grow. I was thinking specifically of a situation where diffusion effects are small.

P. Spalart (reply)

It wouldn't work there.

99 m _ ,r_ _ -

N92-24521

The Present Status

and Future Direction of

Algebraic Reynolds Stress Models

Workshop on Engineering Turbulence Modeling

ICOMP, NASA Lewis Research Center

August 21-22, 1991

Dale B. Taulbee

University at Buffalo

P_CE'II)(AJG P_i._,ti_LANI( NO_" F'@L.I_'_ 101 OUTLINE

• The need for algebraic stress models - deficiencies of the linear gradient model.

Classical algebraic Reynolds stress models.

• Nonlinear stress-strain relations.

• Critique of the models and some new developments.

102 LINEAR GRADIENT MODEL

2 k _ij -- 211_ Sij uiuj - 3

lau t

._ - C. k l - C, k2/c

C_ = 0.09

• Works reasonably well for near-parallel shear flows, however, C, is not a constant.

• Poor representation of the normal stresses and, hence, does not work well for many multidimensional flouts.

103 0 0 0.02

i IJV

1I2 c

0.01 /o

.4{)

0 0 0 I I 1 I O. 1 r/x O. 2

Shear Stress in the Axisymmetric Jet B LDA Date Fit, -- Gradient Hypothesis

104 Normal Reynolds Stress- Round Jet

e°0_"

8,OO I'I o o 0 n o o LDA Data fit a 0.02 o u---_ OU 2 k l£ 2 o _ = - 2ut-_x + 3

8,06 0

fl o

8,84'

8o{_"

8°{_ "

8,81

8 I [ I I 8,1 8.2

105 VARIATION OF C.

MODEL PARAMETERS SHOULD NOT BE TAKEN AS UNIVERSAL CONSTANTS

k 2 u"-_ e og/Oy

.7 /

ROUND WAKE BEHINO /

ROUND JE[ [193 k .4 . PLANE JET Eli k c. \ \

-3 MIXING LAYER £2] _ _ , \ FiPE FLOW [6] .2

.1-

• VC._L

_ 1 -t- f 1 "1 _ 0 ./_ .g 17' l.G 2.

y/a FOR PIPE, Y/Yl/2 FOR JETS AND WAKES

RODI (1975)

106 DEFICIENCIES OF LINEAR GRADIENT MODEL

FOR ltiU j

• Poor Representation of Normal Stresses

o Homogeneous Shear Flow, U = U(y)

_= _- = _= _2k 3

dU uv = -r't dy

• Secondary.Flow in a Non-Circular Duct

Driven by (_ _ _)

o Linear Gradient Model Z'2 = W4 = ,),_k/3I

107 STAGNATION STREAMLINE TURBULENCE

[

I Voo

m u=uoo(1-R_/_ _) =

.,vh ere

model It _ _ •2k/3 - IYt0-_

91J

P=v, \Oz- _

I08 8.9

RSM 1 8.81 / k--E i 8,7 i / 0.6

.....I it/ /_ 8.5 k/ko / 8,4 / I / 8.3

_/(2ko) -- _--_-___

0 I I I l I I I I I -3 -2.6 -2.2 -1.8 -1,4 z/R

Stagnation Streamline Flo_ ;

109 ALGEBRAIC REYNOLDS STRESS MODELS

f (ltiU j," k, _, Sij) -- 0

More accurately describes the anisotropy of the turbulence.

Account for the effects of"

Longitudinal surface curvature

Corner Geometry Swirl

Buoyancy Rotation

Maintains the simplicity of a two-equation (k- e) model calculation

Basic Assumption-- Local Formulation

Reynolds stress depends only on local conditions - turbulence is anisotropic only if maintained by velocity gradient.

2 uiztj -k 5ij if Sij - 0 3 110 DERIVATION OF

ALGEBRAIC STRESS MODELS

Classical ASM

• Derived from modeled Reynolds stress equation

Nonlinear stress-strain models

• Derived from :

o Two-point closure theories

o Continuum mechanics

o Expansion of classical ASM formulation

111 CLASSICAL ASM

Algebraic Stress Model formulated from Mod- eled Reynolds Stress Equation (Rodi, 1972)

D_iuj O_ijk w -_ _Pi j -Jr- _) i j _- ( i j Dt Oxt

1) Convection and Diffusion Neglected

2) Convection- Diffusion proportioned to that of the kinetic energy equation

DuiujDt OTijlcgxl - uiujk ( DkDt cgzlOZl )

UiUj - _ (P-_)

uiuj

112 Algebraic Stress Model Continued

= Oz----_+ Pq + _q + ¢q

u_ujk "( P _ e ) = p_ + _ + e_

Rodi (1976)

#ij = C e - _y( - - -PSij ) 3 3

2 k 1-C2 Pij/e -2 -sP/_Sij = _ _J + 1 k C1 +_(P/¢- _) Shear Flow

u-ff = _k2 + _(14 - C2)-p/,e + C,C1/2 - 1 k3(OU)"£'5 -_y 2

--v 2 = -_k2 - -_2(1 - c_)-p/_ + c.C1/'2 - 1

2 2(1 -C2) P/e + C1/2 - 1 e 2

k 20U -u---_ =C. c Oy

2 C1/2- 1 + C2P/e_ C. = _(1- C2) ( P/_ + c_/2 - _ )_ 113 ALGEBRAIC STRESS MODEL CONTINUED

- c,T(_w - -Sksi_) - c:(P,j - P,_,j)

_ C3( DiJ _ 3PSij ) _ C4k ( OxjOUi F OUjOxi )

C_+8 C3 = 8C_- 2 30C_ - 2 11 11 , c4 = 55

Shear Flow Pope (1975)

2 4C _-7=2k _2 3 3 P/c -t- Ci/2 - 1

2 2 10C+I C#]g31(_UI 2

2 6C + 1 C_ w 2= k-gp/e_--C-i/-2_ 1 --_ -_y

-- -- C_

5 Ci/2- 1-g(llC 2-4C-1)P/C ( P/c + Ci/2 - 1 )2

C ..__ 1 + C 2 111(

114 ALGEBRAIC STRESS MODEL

Comparison with data for Homogeneous Shear Flow

Data from Harris, Graham, and Corrsin (1977)

Neglect Diffusion"

D uiuj uiuj Dk Dt k Dt

u _az--_j _j Ud_ dx k dx -j 1.96 2.09 V 2 0.88 0.83

W 2 1.32 1.24

%LV 0.62 0.62

Stress Values •

Ca = 1.8 C l = 1.5 C1 = 1.8

C2 = 0.6 c; = 0.4 c; : 0.54 Exp.

U2/k 1.195 1.050 0.965 1.004 v2/k 0.402 0.366 0.372 0.398 w2/k 0.402 0.580 0.662 0.598 -uv/k 0.356 0.393 0.305 0.298 C_ 0.068 0.075 0.058 0.057 115 CONVECTION-DIFFUSION ASSUMPTION

Duiuj OTijk D aij OTijl ui?lj OTl + -k Dt Oxl Dt Oxz k Oxl

uiuj + k (p-e)

where: UiUj 2 5ij aij -- k 3

1) OTijk UiUj OTl _ 0 Oxl k Ozl

2) D aij _ 0 Dt

Then the ASM represents the asymptotic solution of the modeled Reynolds stress equation. However, _- = k/c is the local flow value as determined from the k and e transport equation.

116 ' I I 0.4 I I I [ I I i I I "=1 I I I ! I I

O O RSM

U9 O9 (D 0 ASM l-4 UD ...... ASM (Pie = i) ;_ 0 f_ 0 CO) a p,,,_ <

I ! I I I I l I

0 10.0

ASM Prediction for Homogeneous Shear Flow

aijo = 0 - 0 " e dy o

117 ! I ! I I I I l i i _ ! l l I ! l l I ! 0.4 i

[ a?.LU

ASI_

...... As_. (P/_ = 1)

ASM Prediction for Homogeneous Shear Flow

aijo = 0 = 2.5 ' c dy o

118 SOLUTION WITH ASM FOR

MULTIDIMENSIONAL PROBLEMS

Solve set of nonlinear algebraic equations as part of the solution - causes numerical prob- lelTiS.

Simplify the ASM formulation

/ / o

o Neglect certain terms

0 Neglect certain gradient terms for specific problems

Expand into an explicit nonlinear stress-strain form (NLM)

lti_ZJ 2 k = f (T Sij; 7.2 Sij, ... ]

119 EXPLICIT STRESS-STRAIN RELATION

FROM ALGEBRAIC STRESS MODEL 2_ ;f

£ 2 -- -_PSij ) --

P uiujOUi u_uj k OU_ m E E OX i k E OXj

Expansion: ltiltJ 2 _ij -Jc- (1) (2) 7- 2 _ k -_ aij "7-+ aij +...

Ahmadi (1988) Rubinstein & Barten (1990)

Taulbee (1989) Horiuti (1990)

k -- -_ij2 _ 2c k sij ¢ ogj OUl '--"i j --d--- -_- '-C'j l 2 Slrn 5ij axl Oxz 3 Oxm

3 _'-_m

c_ and fl are directly determined Dom the constants of the

RSM. However, if only terms through second order are retained, the expanded formulation does not accurately represent the

original ASM. 12o NONLINEAR STRESS-STRESS MODELS

General form (Speziale, 1991)

aij -- -- 2 C_ 7- Sij

- 4 al "r2( Sil Szy 31 $2 (_ij)

-- 40_ 27"2( SiI _lj + _jl _li )

- 4 _3 __2(t2il _lj 3 )

-- 2 0_4 T 2 D Sij /Dt

v- = time scale (_- -- k/e)

Lumley (1969) Saffman (1974)

Yoshizawa (1984) DIA two-point closure theory Speziale (1987) continuum mechanics Ahmadi (1988) expansion of ASM Rubinstein & Barten (1990) RNG theory

121 NLM COEFFICIENT VALUES

O_1 O_2 C_3

Nisiziwa/Yoshizawa 0.090 0.0709 0.0159 0.0960

(Channel Flow) Speziale 0.090 -0.0138 0.0138 0

(Channel Flow) Rubinstein & Barten 0.0845 -0.0570 0.0120 -0.047

(RNG Theory) Rubinstein & Barten 0.090 -0.0523 0.0198 0

(ASM Expansion) Comparison with RSM 0.090 -0.0015 0.012 0

Homogeneous shear flow-

-_/(_-W)dU

c_1- c_3-- --23( all + a22 )/(TdU-3]])2

OL2 -- 4l(all (t22 ) 1(_)dU 2

Irrotational strain flow

( _-_ _)/ I d r: )

C_ 1 3(a_ + _._._)/(_)_- --,= 122 0.20 , , I , , , I , , , I , ,

-- Homogeneous Shear

--- Irrotational S[rain

0.15-

C# %%%%

0.10-

0.05 2.0 4.0 6.0 8.0 0 "rS

Stress-Strain Model Constants from RSM Solution

123 0.01,5

E_2

0.010-

0.005- NLM Coefficients from RSM Solution

o_...... _._......

o_1

-0.005f'' _ 1 ' ' ' i ' ' i ' ' ' i I 1 l 0 1.0 2.0 3.0 4.0 5.0 TS

Stress-Strain Model Constants from RSM Solution

124 I , , , I , , , I , , , I , , , I ) I

-ca

0 r_) NLM

.¢.a RSM \ O_ 0

.,-4 .4

_o.5onL, , , 1 , , , I , , , i , , , I ,-, , -

0 2.0 4.0 6.0 8.0 10.0 _dU/dy

Nonlinear Stress-Strain Model Solution for Homogeneous Shear Flow"

C,=0.09, c_1 =-0.0015, _2=0-012, a3 =0

125 z_

0.50 - | I | I ! I I 1 I I ! 1 I I

{D 0.25- ¢9

rglit/_ O

iI/J

O cD NLM og 0- RSM k

(/3

O oF--4 -0.25- <

-0.50

0 2.0 4.0 6.0 8.0 "t dUIdx

Nonlinear Stress-Strain Model Solution for Irrotational Strain Flo_,

C. = 0.09, c_1 = -0.0015, c_2 = 0.012, c_3 = 0

126 STRESS-STRAIN RELATION FROM

EXPANSION OF REYNOLDS STRESS MODEL

Reynolds Stress Equation :

Duiuj cgTij k _- Pij -_- _ ij 2 Dt Oxz

( UiUj 3

32p 6ij ) -- 2 ")'k Sij

C2+8 8C2 - 2 30C2 - 2 c_-- 11 , ¢?= 11 , 7 -- 55 A_

Dissipation equation :

Dc O_zlc' ]c 2 + C_k--P- C_:-- Dt Oxl

127 EXPANSION Continued

! i , : 2 : .

UiUj 2 k '7- _ aij -- ]¢, . _(_ij , ..... E

k OXl ( aij n t- -_Sij )OZl 1

.D aij ( CI - 1) aij 8 Sij 15 E 7- 7-

2 - (1 - c_- p)(a_fSfj + _j_X_ 3akISIkSij) - (1 - _ + Z)(_._q_j + _j_%_)

lOutk' 4-=-7 k Ozl

(1) (2)T2 Expansion • aij -- aij 7 + aij + ''.

Requires T( S_zSlk 1/2 be small

Transport terms lead to small second order contributions and are neglected. 128 EXPANSION Continued

• (1) ,, (2)__4_+ ...

where _-- (C_2- 1)- (C_ _ 1) P £

1S26ij ) aij -- -- 2Cff TSij - 4c_1T2( SikSkj 3

4/15 c1 + c,, - 2 + (2- co_)P/(

OL 1 ---_ C1 -_ 2Cc3 - 3 + (3- 2Cq)P/c (1- _+¢_)c. 0:2 --- 2(C1 nt- 2Ce3- 3 -1-(3- 2C_)P/c)

0{ 4 -- 2(C1 -I-- 20_ 3 - 3 -Jr-(3 - 2Ccl)P/6- )

129 " EXPANSION Continued •

• Same general form as nonlinear stress-strain relation.

• Model coefficients are determined from Reynoldg-stress model parameters.

• There isno (f_ilf_li5_26ij)I term. Possibly the model for the rapid part of the pressure strain term in the RSM is incomplete. Stress- strain models derived from DIA or RNG theory contain this term.

• Series is valid for small 7-S. Solution with this model, even with more terms, does not give good results near the asymptotic state in homogeneous shear or irrotational strain

m flows.

130 SOLUTION VALID FOR

SMALL AND LARGE kS/e

(C1 + C_1 T

- (1- _ - Z)(_.s_j + _s_, - sa_,S,_&j)2

For Dr 7- /

• Correct form for small r S

aij _(1) -- cLij -{- 7-

• Right side closely represents the asymptotic solution to the Reynolds stress equation, D aij / Dt - 0

Linear algebraic equations (Pie retained implicitly) solved by the method given by Pope (1975).

131 " ALGEBRAIC STRESS MODEL

IMPROVED:

C1 + c_, - 2 + (2 - c_,) _,j = -_s,j

/ 2 \ --(1 - o_ - fl) |ailSlj + ajlSii - -_ak-ISlkSij) 7" \ o ]

+ (1 - ,_+ p)(a_s + _s_.)

STANDARD:

C!- 1 + aij =-7..5"rSij

. --(1--o_--t_)(ailSlj+ajlSli---_aktSlkS_j)

+(1 - a, + fl)(ail(_2lj + ajt-Qli)w

132 NEW EXPLICIT ALGEBRAIC STRESS MODEL

For Two Dimensions

aij --

4g/15 1

1 cz + c,_ - 2 + (2 - c_)P/c

Model coefficients are determined in terms of the Reynolds-stress Model parameters.

133 . O° !

0 C) New ASM

o0 RSM -l-z r_q

0 o_ <

-0.3-

__ ......

. 1 • J 1 ! 2 • l -0.4 ' t ' ' ' i ' ' ' 1 i I 0 2.0 4.0 6.O 8.0 10.0

7k dU/dy

New Algebraic Stress Model Solution for Homogeneous Shear Flow

134 0.50 ! ! !

_0 0.25-

0 ASM 0

{/l O0 0-

.4.a 0 O0 o ,,,-I -0.2'5- .<

-0.50

_cdU/dx

New Algebraic Stress Model Solution for Irrotational Strain Flow

135 {).3

Standard ARSM New ARSM

C] Exp. Data Fit

e,2 \ \

0 0 0 0 0. t5 \\'x\

o o _ \\

8,1'

o o _ \\ o o o o

0 0 0 O_ o _ o o 0 o 0 0 ! I I G 8.1 8.2

Axisymmetric Jet- Kinetic Energy

136 0.G5

m Standard ARSM New ARSM C] Exp. Data Fit

/ \ / \ / \

/_ \\\

0.[_'

8,81 /o/ ° ° o o /D o o 0 0 I I 1 I 8.1 8.2 ?'/X

Axisymmetric Jet - Shear Stress

137 {).4

Standard ARSM New ARSM v] Exp. Data Fit \ \ \ \ \ \ 0,2 \ \

e,t ' -'-fl_ 0 0 0 0 0 0

I i I 8.1 6.2

Axisymmetric Jet - Eddy Viscosity Coefficient

138 O.B RSM / B.7 New ARSM ,/ / / 0,6 " .t/ ,/

/ B.4 /

0.3 _l(2ko) _..--

0.2

B.i

0 I I I I I I I I I -2.6 -2.2 -1.8 -1.4 z/R

Stagnation Streamline Flow

139 CONVECTIVE EFFECTS

WRITE EQUATIONS IN TERMS OF rS

1 where, r = k/e and S = (SklSlk) _

Daij ,, D (aii_ 1DTS Dt =_-S-_ +

1 D'r-S "r DS _1) P -_Y : s z)t _(c_ - 1)- (c,,

-- 2 "_ (2- C(1) --"_-P aij £

-t-(1 - c_ + fl)T(aitf_lj + ajl_li)

4/15 C/I 2 + (2 - C_,) _P + ._r DSvt C1 + C(2 - I[ - _

140 6.6 /

0.4" RSM New ARSM with Convective Term / 0.5 " / ,/ ..1

_,2" _/(2ko2 __._

O,l

0 i I i I i I i i i -2.6 -2,2 -1.8 -I,4 z/R

Stagnation Streamline Flow

141 SOLUTION FOR THREE DIMENSIONS

Pope (1975), Spencer & Rivlin (1959, 1960)

For aij - aij( Sij, f_ij ) - a(S, f_)

There are Ten Independent Symmetric Tensors.

T (z)= S

T (2) - S_- _S

T(3) _ $2 _ I{S2}/3

T(4) _ ft2_ I{ft2}/3 T(S) _ f'tS 2 _ S2ft

T (_) - ft2S- Sa 2- 2 I{Sa_}/3

o T(r) _ _Sft 2 _ Ft2Sft T(S) _ S_S = _ S2_S T(°)- a's_+ s_a '_- 2I{s"a'}/3 T(10) _ _tS2Ft _ _ ft2S2_

lO Then

n=l

Where G (_) can be functions of the invarients

{s_} {a=}, {s_}, (a=s}, {a's '}

142 3-D SOLUTION Continued

[CI + C_= -2+(2-Ce_ aij =-15 s --Sije

+ (1 -- c_ + _)-_(ailfllj + ajlflu)

k -- (1 -- c_ -- ;9)@(ailSlj -t- ajzSu - -gain2cozk6ij )

1 -- c_+/3 = (1 + 7C_,)/11

1 - _ - Z = (5 - 9C;)/ll small, neglect term

aij = E G(°)) rt--I k 2

1 I - _ [(I- c_+_)9_-]2f_ 2

1 I + [ [(I--c_+ _)g__] 2f/2 + [(i- _ +/3)gA] 4

(I -- _ -P _){I -- 2 [(I -- c_ + _)g_]2f_ 2}

1 4 15 1 + _ [(1 -- o: + _)g_]2_ 2 + [(1 -- o_+,@)g_] g = [C 1 2F 2C_2 - 2 J- (2 --Ce,)P/6] -1

143 CONCLUDING REMARKS

For simplicity and numerical solution purposes, explicit forms are highly desirable.

An Algebraic Reynolds Stress Model and its cor- responding Nonlinear Stress Model can be formulated which closely reproduces the Reynolds Stress Model solution as long as

1) The mean velocity field is not rapidly changing. 2) There are no boundary conditions or imposed flow conditions which give rise to strong non-local effects.

= Coefficients in the stress-strain models are not in general const.ant but depend on the strain field and the time parameter of the turbulence.

A good explicit stress-strain relation which represents the anisotropy of the turbulence should replace the linear relation now used in the k-c mod5l for practical applications.

144 t"q ><

<_oo

¢::10 _Z 0

145 14-6 _T_ i m c_ _<

r,D< r,D oz _____ =l

m _0 2:;

_ m z< z _u_o

•

_ D DD D D

147 148 0

N ° _-,,4

O0 Ch

N 0

C_ .I2 _o •_._ _

_ _- _

Or} J

_e T T r

o.-: _<

149 150 • u

151 >

o ii

,_ E

o U

_ b2

152 _- C 0

153 +

0 r" _; %-- o-- _j

_n ,,1''''!

i... ,...) N_ _ o _o _

"__1

-_,1,,,, I • _',',',',",'_ "z ...... ,;...... ,...... , _-_i...... i_ . ¢., t-. "_ II .__:_.__ "! / _, _+..+_ ._+'p+ <_ d .I I

O_ 2_

,-U o_

154

o o • I ' I ' I ' u_ ' I ' I ' I • if)

o o

| :1 i,.'1

o u3 0

0 o o o ,-., _

° ,r-,,,I v v o _0 0

_ 0 o o r_ i I ' I ' I ' u3 ' i ' I ' i ' 0 0 •_-_ _._ 4-J

o o _ 6 i,'I 0

o u3

0 _ 0 _

u . ! ._ ! o o 2< _ o o I ! I ! I I I I

._==(

v t_ v 4J T 0 o o_ o..-) _m

156 j_

Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

DISCUSSION

A. Demuren

I just want to comment on the k - e model. It appears the reason it performs so poorly is the value of epsilon at the wall. A very simple fix is to eliminate epsilon at the wall and use a simple mixing length. This works very well, and gives the right behavior in adverse pressure gradient, back facing step and separated flow, etc. It is quite an easy fix for the k - e model and yields decent results.

Ronald So

A comment about the compressible calculation with a k- c model. What we have found is that if you do the analysis correctly, you can actually predict compressible flow very well up to Mach 10. What Dave has shown up there about Cy vs. Mach number is not quite correct. You can get the prediction of the adiabatic and cool wall cases very well. We have used the baseline model and Sarkar's correction.

D.C. Wilcox (to B.E. Launder)

When you gave your talk this morning, you said that ASM suffered "frame- invariance". Could you comment on this?

B.E. Launder (reply)

It depends on what hypothesis you use to relate the convective transport of stress to the convective transport of strain. Work attributed to Rodi shows that you get a different answer if your fraane of reference is at rest or rotating at a constant angular velocity. You can devise a scheme, Ahmadi and Speziale have done so, that is frame invariant and Dale Taulbee was talking about these things. At the end of it though, you aren't going to get

157 = \

Cen_er for 7_fodelin 9 of Turbulence and Transition 14Zorkshop on Engineering Turbulence Modeling - 1991

a better model out of it.

D.C. Wilcox (to B.E. Launder)

Are you saying that if I have flow over a curved wall, if you forget to include the Coriolis and centrifugal forces as you go over it, that this is what messes it up?

B.E. Launder (reply)

Yes,

Something that Dale said towards the end of his talk he just slipped in there. You guessed that if you have important diffusive transport, then ASM won't work. There are many free flows where diffusive transport is very important. I just don't know of a good algebraic representation of it. My feeling is that if you haven't already got the software in place for ASM, then you should look beyond ASM for better answers.

D.B. Taulbee (reply)

How about all the people who have k - c programs sitting there. You can easily upgrade them by changing the explicit stress-strain relations. Not everyone has access to RSM's. They just can't buy them because they're too expensive. G. Huang (comment to D. Taulbee) ASM's are just as compficated to code as RSM's.

D. Bushnell

Brian Launder said it very well: if you have a situation that the physics is such that this ASM is fine then it may work. Under the NASP contract, we had a similar workshop about turbulence modelling about two and a half

158 Center for Modeling of Turbulc,ace and Transition Workshop on Engineering Turbulence Modelintl- 1991

years ago. What we asked was, "do you want the wrong answer very easily or the right answer?" In the NASP project, inside the scramjet combustors the flows are such that we need to go to RSM's to get the proper physics.

W. K. George

Since Dennis was free to paraphrase Brian, I feel free to paraphrase my colleague Dale Taulbee. If, for some reason, you don't have the resources to go to I_SM and the physics is bad for ASM, then things will be a hell of a lot worse for a k - e model and you shouldn't be using that either.

Also let me add that there is a lot of beating to death about the difference between 0.98 and 0.95 for the spreading rate of jets. It is probably absolutely impossible to determine this experimentally.

Let me comment on the emphasis on getting the constants in the Millikan formulation. If one goes back and looks at the original data and the compro- mises made in putting those constants there, numbers like 5.1 are an average of numbers that go from (0.5-20.0)! In fact the experiments just aren't that good. And the theory used to interpret them is not that good either.

159 T - 2: =_

E Center for Alodeling of 7}Lrbulence and Transition lVorkshop on Engineering Turbulence Modeling- 1991

Session II

Second Order Closure and PDF Method

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180 600.0 0 0 0 DNS data (C128W of Roger et al) Linear model (LRR) 400.0 Non-linear model (Fu & Launder) Non-linear model (Shih & Lumley) Non-linear model (Shih & Mansour) 200.0 ri (1) 11 0.0

-200.0

-400.0

6O0.O /

/ / i j

400.0 pl IpI /

1 _ /_

200.0 _p_l _ /_

_.. / 0.0 a._.L.0:_'_o _-"

o o--'--=,:_..-°----°-- _'-'---"""'_-_o -200.0

-400.0

0.0 5.0 10.0 15.0 20.0 25.0

Homogeneous Shear flow: Shear rate S = 56.57

181 600.0

400.0

200.0

0.0

-200.0 O o O DNS data (C128W of Roger et al) Linear model (LRR) Non-linear model (Fu & Launder) -400.0 Non-linear model (Shih & Lumley) Non-linear model (Shih ,_ Mansour)

0.0 5.0 10:0 15.0 20.0 25.0

. ._/," 600.0

400.0 J

200.0 t/-_"

0.0

-200.0

-400.0

J__L-U_--Ld__U_I I±LLLLu-u ' ' ' ' ' I " ' U-U--UJ ' ' ' ' ' '" ' 1' ,, , lJ o.o 5.0 io.o 15.o 20.0 25.0 St

Homogeneous Shear flow: Shear rate S = 56.57

182 D',

,.--.,

°_.._ _ "-8 E1 O

O 0 O 00 v-4 0 ./ 0 II 0 °°°."

°°°°.....°.oO.°.°°°'° 0 " - O O oo

CD [0 O. [0 0 tO _" {',l v {",l O P" LD C'_ O C_ O O O O O o o o ,5 ® o

°oX o S

O _"°"-..

O c5

tO O LD O LD

'_,.J v-'-q O O O _ 0 0 0 0 0 I I I

183 ed

0 d UO

0 d Z _ c n;sz 0 -I-

0 0 aO 0 0 d if) II 0 I- ,Iu 0 0

,-_Cq _ 0 kO 0 p'4 0 0 0 d o o I

0 d 0 uO

0 _aml 0 0 ! 0

0 d 140

0 0

_'_ 0 _ 0 "_ _ 0 0 o o d

184 • 0 _0

oo _-I

_, O0 0 C_ 0"_

o °r-4 0 m

0 0 r_

I _o 0 oO o(3 oO _

0 o.) 0

0 "-_

oO 0 _ _ -,--4 -F o oO 0 N m _ a r-_ C_ I LO C_ o_ 4-- 0 O_ 0 •,-_ _ _ 0 0 _1 _ 0

o (D 03 cr_ o,1 ,e._ c_ m 185 I% _ b. II

¢2 "u-'_ c_

Cq I + _2

"e'._ + I1 +

L----..-.J -¢.,_ I1 I "e-_ II °_ to Cq "v-_ I II I oO 0 Cq "¢._ II

C,l ÷

II ¢X) cq <_ II

cq ¢:_

187 oo

188 0.0

o _. -500.0 --.

-i000.0

-1500.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

2000.0

1000.0

0.0

o o o DNS data (C128W of Roger et al) Linear model (Rotta) -1000.0 Linear model (Lumley) i - - _i Non-linear model (S'a_kar & Speziale) I Non-linear model (Shih & Mansour)

Lililiililllllllllillli,lli_--_.b--LLLLLLJ..LLLLIJ-LLiXJJ

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Homogeneous Shear flow: Shear rate S = 56.57

189 2000.0 o o o DNS data (C128W of Roger et al) Linear model (Rotta) Linear model (Lumley) Non-linear model (Sakar ,_ Speziale) 1000.0 Non-linear model (Shih & Mansour)

0.0

-1000.0

LJ_L.I.JL.LLL-L,L I [IIIIIILL_ 0.0 5.0 10.0 15.0 20.0 25.0 30.0

0 0

20OO.Oi000.0 -

O.C

-1000.0

.0.0 5.0 10.0 15.0 20.0 25.0 30.0

Homogeneous Shear flow: Shear rate S = 56.57

190 0 c5

r-t

0 0 0 +

j 0 0 c5 00 tD II

0 e/" 0 ---__:_ u__uj Lu_u_u

C3 tO3 0 t_ C3 tO 0 b- u-O O_ 0 0,2 0 0 0 0 0 0 o d o o o o I

0 r..?

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0 d 0 + z ,r--I

0 0 I I d ! t_ 0 , I

0 I .... •" -),_ 0

LLLLLIIII

U"D 0 U'O 0 uO 0 0 0 _ o d o c5 o I I I

191 0

O d ._ -_-_ l.D

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! I-, 1-4 *_ " (D _ _ _ d O 4- n_z I>--, O O a0 0 .,.-I 0 . I.D I1 o °Oos O O

140 O U"D 0 O O O O O O I

r,.) O O t:I u'D O i w,-,,q

O O O

I O O d uO

11111!11 ! O

O ..-..e,':, _ O O I::zl d o d

192 O0 ,__

@ ÷

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193 -_.,_ It

-t- Jr- I._.I._ "_ i_ _ IS' -4- ._ ,--_ I o9 + + + ! 5" i -t- + :_ _ + i + +

-t-

°_,_ + _ _ i_l"_

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194 +

i% + cq

o_ J% r-_ L_ O0 II ÷ + +

L_ I II c_ ii

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195 . 4111

tF.._

196 7-0to.2 pn_._r_- _e,.,e. _a,_t /,,,brco_..t pl_.,..,¢ flo_

...... ,..=-°_ • ...,.° /* • I % --...... y.°o ,,_#r * I i ...... ,, x_" I

• "\ • "---'" J -_.o . ",, _ / 5A_bb,r

-75.0_5_ , , , ; " " ' i _ ' ' " _ -. 0.00 0.08 0.1 6 u.z_ f/z

0.004 ...;.-;'. \_\.:...... _.,:I.....-:.-'>-'_",.,_, .\ \\ --='---=z-z::-:-::-". .."Z//

- 1 5.0C-

T[f. ;r

-50.00- "..XC_

-45.00 " ' I " ; 0.043 0.08 0.16 0.24

r/z

Figure-7. Compazisonbetweenmodelsfor the pressure temperaLure-gradlenL correlal;ion and the buoyant plume experimenL. .... Launder (1975) model;--- Zeman and Lumley model (1976); .... Shih and Lumley (1985).model;---- Craft ¢1;. al. (1990) model; •-_ presenl; modeI;e experiment of Shabbir mad George (1990) 197 0 o,1 oo 0

• P,,,I

c_ b.O o oo ,-_

0 "-' ;>_ _ c_

• ,r'_l c_ b.O

I,---,I (D ri 0

198 C_

o,_ ÷ ÷

_z_I l_u 0,1

÷ oO t---

199 O0

200 0 {5 UO

0 0 0 +

0 d o LD 0 oO

0 II d

0 _ 0 uO Oa 0 Ca oe Cx2 ,--a 0 0 0 0 0 0I

AAA--

0 0 0 uO t_ .,:--,I oi oi 0

_ a.-_ _ 0 e J,--,I ooool 0 + I 0 I 0 O o/ °1' I 0 • I 0 0 I d I ! tO oo ° _/_'_ { 0

0 - ' 0 _LLLII I_LilllL12 0 LLLLm_Lu_' Lm_

Lf} O _fD O lid O _ 0 0 0 '-4 I I

VIHA--

201 0

0 ooo °°

0 0 aO

o o o o II

t" _ l_l_Lkl kJA_ LLIJ 0

AA[I--

0 0 •" _ _o_ 0 0 r_ ..-/ _ 0 d • ,o/°/ ,: _ _ 3 - 0 + 0 i i o,:1 i 0 0 d I...0

0 d cO 0 u.O 0 LO 0 _ 0 0 0 o o d o d o I I

R_ACA--

202 0 O d tD :o , 0 O 0 0 + i°1 o O "O 0 O

c5 '"... t.O > O. 0 II b- 0 LLI 1_I J I l _LLLIJ_LJ_3J 0 c5 i lie O LD O LO O O O O O O O O O O O o c5 o o o c5 o o 6 I I kG/ana p-- A(-I/AAAp--

O 0 O o o! eP,4 0 iio t.O

O 0 d d O + ep,,i 0 +

! o O 0 (:5 0 LD tD

C} 0 O 0 O LD O LD O tD O LID O O O LO C_ 0 0,2 O O O O Oa '-q O ,-I o c5 c5 o O O CD O I I

A(]/nlIA p- _RG,g_A p-

203 " rI! rII fi1 r_l _5 Vl

I + + + + II

l I i I

! • 0 _q

b II II II II II II

204 0.5

\, 0 Experiment \ (Nagano -Tagawa }

°_.__ o 0 ooo O O Predictions -0.5 O O Present Model O _ Daly-Hariow Hanjati_-Launder Corrr_ck et at. Dek eyser-Launder -1.0 ]11,1 I I , ! t,ltt I 1 t t ,,,,Y 10 102 y+ lO s

Predictions .-\ Present Model Oaly-Harlow \ HanjaliE-Launder \ O o Cormack et al. \ Dek eyser-Launder

0 // j 0 Experiment ", " (Nagano-Tagawa) k../

-1.0 _,,,l x , , , , ,,,l , , t , ,,,,l _ 10 I0 z y+ 10_

(b) v'u 'z

Fig. 5 Comparison of model results for triple velocity correlations in wall turbulence.

205 BUOYANT PLUME

..... I .... _ .... I ..... I ..... 1 .....

O EXPER/MENT OF SHABBIR & GEORGE (1987) MODEL OF NAGANO & TAGAWA (1989) 0'60 l

0 4O

O Ooooo

0,00"

0 0 0

-0.20- .... _C)_OF...... i ...... i.... I ..... | .....

0.00 0.04 0.08 O.12 O.I 8 0.20 0.24 r/z

2.00 -

1.50-

< TAGAWA (I989) __

I% 1:00- 0000000 <

0 0 0 0 0 0.50- 0 0 0 0 O0 0 000

000, ...... I ...... I ..... I ...... 0.00 0.06 O, 12 O. II] 0.24

206 BUOYANT PLUME

0.25- -_ .... ! ..... -I ..... I ..... I ..... I .....

O EXPERIMENT OF SHABBIR & GEORGE (1987) :

MODEL OF NAGANO & TAGAWA (1989) 0.20- 00000

0.15- 0 0

0 _ O0 o

0.10-

0.05-

0.00 0 O0 -0.05

0.04 0.08 O. 12 0.I6 0.20 0.24

I ..... ! ..... I,-.... J_ .... 1 .....

O EXPEKLMENT OF SHABB[R &: GEORGE (1967) 0-25 1 ..... MODEL OF NAGANO _: TAGAWA (1989) 0.20 -J

0.15- i o,o

0.05- O

0.00

0 0 -0.05- O0

-0.10-

0.00 0.04 0.08 0.12 0.16 020 024

207 1.7 | i i i !

x 100 "_ Bradshaw, el ol 'it '-'_,a= ,., z_ ,L Gutmork 8 Wygnaski .9F .8 7 .6 ..5 .4 .:3 .2 J 0 -0.6 -0.4 -0.2 0 2 .4 .6 .8 I 1.2

Y-Y5 DY

Fig. 3 Shear stress profiles in 2-D mixing layer. A: Bradshaw, et al []9], A: Gutmark & Wygnanski {2'1, --- Present model.

,O_*li_J_llli_Jilljl J,

.9 I U| I o ! Brodshaw,i I e! ol i .8 .O4 7/u ,o. Y .6 .0 3 .5 .O2

.2 .01

0 1 0 i .t I I I I i i -0.6 -0.4 -0.2 0 .I .2.3 .4 .5 .6 7 .8 .9 1 I.I1.2 -0.6 -0.4 -0.2 0 .2 .4 .6 ,8 I I.I

Y-Y.5 Y -I".5 DY DY

gig. 4 Normal stress profiles in 2-D mixing layer. O, A. V Pig;. 2 Mean velocity profile in 2-D mixing layer. Um_=: Castro[-'°] O, ak, T: Gutmark & Wygnanski [2q, The the free stream velocity, Ys: the position where U = ! lines represent the present model. 5Um,x. C): Bradshaw, et al {19], --: Present model.

208 .11 _ I I I i I ! I I I

_ U _

D c - -"Lf_ x x X 08 X X .07 .06, .05

4to _ Us .04' .03 .02 .01 0 0 •02 .04 .06 .08 .I .12 .14 .16 .18 2. 22 .24

Fig. 7 Normal stress u_/U_ profile in planar jet. Legend ms in Fig.5.

.II _ i i I I I I I I 0 .02 .04 .06 .08 .I .12 .14 .16 J8 .2 22 .24 I I .I .09 v- X - .Vo .08 U_ Fig. 5 Mean velocity profile in planar jet, Us tile cen- .07 ter line mean velocity. O: Bradbury 1_-'_1,A: tleskestadP-3], .o6! X: Gutmark & Wygnanski [2s], --: Present model. :35

.0 4 X 3 1 I I t I I I I I I I .03 Z_ 2.8 tl t.' 2.6 x 100 .02 2.4 l ol 2.2 .01 2 I I l I I I I 1 ,J oo .02.04 D6 .08 .I .12 .14 .16 .18 .2 .22 .24 1.8 1.6 1.4 Fig. 8 Normal stress v_/U_ profile in planar jet. Legend 1.2 as ill Fig.5. I .II I i _ i _ , , i J _ i , .8 .6 A \.o .09 w- .2 I I __ I | I I l 1 1 _ .08 Ug ,I I 0 0 -02 .04 .06 .08 .I .12 .14 .16 .18 .2 .22 .24 .07

135 Fig. 6 Shear stress profile in planar jet. Legend as in o4 l:ig.5. .06_ .03 .o2F z,-- _ "x,, .o,_ ""x °'_._x _ L 0 L__L..__L__j , I , 1 J _ L_ 0 -_02 .04 .06 .08 .I .12 .14 .16 .18 .2 .22 .24

Fig. 9 Normal stress w2/Us profile in planar jet. Leg- end ;L_ in FigS.

209 •IlF _ _ I i f i ! i i J .I - ,='- o9 .0 8 I _ _X X 0 X Xo .07 - x x

I .06 --: x, x I I | I ! I ! ! | .9 z,5- °° .04 .8 .03 - .7 .O2 .6 .01 .5

.4 .02 +04 .06 .08 el .12 .14 ,16 ol8 .2 ._2

.5 Fig. 12 Normal stress-_'/U_ profile in axisymmetric .2 jet Legend as in Fig.ll. o o o o .I .I I l I I I I I I I I 0 0 .02 .04 .06 .08 .I .12 .14 .16 .10 .2 .22 .09 v- }- .O8 Us X - .¥o .07 -

Fig. 10 Mean velocity U/Us profile in axis.vmmetric jet, .06 _ x t.'s: the centerline mean velocity, O: Abbiss et al [34], .05 X: Wygnanski & £iedler [-%], --: Present model. .04- o _ .03 - _

2 I F 1 I I I I I i =i .01 - x x-,,,,,,,,.,_ 1.8 o L-'--'Vt ,-+.× too 0 , , , ,_ , , , *, ,"I'---_ 1.6 0 •02 .04 .06 .08 .I .12 .14 .16 .18 .2 .1_2 1.4 Fig. 13 Normal stress v--f/U_ profile in &xis)'mmetri¢ 1.2 jet. Legend as in Fig.ll. I

.11I- _ l l t t I I i : .8 .6 .09 .4 .08 U_

.2 .07

I ! . .06 0o •02 .04 .06 .08 .I .12 .14 .16 J8 .2 .22 .05 .04 Fzg. 11 Shear stress uv/U_ profile in axisymmetric jet. .03 Q): Rodi [2_1, X: Wygnanski & Fiedler [2sl, A: Abbiss et .02 al,--: Present model. -- X X .01 0 +L I u I ! i L°X _ 0 02 .04 .06 .08 .I .12 .14 .16 .18 .2 .22

Fig. 14 Normal stress w'-/U_; profile in a-xisymmetric jet. Legend as in Fig.ll.

210 - t,,,t

0 0

,q o

b 0 0

Cb 0 o:O

o 0 i o o "_,°° _

o .d I • , ' 1 " " "r--m' | ' ' ' " d ,5 ,@ ,f_ °_ /.8

;.z.

_o '_ _ _. ._,_

"_o _,.. _: .. "_ _ -_ _

j _ f_ _ .- ,--

0 Q

_0 o

"= o= _ ._-.-_

,',o_ _ ,"R _-, =

_ _ o .__ _ _= _,

"e,/e

211 1.0-

0.8-

0.6-

0.4-

0.2- '_

& A

0.0- • J • , I I ' ' I o.o o.s J.o 1.5 2.o r r.se

Fig. 6 Mean temperature in heated round jet {A, Lockwood & Moneib (1980); o, Becker et al. (1967); --, present model].

C.2

.-,,,

0.!

A 0 0 0

0.0" ).0 0.5 I.O 1.5 2.0 rlr.so

Fig. 7 Square-root of temperature variance in heated round jet [A, Lockwood & Moneib (1980); o, Becker et al. (1967); --, present model].

212 ¢,-o _9 ©

• _--'t r_ (D

0 u_

_9 q9

J_ ¢.) 4--a

¢9 o_

r_ °_ b.O

o_'-t 0 0¢:_ _ ._ © a r_

0 _ O_

213 I J o_ _ .,-_ bO ._m I • _-'._1 • ,.--,I ,-_ o _

•"_ o 0 CO 0 .,--_ 0 o _ •_-_ C_ _ o_--'1 0_ -t- o {t,) _ 0

o ,.----I !

0 _ 0

o _: _ 0 r_ 0 _ °,.--_ o,--_

o,_'1 0 °_,_ _D _ -,_ 0 o 0

o1_.1 °_--'1

0 _ _ • 0

_ _ o _D 4.._ "-_ _

qP _ _ -,_ _ _ 0 o _ m N_o _ N'_o _ _ O_ _N o • • • _

214 • "-" 0 ©

215 0

c_ 0 ©

216 i

m - 0

| -

I ! I

_ 0

O cN 0 (Z) 0 • Q ° ° 0 0 0 0 I I

217 J

!. N9 2-

J.R. RISTORCELLh THESIS TITLE = =

Second order turbulence simulation of the 10./0 rotatinq, buoyant, recirculating convection

in the Czochralski crystal melt

1) PROBLEM STATEMENT

2OM for rotating turbulence

2) THE RAPID PRESSURE MODELS TESTED

LINEAR NONLINEAR

1) The IP model 3) Launder's nonlinear

2) The SSG model 4) Shih & Lumley's model

5) The 2DMFI model

3) THE COMPUTATIONAL PROBLEM

4) SHORTCOMINGS AND FUTURE WORK

PRECEDING PAGE BLANK NO]" FILMED

PRECEDING PAGE BLANK NO]" FILMED 219 THE CZOCHRALSKI PROBLEM

2 2 Res = _sR s/V Res < 105 Rec = c0cRc/V Rec < 105

Gr = gl3ATRc3/V2 Gr s 1012 Re = Grl/2 Re < 106

Pr = v/cz Pr ~ .01 Ar = H/Rc Ar ~ 1.

Bi = ecT4Rc/kAT Bi_4 A = AT/Tm A --'.05

Ma =-_,T_TRsPr/pv2 Ma < 105

1 _

Sb = Re2/Gr _<102

I 1 I # I I I I I H

I i i Ih_ V I R C

220 . THE COMPUTATIONAL PROBLEM

17 NONLINEAR COUPLED PDE's MEAN [ U, V, W, T ]

2OM [.uu.r2 r2r4 r2 uv..uw.IIr2 eu.]< oe> r2

[ _, _:e ]

24 NONLINEAR ALGEBRAIC EQUATIONS

<000>

<00u j> 3OM <(_UiUj>

17 INEQUALITIES (at least) , , dij = -

/ (1/2)

/ (1/2 )

221 _edn temoer_ture meA_ stre_mf;ne_

mean temper4ture _e&n stre4ml|nei

I_/ 1111

I//I

//ll l

Figure 7.5: Temperature, streamfunction, vorticity and angular momentum; Solution with Ristorceili and Lumley's rapid and buoyancy pressure models. Gr = 109 , Re = 3.16104 , Re, = 0.0, Rec = 0.0, Ro = cx_, Ma = 10 3, Pr = 0.01, Bi = 2.5, R,/I_ = 0.5, ar = 1.0, T = 80.

222 , twice turbulent R;net;c e_ergv

' 1" " " "" " " // E_It I/I, ! " " / .,;,,,__' i,'i!' , ! i'i)tL\ _))/IA_I-Jii_ '."..:

IliTli i. _ iiilll ! I . ltllllUll i. e .,iw#iil.-.ll I t. _ i I-i.o _ i • _ • i iNiIl-li iml i_ • • ll.¢lil.,llt i, iIi.i _ iv ,lial

H 0. 000

Figure 7.10: Twice turbulence energy, q2, and scalar variance, < 00 >. second invariant, II, and (i.j=3,3) component of the anisotropy tensor, b_, Solution with R.istorcelli and Lumley's rapid and buoyancy pressure models. Gr = 109, Re = 3.16 10 4, Re, = 0.0, Bee = 0.0, Ro = o_, Ma = 103, Pr = 0.01, Bi = 2.5: R,/R_ = 0.5. ar = 1.0, T = 80.

223 lw;ce turbulent k;net;c energy (tt>, tw;Ce temoer6ture vdr_Ance

L a H I

L I

\. \\ \

b3_ = [i second ;_v_r;ant of _ln;.JotPopy tensor" (..>l(Qa)-i/3. vertlcal _ni$Otrooy

H .646

Figure 7.7: Twice turbulence energy, q2, and scalar variance, < 00 >. second invariant, II, and (i.j=3,3) component of the anisotropy tensor, b33; Solution with the SSG linear rapid pressure model and Ristorcelli and Lumley's buoyancy pressure model. Or = 109, Re = 3.16 104 , Re, = 0.0, Rec = 0.0, Ro = oo, Ma = 103 , Pr = 0.01. Bi = 2.5, R,/I?_ = 0.5, ar = 1.0, T = 70.

224 225 FUTURE WORK / SHORTCOMINGS

1) WALL FUNCTIONS Physics: boundary layers and turbulence

Computations: grid dependence resolution of flow field

Low Re 2OM Models: too many nodes

2) RAPID ROTATION / STABLE STRATIFICATION inadequate parameterization of cascade

3) WHAT IS THE TIME DEPENDENCE ? Joint Realizability couples rapid models What does the averaging mean ? Long time scale "coherent" structures

4) COMPUTATIONAL STRATEGIES FDAs reflecting realizability

Include mean quantities in 3OM eq's FDAs with accurate time evolution

FDAs reflecting turbulent diffusion

226 CONTRIBUTIONS OF THE THESIS

EXPECTATIONS OF 2DMFI MODELS

1) Flows with strong body forces stable stratification rotation magnetic fields

2) Environmental shallow water flows industrial effluents

mixing between bodies of water

3) Quasi two-dimensional geophysical flows large scale ocean mixing regional atmospheric modeling (mesoscale variability)

4) Unsteady flows time scales > integral time scale unsteady separation large scale "coherent structures" ??

227 CONTRIBUTIONS OF THE THESIS

o) THE 2DMFI MODEL _

Models Xpri & Xpsir: f( b, b2, b3), f( b, b2, b3, )

Satisfies Realizability, Joint Realizability

Satisfies 2DMFI

Off-realizability corrections exact

Off-geostrophy corrections exact

Free parameters available

228 N9 2- 24 5_25 _ :_

COMMENTS ON THE PRESENT STATE OF SECOND-ORDER CLOSURE MODELS FOR INCOMPRESSIBLE FLOWS

Charles G. Speziale

ICASE, NASA Langley Research Center Hampton, VA 23665

Workshop on Engineering Turbulence Modeling

NASA Lewis Research Center

August 21 - 22, 1991

229 Second-order closure models account for history and

nonlocal effects of the mean velocity gradients on the

Reynolds stress tensor.

Turbulent flows involving"

• Body forces or curvature

• Reynolds stress relaxational effects i

• Counter-gradient transport i

are usually better described.

230

2-_ Y

homogeneous turbulent shear flow In a rotating frame

231 Rotating Sh, ear Flow

4.0

3.5 3.5 (b) LRR Model

3.0 3.0'

2.5 2.5 --.., "..-2.0 _2.0

f.5 1.5

1.0 1.0

./s ----o._o 0.5 0"5 I-

I 1 I O0 0.0 | .... L 0 2 4 6 8 0 2 4 6 8 t ° t"

d.O

3.5 (¢) LES ( Bardina et al. 198,_)

n/s ,, o.z6 3.0 I t I 2.5 ! ! ! ---, t 2.0 t / ] / 1.5 / , .- O/S -- 0

_ _,...._4_,_d"_ _

n/s -_-o.5o

0.0 I . I I 0 2 4 6 (t f°

232 E REYNOLDS STRESS TRANSPORT EQUATION

07"ij 07ij OUj O-_i -- _- 1-[ij Ot _- _k Oxk --Tik Oxk Tjk Oxk

oeijk

where

! ! Ou_Ou_ C--t/ Tij -- UiU j OXjOXj

lozs t _i _u

e ij-- 2l]OXk OXk -- _e_'3

Cijk --- _i_._k,_,t,,,!,, t + p'u_Sjk + p t UjSikt

_ 7-ii

233 . ISSUES IN SECOND-ORDER CLOSURE MODELING

Models for Ilij:

Typically, it is assumed that

Oqgk I-[iN = e.A_ij(b) + KMijke(b)Oxe where

1 Tij -- -_TkkSij (anisotropy tensor) 7-g_

These models have deficiencies in rotating homogeneous turbulent flows (Reynolds 1989 and Speziale, Sarkar and Gatski 1990).

234 For the return to isotropy problem in a rotating frame (with angular velocity ft), these models predict that the second and third invariants of bij are independent of ft in contradiction of DNS and RDT (Reynolds 1989).

0.08

0.06

0.04

I 0.02 0.00 ,:,LIIZIIIIIIIIIIIIIIZII;,I ,I III I;I,..

0.0 2.0 4.0 6.0 8.0 I0.0 12.0 _T

Fig. 2 Typical RDT solution for the rotation of initially anisotropic homogeneous turbulence (by T.S. Shih).

235 For rotating homogeneous shear flow in the unstable flow regime, these models predict that the growth rate A of the flow defined by

I_" _-' e At £ r" e At is symmetric about its most energetic value (Speziale,

Sarkar and Gatski 1990).

A _ AIH,-IX

1...... 1...... I f ! I

_/s

236 Consequently, if the most energetic state- where

A = Am_×- is placed at f_/S = 0.25, the model will exhibit similarity with respect to the Richardson number Ri =_ 2fl/S(1 - 2f_/S). This is in violation of RDT and LES results (Speziale, Sarkar and Gatski 1990 and Speziale and Mac Giolla

Mh uiris 19 8 9).

LES (Bardina et al 1983) 20

18 - Ri =0

0.8 n/s = 0.5

0.6

0.4

0.2

1 I l J 0.0 L 0 1 2 3 4 5 6

237 Speziale, Sarkar and G atski 1990 recently showed

that the general model is topologically equivalent to

the quadratic model

Ilij -- -Clgbij -t- C2c (bikbkj 3lII6_j)

+C3[(Sij + C4K(bik-_jk + bjk_ik

32bmnSmnSij) + CsK(bikWjk + bjkWik)

in plane homogeneous turbulent flows where

S_j -- l(O_/Oxj-2 + O_j/Ox_) and W_j --- 1 (O_i/OXj

-O_j/Oxi). Based on these ideas, the SSG model was developed.

238 The SSG model yields only modest improvements

on the Launder, Reece and Rodi model. Substantial

improvements will only come if II/j is taken to be a

nonlinear function of the mean velocity gradients.

Two possible approaches are:

(1) The eddy structure model of Reynolds (1990)

(2) Tensor dissipation models (Speziale, Raj and

G atski 1990).

239 NEAR WALL MODELS

We currently do not know how to properly integrate second-order closure models to a solid boundary! The major problem lies in the pressure strain correlation IIij. The commonly used near wall models for IIij have two major deficiencies:

(1) The ad hoc dependence of 1-Iij on the unit normal ni to the wall. This does not allow for the proper treatment of wall bounded flows with corners.

: =:

(2) Asymptotic consistency is satisfied through sin- gular differential equations; for example

02T12 // Oy 2 -- CI_-_T12 + O(Y 2) i for the near-wall behavior of _-12. This can cause problems in numerically recovering an asymptotically consistent solution.

240 Entirely new approaches are needed for the near wall modeling of Ilij!

We are at the end of the road for models of the form

Ogk

Ilij = e.Aij(b) -k I(.£4ijke.(b)_xg

with ad hoc near wall fixes.

241 NEEDED IMPROVEMENTS

• Models for IIij that are nonlinear in the mean

velocity gradients.

• Entirely new methods for the integration of

second-order closures to a solid boundary.

• Incorporation of directional information into

the turbulence length scale (possibly via an

integral tensor length scale).

242 Center for Modelin 9 of Turbulence and Transition Workshop on Engineerin 9 Turbulence Modeling- 1991

DISCUSSION

S. Sarkar (to T.-H. Shih)

I have a question to Dr. Shih about the slow term pressure strain correlation comparisons he showed. It seemed to me that Rotta and our model gave the same results. That was little surprising because the linear term coefficients were different in the two models. On top of that our model had a nonlinear term.

T.-H. Shih (reply)

The nonlinear term is very small. Linear term coefficient for LRR model is 1.5 and for your model is 1.7.

S. Sarkar (to T.-H. Shih)

We have a paper in Physics of Fluids in which we compare the two models and they are completely different.

T.-H. Shih (reply)

Your nonlinear term can also have opposite sign to linear terms thus giving results similar to LRR.

C.G. Speziaie (to T.-H. Shih)

You refered to SSG model as a linear model. There is a coefficient which goes as square root of second invariant and also a term which contains a production term multiplying the anisotropic tensor. In precise mathematical terms it is a quasi-linear model.

243 Center for ]_Yodeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

T.-H. Shih (reply)

If the coefficient is constant the model is linear. In SSG model the coefficient is a function of second invariant and a production term.

J.L. Lumley (to C.G. Speziale)

You refer to your models as being equivalent to all the other models but only in the equilibrium situation. These flows are never in equilibrium. Would you like to comment on that.

C.G. Speziale (reply)

Question is how drastic are the departures. Then there is this issue of cali- brating tile coefficients. My motivation for doing SSG model was that most of the calibration we do is from homogeneous plane flows near equilibrium state. Since all the models are collapsing to this degenerate form, my feeling was to calibrate the model at this sate and see tile differences. It seemed to be reasonable.

J.R. Ristorcelli (to C.G. Speziale)

I have been judging these models from the point of view of computability. SSG model doesn't compute very well. It does better in rotating situations then it does in the non-rotating situations. I imagine it would do well in in homogeneous shear flow situation from which it was calibrated. For me I built the principles of realizability in the computation and I can't compute the flow with SSG model.

C.G. Speziale (to J.R. Ristorcelli)

What happens?

244 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

J.R. Ristorcelli (reply)

I get correlation (coefficients) which are larger than unity or eigenvalues of matrices going to zero.

C.G. Speziale ( to J.R. Ristorcelli)

But no problems with k or e.

J.R. Ristorcelli (reply)

Well k is the sum of these eigenvalues.

C.G. Speziale (to J.R. Ristorcelli)

SSG model satisfies limited realizability. It does guarantee positive k and e.

T.B. Gatski (to J.R. Ristorcelli)

I did various calculations with homogeneous shear flows using some nonlinear models e.g. Shih-Lumley model. It was very difficult to use in homogeneous shear flow because it was very stiffening.

J.R. Ristoreelli (to T.B. Gatski)

What do you mean by stiffening?

T.B. Gatski (reply)

All the equations for these flows are ode's. You are using Runge Kutta

245 Center for J_4odeling of Turbulence and Transition Workshop on Engineerin # Turbulence Modelin 9- 1991

method and you need very very small steps. The only dilemma with making assessments of turbulence models where you have pde's is that you can not be sure unambiguously that there are no problems with the algorithm.

E. Reshotko (to J.R. Ristorcelli)

How do you know that these flows are turbulent

J.R. Ristorcel5 (reply)

A lot of experiments have been done to support this e.g. at AT&T. Also the Gr = 1012 and Re = 10 6 for these flows.

E. Reshotko (to J.R. Ristorcelli)

Will your equations with all the turbulence terms would give a laminar solution?

J.R. Ristorcelli (reply)

Turbulence Would decay indicating a return toward a laminar state.

B.E. Launder (to T.-H. Shill)

Did the channel flow rapid term comparisons you showed include the inhomogeneous part of the rapid term or the wall reflection effects?

T.-H. Shill (reply)

No wall reflection or inhomogeneous effects were included. From the comparisons may be we can see how to include the inhomogeneous effects.

246

-- = Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling - 1991

B.E. Launder (to J.R. Ristorcelli)

Regarding your choice of linear or nonfinear rapid term, it seems that the nature of inter-rinkage between the stress and dissipation equations is crucial in determining if you get a steady state or a periodic behavior.

J.R. Ristorcelli (reply)

Everything was same and just the rapid term model was changed.

B.E. Launder (to J.R. Ristorcelli)

Since some of the e equations you are using are not the ones advocated by the model originators so what you were seeing wasn't the effcct of just a change in the rapid term.

247 i_ iL | | / _92-24S__ _ 7

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Comment on : The Present State and Future Direction of/.__ Second-order Closure M ood Is for Compressible - Flow5 /.

J. R. Viegas NASA-Ames Research Center Moffett Field, CA 94035

P. G. Huang Eloret Inst., NASA-Ames Research Center Mofl'ett Field, CA 94035

Workshop on Engineering Turbulence Modclln$ August 21-22, 1991

ilia_ ___lli,i.'t h l.ltiI_hl, Lt PRECEDING PAGE BLAhlt( NOT FILMED 277 Outline

Viego_

• Opening comments

• An alternative RSE model

• Results f_om VHVRK (FRAME) and LS models

On shear layers- spread rate comparisons

• Results of compressibilitycorrections

Hua_g

• General comments on RSE models

• compressibilitycorrections

• On the l_w of the wall

, Concluding remarks

278 An alternative RSE model: FRAME model (VA_FRK, 1983)

* Developed in collaboration with colleagues from France (A second-order closure model for compressible turbulent boundary layers which is capable of predlc_ing shock boundary layer flows.)

* Builds upon pioneering incompressible second order model of LRR.

* Compressibility effects included by: * Introducing Favre Averaging Reintroducing non-zero divergence terms that w_re eliminated in original models * Accounting for non-zero mass weighted fluctuating velocity - compressibility te_ms.

* Near Wall offects included by using: _ near wall pressure-rats of strain term_ * wall damping of quadratic return to isot=opy (slow) terms and An the dissipation terms * Using the _avre averaged form of the HanJalic-Launder dissipation rate equation with some coefficient modifications to the near wall terms and by using wall damping in the destruction par_ of this equation.

* Uses total energy equation (including k)

* Uses "total" turbulent dissipation rate equation

Successfully applied to:

* Adiabatic flat plate, M-3, to develope model

t Supersonic expansion at M-1.76 - Dussauge

* Transonic shock-boundary layer interaction at M-1.36 - Delery

* Corner flow at M-3 - Settles

Oscillating boundary layer on flat plate at M-0 - Spalart -DNS

* Adiabatic and nonadiabatic flat plate to M-8 - Karman-Shoenherr

279 i

d_ ,1D 280

ALTERNATIVE CORRELATIONS OF SPREAD RATE

IMPACT PRESSURE THICKNESS

a(M,,)/G(O)

Papamoschou and Roshko PrroT PFR3_IJRE I_OFIU_

VORTICITY THICENESS

- i+ _/_

c.(_jlC.(O)

Bagdanoff

_OPlU_ TURBULENCE MODELS APPLIED

EDDY VISCOSITY (mass weighted voHoble=)

2 #==C_ p__ C_ = 0.09

TURBULENCE FIELD EQUATIONS

i (_),= + (_j_)_ = -(,.,.,--,"u_;,_=....j_ - _T + Dk + Ek

(_-),=+ (_r,_ o = -c,, _t,,,._=_j=,,, - c.=-_[ +D. + _,

E_, Ee _ Extro Compressibilify Terms

STANDARD k - e MODEL E_ = 0 and E,=0

5ARKAR El" AL. (SEHK) k - • MODEL E, = 0

ZEMAN k- ¢ MODEL E.=O

E,k=--Cd F(M_)_ ,.o-,=p[.

and F(M,) = O, if M,

Cd - 0.75 Mt ." = _/(1 + _)

283 EFFECT OF COMPRESSIBILITY CORRECTIONS ON PREDICTED FREE-SHEAR SPREAD RATES

1.2 L • ' ' I -' " I " "--' I ' IMPACT-PRESSURE THICKNESS GROWTH RATE 2.4 1.0 _ a

lid . 2.2 a D Q

m O A 0,8 A O =o O MI = 5.6 o.6- A O 4.2 A °°i Matched free stream dens_tfes o 0.4 4.8 Pl " P2 = 2116.8 psf O ml 5.4 Z TI = T= = 500 °R 0.2- i M= - 2.0 i 0.0_ . ! • L .... • 1 . = • • . • t ..... •

0.0 0.5 1.0 1.5

• ' _ ,-- , g VORTICITY THICKNESS GROWTH RATE

--11 O O i Q I [] 0.8 I A A O A i o A U

Model i 0.4 o, k <> " [] Standard - ¢ <3 A SEHK 0.2 ill O Zemon

0,0 |- • i i

0.0 0.5 1.0 1,5 2.O

284 q ,..: ,_ d c_ c:; o c_ (03_/_

285 m 0 o

mLu _Z Lu_ ee_

O 0 u_

0 0 IN !,-, •- c_ c_ c_ o d O W (o)"_/"o 11_ LU

286 Evolution of Turbulence Models

.f (

f /

DN8 & LB8

dlSslpallon I

_;dp/dx l p ' d u"l d x .._._____

287 Modellng

D -U:u". [2 t 3 ...... • • • - _ - dii + pij + _q - peq + Comtrees_,bihty terma

Incompressible Modelin_

• Turbulence diffusion_ d_

• o It. n_f_u_,u ,, j -h -1- _tl_. ''_,'i _L .,..,SI _'_|

: 8u.= O= _* 8a.u. . _0 [_.pT(2 k o=.-+} h _ • + o=a)]s j

• Pressure Interaction,/b ij = _j,1 + _ij,= + 'l_ij,,_

• Slow Term, _ij,1

• -elpebij • -_p_b,_ + _p_(b,kb_ - ]H&i)

• Fast Term, #ij,2

• Launder, Reece and RodJ, 1975

• Fu, Launder and Tselepidakis, 1987

* Cra£t, Fu, Launder a_ad Tselepidakis_ 1991

• Shih and Lumley, 1985

• SpeziMe, Sarkar and Oatski, 1990

288 Incompressible Modeling (continue)

• Wall-Reflection Term, _;i,_

• Hanjalie and Launder, 1976

• Gibson and Launder, 1978

, Dissipation rate, eq

• isotropic, = -_6qe, with constants in _ij functions of II and III (Launder and Shima, 1989 and Launder, 1900)

• anisotropic

Launder, 1976]

• models satisfying asymptotic near-wall behavior [Launder and Reynolds, 1983; So, 1991 and Shill, 1991]

• transport models model for eq[Kollmau, 1991]

• l%at Fluxes,

k_-?J_, n 8T

• ASM type heat-flux equation

• Transport heat-flux equations

289 _Compressible Modeling

• Pressure dilatation, p'u"h,_

. Gatski et al., 1991 (fl(Mt)ek and f2(M,)e)

• Zeman, 1991 (p'Z-equatlon)---

• Rubesln, 1990 (p'2-equatlon -l-polytroplc process)

• Taulbee and Vaa1Osdol_ 1991:(p,2_eq- uation + modeled

f*,"'_k,k Polsson solution)

• Fluctuation velocity average, -"_i

• Gatski et. al., 1991 (denslty-gxadient model)

• Zeman, 1991 (transport equation for _)

• Taulbee and VanOsdol, 1991 (transport equation for u.)

• Rubesin, 1991 (Constant total enthalpy + polytropic process)

• Dilatation dissipation , , = _. + _ _na _d=/(M,)_.

• Zeman, 1990

• Sarkax et. al., 1989

• Rubesin_ "tota/" e-transport equation.

• Rapid Compression Model, pL '_ = Constant (Reynold6, 1980; Morel and Mansour, 1982; Voung and Coakley, 1987; Coakley and Huang, 1991; Rubesin, 1990 and Zeman, 1991) 29o Some Remarks

* There are more models than what has been presented in the position paper. Some have been tested in many "real" flows with success.

• Compax/son o_ the models baaed only on simple homogenous- type flows may be m_sleading.

• Neax-wall modeling is still a chMlen_n_ problem for 2nd moment closure.

• Due to strong coupling sanong governing equations and the absence of numerical stabi]izin 6 turbulence viscosity in the mean-flow equations, the solutionof Reynolds stress equations requires specialattention.

• Currently,LRR, FRAME and Launder-Shima models have been implemented in a N-S code and comparison of models against real-flow experimental data is underway.

The use the total energy is necessary for hypersonic flow calculations: ET -- E -t- k, where E -- c_T ÷ ½_-_

DpE .... - P_ + pe D_

291 Hypersonic Cylinder-Flare 35° flare angle M - 7.05

3O 0 experiments k-e model o

¢'1 with k in E o X w/o k in £ o

0 '20 = 13. 0

10 surface pressure

=: _0 0° 0

-20 -I0 0 10 2O _=

s(cm)

4O F

0 experiments 3O k-co model with k in E w/o k in E O0 0 O0

0 0 0 10 3° surface heat transfer 0

0 -20 -10 0 10 2O

s(cm)

292 Some Remarks (continue)

• Wall flows _ The law of the wall is independent of the Math number if the comparison is made based on the Van Driest transformed variables.

• Models using e-equation produce lower Von Kar- man constant_ n.

• Model constants can be derived as functions of density gradients.

• k- w model is less sensitive to Mach number effects (coincidence ?) -- only for wall flows.

• Need more turbulent energy !l

• Dilatation dissipation concepts make the flow more_ "laminar" -- wrong direction.

• The new pressure dilatation model shown in the position paper also lowers _.

• Rubesin's total e-model approach goes into the right direction.

• Zeman's new pressure dilation model does an ex- ceUent job.

293 U + C

JL 0"I -.,4 C) C) Ln b = _=L:_ _i_ _ - I I I I I I I 'I j I-I--I ''l

---L C)

\ \ II \ 0 \ ===1=, 0 w --=r

0 \ _L II 0 01 0 ==..IN -- 0 I ' m m 0 I I 0 | 0 0 I I

I i I I I I i I i

294 0

0 0 0 0

-t-

\ \

295 Some Remarks (continue)

• Modeling of the e equation is still a challenging problem -- both for incompressible and compressible flOWS" Ex- perience has shown that a 2-equation-level model can be

used to improve the weakness of the e-equation.

compressible mixing layer and the other is the compressible law of the wall. Experience has shown that these two flows display completely different behavior. i

• Mizing la_ler _ As Mach number increases, the spreading rate decreases.

• All unmodified turbulence models fail to predict this behavior.

, This leads to models designed to increase the to- = tal dissipation rate as the turbulence Math n_- ber increases. F

296 Comment on : The Present State and Future Direction of Second-Order Closure Models for Compressible Flows

W. W. Liou

Center for Modeling of Turbulence and Transition ICOMP/NASA Lewis Research Center

Comment 1 : Turbulent Dissipation Rate - _e

--': .ll .ll -pc = -p( E, + co) , *, =

- Incompressible Models - 0.6 _ e, uk,k (in the position paper)

-r_} T".,a S",i + ... (Liou and Shih (1991))

• Preliminary analysis (Liou and Shih (1991)) has shown that the third order moments, r_} T",j S,i" , may be as important as the terms that being retained and may need to be modeled as well.

Comment 2 • Turbulent Mass Flux _-:Tui

Compressibility Effects: .., .., reduced spreading rate, ...

entrainment

• Turbulent mass flux terms in the Favre-averaged equations may have a fair amount of effect on the mean flow development if they are modeled more rig- orously, especially for wall-bounded flows.

297 i

g_ _m l r

L Center for Modeling af Turbulence and Transition Workshop on Engineerin 9 Turbulence Modeling- 1991

DISCUSSION

B. E. Launder (T.B. Gatski)

I must say that I feel preferably unpersuaded by the practice of adding, just multiplying the dissipation rate which comes out of the dissipation rate transport equation by a factor that is proportions to the Mach number. A concept of the dissipation equation is that it is really representing the spectral transfer rate of energy. It is looking at the large scales; the small scales are totally irrelevant. Admittedly, you won't get the right behavior with the so-called standard dissipation equation. But surely one must look at how to improve the transport equation rather than having a quick fix.

A comment on what John Viegas said earlier. He looks at two flows and believes you need a correction of different sign - wrong! I believe we need to look at twenty-two or on hundred and twenty-two. All we are looking at is the desperate sparsity of the compressible flow data base. That is why one can get away with these simplistic ideas. Just by having e, equal to the quantity one plus a function of Mach number, quantity times epsilon won't

work.

S. Sarkar (reply)

First, this was not meant to be the only compressible fix. And, as far as you're saying that the compressible dissipation should not add to the solenoidal dissipation, because after all our conventional wisdom is that this is a fine scale thing, therefore it shouldn't affect epsilon. This does not violate that. What you are saying is that this has an extra irrotational component which has large scales and small scales. It is not as if the compressibility is just changing the small scale end of the spectrum; what it is doing is creating an irrotational component that is both small and large scales. We are just choosing to look at the small scales because it's is simpler to do it.

_EDIPtG PACE BLAN1( NOT FILMED

299 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

G. Huang (S. Sarkar)

You can compose that anyway you like, but only for homogeneous and constant property flow.

S. Sarkar (reply)

Yes, absolutely.

= ie i

, ±

Z R

300 E /

- + ! i N9 2- 24 5 2_8_+zy_

THE PRESENT STATE AND

FUTURE DIRECTIONS OF

PDF METHODS

S. B. Pope

Comell University

301 WORKSHOP OBJECTIVE

"To discuss the present status and the future direction of various levels of engineering turbulence modeling related to CFD computations for propulsion"

Combustion is an essential part of propulsion

Discuss PDF methods for turbulent combustion

3O2 TURBULENT COMBUSTION MODELS

• Essential to integrate the development of:

turbulence model chemical kinetics numerical method

• Turbulent/combustion interactions

• Tractable thermochemistry

303 TURBULENT COMBUSTION MODELS IN USE IN INDUSTRY

Typically:

• k-_

equilibrium/mixing-limited combustion

finite-volume codes

304 IMPROVEMENTS SOUGHT

Finite-rate kinetics NOx, CO, soot extinction, ignition

Generality m beyond idealized premixed and diffusion flames

• PDF Methods can provide these improvements

305 PDF METHODS

Solve modelled evolution equation for a one- point joint pdf

i __(x,t) -- compositions, _=(q_l, _)2,.-.,_)c_} mass fractions, enthalpy

U(x,t) -- velocity

o_(x,t) turbulence frequency = e/k

Hierachy of PDF methods

U,_ - U, co,_

Non-linear reaction rates in closed form

3O6 COMPOSITION JPDF

Need turbulence model

(k-t or - E)

Gradient-diffusion model of turbulent transport of O__

• e.g.J.-Y. Chen et al.

- e 4-step reduced scheme for methane solve for jpdf of 5 compositions Masri/Bilger/Dibble piloted diffusion flame

307 VELOCITY-COMPOSITION JPDF

, etc. obtained from jpdf

Need _ equation (or equivalent)

All convective transport in closed form (no gradient-diffusion modelling)

Connection to Reynolds-stress models e.g. Haworth & E1 Tahry (GM) Anand et al. (Allison GT)

308 FLOW OVER A BACKWARD-FACING STEP

_3 Uref

Y "x

H - 0.0"/62 m W3/(W3-H)- 1.43 N 3 - 0.254 m Ure f - 0.196 mls

MEASUREMENTS" PRONCHICI( &: KLINE (1983) PDF CALCULATIONS" ANAND, POPE &: MONGIA (1990)

309 FLOW OVER A BACKWARD-FACING STEP MEAN AXIAL VELOCITY

x*= -B. 71 -B. 26 -0.01 0.47 3 S I 3.0

2.5

2.0: -r

] 0 I .e .e I .e

/U ref

310 FLOW OVER A BACKWARD-FACING STEP TRIPLE CORRELATIONS

xN- -0.7f -0.26 -0.01 0.47 3.5

3,0

2.S

2.0 I B

1.5

1,0 5 It, .0 I I ,I -2 -I 0 I -2 -I 0 f -2 -f 0 I -2 -f

103 /1Jr3ef

311 VELOCITY-FREQUENCY- COMPOSITION JPDF

Single, self-contained model equation

Describes distribution of _. e.g. plane mixing layer

312 PLANE MIXING LAYER MEAN VELOCITY PROFILE

I I

Data of Lang (1985)

I [ -0.05 0.0 0.05 y/x

313 JPDF CALCULATION OF THE PLANE MIXING LAYER

SCATTER PLOT: AXIAL VELOCIY vs. LATERAL POSITION

2.0

U + AU

1.0

-0.05 0.0 0.05 y+/x

314 JPDF CALCULATION OF THE PLANE MLXING LAYER

SCATTER PLOT: DISSIPATION vs. LATERAL POSITION

I0 £+

(_)maz 1

10 -1

10 -2

10 -3

10 -4 -0.05 0.0 0.05

y+/x

315 PRESENT STATE OF PDF METHODS

Much research and development work remains to be done, but:

Realistic finite-rate kinetics have been incorporated

Applications have been made to complex 2D and 3D flows

Accuracy--should be at least as good as a second-order closure

a16 THE FUTURE OF PDF METHODS

E1 Tahry & Haworth (General Motors):

"...in our opinion, the PDF method is the most appealing of the one-point statistical approaches for in-cylinder reacting flows. Applications to in-cylinder combustion can be expected within a few years."

• Correa (General Electric)"

"The prevalent k-e/assumed shape pdf closure model...must be improved upon or replaced before other quantities can be usefully predicted. An alternative is the Monte-Carlo/pdf approach; although well proven for fully-developed shear flows, this method needs to be adapted to pressure-dominated flow in complex geometries."

From Proposal for PDF research and development by Rolls Royce, SNECMA, MTU ..... to European Community:

...a joint velocity-composition pdf...method allows relatively complex chemistry to be simulated and also fully couples the turbulence with the chemistry. It seems the only way forward from the present position.

317 FUTURE DIRECTIONS OF PDF METHODS

• Improvements and extensions

o Applications to practical combustion devices

318 IMPROVEMENTS

1. Reduced kinetics m Maas & Pope (1991)

1 Mapping closures m Chen, Chen& Kraichnan (1989), Pope (1991), Gao (1991)

319 0 0 0

32O EXTENSIONS

Incorporate V__ in jpdf

Meyers & O'Brien 1981 Pope 1991, Gao & O'Brien 1991, Dopazo 1991

• Represent coupling between reaction and mixing

Contains information on:

jpdf of _ and Z (diffusion flames) and _ (premixed flames)

° Reconciles flamelet and non-flamelet approaches

321 NUMERICAL DEVELOPMENT FOR APPLICATIONS (WORK AT CORNELL)

• Reduced kinetics--automatic generation and tabulation procedures

2. Improved Monte Carlo/particle method

- second-order accurate in space and time - low statistical error

• General, robust pressure algorithm

322 TRANSITION FROM k-e to PDF

Industrial combustor codes:

- complex geometry--grid generation - models for other processesNsprays, soot, radiation - post-processing/integration in design procedures

Incorporate PDF methods within existing codes

Numerical method fundamentally different (particle method vs. finite-volume method)

323 4 STAGE TRANSITION

* Starting point: finite volume code for ,

, k, _,

= i • jpdf of U, _; discard U information (PDF method determines <__>, <9>" incorporate i reduced kinetics)

= 2. jpdf of U, (finite-volume code determines

and _)

= jpdf of U, m, (finite-volume code determines

)

do jpdf of U, m, _)--self-contained particle method

324 FIRST STAGE

finite-volume Monte Carlo/ code particle method

, k, e jpdf of (U),

_ ,k,_ /_

essentially jpdf of

but simple transition (2nd stage) to

jpdf of , _ (avoids gradient-diffusion modelling)

• reduced kinetics can be incorporated

325 CONCLUSIONS

Turbulent combustion modelling: need to integrate

turbulence model chemical kinetics numerical method

PDF methods

m _; u, _; u, co,0__ reaction and convective transport in closed form finite-rate kinetic effects

Future model development:

mapping closures reduced kinetics

Future numerical developments:

more accurate particle methods general pressure algorithm incorporation in combustor codes

326 /

COMMENTS ON

THE PRESENT STATE

FUTURE DIRECTIONS OF

PDF METHODS

E.E. O'Brien

SUNY at Stony Brook

My first comment on the presentation of S.B. Pope is to note that Professor Pope is almost single-handedly responsible for the development of the one-point PDF method to the state in which it can now be reasonably expected to address actual engineering problems.

My second comment is that I am in accord with virtually all of the points he has made including his first, which was to express surprise that a conference on "modeling related to CFD computations for propulsion" should be so thin on combustion modeling. The PDF method he reviewed is relatively complicated, but it appears to be the only format available to handle the nonlinear stochastic difficulties caused by typical reaction kinetics. Turbulence modeling, if it is to play a central role in combustion modeling, as it must, has to be integrated with the chemistry in a way which produces accurate numerical solutions to combustion problems. It is questionable whether the development of turbulent models in isolation from the peculiar statistics of reactant concentrations is a fruitful line of development as far as propulsion is concerned.

There are three issues, two mentioned by S.B. Pope, for which I have prepared additional outlines which are appended to this note. a. The one-point PDF method b. The amplitude mapping closure c. A hybrid stategy for replacing a full two-point PDF treatment of reacting flows by a single-point PDF and correlation (& cross-correlation) functions.

Finally, I would like to appeal for a concerted effort to obtain an adequate data base for compressible flow with reactions for Mach numbers of unity or higher. DNS results have played an important role in aiding the development of PDF models for incompressible flows. A similar role can be played in the efforts to elucidate the many interactions of pressure with other flow variables including species concentrations.

327 PDF Method Outline

From 1) N-S eqn. 2) Energy eqn. 3) Equation of State 4) Species conservation eqn.

Generate an evolution equation for the 1-point PDF (T.S. Lundgren, 1957; C. Dopazo, 1990)

Close the PDF equation where necessary by 'suitable' closures

Use Monte Carlo/particle methods for numerical solution (S. Pope, 1981 +)

Major advantage ÷ linear increase of numerical effort with the number of dimensions

328 _: A 1-.__PointPDF E_uation

single species _(x,t); Statistically homogeneous system

a_=-u. V_ + DV2_p+_(dp) at: -

1-point PDF equation

is expected value of scalar dissipation conditioned by the • DE{ (V(_)21(_} scalar value.

329 General 1-Point.PDF Equation

0 Easily generated, and can include multispecies inhomogeneity, compressibility, etc.

,_pN,_,#;X,_ t) ...... (1)

Closed terms : advection and reaction

unclosed terms • pressure, molecular diffusion of all quantities in (1)

Closure strategies : Satisfy PDF realizability and

a) Reproduce second-order moment closures for physical space terms (Pope, 1985)

b) Approximate density effect by ignoring the coupling with pressure oscillation phenomona, i.e. p = P(_I, "--'_N, Href)

c) Represent molecular diffusion in velocity- composition - enthalpy space by models. Most recently mapping closures.

330 The Amplitude Mapping Closure

(Chen, Chen, Kraichnan 1989; Gao, 1991)

• Attractive for strongly non-Gaussian processes

Simplest example: Non-reacting single scalar 4,(x,t) in statistically homogeneous system.

• 1-Pt PDF

SP($, t) _ S2 cgt _2 [DE{ (V_)) 2]_, t}P($, t) ]

Define 8(Z) time-independent, homogeneous, isotropic, normalized Gaussian r.v. All statistics of 8(z) are known if

0=0,02=1 and f8 (r) :O (z) 0 (z+r)

are given.

Define a scalar field _'(x ,t) generated from 0(z) from the mapping

x=zlJ(t)

and _S(x, t) = X(O(z) , t)

• Demand P(_, t) " p($S, t)

331 Consequences of the Mappinq

Since statistics of O(z) are completely known the statistics of 4_S(x,t) are also completely known if J(t) and X are specified.

N & S condition for p ((_, t) = p ((_s, c) is E{(V_)_15,t}-_ (V_s)_15s, t}

It turns out that substitution of the mapping into the PDF equation produces a solvable equation for X aX _ 8X 82X a.: __ +'--_a4,

DJ 2 (t) dr. is a normalized time scale d_: -

Note: J(t) & Xe (the only parameter of fo that matters)

Appear _OOJ_in the time scale

.', the shape of P((_, _) depends only on the mapping X(8,t).

332 Some Results of the Mapping

@ Symmetric binary mixing (initial double delta PDF) P(@ o) = !8 (,I,+I)+-"8 (@-i) ' 2 2" soln: _(V_)2I_,T}/F(1) = exp{-2 [erf-1(@) ]2}

Unsymmetric binary mixing

P(_,O = a6(_+l) + (l-a)8(_-i), O

Same mapping closure solution for E{ (V@) 2 j_, _}/F ('_)

• General soln. has been obtained (Gao, 1991)

Current status: Formal solutions have been obtained for multispecies cases (Gao & O'Brlen, 1991) But, no reported success in incorporating it in numerical codes for more than one species Also seems to misrepresent asymptotic behavior in time

333 2-Point PDF

Advantages

Spatial structures explicitly included

Self-contained time and length scales as in spectral description of turbulence and unlike 1-Point PDF & K-e or other moment closures

Disadvantages

• Dimensions doubled

• Closures harder to construct

Numerical work so far limited to isothermal reactions of type A+ B--P.

Closure approximations Advection EDQNM Diffusion Linear Mean Square Estimate Reaction None needed

New Wrinkles

• Use 1-point joint PDF of quantities and their gradients

V ;x, :)

• Hybrid closures

334 Hybrid Stra_oeoy

• Aimed at more than 2 species, statistically homogeneous.

• Numerical method is fractional steps (Yanenko, 1971)

ioeo P(C+At)'(I+OAAt) (I+OD_t) (I+Oant) P(t) I is the identity operator

• Replaces the full 2-point PDF method by a correlation function - 1 point PDF approach..

In a cycle of computations in both composition & physical space

a) Advection has no effect on 1-point PDF but it modifies the correlation and cross-correlation functions f(r,t) [EDQNM]

b) Molecular diffusion modifies the 1-point PDF (LMSE or mapping closure, if workable and the correlation functions (known)

c) Chemical reaction effects the 1-point PDF (known) and, inadvertantly, may alter the correlation functions (assume similarity)

Reproduces full 2-point PDF results for

A+B-P

335 f

_ " " " OC-_'o Z I N92- 2_539_./¢ -- Workshop on Engineering Turbulence Modeling

Comments on Pdf Mett_ods

J.-Y. Chert Combustion Research Facility Sandia National Labs Livermore , CA

PRECEDING PAGE BLANK NO]" FILMED337 Grand Challenge of Combustion Engineering

I IT[] I II ii| ..... I I I

Challenge: Significant Reduction of COrn ustion Generated Pollutants

Facts:

[] Pollutant Formation Rate << Fuel Oxidation Rate

[] Small Quantity

[] Highly Sensitive to Interactions between Turbulence and Chemistry

Difficulty;

Not Capable of Solving Navier-Stokes Equations with Detailed Chemistry

Approach:

"Rational" Modeling

338 Research of PDF Methods at Sandia

Simple Geometry (Parabolic Flow)

Reduced Reaction Mechanisms: . Two-step H2 Flames . Three-step COH2 Flames . Four-step CH4 Flames - Five-step CH30H Flames - up to six reactive scalars

Thermal NO Formation in Turbulent Hydrogen Jet Flames

Soot Foramtion in C2H4 flames

339 Experiments of Turbulent Jet Flames (Masri & Dibble, 1988) -- - " I II _ . il _ . _ I I I I I | I (Fuel Jet: 45%CO/15%H2/40%N2) (Pilot Jet: 70%CO/30%H2)

- / / A,,X_,Mer_c

,tor._ 7

70 %CO/30 %H2,_ c_ow_ _ / "_ I! '_

45 % CO/15 % H2/40 % N2 340 Departures From Clremical Equilibrium

,, __ . . . ., r, n --' i . _ "'- " --'_--_

Hydrogen Methanol

40 _11 4k4 Ilvd Idle4 ¢_¢1ie_ !

Carbon Monoxide Methane

N4_,o T ! . cH,-_P._t: / / 2., _tO ,,O..tJ=,,.M I

_.o 21 ':

6J _ e.o o.J O.4 44 0.4 1.9 Mizf_e ._c_. f

341 Modeling Turbulent Reacting Flows

Im n n r St ochastnc" Simul a ti on of The Effects of Turbulence on Chemical Reactions

ReducedMechanismsReaction )

l II I .

! i

342 Superequilibrium OH Radical i - i . t r _2

Measurements Predictions

O.OrS o.O_S - • . :..,"

• • % • = C • ¢,/.. ,p . = t'3 Cese a. xlO=30 _,_.. :_ • <":.a._eR, xlO=30 _:_". ._.:.,_" %.%"• . • . (: .1"% q-q..t.= . * .¢2. a=r • * * . LL f.) U • ..:_.;:...:_2'..--...- q0 " ":'4 _.._. 4.,-,. o • "r_rJ,-. "_ - .¢ _ " "6 | 0 - __,_-. _:_,'_-_. -- , t 0005 - 2: : •._.,-_..:,,,_,_:.-... o .....,_._-_=. _ \',,_,....,.-..,"._ • .. •.j.* , .. U ..-,,_. ./ -_ .. _'_ ...... 6) .. --, / _':-'t_.,,,_.,_ - . • , .,;.-l_r. I \ "'.-3""":.'.-'" " • "'

"_---_.__:_ _ : • • ._t, • 0.000 _2_2"," _'" ":"""'":" OOOO 0.0 0.I 0.2 0 3 0 4 O5 0.0 o I 0.2 o 3 0.4 Or.,

00_5 0 ol,5

" " Case A, x/D-30 0 Case A. x/l::)_30 .._, ' ,:-;',_'.',,.,,,.e. o - • 0 0,0 u o00 U -., ,) ,,,, , in • ' .,_._..-'_ , ' ;_.,-;.,. ; __,.,_'._._.,. . L.'\'-" ,,; ip T :: _'.- .. -; . ....-::;-_;,:._.-.,,._.-,', o 3: 0 005 0 005 o • . • ,', A _/_: <'."i, , °_e • " - . • ._'_ ".i, l_:,:r.'_* ; . u_ • ,," I "_T. ;..-'_.,'_'J;. " •...... -../ .':\..;_._='...,,.;, ..... : . T'I .':W \_-,-. ...r _ _ I • _"_-" -._"":'..'.*'" " " ". •:';-/ \ :::'::,, • "_,' ; / \"_"',_i-_ " ._. ", ' • • ,, .'_ \ "'- _,_ _.L./ _ .... _.. o o6o 0 000 O0 01 02 03 04 0.,5 O0 o_ 0.2 03 04 o$

o 015 o 0_$

IE 0 Ca,se A, xlO=50 Case A. xlO=50 u c: 0 O OIO ,o .. (# °

2_ ""_" ¢ "":*'t-'." 0.005

0 ..,.,., !_,','.. • " .*_ °* ._ -_-_.. . • ..'*, '-

...." 0.000 o o00 o.0 O_ 02 (_3 05 v,o nl 02 03 n41 0_,

M_xtufc [:tact,On. _"

343 S-,-_-- ! __

0 + cO o LL

•" ° • u" -v + m.lnannlq (D I,..

_4 .4 _\@+ _ ,

r--', i . ¢',,& .._., _- o

F.., _'UIAI@>]/J_

w_ o

,4..,a e_ ,4., 0

'"aUIA IO_/J_

344 Scatter Plot for CH4 and 02 (CH4 Turbulent Jet Flames)

]LQ ----

I_\\ _-_ '_-'_"=' \.% f.Z3,i.,17, 1.1_5 .,... 0.o t- O °_ +°'FI!\ ++- (.} lilI:'_:; '\\

+ i 0.4- "... ,-; ;. " • " "% , -i._:::.: " \,,,, I iil /'" " \ >- ...:,';._',._.;., .-, '- °' i!;}_.-+i;i:_'.-- "\ 0.2 i -";'_.::: :..".: _..

: _ _ . .--;_...-_:_-'.,_._,_.._: \.. 0.0 " ( " r • r+r'_-"--'T'_"'+ +'f-_+-_"" o.oo o.o4 o.oo o.lz o.la o.=o 0.2, o.oo 0.04 o.oe o.12 o io 0.20 0,2_, Y02 (M,_._SFre_;tion) Y03 (M_s_ Fr¢ctlon)

|.0 "1-7, Ci44 a;¢ ;lame L: !.O [+, _\\., ,Jo_o. ,,o-l.=e.,.eo i_ " xtO-20. (/O-I.2"f. 1.44 1.6e _. O ff "_;'.'_,\ "% (P & "_ 4-16¢,dp) _+ . % o= 0 (_fill _ + ,, {P & S 4-step)

I_ 0.0 '• '-

:_ . : - ,_.., ".'o . %.

r +ill.-! ' * '- '_-:,;i;,'._ + " i . ::.-:'_.+.. .. >- 02. '_:_.''. - " • _ ,_ .,:-+n.'; '., .+ • ,

: . '_- ..'. i-'.';_,..','." . o.o "-' ,_ . +, ',- ,.... :'-_;"_...,-_ o.oo o.o_ oo0 o.t_ o.16 o.:to o._,_ 0+00 0+04 0,0(_ 0.12 O.1_ 0.20 0.2_ YO2 (Mass Fraction} Y02 (Mass Ftactlort)

1.0 |.O .-'' F J_"", CH4-11fi" Fl_lrne L: J._\ "-. x/0.20 #[3-1.23. 1.66 J:_ x/D-20 ¢AD,. 1.23.i.45.1.66 o.e -_:_ "".. {Experime(ra/O_ta) C .o .-'_ O.II 11!/ (E,t_.,,ed_hlal D,t<=)

:_'_';._", .'... > 02+ ;+i,.t+i-_.+::- ...... , " o 2 :: 1":'_--'-:_,.--'+"-., ...... : .;;_ _:'j:,-+-+_-.-.:,..+.... : .... i, 1" _"..",';*-.'_*_,;':";..::._'._"_<-:'_,::-". " •;,_.j. !__¢_,;=-+..; - .--... ., .-:,.,., _._..-_=_:, -._ ,_,_ .:._,_c:k..-';. ',-.,- +':'"".-:,':.., J--_-_.;_'_..-_. O.OO 0o4 uu_ ol2 0,_(_ O.ZO 02, 0 00 0 04 O.Od 0 tZ 0.16 O._J 0.2,4

YO_ (MaSs Fraction) YOZ {Mass F=auGur0

345 Methanol Turbulent Jet Flames

Joint PDF of (f, C02), (f, CO)

Predictions Measurements

,ti'50"

.o! pO' d;o"

O.O_ 00' _o _ "/_4P__

(f,co)

_0"

+°t_.o

00" 0.0'

(f, C02) -

346 Comparison Between Predictions and Experimental Data

/ I • I r .... _ ...... Experimental Data for H2 Turbulent Jet Flames by Chen & Driscoll (1990)

( Ucoflow/U fuel _ 0.001)

i' "l l I-i I -00 2 ! .= ! !!'!l.]J llJllllllll !1 I II 11 L " L I I IJJ.I I)11t II II1 _I__ HYDROGEN Predict_i_

briment

0.04

3x10 4 10 5 3xl 0 5 10 6

STRAIN RATE UF/d F (l/see)_- Oa -1

347 Turbulent C2H4 Jet Flames

348 Needed Improvements of PDF Methods for Con_ A "ans

[] Compumtionally Expensive

Direct Calculation of Detailed Chemical Kinetics - Not Feasible

Capabilities of Reduced Reaction Mechanisms ??

[] Primitative Status of Mixing Model

Interactions between Turbulence and Chemical Reactions

349 Development of "Rational" Models ---. _' ---- ml m m

_ _. Development i Llnteres! s

Direct Numerical I Combustion in Simulation Compressible ,e Turbulence Simulation Soot in Turbulent Flames

¢ Turbulenc Model Flame Reduced Extinction & anisms Reignition Stock, Simulation NOx in Turbulent Flames Lamit2ar Flames with Detailed Mechanisms

350 m .,J

0---

111I,tl ola.

_.1 I11 Z _ ffl !'-

I1:0 m 0

O0 t.)l_ It. tli

0 |

.J 351 9 I.#CI6£:_SEI-1.6-9_ 352

I-. uJ

W ILl n W r7 W n IM W _J oo LU 0 o: .0 O_ 0

n.O: l.il I-- II:

355 9 I._O v_?,,SEI- _.6 -S ;'_ 356 Center for Modeling of Turbulence and Transition Workshop on Engineerin 9 Turbulence Modeling- 1991

DISCUSSION

D. Wilcox (to Sindir)

I understand that there is some concerns with NASP contractors that RSM is too expensive. I find only a 20% increase in CPU time to compute a full RSM with the newer algorithms, relative to the two equation models.

M. Sindir (reply)

The model that you came up with is not a tool unless it gets into the method- otogy of established codes that contractors use for validation. And that is a major activity.

T. Gatski (to Sindir)

You describe an extremely complex situation and then use a Baldwin Lomax model; I have a confidence level of zero in that calculation!

M. Sindir (reply)

I know and agree, but that is what is being done in industry. But time constraints keep this problem from being handled properly.

A. nsu

I feel that two items ought to be added to the list of tasks to pursue in PDF modeling. First, we have to examine PDF for high speed flows, like flows with shocks. Second, for the particle Monte Carlo method, problems involved in solution over a realistic geometry should be addressed.

357 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modelin 9 - 1991

A. Singhal (to S. Pope)

How do we transition this new and evolving technology into industry? In the first stage of transition, you introduced the joint PDF of velocity and composition, not just the first level of PDF which was shown in the hierarchy to be just composition. I'm curious why.

S. Pope (reply)

The reason for that is that the numerical algorithm for joint PDF of velocity and composition is really simple_aad .more economical than jus t composition alone. It sounds strange, but the reason is that in the PDF for composition, the diffusion terms, turbulent transport,

! !

| !

i ! | ! i !

! ! !

k ! | |

| 358 =

=_ Center for Modeling of Turbulence and 7_ransition Workshop on Engineering Turbulence Modeling- 1991

Session III

Unconventional Turbulence Modeling i

| !

z z

m_ _OP&SItOPON E_GINEEIIING]]JItNJIA_CE H_ELING (Aug./21,22/1991) Center for Modeling of Turbulence and Transition, ICOn, NASALcwis

The Present State of DIA Models and Their Impact on One Point Closures

Akira Yoshizawa Institute of Industrial Science University of Tokyo

I. Objectives A, Outline of DIA B. Outline of TSDIA (two-scale DIA) and some suggestions to turbulence modeling

C| Proposals: Helicity for the study of the effects of swirling and cross flow Density variance for the study of highly compressed flows

361 PR£CEDING PAGE BLANK NOT FILMED I, Basic Laws Compressible fluid

(o/at) p + v. (pu) - o

m (o/at) pu_+ (a/axj) (puju,) i (a/ox,)p + (a/axj) _s_,

(a/at) pe + 7' (;ue) :- pT'u + _ + V' (_VO) where

p: Density; u: Velocity; p: Pressure;

e: Internal energy; O: Temperature;

/_: Viscosity; %: Heat conductivity;

m _=-= s_j = 3uj/ax_ + 3u_/Oxj - (2/3) V'uS_j

= 2 = [ (Ou_/Ox,)=+ (113)(v.u)_] m

362 [Note] Thermodynamic relations for a perfect gas:

- pRO (R: Gas constant) = (_- 1) e (y: Ratio of specific heats)

e- CvO (Cv: Specific heat at constant volume)

Incompressible approximation

_'11 - 0

(O/Or)u_ + (O/Oxj) (uju_) : - (O/Ox_)p + _Au_ where

p/p ,,, p

-t/p: Kinematic viscosity

363 1 BI, Outline of DIA

Premise: Vanishing mean velocity

Homogeneity

- e I

Independence of the coordinate origlni

I '=* Infinite or periodic region

-* Fourier-integral or -series representation

Fourier integral

f (x;t) - ;f (k;t) exp(- ik' x) dk

364 Homogeneous turbulence

k'u (k;t) - 0

(O/Ot) ui (k; t) - ikjff_ (k - p - q)dpdq 2 x uj (p;t) u_ (q;t) - ik_p (k;t) - _k u_(k;t)

[Note] Elim_natlonm i of pressure:

p(k:t) = - (k,k_/k_)#, (k - p - q)dpdq X ui (p;L)uj (q;t)

[Note] Green I s function:w

(O/Ot)G_j' (k;t, t') - iMi,_n (k)ff_ (k - p - q)dpdq 2 X Um(p,t)I Gnj I (q,t,l t') - _k G_j' (k,t,' t') - D_j (k) _ (t - t') where

2 - kikj/k

Mijk (k) [kjD,k (k) + kkDij (k)]/2

365 Fundamental variables

Q,j (k;t,t') - _u_ (k:t) uj (k' ;t')_/_ (k + k')

G_j (k,t,e t') -

2 Difficulties in incorporating inhomogeneity

Necessity of the orthogonal function satisfying the noslip condition

3 [Note] Dannevik' s work.

Turbulent Rayleigh-Benard convection between two parallel plates (no mean flow)

Coexistence of slow modes (mean field) and fast modes (fluctuation)

Simultaneous treatment of different modes

366 IV, Outline of TSDIA (Two-Scale DIA) Departute from a complete two-point scheme

Passive: Difficulty in dealing with a boundary

Positive:, , Difficulty of obtaining, "formulae ]I applicable to general flows

Two scales

(=x), X (: _x); _ (=t), T (=_t) where

_: A small scale-expansion parameter

Then

f - F(X;T) + f' (_,X;v,T), F =

367 Two-scale expressions

V_'u' = - _ (O/aXj)u_I , v_ = (o/at_)

"_ (O/O_i) P' - _]Atui'

- _[- u_' (O/Oxj)U_ (D/DT) u _' - (O/OX_) p (O/OX_)uj' u_'] + _-related terms]

[Note] The effects of slow modes: Direct effects : Through U

Indirect effects: Through X and T lrl U' V

Fourier representation of $

f' (_,X;_,T) - ff' (k,X;_,T) exp[- ik'(_- U_]dk

Scale expansion

co n f' (k, X; _, T) - En=O_ fr, ' (k, X; _, T)

[Note] Lowest-order or basic field' UoI ,o The same system of equations as for homogeneous turbulence, except the X and T dependence f

368 Isotropic and helical field

/_ (k + k' )

- D_j (k)Q_ (k, X; _, _', T)

+ (_/2)' (kJk 2) E_ jmHB (k, k' , X,'_, _', T)

I. ! helicity effect [Note]

i 2 - fQs (k, X; _, _, T) dk

I - fHB (k, X,o r, r, T) dk, _ I - V X UB

Important correlation functions

! Calculation of the Reynolds stress etc, using DIA based on Q_ and G_

Extended eddy-viscosi ty representation for the Reynolds stress

369 V, Main Results from TSDIA: No Helicity

4 Reynolds stress

---I (2/3) K$i5 + _S_j

3 - _n=l In [Tnij - (1/3)rnmm$ij]

- _4 (D/Dt) S_j + ", where

K = i i S_j : OVj/Ox_ + OU_/Oxj

T_j = (OUi/OXm)(OUj/OXm) i

T2ij - [(OUi/OXm) (aUm/OXj)

+ (OUj/OXm) (OUm/axi)]/2

T3_j = (aUm/OXi)(OUm/aXj)

5 6 [Note] See Speziale and Rubinstein and Barton

370 7 Turbulent scalar flux

- _j (O/Oxj)0 where

- r_.(OU_/x,- OU,/xj)]

8 [Note] See Rubinstein and Barton

Some other suggestions

9 Triple correlagTBns

< (u' 2/2) u'> - - _KVK + VK'V_

@ Equations for the dissipation rates of energy and scalar variance. ( and Co) 7

371 VI. Proposal: Helicity Motivation

Explanation of the generation of _white spot x in Sa turn (the spiral vortical structures along 10 the equator)

Fig. l: Saturn's huge white spot (NASA Hubb 1e Space Te 1escope)

What is the helicity?

(O/at) u - u x _ - V(p + u_/2) + _Au

u leo "_ u X 6)-0 (no energy cascade) |

Helicity u'_: A measure of the break of | reflectional symmetry in flow

| | Fig, 2, Hel_c_ty

372 Importance of helicity

A measure of 'smallness n of energy cascade

A 'conserved n quantity in the absence of the inSection and loss due to viscous effects

[Note]

channel flow, jet, wake, mixing layer, etc.

Finite mean helicity: U'fl ¢ 0 Swirling flow, Three-dimendsional mean f1ow

[Note] Swirling flow: Question: Why do the eddy-viscosity models break the swirling motion so fast?

Answer :No consideration of helicity or the decreasing effect of cascade (virtual decrease of eddy viscosity)

373 [Note] Cross-flow effects:

Lag of the hurbulent stresse in response to the cross-flow gradient (also important in 11, 12 aerodynamical flows )

12 Fig, 2 Spinning cylinder

Curvature effects leading to secondary flows

13 Three-equation model

K- : Turbulent kinetic energy

:Energy dissipation rate

H - : Turbulent heliclty

[Mean equation]

V'U-O |

(D/Dt) Uj m (O/axe)p + (a/ax )

374

IE [Reynolds stress]

Rj_ == -

= - (2/3)KS_j + _ (OUj/Ox_ + OU_/Oxj)

[_i _ j + _ j _i -- (2/3) Q, _ 8_ j] where

3 ve - C1K2/_, y - C2 (K4/E)VH

[Note] Symmetry-breaking factor: Reynolds stress: Reflectionally symmetric

Necessity of another symmetry-breaking factor, that is, inhomogeneity such as VH

375 [K equation]

(D/Dt)K- PK- [ J- V' [C3 (K2/[)vm]

PK - Rij (aUj/_xi)

[H equation]

(D/Dt)H- PH- _H + V'TH where

PH : R,j (_j/_Xi) -- _i (_/Xj)Rji

[H -- C4 (e/K)H

TH- Kfi + G5 (Ke/a)VH

[E equation]

2 (D/Dt) E - C6 (_/K) PK - Cv (e /K) + C8 (K2/E)fl.VH

+ V' [G9(K2/E)V_]

376 Swirling flow in a pipe [(r, O, z); z: axial]

- - ,,or (a/ar) (Uo/r) - _9.oaH/ar

- - _e(auz/ar) - _QzaH/ar

- _o(auo/az) - ,jOoaH/az where

,i-c_K 4 /_3 >0

14 Comparison with observation

Fig, 4, Mean velocity F_g,5, -!

(Broken lines: weak swirl region)

uo' > - _,er (a/a r) (Uo/r) 7/_.oaH/O r

I...... I

A t] Around r - O, 7

A>O

B> 0 (_o = - Ou_/0r < 0, 0H/Or< 0)

[Note] H-0 at the wall ,m aH/ar<0 near it 13 Application to SGS modeling

Appearance of steamwise vortices (streaks) _5 dependent on the strength of shear

-.* Nonvanishing helicity on the SGS, but not on the ensemble or time mean

•._ Importance in the SGS modeling

-, Incorporation of the helicity effect

Virtual change of the Smagorinsky constant in channel flow, mixing layer, isotropic flow in accordance with the strength of ! L shear

378

IE _, Proposal: Density Variance Prominent difference between incompressible and

compressible turbulence

Decelerated streamwise velocity effects

Compressible (shock wave): Decrease in turbulence level Increase in temperature etc,

Incompressible: Increase in turbulence level ......

Mass-weighted mean

f - {f} - /F, Y-

f' -f-f

379 [Mean equation]

(a/at) _ + v. (_) - o

(a/at) _ + (a/axj)_-_ = - (a/X i) [(_ - i)F_]

+ (a/axj) Rai

(a/at)-ii + v. (;_) (r- 1)peV.u--/% ._, + _-(E+ z)

+ v. (- H) where

R_j -- -- --p{u_I uj)I

H - - f{u' e'}

z - - (r - 1)/7 - - (r - 1)

R,a: Reynolds stress m H : Internal energy flux

i | Z : Fluctuating d_latat_on effect

380 [Note] Importance of Z: Trace of the pressure-strain correlation (no contribution in the incompressible case)

[Note] Inference of Z: Large positive in a highly compressed region

-* Virtual increase of energy dissipation

16, 17 Three-equation model

K - {u'_/2}, _, Kd - (p' 2)1 Density variance

[Reynolds stress]

A 2 _e - C_K /_

[Internal energy flux]

[Fluctuating dilatation effect]

Z - C3 ([//K) (Kd/O 2)'e

381 [K equation]

(a/at) K + V' (_K) - P.- #(E + Z) + V'TK

[_ equation]

(a/at)E + V' (b%E)- C4 (c/K)PK C_;(_/K)(E+ Z)

+V'T,

[Kd equation]

(a/at)K_ + V. (GK_)-- KaV'G C6 (c/K)Kd+ V'Td

i...... i

exact!

Can the turbulence level decrease behind a shock

wave_ | ==

V'u < 0 near a shock wave m _m Production of Kd (Kd equation)

Larger e, larger Z (than elsewhere) .-m

,-_ Decrease in K and E (K and E equations)

382 REFERENCES

l) R,H. Kraichnan: J. Fluid Mech, 5 (1959) 597, 2) R,H. Kraichnan: Phys, Fluids 7 (1964) 1048, 3)W,P, Dannevik: Ph, D, thesis, Saint Louis U, and NCAR, 1985,

4) NO Yoshizawa: Phys, Fluids 27 (1984) 1377, 5)C, G. Speziale: Annu. Rev. Fluid Mech. 23 (1991) 107, G)R, Rubinstein and J,M. Barton: Phys. Fluids A3 (1990) 1472, 7)A, Yoshizawa: J, Fluid Mech. i95 (1988) 54i. 8)R, Rubinstein and J.M. Barton: Phys. Fluids A4 (1991) 415. g)A. Yoshizawa: J. Phys. Soc. Soc, 51 (1982) 2326. i0) A, Yoshizawa and N, Yokoi: J, Phys, Soc. Jpn. 60 (1991) No.8.

l l 11)J,Marvin: Engineering Turbulence Modeling and Experiments, edited by W.RodJ and E.N. Ganic (Elsevier, New York, 1990), p, 3.

383 12) D°M. Driver and J. P, Johnston: i0th Australasian Flui d Mechanics Conference, Universi ty of Melb ourne, 1989, 1A-3, N. Yokoi and A, Yoshizawa: in preparation for

Phys. Fluids A° P. Mo_n: Near-Wall Turbulence (Hemisphere, 1990) , p. 2. O.K1toh: J, Fluid Mech, 225 (1991)445, 16) A,Yoshizawa: Phys, Fluids A2 (1990)838,

17) A.Yoshizawa: submitted to J, Phys. Soc, Jpi1,

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Kraichnan 1970

= E(k) = _Vo53 2 (k - ko) Not scale invariant

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428 5

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2 Y/H I

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430 BACKWARD-FACING STEP:

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Y O

m H ° o

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(b) Turbulence shear stress Computed turbulence stresses for the new RNG K-_model [E = 1:3;Re = 132,000; 200x100 mesh; computatlons with anlsotropic eddy vlscoslty; o oxperlmonts of KIm et _1, 1980; Eaton & Johnston, 1981] Figure 7 433

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O_-

N92-24536

THE ROLE OF EXPERIMENTS

TURBULENCE MODELLING

William K. George Turbulence Research Laboratory State University of New York at Buffalo

PRECEDING PAGE BLANK NOT FILMEI_63 RECENT REFERENCES

1. W.K. George "Governing Equations, Experiments, and the Experimentalist" Experimental Fluid Thermal Science - 1990

2. W.K. George "Designing Experiments to Test Closure Hypothesis" & D. B. Taulbee to appear Expimental Fluid Thermal Science 1992 (Also Proceedings Synmposfum on Turbulence Modelling Dubrovnik 1990).

Some Experimental Dilemmas -- just a few of many

7 the Round Jet - what is its growth rate? Z the Turbulent Boundary Layer

the Plane Wake

2 the Dissipation

TH____EEROUND JE____T

61/2 = 0.095 X no_it 0.085 X

B Illustrates one of the biggest problems for modeller - Who to beiieve?

• Problem for experimentalists is

How to overturn old results?

- unpublished since disagree with earlier results

- unaccepted since unpublished

- unused since unknown m .... Problem for everyone - Isn't there better way

- to get results out

- to purge old results

464 !_- _" ...... _:_:-

.'_ _OCAL ISOTROPY

- Local Axisymmetry?

- Equi-partition of Dissipation?

465 4 i

._ 7 t

i ! !

\ \

i =

i ! 466 I I .314 P_S

L.W. 8. BrawM, n.A. An_mlaandD. A. ShaA I._,_

' l.i

_ "I_, " 1 :F_'"-:__ ',, i ""°', I .. • -"r ,, I "I i I o...

I , ...... ""_. loI , _ , ! . I _i'l I c o o.4 o.I I.i 1.6 2.o o o.4 o.i 1.2 I.i 20 __ I II .p _ Fill !

_$ _blQ overi_ I the ritio4 Ks to t I ,r.=.,:..X".,..<'' <,,... ,i,-t7_.',:,i>., ., ,..,b1/... _X,,_x.\ :...... ,@ ,, _L With the ex_" "_'. _ all 8iliter tl departurce from i of validating (3)

4 large and the m_ \,,,_. i of the mune eXlx Pioell i. Dtltrflludoo urom the wake oi_ i. the r.m.I, ofdiriviilvel ot velocity auetu,dool, differ_nc_l bctw 6vii tit Iine_: ----, p - v; --. f: ---. v. (a) y - z: O. P" t Item I_1: i. P" u (Ilium I_, correlation r_pe_t_l); V. _ -, (lIF_ I hi: I. _ - • (tliu_ i _l: O. _ -, (Ilium t,. lower X-.ir_): A. #" • to [O] of (4). Bee u I.Smm); Q.p (Igure Ib. lo tr - mi;-t-l-._ wI_Ium lb. upperX.wire): ¢.J- w uaumed the |_lI (fliF_m I¢, X-wire it l I 0); -0-, P - li (figure I¢. X-wire il l - t._ ram). x, j - u. I*iin I two Of the 'expects| dnik. " wires. (b) 1'" T: O. P " v (llgU_ I I): &. J - ,, (_IKueeI i, mp.t_l): T. ,e - i (i_l_m I _, terms in (4 }, t hi,,i Npei_l): I, .8 - _ (Ilium Ib): O. j ,, • (_Ip]re la); L_, .8 -. (t_gurI I¢. repett_l); 1"1.I " le I.re .hown in fig (figure I b); _.. j - ,. traini two single ,ires. (¢) y - l: e. J - v (_m I ¢1: A. J -, (Siure I d): underestimates O. _ i e (tllF_ I_): I"!.._ -, (ttg_m Ic): lit. _ .. i. uslng fro ";qIie wiree, eeatr_lin© to -be Llufer (10541, deriv&tlvel wiih "

467 TH____EPLANE _AKE

-- there is no wake which is independent of generator

i =

i_ ....

|--_F ir

468 l -) -2 -1 • ! ._ )

Flgq_r, 6. Ray_olds stress _¢._llzed by c¢_terlln_ vtloclty deficit for the solid strip and alrfoll (_skl ot al. 19B6).

469 I "+3""++.: l ,.,f_ "f_" + m _k. " I ,_+ .-. + •

-_ -2 -I 4 I ._ ) Fi_ 7. hrmltz.d t.r_1..c.i.t..s+ty _O.s _r t*. tkr_ generators of Ft_rt S (_kS et al. 1_), = : ±± : TN Axxsv_a_raxc K4xx: A FLow _ 0ou _ £voLvt AT Com_AWr Ihnmotos Nu_la

uis_trl¢ wake pres4_ts an +nterestlng contrast to the utsymetrtc jet and plane wake flows described above tn that It does not evolve at coastant Reynolds number (as will be seen). As • consequence, the _ture of the assumptions regarding the dissipation wtll be seen to predict two quite different asysptottc developlnts. There appears to be experlmntal evidence for both form In different expertmonts, Which raises an Interesting question as to how the flow chooses one fore or another. _ +nTe_sttng+_sstb=tll(y +i+i (hatll the _ov evolves from one state to another as the Reynolds number changes. inese possibilities v111 be discussed In more detatl below fol|ovtng the i 11_ equatloes Of notion describing the UtsymMtrtc wake to first order can be shem to reduce to.

vhere U. ts the undisturbed sp4_ of the free streau. This can be Integrated across t_.,f1..ov. 1o,..71¢I.d the letegrel cogstralnt.

21 t U-Ue}rdr = =U._i, - IVp (79) vkere tts defined te be the am_tm tk+ck.qess.

470 z This persistent effect of initial conditions may be widespread!

471 0

_ oql

n I _i l(i)I_ _oI

472 m Th____e_Well-established' Turbulent Boundary ___

Is there really a log layer?

Should comparisons be made with these correlations?

473 _g. 1.1.1 On the determination of the friction coeff_ien_. from _ et al. (1967).

474 Fig. 1.1.2On the determination of the friction coefficient:

from KLLNE et al. (1967).

i _ 4a5 . cz_

A t___ 10

t.____

475 Fig. 2.1.2.1.1Veloci_ in _ variableswith log lay:.

Pro'tellel aI..Rib = 465. 498. 700, I000, 1340, 1370, 1840, 2840. 3480,

4O9O, 5100. ' I ! I I 32.0 • i I I , i I II I I I III I I I I l IIII l i

= 24.0--

16.0--

8.00-- / /

"0 10 0 10 _ 10 _ Y+

c_ = i 476 Ill- I N: Fig. ] .4.1 Veloci o, profiles obtained with a wall slope determined shear stress:

Purtell et al.. Rth = 465,498, 700. 1000, 1340, 1370, 1840, 2840, 3480, 4090, 5100.

i I I I I ]2.0 I I-- I I I I III " I I I I I I lli I I i i i llll i 1 "I I I I III

24.0-

16.0--

od T./

8.00- / I

/ / /

L • I I I I I[I _r_0.E+O0 I l l I I I II' 1 l ! ! l llll ! I I I ! 111 _ I I I 1 10 c 10' 10 2 I0 a 10 " V+ = (Y • U_TAU)/NU

477 Fig. 1.2 Veloci O, derivative from Direct Simulation:

from SPALART (1988)

20 .,e,r--,--:-.-

<. o] ...... , ...... - ...... i

2 10 Y" 10_[S_. _ _ 1000

=.5 (b} _ "1 ! 10 1130 10CK} y. 4' _ -

Flot'xz 5. 51e&n-velocitv profile and its derivatire, f = 300: ---. R, = 6.,70: ----. R, 1 1410 : ---. log law U" = log (9"1/0.41 + 5: +. R e = 617 (Erm eL a/. 19851 : O. Re _ 1368 (._! u rlis el al 1982). (a} U': (b) y*d("/dy'.

_:$7 "

478 Fig. 3.1.2.2.11 Outer variables:

PurteU et al_ Rth = 5100.

J I 0.400 | I | | I i I I I

Rth = 5i00.

03 c_ 0.300-

_ 0.200- Z

v Q

< I-- --I 0.100- . " >. v

L .4 .J,_ _r .z haT"

O.O00E+O0 t _ ; ,,, I I ' I l I I I ' I I i i O.O00E+O0 0.250 0.500 0.750 1.00 U/U_INF

479 Fig. 4.4.3.1.1 Shape factor.

Smith and Walker. x = 15.75 in, 27.75 in, 39.75 in. 51.75 in. , . t . , . 1

+ i + + i F

x = 1 5.75 in _=

x = 27.75 in

x = 39.75 in

x = 51.75 in

J L E l I ! ..._..__.1_ t : , 1 I I

480

IK Fig. 4.2.2.I.l Momentum thickness:.

Smith and Walker. x = !5.75 in,27.75 in,39,75 in,51.75 in.

1 ' ! ! !m ! ! ' ! ! 11 I 'I 0.60DE+05 [ I f

0.450E+05_ ° l _ f < 1.1 -: o.3o_-.,o5-_

. L + L_ Z + I

x = 15.75 in 0.150E+05--

x = 27.75 in

x = 3£.75 in i: x = 51.75 in O.O00E+O0 | .. i , , ! I ' I ' I l I l ! I O.O00Z+O0 0.125E+06 0.250E+06 0.375--_+06 0.500E+06 (U_INF • DELTA) / NU

481 Fig. 4.4.2.1.1 Momentum ,]ficknes_

Smith and Walker. x = 15.75 in, 27.75 in, 39.75 in, 51.75 in.

I I I ' 0.600E+05, ! i ! ! _ I I 1 I I I

0.450E+05-

b.J _ p_.,/_, T 0.300E+05- I""

4 I..2-- Z I --) v

0.150E+05- / _:_?,?,n _ / _-C in

x = 39.75 in

' x = 51.75 in

0.O00E+O0 t I ! ! I 1 1 I I I! ! 1 1 0.O00E+00 O. 125E+ 06 0,250-+06 0.375E+ 06 O.SOOE +06 (U_INF - D-LTA) / NU

482 I_g. 4.4.6.1.IFrictioncoefficie.tas a function o/ r.. Smith and Walker (53 poLnLs).

2.5OO£-O2 , , , , ,,,,[ , , , , , ,,,[ , , ,-

:.500£-02-

l,_ (J

:.400£-02- ..,-% in. <: :m

z.300E-02 -

D.2GOE-02 ! I ! * I I ? l I I, I I 1 ! ! l 1 I ! ' I l I 1 10 5 10 ' 10 "

( U_INF- X )/ NU

483 Th____ePast

Without experiments, there would be no single-point turbulence models•

DNS, LES, etc. have helped augment data base in recent years. Contributions have been important, but relatively minor when taken as part of the whole.

The Future

Question: Is this balance likely to change?

• No question simulations will play increasingly important role

- ability to produce difficult quantities (for expts) (pvi, pressure strainrate, dissipation, etc.)

- increasing Reynolds number (still small)

- National economic agenda of U.S. - we will do what is necessary to maintain market position

- indirect subsidy of CFD

What role will this leave for expts?

Experiments of value to modelling community are very difficult to do

Most efforts, however well-intended, do no___tmeasure up!

Reasons complicated but•., high on the list ....

Inexperience of investigator.

Ignorance of goals.

• Money usually runs out before experimenter learns to do it right•

• These will be increasing problems in next decade.

Biggest Problem for ExperimeBtalist;

• Few sponsors have patience to see an experiment through to its completio n - none in my experience!

As complexity of flows to be investigated increases (Bushnell comment), this problem will be exacerbated!

484 FINAL THOUCHT

However successful our closure efforts may be, we are not solving the turbulence problem - we are only being responsible engineers!

We must therefore be careful not to demean the efforts of those who ar___ee trying to grapple with the real turbulence problem - there may be an underlying physical principle which will very much affect what we do!

485 m

D i R E S) 3--3 _/

g_ ydy .//,3 5 N92-24537

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487 PI:_C_DING PAGE _.A.N_ NOT Fv.'...MED !--

488 0 t- in

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495 STOCHASTIC SHOCKLET DISSIPATION MODEL.

(a) Pt. P_ U!

Pe. Pl

ut u2 . a "z EddysizeL

FIGURE 1. (a) Sketch of shock-llke structure in a turbulent eddy; (b) normal shock relations.

Average dilatation dissipation:

K)dm]

cd = c_c_F(Mt, K)

Z_Z

Total dissipation (for given kurtosis K of m)

etot -- e_{1 -F CdFk(Mt)}

o c5 o o

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498 0

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>4 ,-i,,-)

n,--,,-,,I

0 ,v,,-,-4

o a,,i(

499 o

a 0 0

500

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e- im

e- lm

e" .0 _.. r-

0 _- L.. 0 um

0 0.

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501 LLI

0 tO

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0 _ =i s._ =,=,= II[L _,) ==== ...... "o ..,4==_

z (/') _ .- 0 " LI.

502 0 _q

0 _

_ 0 "_.".\ 0 ,_ ,A,! 0 - 51 _-__ N 0 _ o _--,I o .._ o

I i i 0 _ ,%.

503 i (I) 0 C: 0 m E

m 0 " (1) '=== :_ 0 _ --I o 0

0 o _ m (/) >, g_ _ _ 14. c.

0 "0>" (JO ","-' ILl u_ (/)

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m .-I ,¢1: >, i-- O3 LLI I--

CD ILl "O a t.... o 0 ,.Q c_ O !.._ im :::3 m "0 m ILl wn ";3 O3 wm _3 E O rn X U.I

505 I.- Z ILl

O ._i 1.1.1

ILl C_ a ....I mmmm i I.!.! C.1 ..Q L=.. lira d_ m 0 m "0 (ms im 0 E i.LJ .,l,.-- r- Im m im _.! .4.='P a. E E 0 c_ 0 mm rr' 1.1.. =? x ILl

506 0

507 0

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512 Center for Modelin 9 of Turbulence and Transition Workshop on Englneering Turbulence Modeling - 1991

DISCUSSION

B.E. Launder (to S. Orszag)

You showed us values of C_1 and/S',i' Which would give too high a decay rate of grid turbulence.

S. Orszag (reply)

There is some question about what the decay rate really is.

B.E. Launder (to S. Orszag)

You talked about C_2 but you didn't say how C,1 emerged.

S. Orszag (reply)

That's the same calculation.

S. Pope (to S. Orszag)

Calculations you did with backward facing step, what boundary conditions did you use?

S. Orzsag (reply)

That was not a full RNG calculation. It should have been done using in- terpolation formulation for the various constants all the way to the wall. Instead it was donc using the a fit right to the log layer.

513 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbttlenee Modeling - 1991

T. Gatski (to W.K. George)

Two things modelers are looking for validation and calibration. Bill refered to the kind of experiments we use for validation. The kind of work you (J.H. Ferziger) do with DNS has been building block for calibration. Do you want to design the experiments to validate our model or calibrate?

W.K. George (reply)

I would like to design experiments which would invalidate your models.

T. Gatski (to W.K. George)

But that would be destructive for both of us.

P. Spalart (to J.H. Ferziger)

I differ with your description of DNS as exact solution. I would like to say that my solution are not exact. I spend quite a lot of time thinking how I can keep the error small, if I can double the number of the grid points in each direction I would sleep better but it would take fifty years instead of two years to finish the simulation.

J.H. Ferziger (reply)

I should have said that in my talk that any numerical calculation is approximate and I hope we work hard to keep them small. There are errors due to numerical methods and errors' due to the fact that we have limited computer time.

514 Center for Modeling of :Purbulenee and Transition Workshop on Engineering Turbulence Modeling - 1991

P. Spalart (to W.K. George)

A comment on Bill's theory that it's a power law instead of log law. I think in your original APS abstract you make it sound like it's just a matter of taste if you use defect law or your theory.

W.K. George (reply)

In the original APS abstract I wasn't clear why the theory comes apart asymptotically and I feel much confident now that the existing theory is wrong.

P. Spalart (to W.K. George)

There is very different Galelian invariance to those two theories and its not a matter of test. Defect law says we are coming from the free stream and we don't know how fast the wall is moving; it may be a moving belt. Your theory doesn't do that.

W.K. George (reply)

That's right. There is no question that there is a lot of sorting out to be done,

B.E. Launder (to W.K. George)

It's very interesting that he (or it) brings out into question the universality. If you got flows to decrease rapidly with distance from the wall as you do in low Re channel flows there is data going back to fifties that your log-log constant goes up. So logically you would expect log-log constant would go down in adverse pressure gradients. The implicit faith shown in sectors of fluid community in the universality of the log law I think is misplaced.

515 Center for Modelin 9 of Turbulence and Transition Workshop on Engineering Turbulence Modelin9 - 1991

There is now emerged which I think an excellent paper by Nagano (at the upcoming SFC in Munich) Showing what seems to me a clear dependence of log-log constant on shear stress gradient. In adverse pressure gradients lower log-log constants th_ you find in zero pressure gradients. Bill gets unhappy at unacknowledged at his private discoveries as all of us do, I suggested this in a paper about eight years ago.

W.K. George (reply)

I presented this in 1978.

P. Spalart (to B.E. Launder)

I have results that show that in moderate pressure gradients log-log going down. At y+ = 50 it goes down by almost one wall unit at /3 = 2 which is not very strong at all.

S. Pope (to P. Spalart)

You very quickly mentioned that you use DNS data for guidance and not calibration. Could you expand on that?

P. Spalart (reply)

If I calibrate turbulence model for boundary layer based on flat plate results I'll get too high Re, so ! don't have DNS results which I'll trust within 10% to extrapolate to high Re.

G. Hwang (to J.H. Ferziger)

You talked about P. Durbin model that uses v 2 as damping function. But : : = . ==

516 Center ,for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

problem is we want to use this model in multi-dimensions. Then you are talking about k - v 2 and e equation which is as complicated as Reynolds stress models. I am not disputing the model there are some good points to it.

A. Yoshizawa (to J.H. Ferziger)

You pointed out the possibility of LES as engineering tool but our experience shows we can not perform LES with Smorginsky constant fixed. LES criti- cally depends on Smorginsky constant e.g. we perform LES of channel flow using Cs - 3.1 but using this constant we can not simulate e.g. backward facing step. My opinion is without overcoming this difficulty LES can not become engineering tool.

J.H. Ferziger (reply)

I agree with you. There is a new model which does overcome some of these difficulties. I don't want to say that all the difficulties are overcome but we have hope.

J. Bardina (to A. Yoshizawa)

We have investigated DIA, RDT and LES. LES is much simpler than DIA. You have difficulty with Smorginsky constant but we know that this constant is not right. It's not universal for all flows because only thing it's doing more is dissipating more energy; that's all. At high Re you are putting more energy at small scales and there are many other effects which you have to put there.

517 Center for Modeling of Turbulence and Transition Workshop on Engineerin 9 Turbulence Modelin9 - 1991

E: Reshotko "

Bill brought up many of the problems in looking at experiments. There are many more which try to fulfill some of the desires and expectations that have been brought up here at the workshop. First any self-respecting

experimentalist is not there to design experiments to validate or calibrate a theory. First of all he is there to discover new physics. May be a theoretician wants to sec if the physics is reproduced by ttle model.

But when it comes to doing that particularly in measuring turbulence we come up with the problem of how to measure at a point. We have all these wonderful things at a point when we have probes which are not a point. Recently we had experience with multiple wire probe that showed our probe was not measuring at a point although our probe was less than O.lmm in overall size. And this problem becomes worse if one goes to high speeds. I understood just a few years ago wily al! good turbu!ence measurers were working in large facilities and in low speeds because only in that way you can feel reasonably secure that in terms of wall units you are operating at a point. We tried running some experiments at 100 ft/sec and found that our probe was 100 wall units which typically a spanwise streak size. In compressible flows, aside from increased speeds and increased probe dimensions in wall units, we also have the problem of calibration in transonic regimes. It's not that we don't know how a hot-wire works in transonic regimes. It's so sensitive to Mach number in transonic regime that there is not a way of saying it's r=e!iab!e. I am worried about the double and triple correlations in boundary layers with the present:probes. :One of the things H. Nagib is doing is looking at probe miniaturization and I encourage this but until then I think prospects of getting detailed compressible flow measurements are dim.

518 Center for Modelin 9 of Turbulence and Transition Workshop on Engineerin 9 Turbulence Modeling - 1991

J.H. Ferziger (to all participants)

I thought it would be interesting to throw out at the modelers that what do they think is missing in the experiments, simulations and theory?

B.E. Launder

I ponder from time to time about these nice homogeneous flows which people use to come up with constants in dissipation equation as Steve Orzsag was talking about. Question comes to my mind that its the variation in inhomogeniety which we are interested in looking at. The variation in spatial length scales ought to enter in our closures in ways other than the diffusive like terms. That is to say perhaps if we arc thinking of dissipation equation having adjacent to one another layers of different length scales are going to be promoting spectral transport of energy removed more readily than you'll find in homogeneous flows. So I ask myself if DNS can help clarify this.

J.H. Ferziger (to B.E. Launder)

Are those relatively simple inhomogeneous flows in that regard.

B.E. Launder (reply)

I think simpler inhomogeneous flow you are talking about is channel flow where everything is so dominated by the fascinating structure of the near wall sublayer. If you could do simulation away from wall where low Re dissipation issues are dominated.

J.H. Ferziger (to B.E. Launder)

Then there are mixing layers.

519 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling - 1991

B.E. Launder (reply)

Maybe mixing layer results will be valuable. I don't think they ought to be- long to this question. That is your dissipation equation needs to be different in inhomogeneous from that in homogeneous.

J.H. Ferziger (to B.E. Launder)

We tried doing flow simulation of experiments of Warhaft. May be we can collaborate.

B.E. Launder (reply)

Maybe those experiments themselves Will answer. There are no mean velocity gradients in that experiment John (to J.L. Lumley).

J.L. Lumley (to B.E. Launder)

No mean velocity gradients - just a gradient of scale. I was just going to draw attention to that.

J. Bardina (to B.E. Launder)

Brian are you suggesting that homogeneous flows are not a valid test for k- e model since it would not account for inhomogeneous part in shear flows.

520 Center for Modeling of Turbulence and Transition Workshop on Engineering Turbulence Modeling- 1991

B.E. Launder (reply)

We should look at homogeneous flows as building blocks and may be we shouldn't say going from homogeneous to inhomogeneous flows just adds a diffusion like transport term but may be adds other as well.

J. Bardina (to B.E. Launder)

The k - e model worked good for homogeneous flows we tested and it wasn't tuned for these. We looked at homogeneous shear, and plane strain flows and it did very well. It didn't do well for rotational flows because effect of rotation isn't accounted for.

W.K. George (to B.E. Launder)

I think you are right Brian. If you look at Antonio's dissipation results all but one derivative is way out of line. I have come to believe that it associated with inhomogeniety. And if you look at each term in the equation, if the flow is truly locally homogeneous you can not produce any of those. This question is best resolved by DNS of inhomogeneous situation at low Re.

J. Weinstock (to B.E. Launder)

I wouldn't have any doubt that inhomogeniety would cause changes other then diffusive transport. Nature of the change is such that we can not tell until we do it. I did a calculation where I accounted for strong time variation where the turbulence energy is changing at times of order of eddy circulation time and a simple result came out of that. It wasn't diffusive transport. Eddy circulation becomes function of turbulence time scale. So the coefficients involving damping become function of rate of change of turbulence. I think if we put spatial inhomogeniety we would come up with something related to that. It's obviously not diffusivc.

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Workshop on Engineering Turbulence Modeling

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13. ABSTRACT (Maximum 200 words)

The purpose of this meeting is to discuss the present status and the future direction of various levels of engineering turbulence modeling related to CFD computations for propulsion. For each level of complication, there are a few turbulence models which represent the state of the art for that level. However, it is important to know their capabilities as well as their deficiencies in CFD computations in order to help engineers select and implement the appropriate models in their real world engineering calculations. This will also help turbulence modelers perceive the future directions for improving turbulence models. The focus of this meeting will be one-point closure models (i.e. from algebraic models to higher order moment closure schemes and pdf methods) which can be applied to CFD computations. However, other schemes helpful in developing one-point closure models, such as RNG, DIA, LES and DNS, will be also discussed to some extent.

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