Advances in PDF modeling for inhomogeneous …Advances in PDF modeling for inhomogeneous turbulent ﬂows P. R. Van Slooten, Jayesh, and S. B. Pope Sibley School of Mechanical and

Advances in PDF modeling for inhomogeneous turbulent flowsP. R. Van Slooten, Jayesh, and S. B. PopeSibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853

~Received 8 July 1997; accepted 18 September 1997!

Probability density function~PDF! models which are based on an exact representation of rapidlydistorting homogeneous turbulence are applied to a free shear flow, the temporal mixing layer.These velocity wave-vector models were constructed in a previous paper by Van Slooten and Pope@Phys. Fluids9, 1085 ~1997!#, but were only tested in cases of homogeneous turbulence. At theedges of free shear flows both turbulent–nonturbulent intermittency and pressure transport aredemonstrated to play a significant role in the behavior of the flow. A natural treatment ofintermittency is obtained by extending the PDF formulation to include a model for the turbulentfrequency following a fluid particle. A model for the pressure transport is also constructed via ascaling analysis. The resulting velocity wave-vector turbulent-frequency PDF model with thepressure transport model yields a good comparison with the available direct numerical simulationdata for the temporal shear layer. ©1998 American Institute of Physics.@S1070-6631~98!01301-4#

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I. INTRODUCTION

Despite the rapid advancements in computational powthe direct numerical simulation of complex, turbulent flofields remains intractable, which provides the motivationturbulence modeling efforts. One modeling approach is Pmethods, where the level of closure is one-point, one-tjoint probability density functions, PDF’s. These methoare typically implemented in a Lagrangian frameworkstochastic equations that represent the behavior of fluidticles. With the Lagrangian/particle approach, several advtages over standard moment closures~e.g., k2e and Rey-nolds stress! are readily apparent. Both convection areaction are treated without modeling assumption~Pope1,2!and realizability is assured by construction~Pope;3 Durbinand Speziale;4 and Wouters, Peeters, and Roekaerts5!.

Although the PDF’s contain a tremendous amountstatistical information about the turbulent flow, the primavariables of interest in inert flows are the mean velocitiesReynolds stresses. Also, the modeling of the ‘‘uncloseterms in the PDF equation are best understood through dcomparisons with Reynolds stress models~RSM’s!. Experi-ence has shown that the standard RSM’s often behave uisfactory in complex inhomogeneous turbulence, especifor rotating flows~Launder and Morse;6 Jones and Pascau7

and Ohtsuka8!. The source of the difficulty is traced to modeling of the rapid pressure–rate-of-strain correlation afunction of the Reynolds stress anisotropy, an approwhich has been shown to be fundamentally flawed in rotional flows ~Reynolds9 and Reynolds and Kassinos10!.

A PDF approach has been developed~Van Slooten andPope11! to improve the modeling of the rapid pressure–raof-strain correlation by representing rapidly distorted~or ro-tated! homogeneous turbulence~RDT! without modeling as-sumptions. In the Reynolds stress equation for RDT,rapid pressure–rate-of-strain correlation is the only uncloterm, so the new PDF approach inherently treats this colation exactly in the limit of RDT. The PDF models arbased on a particle representation model for RDT by Kanos and Reynolds12 and add a unit vector, the wave vecto

246 Phys. Fluids 10 (1), January 1998 1070-6631/98/1

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into the PDF formulation. With the added particle propertycorrespondence may be developed between the stochsystem and the directional spectrum, a spectral variable~VanSlooten and Pope11!. Since RDT is closed at the directionaspectrum level, PDF models may be constructed that tRDT exactly. The return-to-isotropy tensor is an additionconsideration for non-RDT homogeneous turbulence, anis modeled in the manner of standard RSM’s~Van Slootenand Pope11!. The extension of these PDF’s models for inhmogeneous turbulence is accomplished by including theticle position and full velocity into the formulation withoucompromising the correspondences to the directional sptrum ~Van Slooten and Pope11!.

Another advantage of PDF methods is their recimplementation with stochastic models for the turbulent fquency following a fluid particle~Pope and Chen;13 Pope;14

and Jayesh and Pope15!. As in other turbulence models, scainformation is required in PDF methods to close the systeand often mean dissipation or turbulence frequency eqtions are used. However, with a velocity turbulent-frequenPDF method more realistic modeling of the turbulent motiis possible. Models for the phenomena of internal and exnal ~turbulent–nonturbulent! intermittency are constructeusing the added information of the particle~instantaneous!values of the dissipation~Pope14 and Norris16! or through aconditional mean of the turbulent frequency~Jayesh andPope15!. Several researchers have applied these modelvarious free shear flows~Norris and Pope;17 Minier andPozorski;18 Anand, Hsu, and Pope;19 and Delarue andPope20!.

In the present work, a velocity wave-vector turbulenfrequency PDF model~WVM ! is applied to the temporamixing layer, which is equivalent to the plane mixing layin the limit as the velocity ratio approaches one. The plamixing layer has been studied by numerous researchersexperimentally ~Bradshaw;21 Wygnanski and Fiedler;22

Champagne, Pao, and Wygnanski;23 Dimotakis and Brown;24

Spencer and Jones;25 Batt;26 Mungal and Dimotakis;27 andBell and Mehta28! and numerically~Riley, Metcalfe, andOrszag;29 Lowery and Reynolds;30 Sandham and Reynolds;31

0(1)/246/20/$10.00 © 1998 American Institute of Physics

P license or copyright, see http://pof.aip.org/pof/copyright.jsp

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Rogers and Moser;32 and Moser and Rogers33!. For the tem-poral shear layer, a thorough DNS study has been perforby Rogers and Moser34 which is referred to throughout thiwork. The DNS data for the plane wake of Moser, Rogeand Ewing70 is an additional case which may be usefulfuture studies.

From an examination of the DNS data and preliminaPDF results for the temporal shear layer, it is found thatadditional model for the pressure transport is requiredPDF methods to properly represent transport at the edgethe layer. In RSM’s a gradient transport hypothesis is usemodel the turbulent transport, while the pressure transpotypically omitted ~Daly and Harlow;35 Hanjalic andLaunder;36 Launder, Reece, and Rodi;37 Launder;38 andSpeziale39,40!. This is equivalent to modeling the combineeffects of the turbulent and pressure transport throughgradient transport hypothesis. Some explicit pressure trport models have been constructed~Mellor and Herring;41

Lumley;42 Fu;43 Demuren, Lele, and Durbin;44 and Demuren,Rogers, Durbin, and Lele45!, but little effort has been madto incorporate these models into standard RSM’s or Pmethods.

It is the purpose of this paper to study the applicationthe WVM to inhomogeneous turbulence. The temporal shlayer was selected as a representative boundary-free sflow for which good DNS data exist. The velocity wavvector model was constructed in Van Slooten and Pope,11 butonly tested in cases of homogeneous turbulence. The tulent frequency model of Jayesh and Pope15 has also seenlimited testing and is demonstrated to be crucial in maintaing the coherency of the shear layer. A surprising prelimnary discovery was the demonstrated importance of the psure transport in PDF methods. A model for presstransport suitable for PDF methods is constructed from Luley’s model for the pressure-velocity correlation~Lumley42!and a simple scaling analysis. The results of the WVM wan explicit pressure transport model are presented and cpared to standard velocity turbulent-frequency models,simplified and generalized Langevin models~Pope3!.

This work begins in Sec. II which contains backgrouinformation on the Reynolds stress equation, a brief intduction to turbulent shear layers, and a description ofbasic concepts involved in PDF methods. In Sec. III,turbulent frequency model of Jayesh and Pope15 is presented.Although it has been used previously in Dreeben and Po46

and Delarue and Pope,20,47 this is the first publication inwhich the basis for the model is fully described. The WVand the generalized Langevin model are described in SIV A and Sec. IV B, respectively, while the pressure tranport model is constructed in Sec. IV D. Preliminary resuthat motivate the pressure transport modeling are presein Sec. IV C, while the results with the pressure transpmodel are presented and discussed in Sec. IV E. Finally,conclusions of this work are summarized in Sec. V.

II. BACKGROUND

A. Reynolds stress equations

The Navier–Stokes equation governs the behaviorturbulent flows, and is used as the basis for turbulence m

Phys. Fluids, Vol. 10, No. 1, January 1998


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eling. The Eulerian velocity,U(x,t), and the modified pressure,P(x,t), are the dependent variable of interest for inflows. The Reynolds decomposition expresses these varias the sum of their mean and fluctuating parts,U5Û&1uand P5^P&1p, and is used with the Navier–Stokes eqution to derive the governing equation for the Reynolds strtensor,ûiuj&

Dûiuj&

Dt[

]ûiuj&

]t1Ûl&

]ûiuj&

]xl

5Pi j 1Ti jt 1Ti j

p 1P i j 2e i j . ~1a!

The symbolic terms are: production,Pi j ; turbulent transport,Ti j

t ; pressure transport,Ti jp ; pressure–rate-of-strain correla

tion, P i j ; and dissipation tensor,e i j ; while the viscoustransport has been neglected due to a high Reynolds numassumption. The definitions of the terms are

Pi j [2ûiul&]Û j&

]xl2ûjul&

]Ûi&]xl

, ~1b!

Ti jt [2

]ûiujul&]xl

, ~1c!

Ti jp [2

]ûip&]xj

2]ûj p&

]xi, ~1d!

P i j [ K pS ]ui

]xj1

]uj

]xiD L , ~1e!

and

e i j [2nS ]ui

]xk

]uj

]xkD52eS ei j 1

1

3d i j D , ~1f!

wheree andei j are the dissipation and the normalized dispation anisotropy, respectively.

A Poisson equation governs the behavior of the flucating pressure

~2!

where the source terms are labeled as ‘‘rapid’’ or ‘‘slowdepending on their response to a step change in the mvelocity gradient. The rapid term is proportional to the grdient, while the slow term is only indirectly related to it. Thallows the fluctuating pressure to be split into rapid and slparts due to superposition in the linear Poisson equation.pressure–rate-of-strain correlation is also split into rapid aslow parts

P i j [P i j~r !1P i j

~s! , ~3!

which are modeled separately.The dissipation anisotropy is often neglected with an

sumption of local isotropy at high Reynolds numbers, bumay be combined and jointly modeled with the slo

247Van Slooten, Jayesh, and Pope


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pressure–rate-of-strain correlation. The combined term isbeled the return-to-isotropy tensor~Lumley and Newman48!

f i j [2P i j~s!/e12ei j . ~4!

The rapid pressure–rate-of-strain correlation and the retto-isotropy tensor are also often combined to yield the retribution term

F i j [P i j~r !2ef i j 5P i j 22eei j , ~5!

which represents the nontransport, intercomponent entransfer. Although the rapid pressure–rate-of-strain corrtion and the return-to-isotropy tensor are modeled in sepamanners, their joint behavior as expressed by the redistrtion term is useful in comparisons with experimental or DNdata.

The gradient of the Reynolds stress tensor occurs inReynolds-averaged Navier–Stokes equation, which ismanifestation of the ‘‘closure’’ problem of turbulence. IRSM’s the Reynolds stress equation is closed using mofor the turbulent and pressure transports, the rapid and spressure–rate-of-strain correlations, and the tensor disstion, while production is in closed form. A difference in thPDF equation is that the term corresponding to turbultransport is in closed form.

In the governing equation for the turbulent kinetic eergy,k[ 1

2ûlul&, there are no terms related to the redistribtion term, but both turbulent and pressure transport havetropic parts. One half the trace of Eq.~1a! yields

Dk

Dt5P1Tt1Tp2e, ~6!

whereP[ 12Pll , Tt[ 1

2Tllt , andTp[ 1

2Tllp are the production,

turbulent transport, and pressure transport of turbulentnetic energy, respectively.

B. PDF formulation

PDF methods are turbulence models where closurecurs at the level of a one-point, one-time joint PDF. Hoever, their implementation is typically achieved through schastic models for the behavior of fluid particles. Trelationships between these ideas is shown schematicalFig. 1, and detailed in the remainder of this section.

The Eulerian PDF of velocity at a specific position atime, f (V;x,t), is the probability density of the Eulerian velocity, U(x,t), being in an infinitesimal volume,dV, at thesample-space locationV, i.e., f (V;x,t)dV[Prob$Vi

<Ui(x,t),Vi1dVi%. The governing equation for this PDunder the assumption of constant fluid properties is derivia standard approaches~Pope1!,

] f

]t1Vl

] f

]xl5n

]2f

]xl]xl1

]^P&]xl

] f

]Vl1

]

]VlF f K ]p

]xlUVL G

2n]2

]Vl]VmF f K ]Ul

]xn

]Um

]xnUVL G , ~7!

where in the final two terms conditional expectationsdefined that are not closed. The notation for the conditioexpectations has been simplified, e.g.,

248 Phys. Fluids, Vol. 10, No. 1, January 1998


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K ]p

]xlUVL [ K ]p

]xlUU~x,t !5VL , ~8!

and similar simplifications are used throughout this workA fluid particle is a notional particle which is specifie

by its location,X1(t), and velocity,U1(t), which equals thelocal Eulerian velocity,U1(t)5U(X1@ t#,t). The particlesmove with their velocity which evolves by the NavierStokes equation

dXi1

dt5Ui

1 ~9a!

and

dUi1

dt52F]^P&

]xi1

]p

]xiG

x5X1~ t !

1nF]2Ûi&]xj]xj

1]2ui

]xj]xjG

x5X1~ t !

. ~9b!

Because the fluid particles are an exact representation oNavier–Stokes equation, the Eulerian PDF of velocity thaderived from the fluid particles is equivalent to Eq.~7!. Thedifficulty in applying either formulation is that neither thgradient of the fluctuating pressure nor the derivatives offluctuating velocity are known, which is another manifestion of the closure problem.

The standard approach in PDF methods is to modelfluid particle’s evolution by stochastic diffusion equatiofor position, X* (t), and velocity, U* (t). The simplifiedLangevin model~SLM! is the basic model

dXi* 5Ui* dt ~10a!

and

dUi* 52]^P&]xi

dt2F1

21

3

4C0G@Ui* 2Ûi&#

dt

t

1AC0k

tdWi , ~10b!

wheret is a turbulence time scale,C0 is a model parameteranddW is an increment of an isotropic Wiener process. T

FIG. 1. Schematic of relationships involved in PDF methods: —, Maematical analysis; ---, turbulence modeling;••• , numerical modeling.

Van Slooten, Jayesh, and Pope


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time scale is commonly defined ast[k/e, which is used inthe remainder of this section. The dissipation is calculathrough either a modeled mean equation or an additioparticle property as in Sec. III.

The modeled Lagrangian PDF of velocity and positiconditional on the initial particle conditions,U* (t0)5V0 andX* (t0)5x0 , is denoted byf L* (V,xuV0 ,x0 ;t). Its evolution isderived from the stochastic system, Eq.~10!, via standardmethods~Pope1,2,49!

] f L*

]t1Vl

] f L*

]xl5

]^P&]xl

] f L*

]Vl1S e

kD F1

21

3

4C0G

3]

]Vl@ f L* ~Vl2Ûl&!#1

1

2C0e

]2f L*

]Vl]Vl.

~11!

This Lagrangian PDF is related to the modeled EulerPDF, f * (V;x,t), through an integral over all possible initiaconditions~Pope1,2!

f * ~V;x,t !5E E f L* ~V,xuV0 ,x0 ;t ! f * ~V0 ;x0 ,t !dV0dx0 .

~12!

Because the integrals overV0 and x0 are independent ofVandx, the governing equation for the Eulerian PDF is idetical to Eq.~11!

] f *

]t1Vl

] f *

]xl5

]^P&]xl

] f *

]Vl1S e

kD F1

21

3

4C0G

3]

]Vl@ f * ~Vl2Ûl&!#1

1

2C0e

]2f *

]Vl]Vl.

~13!

Thus, Eq.~13! is the model for the exact Eulerian PDFvelocity equation, Eq.~7!, as derived from the SLM.

The Reynolds stress equation derived from Eq.~13! is

Dûiuj&

Dt52ûiul&

]Û j&

]xl

2ûjul&]Ûi&

]xl

2]ûiujul&

]xl

2~213C0!ebi j 22

3ed i j , ~14!

where bi j [ûiuj&/2k2 13d i j is the Reynolds stress aniso

ropy. The turbulence modeling of the redistribution termthe SLM is then equivalent to Rotta’s model~Rotta50!. Moresophisticated Reynolds stress models and their relationsto PDF methods are discussed in Sec. IV B and Pope.3

C. Shear layers

Boundary-free shear layers are fundamental flows instudy of turbulence and have received considerable attenboth theoretically and experimentally. One of the ‘‘simplest’’ of these flows is the temporal mixing layer whichequivalent to the plane mixing layer in the limit as the vlocity ratio approaches one. The flow consists of a statically one-dimensional, turbulent mixing layer that grows btween two semi-infinite laminar streams of equal a



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opposite velocities~see Fig. 2!. Despite the geometrical simplicity, the flow contains many important physical phenomena which are representative of more complicated free sflows; such as the generation of turbulence through a mvelocity gradient and a turbulent–nonturbulent interfawhere intermittency plays an important role.

The theoretical analysis that predicts self-similar sotions for mixing layers~Townsend51! is confirmed by experi-mental data in the plane mixing layer case~e.g., Bell andMehta28! and numerical simulations on the temporal shelayer~Rogers and Moser34!. For the temporal shear layer, thconstant mean velocity difference between the two streaDU, and the momentum thickness

dm[2E0

`F1

42S Û&

DU D 2Gdy, ~15!

are used as the characteristic velocity and length scalesspectively. The similarity variablesj, U(j), andt i j (j) aredefined by

j[y/dm~ t !, ~16a!

U[Û&/DU, ~16b!

and

t i j [ûiuj&/DU2. ~16c!

Substituting the similarity variables into the definitiofor the momentum thickness, Eq.~15!, provides a strong in-tegral constraint on the normalized velocity

25E0

`

@124U2#dj, ~17!

which proves useful in understanding the impact of variomodels on the mean velocity.

The similarity form of the mean velocity equation for thtemporal shear layer contains one constant, the layer grorate

r[1

DU

ddm

dt, ~18!

and is expressed as

r jdU

dj5

dt12

dj. ~19!

FIG. 2. Sketch of the initial mean velocity profile in a temporal mixinlayer.



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From the definitions of the layer growth rate, Eq.~18!, themomentum thickness, Eq.~15!, and the similarity variablesEq. ~16!, an additional integral constraint is derived,

r 54E0

`

U*dt12

djdj524E

0

`

t12

dU

djdj

52S dm

DU3D E0

`

P11~j!dj, ~20!

which shows interesting relationships between variablesare useful in discussing model results.

The Reynolds stress and turbulent kinetic energy eqtions in similarity form are

2jrdt11

dj522t12

dU

dj2

1

DU3

dû2v&dj

1S dm

DU3DF11

22

3 S dm

DU3D e, ~21a!

2j r

dt22

dj52

1

DU3

d^v3&dj

22

DU3

d^vp&dj

1S dm

DU3DF22

22

3 S dm

DU3D e, ~21b!

2jrdt33

dj52

1

DU3

d^vw2&dj

1S dm

DU3DF33

22

3 S dm

DU3D e, ~21c!

2jrdt12

dj52t22

dU

dj2

1

DU3

dûv2&dj

22

DU3

dûp&dj

1S dm

DU3DF12, ~21d!

and

2jrdk

dj52t12

dU

dj2

1

2DU3

d^q2v&dj

21

DU3

d^vp&dj

2S dm

DU3D e, ~21e!

whereq2[ulul is twice the energy of the fluctuating velocitand u5(u,v,w) defines this velocity in the stream-wiscross-stream, and span-wise directions, respectively.Reynolds shear stress,ûv&, directly affects the mean velocity through its gradient, while all the other nonzero Reynostresses only indirectly affect the mean velocity. Of thethe cross-stream normal stress has the greatest impacappearing in the production term for the shear stress. Protion also occurs in Eq.~21a! for the stream-wise normastress and Eq.~21e! for the kinetic energy, while pressurtransport occurs in Eq.~21b! for the cross-stream normastress, Eq.~21d! for the shear stress, and Eq.~21e! for thekinetic energy.



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III. TURBULENCE FREQUENCY MODEL

Turbulence modeling requires information on the lengor time scale of the turbulence, which is not providedmodeled evolution equations for the turbulent kinetic enerReynolds stress tensor, or velocity PDF. To overcomedeficiency, these equations are commonly supplementedmodeled evolution equation for the mean dissipation~Laun-der and Spalding52! or the mean turbulence frequenc~Wilcox53 and Menter54!. The extension of velocity PDFmethods to include a particle turbulent frequency model isalternative approach, which has two main benefits overmodels for mean quantities. A closed turbulence modeachieved entirely within the PDF formulation, and the adtional information contained in the joint PDF of velocity anturbulent frequency is available for more realistic modeliin complex turbulent flows.

The velocity turbulent-frequency models that were intrduced by Pope and Chen13 and Pope14 for homogeneous andinhomogeneous turbulence, respectively, are based on cparisons with the statistical behavior of the instantanedissipation. At high Reynolds number Kolmogorov55 andObukhov56 have argued that the instantaneous dissipatiolog-normally distributed. Other researchers have criticizthis hypothesis~e.g., Mandelbrot57!, but with respect to loworder moments, such as the mean velocity and Reynstress tensor, experimental data~see e.g., Monin andYaglom58! and DNS data of stationary isotropic turbulen~Yeung and Pope59! have confirmed that a log-normal distrbution is a reasonable approximation.

In Pope and Chen,13 a stochastic model for the particlturbulent frequency is constructed which yields a log-normdistribution for the modeled instantaneous dissipation. Teffects of internal intermittency are modeled in a natufashion by using the particle turbulent frequency to detmine the turbulent time scale in an altered equation forparticle velocity~refined Langevin model!. Ad hoc modelsare also introduced to treat the entrainment of nonturbufluid and external intermittency~Pope14!. The final form ofthe model has been tested in free shear flows~Pope14 andMinier and Pozorski18! and swirling flows~Anand, Pope, andMongia60!. While these results are fairly good, the turbulefrequency model is quite complicated and computationaexpensive to implement as is the refined Langevin modelto the modeling of the internal intermittency. The long taof the underlying log-normal distribution also hamper tnumerical Monte Carlo solutions through large statistifluctuations. In addition, thead hoc terms are representinimportant physical phenomena.

The model presented in this section is from JayeshPope15 and represents a significant simplification from tprevious particle turbulent frequency models. A computionally efficient model is constructed that treats the entrament of nonturbulent fluid and external intermittency inrealistic manner. A conditional mean frequency is usedspecify the turbulent time scale rather than the individparticle turbulent frequency. This neglects the minor effeof internal intermittency, but a strict adherence to the lonormal distribution is now not necessary allowing the schastic equation to be greatly simplified. Also, with the tim



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scale specified by a mean quantity, alterations to standparticle velocity equations such as in the refined Langemodel are not necessary.

The particle turbulent frequency,v1, is modeled by astochastic turbulent frequency,v* , which in statistically-stationary, homogeneous turbulence is a stationary diffusprocess

dv* 52~v* 2^v&!dt

T1A2s2^v&v*

TdW, ~22!

whereT is a specified time scale ands2 is the normalizedvariance,s2[var(v* )/^v&2. For computational efficiencyand ease of analysis, the drift term has been specifielinear in v* , with a zero mean to give a stationary proceThe determining factor for the shape of the PDF is thendiffusion term which is also chosen to be linear inv* and isdesigned to yield a stationary variance. This selection issimplest model that forces non-negative values ofv* .

From the Fokker–Planck equation written for the diffsion process in Eq.~22!, the stationary PDF of turbulenfrequency,f v(u), is derived

f v~u!51

G~1/s2!@s2^v&#21/s2

u1/s221 expH 2u

s2^v&J ,

~23!

whereG(1/s2) is the gamma function. This PDF is a gammdistribution with mean^v* &5^v& and variance var(v* )5s2^v&2. A modest value of the variances251/4 is speci-fied to minimize numerical errors associated with large fltuations, while at the same time it produces a distributwith a reasonable width. The DNS data of Yeung and Pop59

show thats2 is actually Reynolds number-dependent ws2;1 for a range of Taylor-scale Reynolds numbeRl;38–93. In Fig. 3, the PDF of turbulent frequency for tgamma distribution withs251/4 is compared to log-normadistributions withs251/4 ands251. All three distributionshave the same mean. The gamma distribution is shown ta fair representation of the log-normal distribution of tsame variance, and to have much shorter tails than the o

FIG. 3. PDF’s of turbulent frequency withv&51 and for: —, Gammadistribution with s251/4; ---, log-normal distribution withs251/4; -•-,log-normal distribution withs251.



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log-normal distribution. The purpose of this model is to prvide a time scale to the particle velocity equation whichaccomplished through a conditional mean property. An exrepresentation of the distribution is not necessary.

In nonstationary, homogeneous turbulence, there isource of turbulent frequency,Sv , that corresponds to thesource term in the dissipation equation. To account for tcrucial term, the stochastic model is extended to

dv* 52~v* 2^v&!dt

T2^v&v* Svdt

1A2s2^v&v*

TdW. ~24!

The evolution of the mean turbulent frequency in homogneous turbulence follows from Eq.~24!

d^v&dt

52^v&2Sv . ~25!

If the turbulence time scale is specified using the meanbulent frequency,t[k/e51/ v&, the source of turbulenfrequency is directly related to the source in the standdissipation model. The result is

Sv5~Ce221!2~Ce121!P

k^v&, ~26!

where Ce2 and Ce1 are the standard model constants. Aalternative model is selected for this work and is construcby specifying the ‘‘production’’ in the form used byk-emodels, which yields

Sv5C22C1

Si j Si j

^v&2 , ~27!

where Si j [12@]Ûi&/]xj1]Û j&/]xi # is the mean rate of

strain tensor. The constant model parameters areC150.08and C250.92 in accordance with the standard dissipatmodel.

The ability to model realistically the effects of externintermittency is one of the benefits of particle turbulent frquency models. The intermittency factor,g, characterizesthis behavior by defining the probability of the fluid beinturbulent at a certain position in the flow. The turbulent frquency in nonturbulent fluid is zero which allows the PDFv1 to be expressed as

f v~u!5~12g!d~u!1g f T~u!, ~28!

where f T(u)is the PDF ofv1 conditional on the particlebeing turbulent. In the intermittent region, the mean turblent frequency, v&, and the turbulent-mean turbulent frequency

^v&T[E0

`

u f T~u!du, ~29!

are both relevant and are related by^v&T5^v&/g. It is rea-sonable to suppose that^v&T represents the ‘‘rate’’ of turbu-lent processes in the turbulent fluid better than^v&. Thissupposition is supported by the results found in Norris.16



he

o-

c

owc

ic

re

nthi

st

l-bu

iththth

Inn

tuh

alfrepa

the

ame

ticearin-thetheim-ern-

eshethrgyta,the-nd

fre-,-

In both experimental and modeling applications, tdelta function in Eq.~28! will be smeared. An alternativedefinition for the turbulent-mean turbulent frequency as

^v&T5Ev0

`

u f ~u!duY Ev0

`

f ~u!du, ~30!

wherev0.0 is a small threshold, may be used, but it intrduces the difficulty of specifyingv0 . Instead, a well definedstatistic with a similar behavior is defined

V[CV^v* uv* >^v&&, ~31!

and referred to as the conditional-mean turbulent frequenThe constantCV is chosen so thatV equals^v& in a fullyturbulent [email protected]., when Eq.~23! gives the distribution ofthe PDF#. Applying this definition gives

CV5pG~p,p!

G~p11,p!'0.6893, ~32!

where G(p,p) is the incomplete Gamma Function andp[1/s2 is set to 4.0 as before.

The time scale for the turbulent frequency model is nspecified using the conditional-mean turbulent frequenT21[C3V, which yields the final form of the stochastmodel

dv* 52C3~v* 2^v&!Vdt2^v&v* Svdt

1A2C3C4^v&Vv* dW, ~33!

whereSv is from Eq.~27! and the new model parameters aC351.0 andC4[s250.25. The parameterC3 is specifiedthrough comparisons with the DNS data of Yeung aPope,59 and experience has shown that the results fortemporal shear layer case are relatively insensitive tovalue. ~Dreeben and Pope46 used a value ofC355.0 in anapplication to a wall bounded flow.! In Eq. ~33!, C3V speci-fies the rate of the stochastic process, while^v& maintains thecorrect evolution of the mean and variance of the stochamodel.

The particle velocity models also utilize the conditionamean turbulent frequency to specify the characteristic turlence rate asV instead of v&. The particle velocity equationfor the SLM model becomes

dUi* 52]^P&]xi

dt2VF1

21

3

4C0G~Ui* 2Ûi&!dt

1AC0VkdWi . ~34!

The effects of specifying the turbulent time scales wthe conditional-mean turbulent frequency are tested incase of a temporal shear layer by comparing results fromSLM with t215V, Case A, and witht215^v&, Case B.The complete directory of cases run is listed in Table I.Fig. 4, a scatter plot of the particle turbulent frequency across-stream position for Case A shows the bimodal naof the fluid at the intermittent edges of the shear layer. Tmean and conditional-mean turbulent frequencies areshown. At the edges, the conditional-mean turbulentquency clearly represents the behavior of the turbulent



y.

y,

dets

ic

-

ee

dreeso-r-

ticles better than the mean turbulent frequency, while incenter of the layer these means are nearly equal.

The stream-wise velocity distribution in the cross-stredirection is illustrated in the scatter plot of Fig. 5, while thprofiles of normalized mean velocity and turbulent kineenergy are shown in Fig. 6. From the scatter plots it is clthat the specification of the turbulent time scales greatlyfluences the behavior of the particles. Near the edge ofturbulent layer, the conditional-mean is much larger thanmean turbulent frequency, which increases the relativeportance of the dissipation in this region. The result is fewparticles breaking free from the turbulent region in a nophysical manner. These relatively rare ‘‘rogue’’ particlhardly affect the mean velocity profile, but greatly impact tkinetic energy profile. The mean velocity profiles in boCase A and B are quite good. The turbulent kinetic eneprofile in Case A compares well with the experimental dawhile in Case B there is a poor comparison. Since onlygradients of the shear stress,t12, are fed into the mean velocity equation, it is possible to have Reynolds stress a

FIG. 4. Scatter plot of the normalized particle position (j* [Y* /dm) andturbulent frequency in the case with the conditional-mean turbulentquency model~Case A! at tDU/dm539, compared to the profiles of: —V/^v&m ; ---, ^v&/^v&m ; where^v&m is the maximum mean turbulent frequency across the layer.

TABLE I. Directory of test cases with their growth rates.

CaseVelocitymodel

t21

modelTi j

p

model r

DNSa ••• ••• ••• 0.014Experimental rangeb ••• ••• ••• 0.014–0.022

A SLM V No 0.0187B SLM ^v& No 0.0192C WVM V No 0.0114D LIPM V No 0.0142E LSSG V No 0.0114F WVM V Yes 0.0128G LIPM V Yes 0.0159H LSSG V Yes 0.0131

aFrom Rogers and Moser~Ref. 34!.bTaken from calculations by Rogers and Moser~Ref. 34! of data in Dimo-takis ~Ref. 71!.



iny

Thi

etee

coe

th

Th

bo-ime

e by

y

esenta-T

unitof

in

f

kinetic energy profiles with poor magnitudes and still matain a good mean velocity profile. The profiles of the Renolds stresses and dissipation are shown in Fig. 7.conditional-mean turbulent frequency model in Case Aagain demonstrated to give much better results than Casbut it fails to correctly represent the dissipation or distributhe kinetic energy among the individual Reynolds stressThese problems are due to the simple turbulence modeltained in the SLM and motivate the models presented in SIV.

IV. VELOCITY MODELS

A. Velocity/wave-vector model

The main purpose of this paper is to demonstrateimplementation of the velocity/wave-vector~WVM ! ap-proach to PDF modeling of inhomogeneous turbulence.WVM was first presented in Van Slooten and Pope11 and isbased on an exact particle representation model of RDTKassinos and Reynolds.12 For the energetic scales of homgeneous turbulence, RDT may be applied when the tscale of the turbulence,t, is much greater than that of thimposed mean velocity gradient,S21;i¹Û&i21

FIG. 5. Scatter plot of particle velocity and position with:~a! Theconditional-mean turbulent frequency~Case A! at tDU/dm539; and~b! themean turbulent frequency~Case B! at tDU/dm536.



--e

sB,

s.n-c.

e

e

y

e

St@1. ~35!

If the homogeneous system is expressed in Fourier spacthe complex velocity mode,u(t), and a time evolving unitwave number vector,e(t), a closed and linear set of ordinardifferential equations

dui

dt52

]Ûm&]xn

un~d im22ei em! ~36a!

and

dei

dt52

]Ûm&]xn

em~d in2ei en!, ~36b!

exists for each Fourier mode. A PDF method based on thFourier space variables is an exact Monte Carlo represetion of RDT, but cannot be directly extended to non-RDhomogeneous and inhomogeneous turbulence.

In the PDF approach taken, the wave vector,e* , is in-troduced as a new particle property that represents thewave number vector. The evolution for the Fourier modesvelocity is used to model the particles’ fluctuating velocityphysical space,u* [U* 2Û&. The justification for this pro-

FIG. 6. Self-similar profiles of:~a! Mean velocity; and~b! turbulent kineticenergy; with: ---, the conditional-mean turbulent frequency~Case A!; -•-,the mean turbulent frequency~Case B!; compared to: —, DNS data oRogers and Moser~Ref. 34!; h, experimental data of Bell and Mehta~Ref.28!.



n
FIG. 7. Self-similar profiles of:~a! Reynolds stresses with the conditional-mean turbulent frequency model~Case A!; ~b! Reynolds stresses with the meaturbulent frequency model~Case B!; and ~c! dissipation in both Case A and Case B; with: —, DNS data of Rogers and Moser~Ref. 34!; ---, SLM withV; -•-, SLM with ^v&.
spammis,e

eeo

irn

pe

e-ionoff a

pre-

tove-theser-

cedure is based on an exact correspondence that existRDT between the stochastic system and a spectral svariable, the directional spectrum. The directional spectruG i j (e), is defined as the integral of the velocity spectruover the magnitude of the wave number vector, so that itfunction of the unit wave number vector in a fixed spacee.The stochastic variable that illustrates the correspondencdefined as

L i j* ~h![E ViVj f ue* ~V,h!dV

5ûi* uj* ue* 5h& f e* ~h!↔G i j ~e!, ~37!

whereh is the sample-space variable fore* , f ue* (V,h) is thejoint PDF of velocity and wave vector, andf e* (h) is themarginal PDF of the wave vector. The result of this procdure is an ‘‘exact’’ PDF model for RDT in physical spacwhich is extendible to more complicated flows. The levelclosure for the rapid pressure–rate-of-strain correlationthese models is not at the Reynolds stresses but at the dtional spectrum. A detailed exploration of the correspodences to RDT is presented in both Van Slooten and Po11

and Kassinos and Reynolds.12



force,

a

is

-

finec--

The first extension of the model is to non-RDT homogneous turbulence through the addition of stochastic diffusand drift terms that are designed to model the effectsdissipation and the return-to-isotropy tensor at the level oReynolds stress closure. Four models of this type weresented in Van Slooten and Pope11 with the Langevin velocitymodel being selected for this work. In a further extensioninhomogeneous turbulence, the particle position and fulllocity are required to account for the spatial variations inflow, and additional terms are added to enforce the convation of mean momentum. The final equations are

dXi* 5Ui* dt, ~38a!

dUi* 52]^P&]xi

12]Ûm&

]xn@ei* em* un* 2êi* em* un* &#dt

2VF1

21

3

4auGui* dt1gV@bi j 2bmnbmnd i j #uj* dt

1AauVkdWi , ~38b!

and



alic

h-te

uno

hc

he

ed

ththn,

trapd

mjel

d

or

of

ree

eoft

dof

dei* 52]Ûm&

]xnem* @d in2ei* en* #dt

21

2VFae1au

k

us* us*Gei* dt2gV@d i j 2ei* ej* #

3bjl el* dt2AauVkui* el*

us* us*dWl1AaeV

3Fd i l 2ei* el* 2ui* ul*

us* us*GdWl81Qi , ~38c!

whereV has been used to specify the turbulence time scdW and dW8 are increments of two independent, isotropWiener processes, and (au ,ae ,g) are model parameters witthe values of~2.1, 0.03, 2.0!. These equations for inhomogeneous flow are identical to those developed by Van Slooand Pope11 with the exception of the vectorQ in Eq. ~38c!.The continuity equation in Fourier space states that thewave number vector is orthogonal to the Fourier modevelocity, e–u50. The WVM must maintaine* –u* 50 toprovide realizable models for the directional spectrum. Tvector Q represents a model for the inhomogeneous effein the evolution of the unit wave vector by maintaining torthogonality betweene* andu* through a projection.

The Reynolds stress equation for the WVM is derivfrom Eq. ~38b!,

Dûiuj&

Dt52ûiul&

]Û j&

]xl

2ûjul&]Ûi&

]xl

2]ûiujul&

]xl

22

3ed i j 12

]Ûn&

]xm

@êienujum&1êjenuium&#

1~213au!bi j 24gF S 1

32bmnbmnD bi j

1S bil bl j 21

3bmnbmnd i j D G , ~39!

which is not closed due to the fourth-order correlations infifth term on the right hand side. The difference betweenWVM and the SLM in the derived Reynolds stress equatiooccurs in the model for the redistribution term. In the WVMthere is a spectral model for the rapid pressure–rate-of-scorrelation and a nonlinear model for the return-to-isotrotensor. The redistribution term in the SLM only corresponto Rotta’s model for the return-to-isotropy tensor~see Sec.II B !.

B. Generalized Langevin model

The rapid pressure–rate-of-strain correlation is a donant term in many turbulent flows and has been the subof numerous modeling efforts at the Reynolds stress leveclosure~Naot, Shavit and Wolfshtein;61 Launder, Reece, anRodi;37 Shih and Lumley;62 Haworth and Pope;63 Fu, Laun-der, and Tselepidakis;64 Speziale, Sarkar, and Gatski;65 Jo-hansson and Hallba¨ck;66 and Ristorcelli, Lumley, andAbid67!. By modeling the rapid pressure–rate-of-strain c



e,

n

itf

ets

ees

inys

i-ctof

-

relation, these RSM’s are better than the SLM as sourcecomparison for the WVM. Lagrangian models~i.e., velocityPDF methods! that correspond to some of the RSM’s adeveloped in Pope.3 The particle velocity equation is thgeneralized Langevin model~GLM!, a minor extension ofthat from Haworth and Pope,63

dUi* 52]^P&]xi

dt1Gi j @U j* 2Û j&#dt1AC0VkdWi

~40a!

where

Gi j [V~a1d i j 1a2bi j 1a3bi j2 !1Hi jkl

]Ûk&]xl

, ~40b!

and

Hi jkl 5b1d i j dkl1b2d ikd j l 1b3d i l d jk1g1d i j bkl

1g2d ikbjl 1g3d i l bjk1g4bi j dkl1g5bikd j l

1g6bil d jk . ~40c!

The 12 model coefficients~a,b,g! are allowed to depend onthe scalar invariants ofbi j , Si j , and Wi j [

12@]Ûi&/]xj

2]Û j&/]xi #, the rotation rate tensor.Two forms of the GLM are used for comparison with th

results from the WVM. The Lagrangian isotropizationproduction model~LIPM! is a Lagrangian version of the firsmodel of Launder, Reece, and Rodi37 which is a combinationof the isotropization of production model~IPM! by Naot,Shavit, and Wolfshtein61 and Rotta’s model. The seconform of the GLM is an approximate Lagrangian versionthe Speziale, Sarkar, and Gatski65 model~SSG! and is called

TABLE II. Definition of model coefficients for SLM, LIPM, and LSSG.

Coefficient SLM LIPM LSSG

a1 2(121

34C0) Eq. ~41! Eq. ~42!

a2 0 3.5 4.021.7P

e

a3 0 23a234C2

SSG23a2

b1 0 20.2 20.2

b2 0 12(C2

IPM11)3

8~C3

SSG2C38SSGb8!10.5

b3 0 12(C2

IPM21)3

8~C3

SSG2C38SSGb8!20.5

g1 0 0 0

g2 0 0 0

g3 0 0 0

g4 0 0 0

g5 0 32(12C2

IPM) 322

34C5

SSG

g6 0 232(12C2

IPM) 2321

34C5

SSG

Constants C052.1 C052.1, C2IPM50.6 C052.1, C2

SSG54.2,C3

SSG50.8,

C3SSG51.0,

C5SSG50.4



FIG. 8. Scatter plot of particle velocity and position for:~a! Case C~WVM ! at tDU/dm555.5; ~b! Case D~LIPM! at tDU/dm555.6; and~c! Case E~LSSG!at tDU/dm554.3.

hedi

G

-G

h

n

hees

es

rallds

al-

per-the

nstsonAilarig.

rer-the

to

the LSSG model. The specific coefficients for the SLM, tLIPM, and the LSSG model are listed in Table II. The adtional terms in the LIPM are

a152~ 121

34C0!13a2bll

3 1 12C2

IPM P

kV, ~41!

wherebll3[bl j bjkbkl , and the additional terms in the LSS

model are

a152S 1

21

3

4C0D2

1

4C2

SSGbll2 1S 3a22

3

4C2

SSGDbll3

13

8~C3

SSG2C3*SSGb8!

P

kV, ~42!

wherebll2[bl j bjl and b8[Abll

2 . Also, the value of the parameterC3*

SSG is altered from 1.3 to 1.0, so that the LSSmodel ~an approximation of the SSG model! better matchesthe experimental data in the case of a homogeneous slayer.

C. Preliminary results

The velocity models are studied through comparisobetween DNS data of a temporal shear layer~Rogers and



-

ear

s

Moser34! and experimental data of a plane mixing layer~Belland Mehta28!. The first test of the models is to evaluate tgrowth rates,r , which are presented in Table I. The valufrom the SLM ~Case A!, 0.0187, and the LIPM~Case D!,0.0142, lie within the experimental range, while the valufrom the WVM ~Case C! and the LSSG model~Case E! aretoo low at 0.0114. The growth rate is related to the integacross the layer of the product of the self-similar Reynoshear stress and mean velocity gradient which is the normized stream-wise production term, Eq.~20!. This makes thegrowth rate an insensitive measure of a model’s detailedformance, but rather an indicator of the magnitudes ofReynolds shear and stream-wise normal stresses.

The three scatter plots of stream-wise velocity agaicross-stream location in Fig. 8 show a qualitative comparito that of the SLM in Fig. 5 with a few rogue particles.quantitative comparison is made by studying the self-simprofiles for the mean velocity and Reynolds stresses in F9. In comparison with the results from the SLM~Fig. 6!, themean velocities from the WVM, LIPM, and LSSG model asignificantly worse than that from the SLM! This is a suprising result considering the crude turbulence model inSLM and the constrained nature of the mean velocity duethe integral in Eq.~17!.



FIG. 9. Self-similar profiles of:~a! Mean velocity;~b! Reynolds stresses for Case C~WVM !; ~c! Reynolds stresses for Case D~LIPM!; and ~d! Reynoldsstresses for Case E~LSSG!; with: —, DNS data of Rogers and Moser~Ref. 34!; h, experimental data of Bell and Mehta~Ref. 28!; ---, WVM; -•-, LIPM; ••• ,LSSG.

tatewolssre

eahean

senbud

th

t-utre

dst

ayer.thet,antenen-er

esFig.to

for-mi-ear

ulentear

ortral

ela-e

The Reynolds stress profiles are examined to understhe nature of the problem. As predicted by the growth rathe magnitudes of the peak Reynolds shear and stream-normal stresses from the SLM in Fig. 7 are greater than thfrom the LIPM, WVM, or LSSG model. These three modeshow a more satisfactory comparison with the peak valuethe Reynolds stresses from the DNS data, but they shasimilar problem. They all maintain too much Reynolds shstress and turbulent kinetic energy at the edges of the slayer, which results in a low gradient of the shear stressa poor mean velocity profile.

The relative magnitudes of the Reynolds normal stresare an accurate measure of the performance of the turbulmodel, because the primary source of distributing the turlent kinetic energy among the Reynolds stresses is the retribution term ~i.e., the turbulence model!. For the centralpart of the shear layer, the data in Fig. 9 and profiles ofReynolds stress anisotropy~not shown! illustrate that theWVM performs slightly better than the LIPM and much beter than the LSSG model. The LSSG model fails to distribenough energy from the stream-wise normal stress, wheis produced, to the other normal stresses.

A further examination of the behavior of the Reynolshear stress through its budget is warranted to determine



nds,isese

ofa

rard

esce-

is-

e

eit

he

source of the excess energy at the edges of the shear lSince all three models suffer the same problem, onlyWVM data are shown. In Fig. 10~a!, the budget terms thaare treated explicitly,P12 and T12

t , are shown separatelywhile the modeled terms are combined into one resultmodel,F121T12

p . The sources of the discrepancies betwethe DNS and modeled profiles are now isolated to deficicies in this resultant term and/or to low Reynolds numbeffects in the DNS. The resultant model from the WVM donot match the broadness of the peak of the DNS data. In10~b!, the source of the broad peak in the DNS is shownbe the pressure transport, which is neglected in the PDFmulation. The pressure transport is actually one of the donant terms at the edge of the layer where it removes shstress. The pressure transport nearly balances the turbtransport transport across the entire layer for both the shstresses and the cross-stream normal stresses~not shown!. Inthe following sections the modeling of the pressure transpis demonstrated to be crucial in simulating the temposhear layer via PDF methods.

D. Pressure transport

The modeling of pressure transport has received rtively little attention in the construction of RSM’s. Som



fu

d

d

m

va

d

uddo

port.

eat-

surednglethea aurelingientearbal-e ofral

atin

od-port

tedn,

in

s

e

ar-

omanki-

ts abu-

yeress

eticre-by

bu,

researchers have derived models based on an assumedfor a transport term at the Reynolds stress level of clos~Daly and Harlow;35 and Mellor and Herring41!, but theyeither neglect the term in their calculations or fail to aequately test the performance of the model. Lumley42 de-rived an expression for the velocity-pressure correlation

^pui&52 15ûiukuk&, ~43!

which implies a model for the pressure transport

Ti jp 5CptF]ûiukuk&

]xj1

]ûjukuk&]xi

G , ~44!

where the numeric constant is here allowed to be a moparameter,Cpt . For Lumley’s modelCpt is 1/5. Fu43 usedthis formulation and models for the triple correlation froDaly and Harlow35 and Hanjalic´ and Launder36 to constructpressure transport models for a RSM. The models achieimproved values for the growth rate of the half-width inshearless turbulence mixing layer~experimental data fromVeeravalli and Warhaft68! and the spreading rate in a rounjet ~experimental data from Taulbee, Hussein, and Capp69!.However, without comparison of the Reynolds stress bgets, it cannot be determined if these results are merelyto an improved representation of the turbulent transp

FIG. 10. Self-similar profiles of the normalized Reynolds shear stressget for: ~a! Full budget; and~b! pressure transport examination; with: —DNS data of Rogers and Moser~Ref. 34!; and ---, WVM.



ormre

-

el

ed

-uert

rather than a consequence of modeling the pressure transDemuren, Rogers, Durbin, and Lele45 have proposed ellipti-cal models for pressure transport that show promise in tring the nonlocal nature of these terms.

Other researchers have merely neglected the prestransport~Hanjalic and Launder;36 and Launder, Reece, anRodi37! with an acknowledgment that there is no convincievidence for doing so. More recent reviews by Spezia39

and Launder38 have supported this approach. Therefore,standard RSM’s currently model the turbulent transport vigradient transport hypothesis while ignoring the presstransport. This approach is equivalently viewed as modea combined turbulent and pressure transport via the gradtransport hypothesis. The DNS data for a temporal shlayer show that turbulent and pressure transport nearlyance across the entire layer, which lessens the importancthe combined transport. If this result is applicable to genefree shear flows, it would justify the lack of attention thtransport has received in the RSM literature. However,PDF methods the turbulent transport is treated without meling assumption so that an appropriate pressure transmodel is required to maintain the near balance.

A PDF model for the pressure transport is construcusing Lumley’s model for the velocity–pressure correlatioEq. ~43!. The model is not identical to Eq.~44!, becausethere is no direct way to apply nonconvective transportLagrangian methods~see Appendix!. As an alternative, Eq.~44! is simplified by approximating the partial derivativeusing a length scale,L, and an unit vector,n,

Ti jp 5Cpt@ûiukuk&n j1ûjukuk&ni #/L. ~45!

The new model is fully defined by specifying the ratio of thunit vector to the length scale

ni

L5

1

2k

]k

]xi. ~46!

The corresponding term in the stochastic model for the pticle velocity is

dUi* 5CptFus* us*

2k21G ]k

]xidt1••• , ~47!

which maintains the conservation of mean momentum. Frthis formulation, the pressure transport is modeled asadded force that pushes the high energy particles up thenetic energy gradient, which in a free shear flow represenforce reacting against turbulent excursions into a nonturlent region.

Applying the model in the case of a temporal shear lacorrectly yields the two nonzero terms in the Reynolds strequation

T22p 5Cpt

^vq2&k

]k

]y, ~48a!

T12p 5Cpt

ûq2&2k

]k

]y. ~48b!

At the center line both terms are zero, because the kinenergy gradient is zero by symmetry while the triple corlation is finite and the kinetic energy is nonzero. Also,

d-



FIG. 11. Scatter plot of particle velocity and position transport for:~a! Case F~WVM ! at tDU/dm550.9; ~b! Case G~LIPM! at tDU/dm548.4; and~c! CaseH ~LSSG! at tDU/dm549.8.

io

oheoimor

urhe

Na

nidnddge

is

,

portreeticlesortof

fareenlots

rehet ofs inthe

ayer,yer.earle.s inoth

assuming gradient transport for the velocity triple correlatit is shown that across the layerT22

p >0 andT12p <0. There-

fore, the model acts as an additional dissipation away frthe centerline in the Reynolds cross-stream normal and sstress equations rather than as a true transport term. Hever, as is shown in the next section the model greatlyproves the performance of the PDF models in the tempshear layer.

E. Results and discussion

The velocity models are combined with the presstransport model and tested in the case of the temporal slayer. To complete the model, the parameter,Cpt , is speci-fied through comparisons with the experimental and Ddata for the mean velocity profile. The values chosenCpt

WVM50.12 andCptLIPM5Cpt

LSSG50.2. The SLM is not testedwith the pressure transport model, because the fundameproblem with the SLM is the lack of a model for the rappressure–rate-of-strain correlation. Also, the WVM is fouto be relatively sensitive to the added dissipation at the eof the layer, and a parameter inSv is altered to compensatwith a lower dissipation at the center of the layer,C1

WVM

50.06.



n

marw--

al

ear

Sre

tal

e

The growth rates are compared in Table I, where itshown that the values from the WVM~Case F! and theLSSG model~Case H! are still too low at 0.0128 and 0.0131respectively. The value from the LIPM~Case G! falls withinthe experimental range at 0.0159. The pressure transmodel increases the value for the growth rate in all thcases despite acting as a force that pushes energetic partoward the centerline. This effect of the pressure transpmodel is noticeable in Fig. 11, where the scatter plotsparticle velocity against cross-stream position containfewer rogue particles and a more distinct interface betwthe turbulent and nonturbulent regions than the scatter pin Fig. 8.

The mean velocity profiles as illustrated in Fig. 12 amuch improved over the results from Sec. IV C, but tmodel parameter is selected for this purpose. A better testhe pressure transport model is the Reynolds stress profileFig. 13. The pressure transport model has eliminatedproblem of excess Reynolds stresses at the edge of the lwhile increasing the shear stress at the center of the laThe combination yields an increased gradient of the shstress and consequently an improved mean velocity profi

The increased growth rates are the result of changethe mean velocity and Reynolds shear stress profiles. B



re

ucaa

redtnndtheta,ent

y-theas

ess

anw-istri-this

gresseuc-nyrms

the mean velocity gradient and the Reynolds shear st~i.e., P11! are increased in the rangeujuP(2,4), which by Eq.~17! results in a larger growth rate. The increase in prodtion also yields higher values of the Reynolds normstresses for the LIPM and LSSG models, while the norm

FIG. 12. Self-similar profiles of the mean velocity in: ---, Case F~WVM !;-•-, Case G~LIPM!; ••• , Case H~LSSG!; with: —, DNS data of Rogers andMoser ~Ref. 34!; h, experimental data of Bell and Mehta~Ref. 28!.



ss

-ll

stresses in the WVM were unexpectedly lower. This requithe altered parameter inSv to maintain the correct turbulenkinetic energy for the WVM. The profiles for the dissipatioand turbulent kinetic energy are provided in Fig. 14 acompare with the DNS data quite well. The dissipation atedge of the layer compares very well with the DNS dawhich is strong evidence that the conditional-mean turbulfrequency model is performing correctly.

The distribution of the kinetic energy amongst the Renolds normal stresses is a key test for the models ofredistribution term. The results follow the same patternthose in Sec. IV C. From Fig. 13 and the Reynolds stranisotropy profiles~not shown!, the WVM provides aslightly better representation of the redistribution term ththe LIPM, which is much better than the LSSG model. Hoever, the pressure transport also has an impact on the dbution of the normal stresses and is discussed later insection.

To study fully the behavior of the turbulence modelinat the level of the Reynolds stresses, the Reynolds stbudgets are shown in Fig. 15 for the WVM. Overall thcomparisons with the DNS data are quite good. The prodtion and turbulent transport are treated explicitly and adifferences between DNS are due to other modeled te

FIG. 13. Self-similar profiles of Reynolds stresses in:~a! ---, Case F~WVM !; ~b! -•-, Case G~LIPM!; and ~c! ••• , Case H~LSSG!; with: —, DNS data ofRogers and Moser~Ref. 34!.



FIG. 14. Self-similar profiles of the turbulent kinetic energy and dissipation in:~a! ---, Case F~WVM !; ~b! -•-, Case G~LIPM!; and~c! ••• , Case H~LSSG!;with: —, DNS data of Rogers and Moser~Ref. 34!.

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~low Reynolds number effects in the DNS are also an iss!.The differences between the modeled and DNS data foe,F11, and F22 are relatively minor and likely secondary tthe discrepancies in theF i j 1Ti j

p terms of the Reynoldscross-stream normal and shear stress budgets.

The models in the Reynolds cross-stream normalshear stress budgets are examined independently in FigAs theorized in Sec. IV D, the pressure transport model oacts as a small added dissipation at the edges of the laThe peaks in this dissipation fail to match the DNS datamagnitude, but are at the correct location where presstransport is a dominant term. The pressure pressure tranis not a dominant term in the center of the layer, so tfailing to model the transport to the center is only a mindeficiency. However, a true transport model would increthe relative magnitude of the cross-stream normal Reynstress with a corresponding increase in the productionshear stress. These are both desirable effects to improvcomparisons with the DNS data. It is interesting to note tthe relatively small amount of added dissipation at the eof the shear layer caused by the pressure transport modea tremendous impact on not only the Reynolds stress probut also the mean velocity profiles across the entire laye

Two additional plots that are of interest are the profiof the normalized shear rate parameter,Sk/e, and the mod-



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ofthete

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eled intermittency factor,g, in Fig. 17. The normalized shearate parameter is shown for both the DNS data and the Wmodel and represents the level of distortion in the fluid.peak value of less than six does not indicate rapid distortwhich illustrates that the WVM performs quite well awafrom the RDT limit. The intermittency factor indicates thfraction of time that the fluid at a particular location is tubulent. With V as a surrogate forv&T , the intermittencyfactor is modeled as

g5^v&V

. ~49!

The profile of the intermittency factor from the plane mixinlayer data of Wygnanski and Fiedler22 has been centered anboth the high and low stream sides are presented. The refrom the WVM contain a broader peak than the experimendata, but illustrate an added ability of velocity turbulenfrequency PDF formulations in modeling the intermittent bhavior of the turbulent fluid.

V. CONCLUSIONS

A velocity wave-vector–turbulent frequency PDF mod~WVM ! has been implemented for inhomogeneous turlence and compared with velocity turbulent-frequency P



FIG. 15. Self-similar Reynolds stress budget profiles for:~a! Stream-wise normal stress;~b! cross-stream normal stress;~c! span-wise normal stress; and~d!shear stress; in Case F~WVM ! with: —, DNS data of Rogers and Moser~Ref. 34!; ---, WVM.

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models using the SLM, LIPM, and LSSG model for the pticle velocity. A representative boundary-free shear flow,temporal shear layer, is the test case studied. A detailed cparison of the self-similar profiles of the mean velocity, tubulent kinetic energy, dissipation, Reynolds stresses,Reynolds stress budgets between the models and DNS~Rogers and Moser34! indicates that an additional model fothe pressure transport is required in PDF methods toequately model the flow. A pressure transport model is cstructed which greatly improves the results for all statistmeasured.

The conditional-mean turbulent frequency modelJayesh and Pope15 is an efficient model for the particle turbulent frequency that represents the processes of exteintermittency and entrainment of nonturbulent fluid innatural manner. The model is based on specifying the tulent time scale through a stable representation ofturbulent-mean turbulent frequency, the conditional-meturbulent frequency. In the temporal shear layer,conditional-mean turbulent frequency model with the SLfor the particle velocity is demonstrated to maintain theherency of the layer while improving the profiles for thturbulent kinetic energy and Reynolds shear stress overtest case run without the conditioning. However, the failuof the turbulence modeling in the redistribution term are e



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denced by the poor distribution of energy among the Rnolds normal stresses and by the poor profile for the disstion. The representation of the redistribution term for tSLM corresponds to Rotta’s model for the return-to-isotrotensor and no model for the rapid pressure–rate-of-stcorrelation.

The LIPM and LSSG model represent the redistributiterm with more advanced turbulence models at the Reynstress level of closure, while the WVM models the return-isotropy tensor via a nonlinear RSM and the rapid pressurate-of-strain correlation via a spectral model. The specmodel is based on an exact representation of rapidly distohomogeneous turbulence which is extended to inhomoneous turbulence in Van Slooten and Pope11 ~see also Kassi-nos and Reynolds12 for their particle representation model oRDT which motivated the later work!. In testing these mod-els with the conditional-mean turbulent frequency model ifound that the results for the mean velocities and Reynostresses are worse than those of the SLM due to a failurmodel the pressure transport. In particular, the Reynostresses at the edge of the layer contain too much energ

From the DNS data, the pressure and turbulent transnearly cancel one another across the entire layer anddominant processes at the edge of the layer. In RSM’sjoint pressure and turbulent transport is modeled throug



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gradient transport hypothesis, but due to the cancellatiothe transport the model is of lesser importance for this floIn PDF methods the turbulent transport is treated explicwhich necessitates a model for the pressure transpormaintain the balance. A simplified pressure transport mobased on a model for the pressure-velocity correlation frLumley42 is constructed for PDF methods. Due to the difculty in modeling nonconvective transport in particlLagrangian methods, the model acts as an added dissipat the edges of the shear layer and not as a true transterm. However, the pressure transport is only a dominterm at the edges of the layer, and the results for the mvelocity, dissipation, and Reynolds stresses are greatlyproved for all models.

A good test for comparing the performance of the moels is examining the distribution of the turbulent kinetic eergy among the Reynolds normal stresses. The WVM pvides slightly better results than the LIPM, but both anoticeably better than the LSSG model. However, due tosimple mean velocity gradient and relatively low normalizshear rate involved in the temporal shear layer the expebenefits of the WVM have not been demonstrated. A mcomplex turbulent flow such as swirling jets would be a bter test of the model.

FIG. 16. Modeled terms in the self-similar Reynolds stress budget profor: ~a! Cross-stream normal stress; and~b! shear stress; in Case F~WVM !with: —, DNS data of Rogers and Moser~Ref. 34!; ---, WVM.



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The conclusions involving the importance of the presure transport term are based on comparisons with DNSfor the temporal shear layer. It seems reasonable to expesimilar importance of the pressure transport in other fshear flows, but to conclusively extend the arguments worequire DNS data in other flows at increased Reynolds nubers. In addition, the pressure transport model presentedonly been applied in the temporal shear layer test case.portunities for future work are the applications of this modto more complicated free shear flows and the implementaof elliptical methods into the pressure transport modelingPDF methods.

ACKNOWLEDGMENTS

This paper is dedicated to the memory of Jayesh whwork on the conditional-mean turbulent frequency modepresented in Sec. III. This work was supported in part byDepartment of Defense through the Air Force Graduate Flowship Program, by NASA through the New York SpaGrant Graduate Fellowship Program, and by the Air FoOffice of Scientific Research, Grant F49620-97-1-0126.

sFIG. 17. Self-similar profiles of:~a! Normalized shear rate parameter; an~b! intermittency factor and mean velocity; with: ---, Case F~WVM !; —,DNS data of Rogers and Moser~Ref. 34!; ~h,n!, experimental data ofWygnanski and Fiedler~Ref. 22! for the high and low speed side of a planmixing layer, respectively.



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APPENDIX: PRESSURE TRANSPORT IN PDFMETHODS

The purpose of this appendix is to demonstrate theficulty encountered in the construction of stochastic presstransport models that truly behave as transport terms.fluctuating pressure term in the transport equation forEulerian PDF of velocity, Eq.~7!, is expanded~Pope49! toyield

] f

]t1Vm

] f

]xm2

]^P&]xm

] f

]Vm

5n]2f

]xm]xm1

]2

]Vm]xm@ f ^puV&#1

]2

]Vm]Vn

3F f H K p]Um

]xnUVL 2n K ]Um

]xs

]Un

]xsUVL J G . ~A1!

The terms on the right hand side generate the viscous dsion, the pressure transport, the pressure-rate-of-strain clation, and the dissipation terms in the Reynolds stress etion, respectively. Lumley’s model corresponds to^puV&5 1

5v–v, wherev[V2Û&.In PDF methods, stochastic diffusion equations are c

structed for a Lagrangian particles’ position and velocity.general formulation is

dXi* 5ai~X* ,U* !dt1Ail ~X* ,U* !dWl ~A2a!

and

dUi* 5bi~X* ,U* !dt1Bil ~X* ,U* !dWl

1Cil ~X* ,U* !dWl8 . ~A2b!

The evolution equation for the Lagrangian PDF of positiand velocity,f L* (x,V;t), is derived via standard approach

] f L*

]t52

]

]xm@amf L* #2

]

]Vm@bmf L* #

11

2

]2

]Vm]Vn@~BmlBnl1CmlCnl! f L* #

1]2

]xm]Vn@AmlBnl f L* #1

1

2

]2

]xm]xn@AmlAnl f L* #.

~A3!

The last term in Eq.~A3! and the first term on the right-hanside of Eq.~A1! represent diffusion in physical space. Tmatch the coefficients of these terms requiresA to satisfy

AmlAnl52ndmn . ~A4!

At high Reynolds numbers, whenn is negligible, this forcesA to be zero. Lumley’s model for pressure transport corsponds to

AmlBnl52 15v iv idmn . ~A5!

Clearly, sinceA is forced to be zero, this equation cannotsatisfied. Similar difficulties exist for all other pressure tranport models that are truly transport terms. Therefore,simple extension of the stochastic diffusion equations exthat precisely models the pressure transport.



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Advances in PDF modeling for inhomogeneous …Advances in PDF modeling for inhomogeneous turbulent ﬂows P. R. Van Slooten, Jayesh, and S. B. Pope Sibley School of Mechanical and