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Progress in probability density function methods for turbulent reacting ?ows

D.C.Haworth *

Department of Mechanical and Nuclear Engineering,The Pennsylvania State University,232Research Building East University Park,PA 16802-2321,USA

a r t i c l e i n f o

Article history:

Received 25January 2009

Accepted 9September 2009

Available online 15October 2009

Keywords:

Probability density function method

Filtered density function method

Turbulent combustion modeling a b s t r a c t Probability density function (PDF)methods offer compelling advantages for modeling chemically reacting turbulent ?ows.In particular,they provide an elegant and effective resolution to the closure problems that arise from averaging or ?ltering the highly nonlinear chemical source terms,and terms that correspond to other one-point physical processes (e.g.,radiative emission)in the instantaneous governing equations.This review is limited to transported PDF methods,where one models and solves an equation that governs the evolution of the one-point,one-time PDF for a set of variables that determines the local thermochemical and/or hydrodynamic state of a reacting system.Progress over the previous 20–25years (roughly since Pope’s seminal paper [24])is covered,with emphasis on developments over

the past decade.For clarity and concreteness,two current mainstream approaches are adopted as

baselines:composition PDF and velocity–composition PDF methods for low-Mach-number reacting

ideal-gas mixtures,with standard closure models for key physical processes (e.g.,mixing models),and

consistent hybrid Lagrangian particle/Eulerian mesh numerical solution algorithms.Alternative formu-

lations,other ?ow regimes,additional physics,advanced models,and alternative solution algorithms are

introduced and discussed with respect to these baselines.Important developments that are discussed

include velocity–composition–frequency PDF’s,PDF-based methods as sub?lter-scale models for large-

eddy simulation (?ltered density function methods),PDF-based modeling of thermal radiation heat

transfer and turbulence–radiation interactions,PDF-based models for soot and liquid fuel sprays,and

Eulerian ?eld methods for solving modeled PDF transport equations.Examples of applications to

canonical systems,laboratory-scale ?ames,and practical combustion devices are provided to emphasize

key points.An attempt has been made throughout to strike a balance between rigor and accessibility,

between breadth and depth of coverage,and between fundamental physics and practical relevance.It is

hoped that this review will contribute to broadening the accessibility of PDF methods and to dispelling

misconceptions about PDF methods.Although PDF methods have been applied primarily to reacting

ideal-gas mixtures using single-turbulence-scale models,multiple-physics,multiple-scale information is

readily incorporated.And while most applications to date have been to laboratory-scale nonpremixed

?ames,PDF methods can be,and have been,applied to high-Damko

¨hler-number systems as well as to low-to-moderate-Damko

¨hler-number systems,to premixed systems as well as to nonpremixed and partially premixed systems,and to practical combustion devices as well as to laboratory-scale ?ames.It is

anticipated that PDF-based methods will be adopted even more broadly through the 21st century to

address important combustion-related energy and environmental issues.

Contents

Introduction .......................................................................................................................1702.Preliminary examples .. (173)

2.1.An idealized premixed turbulent flame (173)

2.2.An idealized nonpremixed turbulent flame (175)

2.3.Real nonpremixed turbulent jet flames (176)

2.4.Preview of the remainder of the article .........................................................................................177*Tel.:t181********.

E-mail address:

dch12@46254ffbf111f18582d05a6d

Contents lists available at ScienceDirect

Progress in Energy and Combustion Science

journal homepage:

46254ffbf111f18582d05a6d/locate/pecs

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259169

46254ffbf111f18582d05a6derning equations for turbulent reacting flows (178)

3.1.Instantaneous PDE’s (178)

3.2.Thermochemistry (178)

3.3.Reaction mechanisms (178)

3.4.Molecular transport (179)

3.5.Thermal radiation (179)

46254ffbf111f18582d05a6dposition variables (179)

3.7.Reduced descriptions (180)

4.PDF methods for turbulent reacting flows (180)

4.1.Mean quantities and probability density functions (180)

4.2.PDF transport equations (181)

4.2.1.Velocity–composition PDF (181)

46254ffbf111f18582d05a6dposition PDF (181)

4.3.Equations for mean quantities (182)

46254ffbf111f18582d05a6dgrangian particle equations (183)

4.4.1.Velocity–composition PDF (183)

46254ffbf111f18582d05a6dposition PDF (184)

4.4.3.Consistency (184)

4.5.Eulerian field equations (184)

4.6.Regimes of applicability (184)

5.Filtered density function(FDF)methods for large-eddy simulation (184)

5.1.Spatially filtered quantities and filtered density functions (185)

5.2.FDF transport equations (186)

5.3.Equations for filtered quantities (186)

46254ffbf111f18582d05a6dgrangian particle equations (187)

5.5.The FDF as a basis for modeling (187)

5.6.General comments on LES and RAS (188)

6.Physical models (189)

6.1.Turbulence scale specification (189)

6.1.1.RAS/PDF formulation (189)

6.1.2.LES/FDF formulation (189)

6.2.Turbulent transport in velocity–composition PDF and FDF methods (190)

6.2.1.RAS/PDF formulation (190)

6.2.1.1.A stochastic mixing model (190)

6.2.1.2.A stochastic reorientation model (190)

6.2.2.LES/FDF formulation (192)

6.3.Turbulent transport in composition PDF and FDF methods (192)

6.3.1.RAS/PDF formulation (192)

6.3.2.LES/FDF formulation (192)

6.4.Molecular transport and scalar mixing models (192)

6.4.1.RAS/PDF formulation (193)

6.4.2.LES/FDF formulation (194)

6.5.Turbulence scale specification revisited:PDF’s of gradient quantities (195)

6.6.Flamelet combustion (195)

6.7.Wall modeling (196)

6.8.High-speed flows (196)

6.9.Summary (197)

6.9.1.A baseline velocity–composition PDF method (197)

6.9.2.A baseline composition PDF method (197)

7.Numerical algorithms (197)

7.1.Hybrid Lagrangian particle/Eulerian mesh methods (198)

7.1.1.Operator splitting and higher-order methods (199)

7.1.2.Interpolation and estimation of means from particle values (199)

7.1.3.Particle initialization,tracking,and boundary conditions (200)

7.1.3.1.Inflow/outflow boundaries (201)

7.1.3.2.Solid walls (201)

7.1.3.3.Symmetry planes (201)

7.1.3.4.Periodic and cyclic symmetry boundaries (201)

7.1.3.5.Processor boundaries (201)

7.1.4.Particle number density control (201)

7.1.5.Mixing and chemical reaction (202)

7.1.6.Redundancy,consistency,and Lagrangian/Eulerian coupling (202)

7.1.7.Statistically stationary or homogeneous flows,and generalized coordinates (202)

7.1.8.Numerical accuracy and convergence (203)

46254ffbf111f18582d05a6dputational efficiency and modularity (203)

7.1.9.1.Sorting (203)

7.1.9.2.Parallelization (203)

7.1.9.3.Chemistry acceleration (203)

7.1.9.4.Adaptive particle timestepping (204)

7.1.9.5.Modularity (204)

7.1.10.Additional considerations for velocity–composition PDF methods (204)

7.1.11.Additional considerations for LES/FDF methods (205)

7.2.Sparse-Lagrangian particle methods (205)

7.3.Eulerian particle methods (205)

7.4.Stand-alone Lagrangian particle methods (205)

7.5.Eulerian field methods (206)

7.5.1.Stochastic Eulerian field methods (206)

7.5.2.A deterministic Eulerian field method with DQMOM closure (207)

8.Multiphase flows and thermal radiation (207)

8.1.Soot (208)

8.2.Sprays (208)

8.3.Radiation and turbulence–radiation interactions (209)

8.3.1.Spectral radiation properties (209)

8.3.2.RTE solution methods (210)

8.3.3.Turbulence–radiation interactions (210)

8.3.4.The effects of radiation and TRI in turbulent flames (210)

8.3.5.Incorporating radiation and TRI in PDF methods (212)

9.Canonical configurations (213)

9.1.A statistically homogeneous system with imperfect mixing (213)

9.2.A statistically one-dimensional system with imperfect mixing (214)

9.3.Statistically one-dimensional premixed turbulent flames (214)

46254ffbf111f18582d05a6dboratory-scale flames (219)

10.1.Nonluminous,nonpremixed flames (219)

10.1.1.Piloted methane–air flames:Sandia flames D,E,and F (220)

10.1.2.Bluff-body-stabilized flames (225)

10.1.3.Swirl burners (231)

10.1.4.H2/N2jets in vitiated co?ow (231)

10.1.5.Autoignition of H2/N2jets in heated co?ow (232)

10.2.Premixed flames (233)

10.3.Luminous flames (238)

10.4.Spray flames (242)

11.Device-scale applications (242)

11.1.Gas-turbine combustors (243)

11.2.Reciprocating-piston engines (244)

11.2.1.Flame propagation in spark-ignition engines (245)

11.2.2.Autoignition and emissions in HCCI and quasi-HCCI engines (246)

11.2.3.Autoignition and emissions in direct-injection diesel engines (247)

12.Summary,conclusions,and closing remarks (248)

12.1.Key accomplishments (248)

12.2.Outstanding issues (248)

12.3.Promising directions (249)

Acknowledgments (249)

Appendix A.Derivation of PDF transport equations (249)

A.1.Fine-grained-PDF derivation (250)

A.2.Test-function derivation (251)

Appendix B.Moment equations from PDF equations (252)

Appendix C.Derivation of FMDF transport equations (252)

References (253)

1.Introduction

Turbulent combustion remains an important and timely subject in engineering science.Many of the most urgent energy ef?ciency, climate change,and pollutant emission issues worldwide are related to the conversion of chemical energy to sensible energy (heat)via a combustion process in a turbulent?ow environment. Combustion devices of practical interest include stationary and ground-vehicle reciprocating-piston internal-combustion engines, stationary and aircraft gas-turbine combustors,and industrial burners.The combustion process in such devices often is charac-terized by complex turbulence–chemistry interactions that span multiple combustion regimes:premixed?ame propagation,mix-ing-controlled burning,and chemical-kinetics-controlled processes may occur simultaneously within a single device.A wide range of ?ow speeds(Mach numbers)may be relevant.Multiphase?ows (liquid fuel sprays,solid particles),heterogeneous combustion (walls,catalysts),and radiation heat transfer(high-pressure and/or large-scale systems,sooting?ames)often are important.This complex turbulent aero-thermo-chemistry typically occurs in tortuous three-dimensional geometric con?gurations.And the prediction of fuel composition effects,local extinction/ignition,and key trace species(reaction intermediates,pollutants,and/or signature species)may require consideration of tens-to-hundreds of chemical species and hundreds-to-thousands of chemical reactions.

Computational?uid dynamics(CFD)-based tools for turbulent combustion modeling have improved dramatically over the past 10–20years,spurred by rapid advances in physical models, numerical algorithms,and computational power.Still,hydrody-namic turbulence and chemical kinetics remain among the most challenging fundamental and practical problems of computational

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259 170

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259171

science and engineering.In chemically reacting turbulent ?ows,these are coupled in interesting and highly nonlinear ways,leading to entirely new classes of phenomena;turbulence–chemistry interactions and turbulence–radiation interactions are discussed in subsequent sections.Chemically reacting turbulent ?ows are characterized by a wide dynamic range of spatial and temporal scales.For that reason,direct numerical simulation (DNS)of the instantaneous governing equations can be carried out only in limited and highly idealized cases.In general,to bring CFD to bear in modeling and/or simulating chemically reacting turbulent ?ows,it is both desirable and necessary to limit the dynamic range of scales.This can be accomplished by averaging or ?ltering the instantaneous governing equations;averaging or ?ltering of nonlinear terms gives rise to new terms that represent the effects of turbulent ?uctuations about the local averaged or ?ltered values.To close the system,these new terms must be expressed in terms of the averaged or ?ltered quantities for which one is solving;this is the essence of turbulent combustion modeling.

Turbulent combustion has been the subject of several books and review articles.The recent book by Cant and Mastorakos [1]provides an accessible introduction and entry points to the litera-ture.Several noteworthy early contributions are contained in the collection edited by Libby and Williams [2].Other books include the texts by Peters [3](emphasis on ?amelet models:level-set methods,in particular),Fox [4](emphasis on probability density function methods,in a chemical engineering context),and Poinsot and Veynante [5](compilation of theoretical results for laminar ?ames;reviews of direct numerical simulation,Reynolds-averaged simulation,and large-eddy simulation physical modeling and numerical tools for turbulent combustion).Review articles include those by Veynante and Vervisch [6](reviews of models for turbu-lent premixed and nonpremixed combustion;emphasis on common links among models)and Westbrook et al.[7](progress in computational combustion over 50years).The biennial (since 1999)von Karman Institute for Fluid Dynamics lecture series on turbulent combustion [8]is a good source of up-to-date information.

Probability density function (PDF)methods have emerged as one of the most promising and powerful approaches for accommodating the effects of turbulent ?uctuations in velocity and/or chemical composition in CFD-based modeling of turbulent reacting ?ows.In this review,the term ‘‘PDF method’’refers to an approach based on solving a modeled transport equation for the one-point,one-time Eulerian joint PDF of a set of variables that describe the hydrody-namic and/or thermochemical state of a reacting medium:that

is,

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259

172

a transported(versus presumed)PDF method.Turbulence closure based on the solution of a modeled transport equation for the joint PDF of the velocity components originated with the work of Lundgren[9,10].Dopazo and O’Brien[11,12]were the?rst to consider a modeled equation for the PDF of a set of scalar variables that describe the thermochemical state of a reacting medium(a composition joint PDF)to model mixing and chemical reaction in turbulent reacting?ows.PDF methods subsequently were devel-oped and elucidated by these researchers[13–17]and by others including Pope[18–20],Janicka and Kollmann[21],and Borghi[22]. The relationship between particle models and PDF methods was established by Pope[23],and particle methods have become the dominant approach for modeling and solving PDF transport equa-tions.The point of departure for PDF methods in their modern form, including the associated Lagrangian-particle-based solution algo-rithms,often is taken to be Pope’s1985paper[24].Reviews pub-lished since then include those by Pope[25],Kollmann[26],and Dopazo[27].Detailed treatments of PDF methods are provided in the books by Kuznetsov and Sabel’nikov[28],Fox[4],and Pope[29]; the latter is limited to constant-property turbulent?ows without chemical reaction.The volume edited by Launder and Sandham[30] includes several chapters on PDF methods,and the book by Heinz [31]emphasizes PDF methods.

The present review emphasizes the progress in PDF methods that has been made over approximately the past20–25years.Over that period,PDF methods have evolved from being a research tool for science discovery limited to canonical and laboratory-scale systems,to being a mainstream turbulent combustion modeling approach that can be applied to practical devices.The focus is primarily(although not exclusively)on one-point,one-time trans-ported composition PDF and velocity–composition PDF methods for low-Mach-number reacting ideal-gas mixtures in a Reynolds-averaged context,and on the associated hybrid Lagrangian-particle/ Eulerian-mesh-based CFD solution algorithms;these are the current predominant approaches.However,many important extensions and variants have been introduced in the mathematical formulations,physical models,and numerical algorithms.These include PDF-based approaches for sub?lter-scale modeling in large-eddy simulation,consideration of multiphase?ows(e.g.,liquid fuel sprays and/or soot),thermal radiation heat transfer,stand-alone particle-based solution methods,and nonparticle solution methods,among others.These developments also are discussed.

The level of coverage is intended to strike a balance between rigor and accessibility,and between fundamentals and practice. The material is organized as follows.Introductory examples of PDF’s in turbulent reacting?ows are provided in the following section.A review of the underlying governing equations is provided in the third section.PDF methods are introduced in Section4.The extension of PDF-based methods to sub?lter-scale modeling for large-eddy simulation(?ltered density function–FDF–methods) is the subject of Section5.Physical models and numerical algo-rithms are the subjects of Sections6and7,respectively.Additional considerations for multiphase?ows(soot and liquid sprays)and thermal radiation including turbulence–radiation interactions are discussed in Section8.Sections9–11provide examples of appli-cations of PDF methods to canonical systems,laboratory-scale ?ames,and device-scale systems,respectively.In the?nal section, observations are made on the progress of PDF-based modeling since approximately1985[24];key successes and outstanding issues and highlighted,and predictions are offered for the next20–25years.The Appendices provide skeletal derivations of PDF and FDF transport equations,and derivations of moment equations from PDF equations.

2.Preliminary examples

This section provides a somewhat heuristic introduction to PDF’s in turbulent combustion;readers who are familiar with the basic concepts of PDF methods can skip this section without loss of continuity.The formal equations and de?nitions will be introduced starting in Section3.The reader is referred to Section2of[24]and to Chapter3of[29]for a review of those elements of probability theory that are most germane to PDF methods.

2.1.An idealized premixed turbulent?ame

We consider a rod-stabilized premixed turbulent?ame,as shown schematically in Fig.1.A low-Mach-number,high-Reynolds-number turbulent?ow of premixed fuel and oxidizer impinges on a stationary rod whose axis is parallel to the z-axis(out of the page);the rod serves to anchor the?ame.The in?ow conditions are constant and are uniform in z(in a time-averaged sense).Such con?gurations have been explored experimentally and computa-tionally(e.g.,[32]),although only at low-to-moderate Reynolds numbers.

In the analysis and modeling of premixed systems,it is conve-nient to introduce a reaction progress variable,c?c(x,t),that increases monotonically from zero in unburned reactants to unity in fully burned products.Here c can be identi?ed with a normalized major-product-species mass fraction or with normalized temper-ature:c h(T–T u)/(T b–T u),for example,where subscripts u and b refer to the unburned reactants and to the burned products, respectively.The?ame sheet then can be identi?ed as an instan-taneous three-dimensional isocontour of c:c?0.5,say.Represen-tative?ame contours on a z?constant plane at three instants in time are shown in Fig.1.These contours are highly irregular, because of the turbulent nature of the system.

An appropriate choice of a chemical time scale to characterize this system is s chem?d L/s L,where d L and s L are the thickness and speed of a steady,unstrained,one-dimensional laminar premixed ?ame corresponding to the inlet thermochemical conditions.An appropriate hydrodynamic(turbulence)time scale is s turb?l T/u0, where l T and u0are the integral length scale and rms?uctuating turbulent velocity in the incoming reactant stream,respectively.A key dimensionless parameter(in addition to the Reynolds number Re)that characterizes turbulent reacting?ows is the Damko¨hler number:Da h s turb/s chem.In practical premixed turbulent?ames,

premixed fuel + air

Fig.1.A statistically stationary,statistically two-dimensional rod-stabilized turbulent

premixed?ame.

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259173

the values of Re and Da sometimes correspond to a regime where the ?ame is thin compared to all hydrodynamic scales;that is the regime that is considered 46254ffbf111f18582d05a6drmation on the regimes of premixed turbulent combustion can be found in Section 5of [6]and Chapter 5of [5],for example.We shall also idealize the system as being adiabatic,so that T b is the constant-pressure adiabatic ?ame temperature corresponding to the inlet thermochemical conditions.

Consider that a nonintrusive device with in?nitesimally small probe volume and in?nitely fast time response is available to measure the local instantaneous temperature T (hence c )at any point in the turbulent ?ame.At location a (Fig.1),unburned gases are always present so that the measured value of T is constant at T ?T u (c ?0).At location e ,burned gases are always present so that the measured value of T is constant at T ?T b (c ?1).Locations b -d sometimes lie in the unburned gases and sometimes in the burned gases,depending on the instantaneous ?ame-sheet loca-tion.For an in?nitesimally thin ?ame,the instantaneous temper-ature at locations b –d is either T ?T u or T ?T b ;there is negligible probability of the measurement point lying inside the ?ame sheet.

The fraction of the total measurement time t meas that is spent in burned gases increases from b to c to d .For suf?ciently long t meas et meas [s turb [s chem T,the fraction of time that is spent in burned gases tends toward a constant value at each location;that value is the local time-averaged value of c :

t h

t meas

Z t 0tt meas t 0

c ex ;t T

d t :(1)

For concreteness,we will take locations b ,c ,and d to correspond

to locations where burned gases are present 25%,50%,and 75%of the time,respectively.The same value of c t would be found for any ?xed values of x and y ,independent of the value of z .This system is both statistically stationary (time-averaged one-point statistics are independent of t 0)and statistically homogeneous in the z direction (time-averaged one-point statistics are independent of z ).

The fraction of time that c (x ,t )takes on a particular value can be analyzed further.Consider N uniform bins of width D j N ?1/N ,indexed i ?1,2,.,N ,such that the ith bin is centered at j ei ;N T?ei à1=2TDj N and spans (i à1)D j N c (x ,t )

bin includes c (x ,t )?1.Then h c

(i ,N )

?h c (j (i ,N );x )is de?ned such that the value of h c (i ,N )is constant within the i th bin and h c

(i ,N )

D j N is the fraction of time that c (x ,t )is measured to have a value that lies in

the i th bin.For the idealized ?ame of Fig.1,only h c (1,N )and h c

(N ,N )can be nonzero.At measurement location b ,for example,h c

(1,N )

?0.75N and h c

(N ,N )

?0.25N .As N /N eDj N /0T,h e1;N Tc /N and h eN ;N Tc /N ,while h c

(1,N )D j N ?0.75and h c (N ,N )

D j N ?0.25for any N including N /N .The function h c

(i ,N )?h c (i ,N )

(x )can be used to estimate one-point time-averaged statistics of c .For example,the N -bin average of

c (x ,t )(denote

d c ex TN )and th

e mean-square o

f the ?uctuations in c (x ,

t )about c ex TN (denoted c ex TN )are c ex TN ?P N i ?1j

ei ;N Th ei ;N T

c ex TDj N an

d c 02N ?P N

i ?1ej

ei ;N Tàc ex TN T2h ei ;N Tc ex TDj N ,respectively.In general,the N -bin-based n th-order central moment is,

c 0n ex TN ?

X N i ?1

j ei ;N Tàc ex TN

h ei ;N T

ex TDj N :

(2)

Passing

the continuous limit

(N /N ,

Dj N /d j ,

h ei ;N T

c ex T/h c ej ;x T),

formally:

Z 1

j h c ej ;x Td j ;0n ?

Z 1

j àc ex Tá

n h c ej ;x Td j :

(3)

The quantity h c (j ;x )is non-negative,integrates to unity (R N

àN h c ej ;x Td j ?1;here h c (j ;x )?0for j <0and for j >1),has dimensions of j à1(here dimensionless),and varies with spatial location x in the ?ame.At this point,one might identify h c (j ;x )as an experimentally measured PDF of c and the overbarred quan-tities as mean quantities.We will use that terminology (some-what loosely)in the remainder of this section.However,a separate notation will be maintained from that introduced in Sections 3and 4below to emphasize the distinction between the formal de?nitions and manipulations of PDF’s and mean quanti-ties (based on axiomatic probability theory)from the approaches that are used to estimate these quantities from experimental measurements:in this example,time averaging in a statistically stationary system.Here the same result could have been obtained by considering spatial averaging and binning based on the frac-tion of locations in the statistically homogeneous (z )direction where c (x ,t )takes on a particular value (excluding end effects),and h c (j ;x )?h c (j ;x ,y ).

In this case,h c ej ;x T?e1àd ej Ttd ej à1Twhere is the fraction of time spent in burned gases at location x and d (j àc )denotes a Dirac delta function at j ?c .From Eq.(3),c 02ex T?e1à$.A review of the properties of Dirac delta

h c

)

00.20.40.60.81

Fig.2.The function h c (j ;x ,y )at location b in Fig.1.

air

Fig.3.An axisymmetric nonpremixed turbulent jet ?ame.

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259

174

functions can be found in Appendix C of [29].The function h c (j ;x )is plotted for location b in Fig.2.It has the same general shape at any spatial location;only the magnitudes of the delta functions vary with spatial location in the ?ame.It is important to understand that the double-delta-function PDF is only an approximation,and that it leads to an apparent contradiction.All chemical source terms are equal to zero in cold unburned reactants and in hot fully burned products;therefore this PDF implies a zero mean rate of reaction,which clearly is incorrect.The contradiction can be resolved by adding a small,nonzero probability of encountering burning gases [33–35].This is discussed further in Section 6.6.

The regime of in?nitesimally thin premixed ?ame sheets is not one where transported PDF methods often have been applied.However,PDF methods can be and have been used in this regime to provide valuable new physical insight.Examples are provided in Sections 9and 10.

2.2.An idealized nonpremixed turbulent ?ame

Nonpremixed turbulent jet ?ames have been the subject of much analysis,measurement,and modeling;see,for example,Chapter 13of [36].A particularly relevant con?guration is a high-Reynolds-number jet of fuel that issues from a tube of circular cross section into co?owing or stationary air (Fig.3).

Mixture fractions are useful in the analysis and modeling of nonpremixed systems.A mixture fraction x ?x (x ,t )can be de?ned as a linear combination of species mass fractions in such a way that it quanti?es the local mass fraction of material that originated from the fuel jet (versus oxidizer stream [36]).Mixture fractions are conserved scalars;that is,the transport equation for x (x ,t )contains no chemical source term.In cases where one or more elements (e.g.,carbon and/or hydrogen)are present only in the fuel stream,normalized element mass fractions are appropriate as mixture fractions.De?nitions based on a small number of measured major species can be used to make quantitative comparisons between models and experiment [37,38].The de?nition and utility of mixture fractions become problematic in situations where differ-ential diffusion is important and/or there are multiple fuel streams.A general treatment is provided in Chapter 5of [4].

Here we consider a one-step irreversible reaction of fuel F and oxidizer O to form product P.With s denoting the mass of oxidizer required per unit mass of fuel,the reaction can be written on a mass basis as,

F ts O /e1ts TP :(4)

A chemical time scale s chem can be de?ned based on the

chemical kinetics:s chem ?r F =eW F _u

T,where r F ,W F ,and _u are the mass density of fuel,the molecular weight of fuel,and the molar rate of reaction for Eq.(4),respectively.In the fast-chemistry limit eDa ?s turb =s chem /N T,fuel and oxidizer react immediately to form product as soon as they are mixed at the molecular level.For a low-Mach-number,adiabatic system with equal and constant mass densities,speci?c heats,and molecular transport coef?cients for all species (a highly idealized con?guration),the species mass fractions,mixture enthalpy,and temperature are piecewise linear functions of a mixture fraction x (x ,t )that is de?ned such that x ?1corresponds to the fuel stream,x ?0corresponds to the oxidizer stream,and x ?x st ?1/(1ts )corresponds to fuel and oxidizer in stoichiometric proportion.A suitable de?nition is x h Y F tY P /(1ts ),where Y F and Y P are the mass fractions of fuel and product,respectively.A typical value of x st for a hydrocarbon-air system is x st z 0.05–0.07.For this idealized system,fuel and oxidizer cannot coexist,and the peak temperature is the constant-pressure adiabatic ?ame temperature T P ,which occurs where fuel and oxidizer are in stoichiometric proportion.Species mass frac-tions and normalized temperature are plotted as functions of

p d f

00.20.4

0.60.8

510

1520

a:0.987,0.00137b:0.908,0.00396c:0.657,0.0134d:0.387,0.00955e:0.220,0.00397f:0.139,0.00164

p d f

00.20.4

0.60.81

b:0.908,0.00396g:0.621,0.0275h:0.150,0.0101

i:0.00810,0.000272

Fig.5.Beta mixture-fraction distributions at several locations (indicated in Fig.3)for an idealized nonpremixed turbulent jet ?ame.Left:Distributions at six axial locations (x /d ?3,15,30,45,60,and 75)along the jet centerline.Right:Distributions at four radial locations (r /d ?0,0.83,1.67,and 2.78)for x /d ?15.Here d is the fuel-jet diameter.The local values of x and x 02are given in the legend.These values have been taken from experimental measurements for Sandia ?ame D (x st ?0.351)at the speci?ed locations [41].

Y F ,Y O ,Y P ,Θ

00.20.4

0.60.81

0.2

0.4

0.6

0.8

Y F Y O

Y P =Θ

Fig.4.Species mass fractions Y F ,Y O ,and Y P and normalized temperature Q as functions of mixture fraction x for an idealized nonpremixed system with x st ?0.06.

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259175

mixture fraction in Fig.4for the case where the temperatures of the fuel stream T F and the oxidizer stream T O are equal.In that case,a normalized temperature Q can be de?ned as Q h (T –T F )/(T P àT F ),and the pro?le of Q versus x is identical to the pro?le of Y P versus x .

In this case,it is appropriate to consider the PDF of the mixture fraction h x (j ;x ,t ),where h x (j ;x ,t )can be determined operationally following a procedure similar to that used in Section 2.1for h c (j ;x ,t ).Because this con?guration is statistically stationary,h x (j ;x ,t )?h x (j ;x ).The system also is statistically homogeneous in the azimuthal direction,with respect to the polar cylindrical coordinate system of Fig.3.Measurements in nonreacting turbulent jets and in nonpremixed turbulent jet ?ames show that h x (j ;x )corresponds approximately to a beta distribution [39],and that is suf?cient for the present discussion.In that case [40],

h x ej ;x T?

G ea tb TG ea TG eb T

j a à1

e1àj Tb à1;

(5)

where G (a )denotes the Gamma function.Here 0 j 1and the parameters a and b (a ?a (x )>0and b ?b (x )>0)are related to the mean and variance by,

x ex T?

a a tb

;x 02

ex T?

ea tb T2

ea tb t1T

:(6)

The beta distribution can approximate a variety of PDF shapes that include uniform distributions (h x (j ;x )?1for 0 j 1),double-delta-functions at j ?0and j ?1,and continuous distri-butions with a single maximum between j ?0and j ?1.Beta mixture-fraction distributions corresponding to six locations along the jet centerline and to four radial locations at a ?xed axial location are shown in Fig.5.

For any Q ?Q (x ),Q ex T?R 1

0Q ej Th x ej ;x Td j .By virtue of the relationships illustrated graphically in Fig.4,knowledge of h x (j ;x )then is suf?cient to determine any one-point joint statistic of the species mass fraction and temperature ?elds.For example,the mean fuel mass fraction,mean temperature,and fuel mass fraction-temperature covariance are

Y F ex T?R 10Y F ej Th x ej ;x Td j ,T ex T?R 1

0T ej Th x ej ;x Td j ,and

0F 0ex T?R 1

0eY F ej TàY F ex TTeT ej TàT ex TTh x ej ;x Td j ,respectively.2.3.Real nonpremixed turbulent jet ?ames

In contrast to the idealized system considered in Section 2.2,it generally is not the case that a single quantity (there a mixture

fraction)suf?ces to specify uniquely the local instantaneous ther-mochemical state of the reacting mixture in a turbulent ?ow.It then is useful to consider joint statistics of multiple physical quantities.Fig.6shows scatter plots of temperature T and NO mass fraction Y NO versus mixture fraction x obtained from laser-based measure-ments for two piloted nonpremixed methane–air turbulent jet ?ames (Sandia ?ames D and F)at a ?xed spatial location (x /d ?30,r /d ?1.67)[38,41–43].In these ?ames,the stoichiometric value of the mixture fraction has been increased (compared to its value for pure methane fuel and air oxidizer)to x st ?0.351by diluting the fuel jet with air,and the ?ame continues to burn as a nonpremixed ?ame.At this measurement location,there are very few samples for x <0.1or x >0.7.The fuel-jet and pilot velocities (hence Re and Da )are different for the two ?ames;the fuel-jet Reynolds number increases from 22,400for ?ame D to 44,800for ?ame F,while the

Damko

¨hler number (the more relevant parameter)decreases from ?ame D to ?ame F.These ?ames will be discussed further in Section 10.1.Each point in the scatter plot corresponds to a single essen-tially instantaneous point measurement where the local tempera-ture and species mass fractions (CH 4,O 2,N 2,H 2O,CO 2,CO,H 2,OH,and NO)all are measured simultaneously;the local

mixture

ξT [K ],107

Y N O

T [K ],107

Y N O

Fig.6.Scatter plots of T (red squares)and Y NO (blue diamonds)versus mixture fraction x for Sandia ?ames D and F at x /d ?30and r /d ?1.67[41].

c o n

d i t i o n a l r m s /c o n d i t i o n a l m

e a n

0.10.20.30.40.50.60.70.80.9

00.10.20.30.40.50.60.70.80.91Sandia D,T Sandia D,NO Sandia F,T Sandia F,NO

Fig.7.Ratios of conditional rms to conditional mean temperatures and NO mass fractions (conditioned on the value of the mixture fraction)for Sandia D and F ?ames at x /d ?30and r /d ?1.67[41].

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259

176

fraction then is determined from the major species mass fractions [37,38].Clearly,there is no unique value of T or of Y NO for each value of mixture fraction x .Moreover,the level of variation in the values of T and Y NO for a given value of x is greater for ?ame F than for ?ame D.

These notions can be quanti?ed by introducing joint PDF’s.For concreteness,we consider joint statistics of T and x ,and introduce the joint PDF of temperature and mixture fraction,h T x (s ,j ;x ).This quantity can be determined operationally using a two-dimensional analogue of the binning and limiting procedure that was used in Section 2.1.Here h T x (s ,j ;x )has the following properties:it is non-negative,integrates to unity (R N s ?àN R N

j ?àN h T x es ;j ;x Td s d j ?1;here h T x (s ,j ;x )?0for j <0and for j >1),has dimensions of s à1j à1(inverse temperature),and varies with spatial location x in the ?ame (this con?guration is statistically stationary).The PDF’s of temperature and of mixture fraction (the ‘‘marginal PDF’s’’)can be

determined from the joint PDF:h x ej ;x T?R N

àN h T x es ;j ;x Td s ;

h T es ;x T?R N

àN h T x es ;j ;x Td j .The mean of any function of T and x ,

Q ?Q (T ,x ),is given by Q ex T?R N àN R N

àN Q es ;j Th T x es ;j ;x Td s d j .

From data of the kind shown in Fig.6,one can determine the average value of T (or of Y NO )for any speci?ed range of values of x ,j x

Q ?Q (T ,x ),j ?R N àN Q es ;j Th T j x es j j Td s and Q ?R N

àN j h x ej Td j .In general,it is not possible to reconstruct the joint PDF from the marginal PDF’s alone.An exception is the case where the random variables are statistically independent.If T and x were statistically independent (that is not the case here),then by de?nition,h T x ?h T h x ,

and therefore h T j x ?h T and h x j T ?h x .Although here we have considered the joint statistics of two variables,joint PDF’s and conditional PDF’s can be de?ned for any number of random variables.

The bin-based conditional rms temperature and rms NO mass fraction corresponding to Fig.6,normalized by their respective conditional mean values,are plotted in Fig.7.There the mixture fraction bin width is 0.05,and the statistics are noisy because of the limited sample size.Nevertheless,this veri?es quantitatively the observation from Fig.6that the magnitudes of the ?uctuations in T and in Y NO for a given value of x are higher in ?ame F compared to ?ame D.In general,the magnitude of the ?uctuations is reduced by conditioning.The ratios of the unconditional rms temperatures to the unconditional mean temperatures for ?ames D and F are 0.189and 0.228,respectively;and the ratios of the unconditional rms NO mass fractions to the unconditional mean NO mass fractions for ?ames D and F are 0.394and 0.702,respectively.

Laboratory-scale,atmospheric pressure nonpremixed turbulent jet ?ames probably are the single con?guration to which PDF methods (and many other turbulent combustion models)have been applied most often.The quantitative performance of PDF methods for this and other laboratory-scale ?ame con?gurations will be discussed in Section 10.

2.4.Preview of the remainder of the article

In this section,mean quantities,PDF’s,and other concepts and notation have been introduced in a heuristic manner motivated by physical arguments for statistically stationary turbulent ?ames.A more formal and quantitative development begins in Section 3.It is emphasized that PDF methods are not limited to statistically stationary con?gurations or to con?gurations having one or

more

46254ffbf111f18582d05a6dputed (lines)[44]and measured (symbols)[38]radial pro?les of mean temperature (upper row)and mean O 2mass fraction (lower row)at three axial locations in Sandia ?ame D.Here D j is the fuel-jet diameter.Solid lines are from a PDF-based model.Dashed lines are from a model that ignores the in?uence of turbulent ?uctuations on mean chemical source terms.Courtesy of Dr.Y.Z.Zhang,CD-adapco,Melville,NY.

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259177

directions of statistical homogeneity.Applications to statistically nonstationary systems and to statistically three-dimensional systems are provided in Sections9–11.

The aim of(transported)PDF methods is to predict the spatial and temporal evolution of PDF’s of one or more quantities of interest in a turbulent reacting?ow,and that is the subject of the remainder of this article.In general,transported PDF methods are most bene?cial in situations where the shape of the PDF is not known a priori and/or cannot be parameterized simply(in contrast to the double-delta-function of Section2.1,or the beta distribution of Section2.2).In practice,one rarely compares measured and computed PDF’s directly.Instead,mean quantities(typically only a few lower-order moments)are used as the basis for comparisons between experimental measurements and PDF-based modeling studies.The bene?ts of accounting explicitly for the in?uence of turbulent?uctuations then can be seen by comparing the results of PDF-based models with those from models that do not account for turbulent?uctuations or that treat them in an oversimple manner.

An initial example is provided in Fig.8.There mean temperature and mean O2mass-fraction pro?les computed using a PDF method are compared with pro?les computed using a model that ignores turbulent?uctuations altogether(the local mean chemical source terms are computed based on the local mean species composition and temperature)for Sandia?ame D(44).These models will be discussed in subsequent sections.Here it suf?ces to note that a simple one-step,?nite-rate global methane–air chemical mech-anism has been used;good quantitative agreement with experi-ment therefore is not expected.Nevertheless,this serves to illustrate the bene?ts of accounting explicitly for turbulent?uctu-ations in species composition and temperature using a PDF method. The model that ignores turbulent?uctuations grossly overpredicts the local mean heat-release rate and overpredicts local mean temperatures by as much as several hundred Kelvin.Results are signi?cantly improved with the PDF-based model,in spite of the oversimpli?ed chemistry.

46254ffbf111f18582d05a6derning equations for turbulent reacting?ows

The principal unaveraged,un?ltered equations that govern chemically reacting turbulent?ows are reviewed in this section. This material is provided to introduce physical quantities and notation that will be used throughout the remainder of this review, and to clearly distinguish the models and approximations that must be introduced as a result of averaging or?ltering the gov-erning equations(Sections4and5)from those that are implicit in the underlying formulation.

3.1.Instantaneous PDE’s

The partial differential equations(PDE’s)governing a gas-phase multicomponent reacting system comprising N S chemical species are expressed here using Cartesian tensor notation.A Roman index denotes a component of a three-dimensional vector(e.g.,i?1,2,3), a Greek index denotes a chemical species(e.g.,a?1,2,.,N S),and the usual summation convention applies over repeated Roman indices within a term:

v r v t t

v r u i

v x i

?0;

v r u j

v r u j u i

i ?

v s ij

v p

tr g j;

v r Y a

v r Y a u i

i ?à

v J a

tr S a;

v r h v t t

v r hu i

v x i

?à

v J h

v x i

ts ij

v u j

v x i

à_Q rad:

(7)

Here u denotes the velocity vector,Y is the vector of mass frac-

tions of the N S chemical species,and h is the mixture speci?c

enthalpy.Mixture mass density is r,pressure is p,body force per

unit mass(constant)is g,and s,J a and J h,denote,respectively,the

viscous stress tensor and the molecular?uxes of species and

enthalpy.The molar chemical production rate and molecular weight

for species a are_u a and W a.The volume rate of heating due to

radiation(absorption minus emission)isà_Q rad,and r S a?W a_u a.

The enthalpy(or energy,or temperature)equation can be written in

several alternative forms[45].The present form is appropriate for

the absolute enthalpy(sum of sensible and formation enthalpies).

Additional terms to accommodate multiphase mixtures(e.g.,liquid

fuel sprays and soot)will be introduced in Section8.Derivations of

Eq.(7)can be found in combustion textbooks(e.g.,[45]).

3.2.Thermochemistry

The PDE’s are supplemented by thermal(p?p(Y,T,r))and caloric

(T?T(Y,h,p))equations of state and by speci?cation of?uid trans-

port properties and constitutive relations for stresses and?uxes.

For an ideal-gas mixture,the state equations can be written as:

p?r RTeR?R u=WT;h?

X N S

a?1

Y a

B@D h0

f;a

Z T

c p;a

d T0

C A:(8)

The second equation can be used to obtain the temperature,

given the species mass fractions,mixture speci?c enthalpy,and?uid

properties.Here R u is the universal gas constant,W is the mixture

molecular weight,D h0f;a is the enthalpy of formation of species a at

reference temperature T0,and c p,a is the species-a constant-pres-

sure speci?c heat.Requisite?uid properties include the speci?c

heats c p,a and/or c v,a(for ideal-gas mixtures,c p,a and c v,a are func-

tions of temperature only,and c p,aàc v,a?R a?R u/W a),and the

molecular transport coef?cients(Section 3.4).For low-Mach-

number?ows,it often is convenient to solve a PDE for pressure p

rather than for mixture density r and to use the thermal equation of

state to obtain r.This is discussed further in the context of the

equations for mean quantities in Section4.3.

3.3.Reaction mechanisms

An arbitrary elementary chemical reaction mechanism involving

the N S species can be written as a set of L reversible reactions:

X N S

a?1

l a

M a!

X N S

a?1

n00

l a

M ael?1;2;.;LT:(9)

Here M a denotes a chemical species symbol,and n0l a and n00l a are

the stoichiometric coef?cients.The corresponding molar rate of

production of species a is given by the law of mass action:

_u a?

X L

l?1

&à

n00

l a

àn0l a

k l;feTT

Y N S

b?1

l b

àk l;reTT

Y N S

b?1

n00

l b

;(10)

where k l,f(T)and k l,r(T)are,respectively,the forward and reverse

rate coef?cients for the l th reaction(the two are related through an

equilibrium constant[36,45])and c b denotes the molar concen-

tration of species b.The rate coef?cients usually are expressed in

modi?ed Arrhenius form as,

k l;feTT?A l;f T b l;f exp

àE A

l;f

R u T

áo

;(11)

where the three coef?cients for each reaction(pre-exponential A l,f,

temperature exponent b l,f,and activation energy E Al,f)may be

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259

178

determined from?rst principles(in the case of dilute binary elementary reactions)or empirically.An explicit pressure depen-dence sometimes is included in Eq.(11),as well.

Detailed chemical mechanisms(that include all species and reactions for a‘‘complete’’representation at the molecular level) and skeletal chemical mechanisms(obtained through systematic elimination of unimportant species and reactions starting from a detailed mechanism)generally contain only elementary reac-tions and can be expressed in the form of Eqs.(9)–(11).Reduced chemical mechanisms(obtained by further systematic reduction or empirically),on the other hand,may contain nonelementary reactions and/or algebraic constraints so that Eq.(10)does not follow from Eq.(9);the modi?ed Arrhenius form still generally is used for the reaction rates.Up-to-date entry points to the liter-ature on mechanism reduction(including approaches that go beyond the chemical kinetics in isolation,to deal with the dynamics of coupled reactive-diffusive systems)include Refs.[46] and[47].

In the remainder of this review,it is assumed that an appro-priate detailed,skeletal,or reduced chemical mechanism is avail-able as needed to provide the chemical source terms in the instantaneous species mass fraction equations(Eq.(7))as functions of the local species mass fractions,temperature,and pressure: S?S(Y,T,p).

3.4.Molecular transport

The speci?c forms of the molecular transport terms(s ij,J i a,and J i h)are not central to most of what follows;turbulent transport dominates in most cases.Exceptions are discussed in Section6 below.For concreteness,we can consider that standard formula-tions have been adopted(e.g.,[48]).For example,the viscous stress normally is written in a form appropriate for a Newtonian?uid: s ij?mev u i=v x jtv u j=v x iTà2=3m v u l=v x l d ij,with a mixture-aver-aged viscosity m;and species and enthalpy transport usually are modeled using multicomponent forms of Fick’s and Fourier’s laws, respectively.Thermal diffusion(Soret and Dufour effects)may or may not be included;these usually are neglected in modeling turbulent reacting?ows,as are the effects of differential species diffusion and nonunity-Lewis-number effects.

3.5.Thermal radiation

Thermal radiation often is an important mode of heat transfer in combustion systems,by virtue of their high temperatures[49];peak temperatures are typically w2000K.Even the global properties of canonical steady one-dimensional laminar premixed?ames(?ame speeds and?ammability limits)depend strongly on radiation heat transfer and on the spectral radiation properties of the participating gases[50].And recently it has been shown that soot radiation can suppress?ame?icker in axisymmetric laminar diffusion?ames [51].Accurate predictions of temperature,heat transfer rates,and pollutant emissions in turbulent?ames often require consideration of participating medium effects and spectral radiation properties. Interesting and fundamentally new phenomena arise in turbulent reacting?ows through nonlinear interactions among?uctuations in temperature,species composition,and radiative intensity.This is discussed further in Section8.3,and examples are provided in Section10.

The radiation source term_Q rad in the instantaneous energy (enthalpy)equation(Eq.(7))can be expressed as the pergence of the radiative heat?ux q rad,_Q

rad

?V$q rad?_Q rad;emà_Q rad;ab

Z N

k h

@4p I b hà

I h d U

A d h

?4k P s T4à

Z N

k h I h d U d h;(12) where

k P h

Z N

k h I b h d h

Z N

I b h d h

s T4

Z N

k h I b h d h(13)

is the Planck-mean absorption coef?cient and s is the Stefan–Boltzmann constant.The?rst term on the right-hand side of Eq.

(12)e_Q rad;emTcorresponds to emission and the seconde_Q rad;abTto absorption.Here h denotes wavenumber,U is solid angle,k h is the spectral absorption coef?cient,I b h is the Planck function(a known function of local temperature and wavenumber:I b h?I b h(T)),and I h is the spectral radiative intensity.Intensity is determined from the radiative transfer equation(RTE)[49]:

dI h

?b s$V I h?k h I b hàb h I ht

s s h

I h

b s0

F h

b s0;b s

d U0:(14)

Here b s and b s0denote unit direction vectors(a prime is used to distinguish the incident direction in the scattering integral),s s h is the spectral scattering coef?cient,b h?k hts s h is the spectral extinction coef?cient,and F heb s0;b sTdenotes the scattering phase function;the latter describes the probability that a ray from inci-dent direction b s0is scattered into direction b s.The local value of I h depends on nonlocal quantities,on directioneb sT,and on wave-number.The RTE is an integral-differential equation in?ve inde-pendent variables(three spatial coordinates and two direction coordinates)whose structure is quite different from that of the other governing equations for chemically reacting turbulent?ows: special approaches are required to solve the RTE(Section8.3).

The absorption coef?cient k h exhibits strong and erratic varia-tions with wavenumber h for the participating species that are of interest in combustion(principally H2O,CO2,and CO).For present purposes,it is suf?cient to note that k h is(in principle)a known function of local species mass fractions,temperature,and pressure: k h?k h(Y,T,p).

46254ffbf111f18582d05a6dposition variables

A set of scalar variables from which one can deduce the local thermochemical properties(e.g.,mixture mass density,molecular weight,and speci?c heats),species chemical production rates(S), molecular transport properties,and radiation properties(k h,where required)will be referred to as the‘‘composition variables.’’This set will be denoted by f,a vector of dimension N f.For a low-Mach-number,ideal-gas,single-phase,multicomponent reacting mixture,all necessary quantities normally can be determined from the species mass fractions,enthalpy(or temperature),and a refer-ence pressure p0that is,at most,a function of time:

r?reY;T;p0T;S a?S aeY;T;p0T;k h?k heY;T;p0T;etc:;

(15) with T?T(Y,h)(Eq.8).Our default set of composition variables, therefore,is the N S species mass fractions Y and the mixture speci?c enthalpy h(N f?N St1):

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259179

f?è

Y1;Y2;.;Y N

:(16)

3.7.Reduced descriptions

It often is convenient and appropriate to work with a reduced set of composition variables.Reaction progress variables and mixture fractions are especially useful,and have been introduced in Sections2.1and2.2,respectively.Joint progress variable/mixture fraction descriptions have been developed for modeling partially premixed turbulent systems(e.g.,[52]),and PDF’s of mixture fractions and/or reaction progress variables have been considered in transported PDF methods(Sections9–11).

Flamelet models provide another form of reduced description, and have been formulated for premixed,nonpremixed,and partially premixed systems(see Refs.[3]and[5],for example).In this case, the local thermochemical state is parameterized in terms of a small number of variables(e.g.,a mixture fraction and a scalar dissipation rate,in the case of nonpremixed systems)and the mapping between the full set of variables and the reduced set is established by considering a speci?c canonical laminar?ame con?guration. Flamelets also can be accommodated in PDF methods(Section6.6).

More complete discussions of reduced descriptions can be found in the references that were cited earlier in Section3.3[46,47].Here we presume that an appropriate level of description is available to de?ne the local instantaneous state at any point in the turbulent reacting?ow(Eq.(15)).

4.PDF methods for turbulent reacting?ows

The equations presented in Section3,together with appropriate initial and boundary conditions,are complete.In principle,these equations are suf?cient to determine the spatial and temporal evolution of all dependent variables of interest for a multicompo-nent gas-phase reacting mixture in either laminar or turbulent?ow (e.g.,u?u(x,t),Y?Y(x,t),T?T(x,t),etc.).In the case of turbulent ?ows,however,this rarely is feasible,even for relatively simple geometric con?gurations and using the largest available computers.

This highly nonlinear set of coupled PDE’s exhibits extreme sensitivity to small variations in initial and boundary conditions. One problem,then,is that the initial and boundary conditions corresponding to a real combustion device cannot be speci?ed with suf?cient precision.Even if they could be speci?ed,the equations could not be solved.The solutions are characterized by a wide dynamic range of spatial and temporal scales.For constant-property turbulent?ows,scaling arguments[53]show that the ratio of the largest to the smallest hydrodynamic length scales increases as the Reynolds number to the power3/4;and scales smaller than the smallest hydrodynamic scales may be relevant in the case of chemically reacting?ows at high-Dam-ko¨hler-number.There is a commensurately broad range of velocity and time scales.Finally,even if the equations could be solved,the resulting fully resolved spatial and temporal?elds for all dependent variables would simply be too much data for most practical purposes.

The fundamental premise underlying turbulent combustion modeling is that it should not be necessary to resolve this level of detail to predict key global quantities of interest,such as the overall rate of energy conversion and the net rate of formation of pollutant species.In practice,then,it is both necessary and desirable to limit the dynamic range of length and time scales in the problem.This can be accomplished using probabilistic approaches(the subject of this section)or by?ltering(the subject of Section5).4.1.Mean quantities and probability density functions

The notions that were introduced in Section2now are made precise by introducing the probabilistic mean or expected value of a random variable and the associated PDF’s[24].Here Q?Q(x,t)is a random variable that corresponds to any of the physical quantities of interest in turbulent combustion:velocity,species mass fractions, enthalpy,etc.In a physical device,estimates of the mean would be obtained by ensemble-or phase-averaging(e.g.,measuring the in-cylinder velocity at a speci?ed location for a given piston position over many engine cycles in a reciprocating-piston internal combustion(IC)engine[54]),by long-time averaging in a statisti-cally stationary?ow(e.g.,measuring time series of the species mass fractions at a?xed radial and axial position in an axisymmetric jet ?ame[38]),or by spatial averaging in con?gurations having direc-tions of spatially homogeneity(e.g.,averaging parallel to the rod in a ducted rod-stabilized?ame[32]).Examples of the latter two have been given in Section2.This approach to reducing the dynamic range of scales is the basis for Reynolds-averaged simulation(RAS) of turbulent reacting?ows.The acronym‘‘RANS’’(Reynolds-aver-aged Navier–Stokes)also is used.In the case of reacting?ows,the system of equations goes beyond the Navier–Stokes equations,and ‘‘RAS’’(in the opinion of the author)is a more precise descriptor.It is emphasized that RAS is not limited to statistically stationary?ows.

The mean of Q,denoted C Q D,can be expressed in terms of the PDF of Q,f Q(j;x,t):C Q D?C Qex;tTD?

R N

àN

j f Qej;x;tTd j.The PDF quan-ti?es the probability of Q taking on a particular value:f Q(j;x, t)d j?Prob{j Q(x,t)

R N

àN

f Qej;x;tTd j?1.The PDF is a density in the sample space j correspondin

g to random variable Q, and its dimensions are those of Qà1;the PDF is a function of position x and of time t,re?ecting the fact that the statistics of Q can vary in space and in time.Joint PDF’s of multiple random variables are de?ned analogously.

In turbulent combustion,a central role is played by the joint PDF of the velocity u(x,t)and of the composition variables fex;tT(Section 3.6)[24].With the simpli?cations introduced in Section3.6,a suitable set of composition variables is composed of the species mass fractions and enthalpy(Eq.(16)),and further reductions often can be made (Section3.7).We then introduce the velocity–composition joint PDF f u feV;j;x;tT;this is the one-point,one-time Eulerian joint PDF of the event f uex;tT?V;fex;tT?j g.Properties of this PDF include f u feV;j;x;tT!0and

R R

f u feV;j;x;tTd V d j?1.Here and in the following,integration over the entire sample space is implied,unless speci?ed otherwise.The most important property of the PDF for the present discussion is that the mean of any function of u(x,t)and fex;tT,Q?Qeu;fT,can be expressed as an integral over the PDF:

h Q i?h Qex;tTi?

Z Z

QeV;jTf u feV;j;x;tTd V d j:(17)

For variable-density?ows,it is advantageous to work with density-weighted(Favre-averaged)quantities,as the resulting equations have a simpler form that is close to that of the original instantaneous equations.The Favre PDF~f u feV;j;x;tT,mass density function F u feV;j;x;tT,and conventional PDF f u feV;j;x;tTare related by:

F u feV;j;x;tT?

rex;tT

u feV;

j;x;tT?rejTf u feV;j;x;tT;(18)

where C r D is the mean mixture mass density.Conventional(CD)and density-weightedeeTmeans of any function of u and f are readily computed from the PDF.For any Q?Qeu;fTwe have,

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259 180

Z Z Q eV ;j Tf u f eV ;j ;x ;t Td V d j ?h Q ex ;t Ti ;Z Z

Q eV ;j T~f u f eV ;j ;x ;t Td V d j ?~Q

ex ;t T:(19)

It follows that,

ex ;t T?D

r ex ;t TQ ex ;t TE.D

r ex ;t TE

;

(20)

and hence C r D ?eg r

à1Tà1.Fluctuations about the conventional mean are designated by a single prime,while double primes are used for ?uctuations about the Favre mean.Instantaneous quanti-ties then are decomposed into their mean and ?uctuating compo-nents (a Reynolds or Favre decomposition)as,

Q ex ;t T? Q ex ;t T

tQ 0ex ;t T;or

Q ex ;t T?~Q

ex ;t TtQ 00ex ;t T:(21)

It follows that,

Q 0ex ;t T

h 0and f Q

00ex ;t Th 0;while in general

Q 00ex ;t T s 0and f Q

0ex ;t Ts 0:(22)

Also,

DD Q ex ;t TEE ?D Q ex ;t TE

and e ~Q

ex ;t T?~Q ex ;t T:(23)

Important quantities including the Favre-averaged velocity

e~u T,Favre-averaged mass fractions e~Y

T,velocity covariances or Reynolds stresses eg u 00i u 00j

T,mass-fraction variances eg Y 002a Tand turbulent scalar ?uxes eg u 00i Y 00a

Tcan be extracted from the PDF using Eqs.(19)and (21).

Marginal PDF’s,conditional averages,and conditional PDF’s were introduced in the context of an example in Section 2.3,and are useful in the development of PDF methods.Key relationships are summarized here.The marginal PDF’s of velocity,f u ?f u (V ;x ,t ),and of composition,f f ?f f ej ;x ;t T,can be obtained from the joint velocity–composition PDF by integration,

f u eV ;x ;t T?Z f u f eV ;j ;x ;t Td j ;f f ej ;x ;t T?

f u f eV ;j ;x ;t Td V :

(24)

For any Q ?Q (x ,t ),the mean value of Q ,conditioned on the velocity u ?u (x ,t )at location x at time t having the particular value V and the composition f ?f ex ;t Tat location x at time t having the particular value j ,is denoted by C Q ex ;t Tj u ex ;t T?V ;f ex ;t T?j D ?C Q ex ;t Tj V ;j D .If Q is a known function of velocity and the composition variables eQ ?Q eu ;f TT,then

h Q ex ;t Tj V ;j i ?Q eV ;j Tfor Q ?Q eu ;f T:

(25)

On the other hand,if Q ?Q (x ,t )is independent of u and of f ,then

h Q ex ;t Tj V ;j i ?h Q ex ;t Ti for Q independent of u and f :

(26)

Conditional PDF’s can be introduced.For example,

f u j f ?f u j f eV j j ;x ;t Th f u f eV ;j ;x ;t T

f f ej ;x ;t T

(27)

Then

h Q ex ;t Tj j i ?

Q eV ;j Tf u j f eV j j ;x ;t Td V ;

(28)

and

h Q ex ;t Ti ?

h Q ex ;t Tj j Ti f f ej ;x ;t Td j :

(29)

If u and f are statistically independent,then

f u j f ?f u and f u f ?f u f f eu and f statistically independent T:

(30)

One-point,one-time statistics (and hence the PDF’s considered

here)provide no explicit information on spatial structure,length scales,or time scales in turbulent reacting ?ows.Both the formal axiomatic development and the methods that are used to estimate mean values in physical systems (e.g.,ensemble averaging,time averaging,and/or spatial averaging)suggest that mean quantities and PDF’s in general should evolve on length and time scales that are not smaller than the largest turbulence scales (the integral scales).Therefore,the spatial derivatives and temporal derivatives that appear in PDF transport equations and in any equations that are derived from them (e.g.,moment equations)in most cases are expected to be smooth at scales that are of the order of the integral scales.

4.2.PDF transport equations

4.2.1.Velocity–composition PDF

An outline of the derivation of the velocity–composition PDF equation corresponding to Eq.(7)is provided in Appendix A .The result (in terms of the conventional PDF,with f as in Eq.16,and neglecting the D p /D t and dissipation terms in the enthalpy equa-tion)is:

v r f u f v t t

v r V i f u f v x i tàr g i àv h p i v x i áv f u f

v V i

tv r S a f u f j a àd a eh T

v _Q rad ;em f u f

j a ?v v V i " ààv s ji

v x j tv p 0v x i á V ;j f u f #

v v j a ? v J a i v x i V ;j f u f ?àd a eh T

v v j a

? _

Q rad ;ab V ;j f u f ?:

(31)

Here r ?r ej T,summation is implied over repeated indices i ,j ,or a within a term,and a ?1,.,N f (N f ?N S t1,to include enthalpy).The notation d a (h )indicates that that radiation source terms appear only the the enthalpy equation (here a ?N f ?N S t1corresponds to h ).Transport in physical space by the velocity V (including turbulent velocity ?uctuations),transport in velocity space due to body forces and the mean pressure gradient,transport in composition space by chemical reaction,and transport in composition space due to radiative emission appear in closed form (left-hand side of Eq.(31)).On the right-hand side are terms to be modeled.These represent transport in velocity and composition space due to molecular processes (terms involving s ji and J i a ,respectively),transport in velocity space due to pressure ?uctua-tions (term involving v p 0/v x i ),and transport in composition space due to radiative absorption.

46254ffbf111f18582d05a6dposition PDF

The composition PDF f f ej ;x ;t Tis recovered from the velocity–composition PDF by integrating over velocity space:f f ej ;x ;t T?R

f u f eV ;j ;x ;t Td V .Applyin

g this to Eq.(31)yields Eq.(A.15):

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259181

v r f f v t t

v r ~u i f f v x i tv r S a f f v j a àd a eh Tv _Q rad ;em f f

v j a

?àv v x i hD u 00i j E r f f i tv v j a (v J a i v x i j )f f !àd a eh Tv j a

hD _Q rad ;ab

j E f f i :

e32T

In this case,only the mean velocity appears in a closed form,while the effect of turbulent velocity ?uctuations (the so-called ‘‘turbulent diffusion’’)appears as a term to be modeled (?rst term on the right-hand side of Eq.(32)).The other terms to be modeled again represent molecular transport and radiative absorption.

4.3.Equations for mean quantities

Equations for mean quantities can be derived from the PDF transport equations by applying Eq.(19);see Appendix B .The principal equations of interest here are those for the mean density (continuity),mean velocity (momentum),mean mass fractions (chemical species),and mean enthalpy (energy).The mean conti-nuity,mean species mass fractions,and mean enthalpy equations can be derived from either Eq.(31)or Eq.(32)while the mean momentum equation requires Eq.(31).In Favre-averaged form,and retaining the D p /D t and dissipation terms in the enthalpy equation,these equations are:

v h r i v t tv h r i ~u

i v x i ?0;

v h r i ~u

j v t t

v h r i ~u j ~u i v x i

?v h r i ~u j ~u i àh r i g u j u i

v x i

tv s ij

v x i àv h p i v x j

th r i g j ej ?1;2;3T;v h r i ~Y

a v t t

v h r i ~Y a ~u i v x i

h r i ~Y a ~u i àh r i g Y a u i v x i àv J

a i v x i th r i ~S a ea ?1;2;.;N S T;v h r i ~h v t t

v h r i ~h ~u

i v x i

h r i ~h ~u i àh r i f hu i v x i àv J h i

v x i

tD h p i Dt tF à _Q rad ;(33)

where F is the mean viscous dissipation rate of kinetic energy to

heat,

F h (

s ij v u j

v x i

(34)

The form of the equations is similar to that of Eq.(7),with the exception of the ?rst term on the right-hand side of the mean momentum,species,and enthalpy equations.The quantity in parentheses in each of those terms can be written as,

h r i ~u j ~u i àh r i g u j u i ?àh r i g u 00j u 00i ?s Tij ;h r i ~Y a ~u i

àh r i g Y a u i ?àh r i g Y 00a u 00i ;h r i ~h ~u i àh r i f hu i ?àh r i g h 00u 00i

:(35)

These represent turbulent ?uxes of momentum,species,and

enthalpy,respectively.The ?rst of these is the apparent turbulent stress,s Tij .

Eq.(33)can be derived directly from Eq.(7)without using the PDF equation as an intermediary.The principal requirements are that instantaneous quantities can be decomposed as indicated in Eq.(21),that the CD (hence e ,by virtue of Eq.(20))operator commutes with temporal and spatial differentiation,and that Eqs.(22)and (23)are satis?ed.These formal manipulations do not require the assignment of any additional physical meaning to the CD operator.The derivation based on the PDF equation has several advantages for present purposes.The PDF equation is established as the point of departure for turbulent combustion modeling:PDF’s play a central role in essentially all turbulent combustion models.The approach establishes the connections among the instantaneous governing PDE’s,PDF equations,and moment equations.It eluci-dates the physics of the terms that require modeling.And it mini-mizes confusion between formal manipulations of the governing equations and the techniques that are used to approximate mean quantities in experimental measurements (e.g.,ensemble aver-aging,time averaging,or spatial averaging).

An equation governing the mean pressure ?eld can be derived by taking the pergence of the mean momentum equation,and invoking the mean continuity equation:

v 2D p E j j

?v 2h r i v t àv 2h r i ~u i ~u j i j tv 2à s ij ts Tij á

i j tg j

v h r i

j :(36)

This equation involves the mean density and one-point statis-tics of the velocity ?eld,thus con?rming that the mean pressure ?eld is closed in terms of the velocity–composition joint PDF.CFD algorithms for low-Mach-number ?ows often are based on solving a PDE for pressure rather than for density,and density is obtained from an equation of state [55].This is discussed further in Section 7.

Commonly invoked simpli?cations to Eqs.(33)and (36)include neglecting the mean molecular transport terms compared to the turbulent transport terms (appropriate for high Reynolds number),neglecting the molecular dissipation term F in the mean enthalpy equation,and neglecting the spatial variations in pressure in the mean enthalpy equation so that D C p D /D t z d p 0/d t (appropriate for low-Mach-number).The d p 0/d t term may also be negligible in open ?ames,but is important in applications such as reciprocating-piston IC engines.

Equations governing any one-point statistic of the composition ?eld can be derived from Eq.(31)or (32),and equations governing any one-point joint statistic of the velocity and composition ?eld can be derived from Eq.(31)(Appendix B ).These equations also can be derived directly from Eq.(7)without a PDF equation as an intermediary.The velocity covariance (Reynolds-stress),species covariance,and velocity-species covariance (turbulent scalar ?ux)equations are provided in Appendix B ,for reference.

The closure problem appears in different guises in PDF equa-tions (e.g.,Eqs.(31)and (32)and Eqs.(A.12)–(A.17))compared to moment equations (e.g.,Eqs.(33)and (36)and Eqs.(B.1)–(B.3)).In a velocity–composition PDF method,models are required to represent the effects of molecular transport,the ?uctuating pres-sure gradient,and radiative absorption;whereas turbulent trans-port,the mean pressure gradient,body forces,chemical reaction,and radiative emission appear in closed form.In approaches based on solving any ?nite number of moment equations,on the other hand,all of these processes must be modeled.A one-point,one-time Eulerian PDF essentially contains the same information as an in?nite set of moments.

The single most compelling reason for bringing PDF methods to bear in the analysis and modeling of turbulent reacting ?ows is the advantage that they offer in dealing with the chemical source term.Although S ?S ef Tis in principle known (Sections 3.3and 3.6),

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259

182

ef Ts S ~f :(37)

This is a manifestation of ‘‘turbulence–chemistry interactions’’

(TCI).The problem arises because of the strong nonlinearity of the

chemical source term.Modeling ~S

in terms of any ?nite number of moments of f generally is not viable [56].On the other hand,for an arbitrary detailed,skeletal,or reduced chemical mechanism

(Section 3.3),the mean chemical source term ~S

is closed in terms of the velocity–composition PDF or the composition PDF:

?Z Z

S ej T~

f u f eV ;j ;x ;t Td V d j ?Z

S ej T~f f ej ;x ;t Td j :

(38)

Thus in PDF methods,the emphasis shifts from directly modeling the effects of turbulent ?uctuations on the mean chem-ical source terms to the modeling of molecular transport processes (‘‘mixing models’’–Section 6).The consequences of replacing the inequality in Eq.(37)with an equality have been illustrated in Fig.8.Similar arguments can be made for the mean radiation emission source term (see Section 8.3)46254ffbf111f18582d05a6dgrangian particle equations

The instantaneous Eulerian PDE’s (Eq.(7))can be recast in Lagrangian form as equations governing the motion and property evolution of ?uid particles in a chemically reacting turbulent ?ow [24].The transformation from an Eulerian frame to a Lagrangian frame does not,by itself,resolve the inherent dif?culties that were listed in the second paragraph of Section 4.It does,however,provide an alternative viewpoint for devising physical models (Section 6)and numerical solution strategies for PDF transport equations (Section 7)that has proved to be quite powerful.

The general idea is to devise a system of ‘‘notional’’particles whose evolution yields the same one-point,one-time Eulerian PDF as the real ?uid particle system.Models for notional particle behavior then effectively provide closure models for unclosed terms in the PDF equations,and solving for the evolution of notional particles effectively corresponds to solving a modeled PDF transport equation.For concreteness,we consider the turbu-lent reacting ?ow to be represented by a large number N P of notional particles.The i th particle is assigned a mass m (i );in the simplest case,m (i )?m /N p where m is the total system mass.In addition,each particle is characterized by three position coordi-nates x (i )(t )and N f scalar variables f ei Tet T(species mass fractions and mixture enthalpy being the default set –Eq.16);in the case of a velocity–composition PDF,each particle also carries three velocity components u (i )(t ).

4.4.1.Velocity–composition PDF

Formally,a discrete mass density function F *is introduced,

F *u f x eV ;j ;y ;t Th

X N P i ?1

m ei Td V àu ei Tet T d j àf ei Tet T d y àx ei T

et T ;(39)where d ey àx ei Tet TTis a three-dimensional delta function at the particle location,and similarly for d eV àu ei Tet TTand d ej àf ei Tet TT.For a consistent discrete representation,we require that,

D F *u f x

eV ;j ;y ;t TE ?F u f eV ;j ;x ;t T?D r ex ;t TE

~f u f eV ;j ;x ;t T?r ej Tf u f eV ;j ;x ;t T

(40)

(see Eq.(18)).This representation is consistent with the ?ne-grained PDF approach that is one of the avenues that can be fol-lowed to derive the PDF transport equations (Appendix A ).

In an in?nitesimal time increment d t ,the position,velocity,and

composition of each notional particle evolve according to,

d x *i ?u *i d t ei ?1;2;3T;d u *i ?

g i à1r ef *Tv h p i v x i

* d t ta *i ;p 0d t ta *i ;vis d t ei ?1;2;3T;d f *a

?S a f * d t àd a eh T_Q rad ;em f * d t td a eh Tq *rad ;ab d t tq *

a ;mix d t

a ?1;2;.;N f :

(41)Here the superscript *refers to any particle,1 i N P ,and a mean quantity with superscript *refers to the mean value eval-uated at the particle location:e.g.,v C p D */v x i ?v C p D /v x i j x ?x *(t ).The quantity a *p 0denotes a particle acceleration due to the ?uctuating pressure gradient,a *vis is a particle acceleration due to molecular viscosity (velocity mixing),q *rad,ab d t is the increment in enthalpy

due to radiative absorption,and q *

mix d t is the increment in composition due to molecular diffusion (scalar mixing).The mean density at the particle location C r D *can be used instead of the particle density r ef *Tthat appears in the mean pressure gradient term in the particle velocity equation.The signi?cance of this density speci?cation will be discussed in the context of an example in Section 9.3.

The transport equation for the one-point,one-time Eulerian velocity–composition PDF g u f eV ;j ;x ;t Tcorresponding to Eq.(41)is (Appendix A ),

v r g u f v t t

v r V i g u f v x i

r g i àv h p i v x i v g u f

v V i tv r S a g u f v j a àd a eh T

v _Q rad ;em g u f

v j a

?à

v v V i hDD a *i ;p 0ta *

i ;vis V ;j EE r ej Tg u f i àv v j a

d a eh Tq *rad ;ab

tq *

a ;mix

V ;j E r ej Tg u f i

(42)

Equations governing the evolution of the one-point statistics

implied by the notional particle system then can be derived in the same way that Eq.(33)and Eqs.(B.1)–(B.3)were derived from Eq.(31),for example.

Comparing Eqs.(31)and (42),it can be seen that a necessary condition for g u f eV ;j ;x ;t Tfrom the notional particle system to evolve in the same manner as f u f eV ;j ;x ;t Tfrom the real turbulent

?uid system is that the conditional expectations of a *i ;p 0,a *i ;vis ,q *rad ;ab ,and q *

a ;mix must (in some sense)mimic the behavior of the condi-tional expectations of àr à1v p 0/v x i ,r à1v s ji /v x j ,r à1_Q rad ;a

b ,and

àr à1v J i a /v x i .This must be accomplished by modeling a *p 0,a *vis ,q *rad ;ab ,and q *

mix as functions of quantities that can be deduced from the PDF (any one-point joint statistic of the velocity and/or composition ?elds)using either stochastic or deterministic models,together with scale information.These models for notional particle behavior provide closure models for the terms that appear on the right-hand side of Eq.(31).To have made progress,the quantities that are used to close the notional particle equations generally should be selected such that the resulting modeled PDF transport equation does not develop scales that are smaller than the integral scales.It is important to note that this does not exclude processes that are discontinuous in time at the notional particle level.In fact,discontinuous-in-time processes often are used in PDF-based modeling (Section 6),and that is the reason for writing Eq.(41)in terms of in?nitesimal increments rather than as differential equa-tions.By virtue of the statistical equivalencies that have been

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259183

established between the instantaneous Eulerian PDE’s,?uid particle equations,PDF transport equations,moment equations, and notional particle systems,model development can be guided by information obtained from any or all of these representations (Section6).

46254ffbf111f18582d05a6dposition PDF

In the case of a composition PDF,notional particles do not carry velocity as a transported quantity.The equations analogous to Eqs.

(39)–(42)are:

F*f xej;y;tTh

X N P

i?1meiTd

jàfeiTetT

yàxeiTetT

;(43)

F*f xej;y;tTE

?F fej;x;tT?

rex;tT

fej;x;tT

?rejTf fej;x;tT;(44)

d x*

i ?~u*i d ttdx*

i;turb

ei?1;2;3T;

d f*a?S a

d tàd aehT_Q rad;em

d t

td aehTq*rad;ab d ttq*a;mix d t

a?1;2;.;N f

;(45)

and

v r g f

v r~u i g f

i t

v r S a g f

j aà

d aehTv

rad;em

g f

j a

?àv

v x i hD

u00

r g f

àv

j a

d aehTq*

rad;ab

tq*a;mix

rejTg f

;(46)

where g fej;x;tTis the one-point,one-time Eulerian composition PDF corresponding to Eq.(45).

Here d x*

turb is the increment in particle position resulting from

turbulent velocity?uctuations about the local mean velocity~u* (‘‘turbulent diffusion’’),and usually is modeled using a random walk model that is consistent with a gradient transport approximation. Models for the unclosed scalar terms generally are the same as those that are used for the scalar terms in the velocity–composition PDF equation(Section6),although models that explicitly require a particle velocity generally cannot be used in the composition PDF method.

4.4.3.Consistency

In the representation outlined above,each notional particle represents a speci?ed mass of?uid:this is a mass density function formulation.Therefore,the distribution of particle mass(or volume)in physical space is not arbitrary;it must remain consistent with the?uid mass(or volume)distribution.

The conditions under which an initially valid particle distribu-tion remains valid as the system evolves in time have been enumerated in[24].An evolution equation for the particle mass density distribution in physical space has been derived in[57].And an analysis for constant-property turbulent?ow is presented in Chapter12of[29].Essentially the requirement is that the expected (mean)density of particles in physical space must remain propor-tional to the local mean?uid density.

The initial distributions of particle masses and of particle loca-tions in physical space are readily prescribed in a manner that is consistent with the initial spatial distribution of mean?uid mass (Section7).Then in order for the discrete particle representation to remain a valid discretization of the modeled PDF transport equa-tion,it is necessary that the spatial distribution of mean particle mass remain consistent with the distribution of mean?uid mass for all time.This will be satis?ed provided that the particle system evolves in a manner that is consistent with the mean continuity equation.The mean continuity equation,in turn,is satis?ed if and only if the mean pressure?eld satis?es the Poisson equation that results from taking the pergence of the mean momentum equa-tion(Eq.(36)).

It is emphasized that this is an intrinsic requirement for the particle representation to be valid;it provides guidance and places constraints on the construction of numerical algorithms,including both stand-alone particle methods and hybrid Lagrangian particle/ Eulerian mesh methods.Additional consistency constraints that are speci?c to particular algorithms are discussed in Section7.

4.5.Eulerian?eld equations

In Section4.4,the concept of systems of equivalent notional Lagrangian particles was introduced.The utility of particle systems for physical modeling and numerical solution will be discussed in Sections6and7,respectively.

Systems of equivalent notional Eulerian?elds also can be devised.Starting from a system of Eulerian?eld equations,the corresponding velocity–composition PDF and/or composition PDF transport equation can be derived following the approaches outlined in Appendix A.The general idea is to devise a set of Eulerian?eld equations whose one-point,one-time Eulerian PDF equation evolves in the same way as the one-point,one-time Eulerian PDF equation that corresponds to Eq.(7),while cir-cumventing the issues that were raised in the second paragraph of Section4.

Both stochastic[58,59]and deterministic[4,60]Eulerian?eld PDF methods have been developed over the past10years.This line of research is relatively new compared to particle-based PDF methods.Further discussion is deferred to Section7below;the point of departure there will be a closed(modeled)form of a PDF transport equation.

4.6.Regimes of applicability

At this point,no approximations have been introduced in the PDF transport equations(Eqs.(31)and(32))or in the notional-particle-based representations(Eqs.(41)and(45)).These equations are valid for premixed,nonpremixed,and partially premixed systems in any regime of turbulent combustion;see Chapters5and6of[5]or Section5of[6]for modern discussions of regimes of premixed and nonpremixed combustion.Restrictions to speci?c combustion regimes are introduced in the modeling stage(Section6).

PDF methods are most valuable in situations where either chemical kinetics or turbulence may be rate controlling.In practice, they have been applied primarily to nonpremixed systems,and operator-splitting algorithms have been employed(Section7)that imply weak coupling between molecular mixing and chemical reaction.These are not intrinsic limitations,however.PDF methods can be and have been applied to premixed systems and to?amelet regimes where molecular transport and chemical reaction are tightly coupled.The treatment of?amelets in PDF methods is dis-cussed in Section6.6,and examples of applications of PDF methods and of hybrid PDF/?amelet models to premixed and partially pre-mixed systems are included in Sections9and10.

5.Filtered density function(FDF)methods for large-eddy simulation

An alternative to the probabilistic(Reynolds-averaged)approach for reducing the wide dynamic range of scales that is inherent in the

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259 184

instantaneous governing equations at high Re and/or high Da is to ?lter the equations in space and/or in time.This is the basis for large-eddy simulation (LES)of turbulent reacting ?ows.The usual practice is to apply spatial ?lters only,and that is the approach that is fol-lowed here.

In LES,the dynamics of scales that are larger than the ?lter width (the resolved scales)are captured explicitly while the effects of sub?lter-scale ?uctuations are modeled.This is in contrast to RAS,where the effects of all turbulent ?uctuations about the local mean at all scales must be modeled.Therefore,it long has been argued that LES should be advantageous compared to RAS in accuracy and in generality [61].However,there are signi?cant differences between LES for nonreacting ?ows (where most of these arguments have been made)and LES for reacting ?ows.In constant-density,high-Reynolds-number hydrodynamic turbulence,remote from walls and without chemical reactions,the rate-controlling processes are determined by the resolved large scales [29].In contrast,in chemically reacting turbulent ?ows the essential rate-controlling processes (molecular transport and chemical reaction)occur at the smallest (usually unresolved)scales [25,62].In that case,the LES-based models that are required to parameterize the effects of sub?lter-scale ?uctuations on local ?ltered reaction rates in terms of resolved-scale quantities must be essentially the same as the RAS-based models that are required to parameterize the effects of turbulent ?uctuations on local mean reaction rates in terms of mean quantities.In this sense,the arguments favoring LES over RAS may be less compelling in the case of turbulent reacting ?ows.Nevertheless,a wide and rapidly growing body of evidence demonstrates quantitative advantages of LES in modeling studies of laboratory ?ames and in applications to gas-turbine combustors,IC engines,and other combustion systems.Examples are provided in Sections 10and 11,and recent reviews include [63–66].

The analogue of a PDF method in the LES context is a ?ltered-density function (FDF)method.The use of PDF-based approaches for sub?lter-scale modeling in LES was suggested by Givi [67].The FDF formulation was proposed by Pope [25],and a transport equation for a composition FDF was derived and modeled by Gao and O’Brien [68].A Lagrangian Monte Carlo method for the numerical solution of a modeled composition FDF equation in constant-property turbu-lent ?ow was developed and exercised in [69];this was extended to variable-density ?ows in [70].The velocity FDF and velocity–composition FDF for constant-property turbulent ?ows were introduced in [71]and in [72],respectively,and the velocity–composition FDF was extended to variable-density ?ows in [73].FDF methods currently are under intense development by several research groups [74–82].An experimental study targeting the velocity FDF is reported in [83].Recent reviews include [84]and [85].5.1.Spatially ?ltered quantities and ?ltered density functions The local spatially ?ltered value of a physical quantity Q ,deno-ted C Q D D ,is de?ned as [25,84,86],

h Q i D ?h Q ex ;t Ti D h Z

Q ey ;t TG ej x ày jTd y ;

(47)

and integration is over the entire ?ow domain.Here the low-pass

spatial ?lter function,G (j x ày j ),satis?es R

G ex Td x ?1,has a char-acteristic ?lter width D ,is independent of spatial location,and is isotropic.The latter two properties often are not satis?ed by the ?lters that are used in practice,but simplify the exposition.Prop-erties of ‘‘well-behaved’’?lter functions for LES are discussed in [84]and in Chapter 13of [29],for example.The instantaneous value of any physical quantity Q ?Q (x ,t )in a turbulent ?ow can be decomposed into a ?ltered component (the resolved ?eld)and

a ?uctuation about the ?ltered component (the residual ?eld):Q (x ,

t )?C Q (x ,t )D D tQ 0

(x ,t ).Density weighting is useful in variable-density ?ows:

b Q h h r Q i D =h r i D

r ey ;t TQ ey ;t TG ej x ày jTd y =h r i D ;

(48)

and the corresponding ?uctuation is denoted using a double prime:

Q ?b Q

tQ 00.To reduce the proliferation of nomenclature,the same notation is used here for sub?lter-scale ?uctuations about local spatially ?ltered values as that used earlier for ?uctuations about local mean values;the interpretation will be clear from the context (RAS/PDF versus LES/FDF).The properties of spatial ?lters (LES)differ from those of probabilistic means (RAS).For example,in contrast to Eqs.(22)and (23),in general CC Q D D D D s C Q D D and C Q 0D D s 0.

The velocity–composition FDF is denoted f D ;u f ?f D ;u f eV ;j ;x ;t T,and is de?ned as,

f D ;u f eV ;j ;x ;t Th

d eV àu ey ;t TTd ej àf ey ;t TTG ej x ày jTd y ;

(49)

where multidimensional delta functions are used,as in Eq.(A.2).Thus f D ;u f is the G -weighted spatial average of the ?ne-grained PDF in a neighborhood of x ,and f D ;u f d V d j is the G -weighted fraction of the ?uid in a neighborhood of x whose velocity and composition are in the range V u

Important properties of the FDF are that it is non-negative (positive ?lters are required for FDF methods,although not for LES in general),it integrates to unity over V àj space,and the local spatially ?ltered value of any function of u and of f ,Q ?Q eu ;f T,can be expressed as an integral over the FDF:

f D ;u f eV ;j ;x ;t T!0;Z Z

f D ;u f eV ;j ;x ;t Td V d j ?1;Z Z

Q eV ;j Tf D ;u f eV ;j ;x ;t Td V d j ?h Q i D ?h Q ex ;t Ti D :

(50)

These are analogous to the corresponding properties of a PDF.However,there are important differences between FDF’s and PDF’s.The FDF varies on length scales down to D ,versus the turbulence integral scale l T in the case of the PDF;normally D

f u f ?lim

D /0

f D ;u f Eo :

(51)

That is,the PDF is the limit of the expected value of the FDF in the limit as the ?lter width D shrinks to zero.

Favre (density-weighted)FDF’s and ?ltered mass density func-tions (FMDFs)can be de?ned that are analogous to the corre-sponding PDF-based quantities (Eq.(18)).Filtered mass density functions usually are taken as the basis for modeling variable-density turbulent reacting ?ows,although we shall continue to refer to the generic approach as an FDF method.For example,the velocity–composition FMDF is,

F D ;u f eV ;j ;x ;t Th Z

r ey ;t Td eV àu ey ;t TTd ej àf ey ;t TTG ej x ày jTd y ;

(52)

and has the following properties:

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259185

F D ;u f eV ;j ;x ;t T!0;Z Z

F D ;u f eV ;j ;x ;t Td V d j ?h r i D ;Z Z

Q eV ;j TF D ;u f eV ;j ;x ;t Td V d j ?h r Q i D ?h r i D b Q

ex ;t T:(53)

Composition FDF’s and composition FMDF’s also can be intro-duced,either starting from the de?nitions,

f D ;f ej ;x ;t Th Z

d ej àf ey ;t TTG ej x ày jTd y ;

F D ;f ej ;x ;t Th Z

r ey ;t Td ej àf ey ;t TTG ej x ày jTd y ;

(54)

or by integrating the corresponding velocity–composition FDF or

FMDF over velocity space,

f D ;f ej ;x ;t T?

Z f D ;u f eV ;j ;x ;t Td V ;F D ;f ej ;x ;t T?

F D ;u f eV ;j ;x ;t Td V :

(55)

5.2.FDF transport equations

Skeletal derivations of the velocity–composition FMDF equation and the composition FMDF equation corresponding to Eq.(7)are provided in Appendix C .With the same set of simpli?cations as was adopted for Eq.(31),the results are Eqs.(C.7)and (C.9),respectively:

v F D ;u f t

v V i F D ;u f i tg i v F D ;u f i àv i

?r à1v h p i D

i F D ;u f ?tv S a F D ;u f v j a àd a eh T

v r à1_Q rad ;em F D ;u f

v j a

?v v V i ?à r à1v p v x i j V ;j D

àr à1v h p i D v x i á

F D ;u f

?à

v v V i

? r à1v s ji

v x j j V ;j D

F D ;u f ?tv v j a ?

à1v J a

v x i

j V ;j

F D ;u f

àd a eh T

v j a

r à1_Q rad ;ab j V ;j

D F D ;u f ?

;

(56)

and

v F D ;f v t t

v b u i F D ;f v x i tv S a F D ;f v j a àd a eh Tv r à1_Q rad ;em F D ;f

v j a

?v v x i h b u

i àh u i j j i D F D ;f i tv v j a (r à1v J a i v x i j j )D

F D ;f !àd a eh Tv v j a

hD r à1_Q rad ;ab j j E D F D ;f i ;(57)

where r ?r ej T.

Eq.(56)for the velocity–composition FMDF has essentially the

same structure as Eq.(A.14)for the velocity–composition PDF (expressed as a mass density function);similarly,Eq.(57)for the composition FMDF has essentially the same structure as Eq.(A.17)for the composition PDF.The principal difference is that mean quantities in the PDF equations are replaced by spatially ?ltered quantities in the FMDF equations.Another difference is that for a Reynolds decompo-sition,C Q j V ;j D ?C eC Q D tQ 0Tj V ;j D ?C Q D tC Q 0j V ;j D ,whereas for a spatial ?lter,C Q j V ;j D D ?C eC Q D D tQ 0Tj V ;j D D s C Q D D tC Q 0j V ;j D D .

5.3.Equations for ?ltered quantities

By virtue of Eq.(53),equations for density-weighted,spatially ?ltered quantities can be obtained by taking moments of the FMDF equation in V àj space.They also can be derived directly by spatially ?ltering Eq.(7)without using the FMDF equation as an intermediary,and identical results are obtained as long as the ?lter that is used has the same properties in both cases.The equations for ?ltered density,velocity,species mass fractions,and enthalpy have the same forms as Eq.(33)(replacing CD with CD D ,and e with ˇ

),provided that the ?ltering operation commutes with temporal and spatial differentiation.Retaining the D p /D t and dissipation terms in the enthalpy equation,the equations are:

v h r i D v t t

v h r i D b u

i v x i ?0;v h r i D b u

j t

v h r i D b u j b u i i

h r i D b u j b u i àh r i D d u j u i i tv s ij

D i àv h p i D j

th r i D g j ej ?1;2;3T;

v h r i D b Y a v t t

v h r i D b Y a b u i v x i

?v h r i D b Y a b u i àh r i D d Y a u i

v x i àv J

a i D i

th r i D b S a ea ?1;2;.;N S T;

v h r i D b h v t t

v h r i D b h b u

i v x i

h r i D b h b u i àh r i D d hu i v x i àv D J h i E D v x i tD h p i D Dt

tF D à _Q rad D ;(58)

where F D is the ?ltered viscous dissipation rate of kinetic energy to heat,

F D h (

s ij v u j

v x i )D

(59)

Although Eqs.(58)and (33)have the same form,there are important differences that follow from the differences between PDF’s and FDF’s that were noted earlier.The ?ltered quantities whose evolution is governed by Eq.(58)are random variables.The PDE’s for ?ltered quantities (Eq.(58))involve length scales down to D ,while the PDE’s for mean quantities (Eq.(33))involve length scales down to l T (the turbulence integral scale),and normally D

The ?rst terms on the right-hand sides of Eq.(58)also are more complicated than their counterparts in Eq.(33).Because ?ltering of a conventional or density-weighted ?ltered quantity does not

recover the original ?ltered quantity eCC Q D D D D s C Q D D ;b b Q

s b Q Tand sub?lter-scale ?uctuations do not ?lter to zero eC Q 0D D s 0;c Q

00s 0T,expressions analogous to Eq.(35)generally are not valid for LES.The corresponding expressions are,

h r i D b u j b u i àh r i D d u j u i

?h r i D b u j b u i àh r i D d b u j b u i àh r i D d b u j u 00i àh r i D d u 00j b u i àh r i D d u 00j u 00i ;

h r i D b Y a b u i àh r i D d Y a u i

?h r i D b Y a b u i àh r i D d b Y a b u i àh r i D d b Y a u 00i àh r i D d Y 00a b u i àh r i D d Y 00a u 00i ;

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259

186

h r i D b

h b u i àh r i D d hu i ?h r i D b h b u i àh r i D d b h b u i àh r i D d b hu 00i

àh r i D d h 00b u i àh r i D d h 00u 00i :(60)

These terms are closed in a velocity–composition FDF method,but require modeling in a composition FDF method.The ?rst two terms on the right-hand side of each expression involve only the resolved ?elds,and therefore do not require modeling in a composition FDF method or in a more conventional LES approach based directly on the solution of Eq.(58).The three remaining terms require modeling;these represent the effects of correlations between ?uctuations in resolved-scale and sub?lter-scale quanti-ties (third and fourth terms),and the effects of correlations between ?uctuations in sub?lter-scale quantities (?fth term).

As was the case for PDF methods,the single most compelling reason for pursuing FDF methods in turbulent reacting ?ows is that the chemical source terms (and other important one-point processes,including radiative emission –Section 8.3)remain in closed form even as the dynamics of the small scales are removed by spatial ?ltering.By analogy with Eqs.(37)and (38),while S ?S ef Tis in principle known (Sections 3.3and 3.6),

b S ef Ts S b f

;(61)

and the ?ltered chemical source term b S is closed in terms of the

velocity–composition FMDF or the composition FMDF:

h r i D b

S ?Z Z S ej TF D ;u f eV ;j ;x ;t Td V d j

S ej TF D ;f ej ;x ;t Td j :

(62)

Thus as in PDF methods,the emphasis shifts from directly modeling the effects of turbulent ?uctuations on the (now ?ltered)chemical source terms to the modeling of molecular transport processes (‘‘mixing models’’–Section 6)46254ffbf111f18582d05a6dgrangian particle equations

The discrete particle representation for a LES/FDF method is essentially the same as that introduced in Section 4.4for RAS/PDF methods.Those are Eq.(39)for a velocity–composition method or Eq.(43)for a composition method.The consistency conditions analogous to Eq.(40)for a velocity–composition method and Eq.(44)for a composition method,respectively,now become,

hF *u f x eV ;j ;y ;t Ti D ?F D ;u f eV ;j ;x ;t T;

(63)

and

hF *f x ej ;y ;t Ti D ?F D ;f ej ;x ;t T:

(64)

The particle equations for velocity–composition and composi-tion FDF methods have the same form as Eqs.(41)and (45),

respectively,but with local ?ltered values replacing local mean values throughout.FDF transport equations corresponding to these particle equations then can be derived (the analogues of Eqs.(42)and (46)).The goal of the modeling then is to have the FDF equa-tions corresponding to the notional particle system evolve in the same manner as the FDF equations corresponding to the real ?uid system.

At the particle level,then,the key difference between PDF and FDF methods lies in the speci?cation of the models for the ?uctu-ating pressure gradient,molecular transport,and radiative absorption in Eq.(41)or for the molecular transport,turbulent transport,and radiative absorption in Eq.(45):the speci?cation of the turbulence scales,in particular.In PDF methods,the models can use any one-point,one-time Eulerian statistic and (in general)turbulence scales down to the integral scale l T .In FDF methods,the models can involve any resolved-?eld quantities and turbulence scales down to the ?lter width D .Molecular transport (mixing)models usually are the most critical element for turbulent reacting ?ows.Speci?c models are discussed in Section 6.5.5.The FDF as a basis for modeling

As has been pointed out throughout this section,there are important differences between PDF’s and FDF’s,in spite of the fact that their evolution equations have essentially the same form.Deeper conceptual issues with FDF-based modeling for LES have been raised by Pope [29,62],Fox [4],and Pitsch [65],among others.

In Section 13.5.6of [29],Pope points out that there is a distri-bution of sub?lter-scale (residual)?elds corresponding to any ?ltered (resolved)?eld,and the resolved ?eld will evolve differ-ently for different realizations.It is argued that before modeling the unclosed terms in the FDF equation,one should ?rst,in principle,take their conditional mean.

Fox [4]notes that the FDF is a quantity that essentially is conditioned on the full (nonlocal)velocity ?eld,and that it is dependent on the choice of ?lter.He introduces a LES velocity PDF that is different from the FDF,and is a ‘‘true conditional PDF.’’It is

Crankangle Degrees T u m b l e R a t i o

4000

800012000160002000024000

-0.3

00.30.6

0.91.2

Crankangle Degrees

T u m b l e R a t i o

20160

20880216002232023040

-0.30

0.3

0.6

0.9

1.2

Fig.9.LES-computed in-cylinder tumble ratio through multiple consecutive 720-degree four-stroke motored engine cycles for a simple engine con?guration [90].Left:32consecutive engine cycles.Right:expanded view of the last four cycles.

D.C.Haworth /Progress in Energy and Combustion Science 36(2010)168–259187

argued that this conditional PDF should provide a sounder basis for modeling compared to the FDF.

Pitsch[65]discusses a‘‘marginal density-weighted?lter PDF’’(FPDF)(a conditional PDF)that is different from the FDF,and argues that the FPDF(not the FDF)should be the basis for sub?lter-scale modeling in LES,for reasons similar to those raised by Pope and by Fox.

These are important conceptual issues that,unfortunately, remain unresolved at the time of this writing.Their practical implications for modeling are not clear.As we shall see in Section6 below,current practice in LES/FDF-based modeling is to use essentially the same models that have been developed and used in the RAS/PDF context.The local mean quantities that appear in the RAS/PDF models are replaced with local spatially?ltered values in the LES/FDF models,and the speci?cation of the turbulence scales that drive the models for molecular transport and for transport by unresolved velocity?uctuations(the latter in the case of a composition FDF method)is changed to re?ect the fact that in LES/FDF,the model represents only the effects of the sub?lter-scale ?uctuations.

5.6.General comments on LES and RAS

LES can be expected to have the greatest advantage over RAS in situations where large-scale?ow dynamics are important.This is the case,for example,for combustion instabilities in gas-turbine combustors and for cycle-to-cycle variations in reciprocating-piston IC engines.The piston engine is a good candidate for LES from several points of view[87]:the Reynolds number is relatively low compared to many other practical turbulent combustion systems;transients and cycle-to-cycle variations are of direct interest;and three-dimensional time-dependent CFD is required even for RAS.This last point sometimes has led to confusion in the distinction between RAS and LES,and in the application of RAS to statistically nonstationary and nonhomogeneous?ows.The piston engine is therefore also a good con?guration for bringing out important issues in interpre-tation of LES and distinctions between RAS and LES.

In a reciprocating-piston IC engine,the result of a consistently formulated RAS-based computation can be interpreted unambigu-ously as the ensemble-average mean cycle at a?xed steady-state operating condition;see the?rst paragraph of Section4.1here,and [88].That is,a three-dimensional time-dependent simulation through a single induction/compression/combustion/exhaust event represents the average over a large number of engine cycles,all ?uctuations about the ensemble mean are represented by the turbulence model,and the simulation results can be compared with ensemble-averaged experimental data.In practice,it usually is necessary to carry a RAS-based calculation through more than a single engine cycle with appropriate time-periodic boundary conditions to ensure that the result corresponds to a cyclic steady state;a small number of cycles(two to?ve)usually are suf?cient. There is no guarantee that a turbulence model that has been adapted from other applications(e.g.,statistically stationary?ows)will yield identical results on successive computed cycles in an engine, although experience shows that a standard k–3model does behave this way.If a different result is obtained on every engine cycle in what is intended to be a RAS-based modeling study,then the interpreta-tion of the results is ambiguous.

In applications of LES to?ows in reciprocating-piston engines, a RAS-based turbulence model(e.g.,k–3)is replaced with a sub-?lter-scale turbulence model(e.g.,Smagorinsky),and other models are modi?ed appropriately to re?ect the fact that only the effects of sub?lter-scale?uctuations must be modeled.As discussed in Section5.5,the relationship between the resolved?elds in LES and the underlying turbulent?elds is a statistical one.It is not appro-priate or meaningful to compare a single computed LES engine cycle with experimental data from a single measured engine cycle or with ensemble-averaged engine data.One can compare ensemble-averaged LES results with ensemble-averaged engine data, although clearly one would like to go beyond that.Making mean-ingful comparisons between LES and experimental measurements for in-cylinder?ows will require simulations through multiple engine cycles,statistical analysis of the results,and judicious conditional averaging.Proper orthogonal decomposition[89,90] has been proposed as one approach to make objective quantitative comparisons between LES results and experimental data obtained using modern high-speed,multidimensional optical diagnostics techniques.

LES through32successive motored(no combustion)four-stroke engine cycles were analyzed to shed new insight into in-cylinder ?ow dynamics in[90].An example from those simulations is shown in Fig.9.There the computed in-cylinder global tumble ratio is plotted as a function of crankangle degrees of rotation(time) through32consecutive engine cycles(after discarding the?rst cycle to avoid contamination by the arbitrary initial conditions). Here the tumble ratio is de?ned as the angular momentum of the in-cylinder?ow about an axis(the direction corresponding to maximum angular momentum)normal to the cylinder axis, pided by the moment of inertia about that axis,and normalized by the crankshaft rotational speed.For this con?guration, substantial cycle-to-cycle variations are evident in the global in-cylinder?ow structure.These would be represented by the turbulence model,or ignored altogether,in a RAS-based compu-tation.A RAS-based computation(not shown)gives the same result for every cycle,once a cyclic steady state has been reached.

Another important difference between RAS and LES is in their behavior as the computational mesh is re?ned.A properly formulated Reynolds-averaged simulation converges to a grid-independent solu-tion of the modeled mean equations(e.g.,Eq.(33),with appropriate models for unclosed terms)–not to the unaveraged instantaneous equations(Eq.(7)).Similar convergence to the modeled?ltered equations can be established for physical LES(LES where explicit sub?lter-scale models are used[62])where the?lter width remains ?xed as the mesh spacing decreases.It is more often the case in LES that the?lter width is proportional to the mesh size.In that case,a properly formulated LES-based model may converge to direct numerical simulation of the unaveraged/un?ltered underlying equations as the mesh is re?ned,if the numerical algorithms are consistent[91].

Relatively few LES modeling studies have addressed the issues of convergence,consistency,and quality of the solution.Consistent sub?lter-scale modeling requires that the sum of the effects of resolved and unresolved scales on statistics of interest should be independent of the?lter scale D and other numerical parameters. Normally the?lter scale D is smaller than the turbulence integral length scale l T.In the limit as the?lter scale D approaches l T, essentially all?uctuations are at sub?lter scale and LES approaches (operationally)unsteady RAS.Assessing the quality of LES results also has proved dif?cult.Proposals have included comparing the local mesh size(or?lter scale)with estimates of local turbulence scales,or keeping the energy in the sub?lter-scale?uctuations less than a small fraction of the energy in the resolved-scale?uctua-tions[62].The former criterion is dif?cult to apply in complex?ows where the scales vary widely in space and in time.Quality estimates based on Richardson extrapolation also have been proposed;those require LES solutions on multiple grids[92,93]

Table1

D.C.Haworth/Progress in Energy and Combustion Science36(2010)168–259 188

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