Survey of Turbulent Combustion Models for Large Eddy Simulations of Propulsive Flowfields

Transcription

1 AIAA SciTech 5-9 January 2015, Kissimmee, Florida 53rd AIAA Aerospace Sciences Meeting AIAA Survey of Turbulent Combustion Models for Large Eddy Simulations of Propulsive Flowfields Downloaded by Richard Miller on August 28, DOI: / Justin W. Foster Corvid Technologies, Mooresville, NC Richard S. Miller Department of Mechanical Engineering, Clemson University, Clemson, SC AIAA Science and Technology Forum and Exposition January 5-9, 2015, Kissimmee, FL (Session: Turbulent Combustion Models, Their Foundations and Major Assumptions) A general review of turbulent combustion modeling closures applicable to large eddy simulations (LES) is provided. The focus is on regime-independent models able to provide turbulent combustion closures ranging from purely premixed to purely non-premixed and all regimes between these two limits. Special emphasis is placed on primary propulsion applications, including liquid rocket engines, diesel engines, gas turbines, and scramjets. These applications span a large range of physical phenomena including both ideal- and real-gas behavior, single-phase and multi-phase combustion, relatively low Mach number to supersonic and hypersonic combustion, and relatively simple geometries to highly complex geometries. Three classes of models are identified as possibly providing such broad based modeling closures: flamelet-library/presumed probability density function (PDF) models, linear eddy based models (LEM), and transported PDF or filtered density function (FDF) based models. This review focuses both on fundamental physical assumptions that apply across all of the models and assumptions that apply to each of the models individually. Namely, assumptions regarding the presumed size of the large dimensional turbulent scalar manifold apply to all of the models; however, flamelet models almost always presume only a few dimensions are necessary to yield adequate representation of the larger, turbulent manifold. In contrast, LEM and FDF models are not, in theory, bound by any manifold size assumptions (i.e. direct calculation of the turbulent scalar manifold is possible); however, due to current computational limitations, these models often employ manifold reduction techniques which are usually not as restrictive as those used by flamelet models. Individual assumptions associated with the specific formulation of each model are also analyzed. From these discussions, additional novel results testing some of the fundamental physical assumptions associated with each model are provided from a unique database of DNS of high pressure turbulent reacting temporally developing shear flames. The DNS database includes simulations of H 2/O 2, H 2/Air, and Kerosene/Air flames with both detailed and reduced chemistry. The DNS include real property models, a real-gas equation of state, and generalized heat and mass diffusion derived from non-equilibrium thermodynamics. The simulations span a large range of Reynolds numbers and pressures (up to 125 atm), with resolutions approaching 1 billion grid points. Finally, some general comments towards the future challenges related to LES combustion modeling are offered. Computational Analyst, 145 Overhill Drive. Associate Professor, Department of Mechanical Engineering, Senior Member AIAA 1 of 21 Copyright 2015 by Corvid Technologies LLC. Published by the, Inc., with permission.

2 Nomenclature Downloaded by Richard Miller on August 28, DOI: / A m, B m = mixture parameters for the equation of state C Ω = modeling constant dw = differential increment of the Wiener process D = molecular diffusivity D i = drift coefficient D t = subgrid turbulence diffusivity e t = total energy (internal plus kinetic) E = modeling term F L = filtered mass density function F stir = triplet mapping advection model G = filter kernel H,α = partial molar enthalpy of species α J i,α = mass flux vector of species α k sgs = subgrid turbulence kinetic energy M = total number of represented species N s = total number of species P = pressure Q = generic variable being filtered Q i = heat flux vector R = universal gas constant Re 0 = initial Reynolds number s = one dimensional spatial coordinate S e = chemical source term for energy S α = chemical source term for partial densities Sc t = subgrid Schmidt number t = time T = temperature T ij = subgrid stress tensor u i = velocity vector v = specific mixture volume x i = spatial coordinate vector X i = Lagrangian spatial coordinate vector Y α = mass fraction of species α Y m = mapping function from represented to unrepresented subspaces δ = delta function δ ij = Kronecker delta tensor G = filter width t = time step ɛ = additional terms in the energy equation ζ = the fine grained density Θ = generic conditional filtered diffusion model Π = product operator µ t = subgrid turbulence viscosity ξ = Gaussian probability density function ρ = density τ ij = viscous shear stress tensor φ = mixture fraction φ α = set of all scalars transported by the filtered mass density function Φ = generic variable Φ = generic filtered variable ψ = sample space of the scalar array Ψ = the set of primitive variables Ψ = the set of filtered primitive variables available in LES Ω = filtering domain 2 of 21

3 Ω m = subgrid mixing frequency = standard filter = Favre density weighted filter < > = conditional standard filter = conditional Favre filter Superscript r = represented R = resolved scale S = unresolved subgrid scale sgs = subgrid scale u = unrepresented = denotes quantity evaluated at particle level = fluctuation with respect to the standard filtered variable = fluctuation with respect to the Favre filtered variable Downloaded by Richard Miller on August 28, DOI: / Subscript i, j = free indices I, L = represented, unrepresented quantity indices respectively h = enthalpy sgs = subgrid scale m = mixture α = scalar index I. Introduction This review of multi-regime large eddy simulation (LES) approaches is motivated by propulsive applications. Such applications cover a broad range of combustion regimes with primary applications being liquid rockets, diesels, gas turbines, and scramjets. These applications cover a broad range of thermodynamic and chemical physics, including both liquid and gaseous fuels, ideal and real thermodynamic states, and low speed to supersonic speed combustion. Although many LES approaches are available in the literature, only relatively few are capable of capturing the multi-regime combustion phenomena associated with such applications. 1 Such a general purpose model must be able to capture all combustion regimes from purely non-premixed, to purely premixed, to partially premixed, from low speed to high speed, as well as from ideal to real thermodynamic behavior. Three classes of models are identified as possibly providing such broad based modeling closures: flamelet-library/presumed probability density function (PDF) models, linear eddy based models (LEM), and transported PDF or filtered density function (FDF) based models. The primary objective of the current paper is to re-examine the fundamental physical assumptions associated with each of these modeling approaches and to compare their limitations to use in general propulsive applications. This is by far not the first such work to review these LES models and many recent reviews exist including Refs. 1 8 Nevertheless, some fresh perspective and novel results are presented in what follows. In addition, other potentially multi-regime geometric models such as the flame surface density, flame wrinkling, and thickened flame models are not addressed in this review. A recent review of these models can be found in Ref. 1 Despite the primary emphasis on reviewing the fundamental physical assumptions of LES combustion techniques, several novel results are also presented to test the validity of these assumptions under conditions relevant to today s common propulsion applications. The authors have at their disposal a database of direct numerical simulations (DNS) of turbulent temporally developing shear layer flames that include detailed and reduced mechanisms for H 2 /O 2, H 2 /Air, and Kerosene/Air combustion (additional details below). 9, 10 The formulation is based on the general compressible Navier Stokes equations, a cubic Peng-Robinson equation of state, generalized molecular diffusion derived from non-equilibrium thermodynamics and fluctuation theory, and realistic property models. The simulations are conducted with eighth order central finite differencing for 3 of 21

4 all spatial derivatives and fourth order Runge Kutta time integration. Initial Reynolds numbers based on the initial vorticity thickness, free stream velocity difference, and mean free stream densities and viscosities range from 850 to 4, 500. Pressure range from 1 atm to 125 atm. Resolutions up to 3/4 billion finite difference grid points are employed. These flames represent very different thermochemical conditions with the H 2 /O 2 and Kerosene/Air exhibiting fast chemistry with no local extinction, whereas the H 2 /Air flame is characterized by large amounts of scatter due to strong local extinction. Figure 1 portrays these local extinction events by showing instantaneous center plane temperature contours and temperature scatter plots on the stoichiometric surface as a function of normalized scalar dissipation for the H 2 /Air flame at Re 0 = This simulation utilized numerical grid points and achieved a final time Reynolds number based on instantaneous vorticity thickness of 26, 000 and a centerplane Taylor Reynolds number of 240 based on the streamwise rms velocity and Taylor length scale. 9 Downloaded by Richard Miller on August 28, DOI: / (a) Figure 1. Temporally developing H 2 /Air shear flame at Re 0 = 4500 and P = 35atm showing long time (a) instantaneous center plane temperature contours and (b) scatter plots of temperature as a function of normalized scalar dissipation on the stoichiometric surface The outline for this document includes an overview of the governing equations for LES in Section II followed by a discussion of the assumptions associated with the reduction of turbulent scalar manifolds in Section III. Sections IV, V, and VI discuss the particular physical assumptions associated with LES-Flamelet, LES-LEM, and LES-FDF closure models, respectively. Finally, Section VII offers conclusions and comments about the remaining challenges associated with LES of turbulent combustion modeling going forward. II. LES Governing Equations We begin this section by offering the exact (unclosed) LES equations that describe the applications of interest for this study. Consistent with traditional LES formulations, each instantaneous variable is decomposed into a resolved scale plus a subgrid scale fluctuating component, mathematically represented by, Φ = Φ + Φ. The filtered variable, Φ, is mathematically defined by the convolution integral: Φ(x) = Φ(x j)g(x j x j)dx j, (1) Ω (b) where G(x j ) represents the filter kernel defined over the domain, Ω. Similar to a PDF, G(x j ) must integrate to unity to maintain the conservation of constants. Compressible flows use the concept of Favre filtering, which is a density-weighted filter related to the standard filter represented by the expression, Φ = ρφ/ ρ. Filtered variables can be decomposed by Favre filter as Φ = Φ + Φ. In a rigorous LES approach, the set of governing equations is filtered term by term to derive the exact (albeit unclosed) LES equations: ρ t + ρũ j x j = 0, (2) 4 of 21

5 ρũ i + ρũ iũ j t x j ρẽ t t + ρẽ tũ j x j P ( Ψ) = + τ ij( Ψ) x i x j = P ( Ψ)ũ j x j + ũ iτ ij ( Ψ) x j x j [Q j (Ψ) Q j ( Ψ)] + x j [P (Ψ)u j P ( Ψ)ũ j ] + x j [P (Ψ) P ( x Ψ)] + [τ ij τ ij ( i x Ψ)] T ij, (3) j x j [ Ns ] H,α ( Ψ)J j,α ( Ψ) + S e Q j( Ψ) x j + x j α=1 [u i τ ij ũ i τ ij ( x Ψ)] [ ρẽ t u j ρẽ t ũ j ] j x j ] N s H,α (Ψ)J j,α (Ψ) H,α ( Ψ)J j,α ( Ψ), (4) [ Ns α=1 ρỹα + ρỹαũ j = J j,α( Ψ) + S Yα [ ρ Y t x j x j x α u j ρỹαũ j ] [J j,α (Ψ) J j,α ( j x Ψ)], (5) j where gravity and radiation have been neglected and a single phase gaseous or supercritical fluid has been assumed for the sake of simplicity. In the above, we define Ψ to represent the set of primitive variables (eg. density ρ, velocity vector u i, energy e t, and mass fraction Y α ). In contrast, Ψ represents the set of appropriately filtered primitive variables (eg. ρ, ũ i, ẽ t, Ỹα), while Φ( Ψ) represents a variable calculated from the filtered primitive variables (eg. P ( Ψ), T ( Ψ)). As such, P (Ψ) is the filtered pressure. The resolved pressure P ( Ψ) is the pressure obtained from the chosen equation of state as a function of only the available resolved field LES variables (i.e. that can be calculated in an actual LES). The above set of equations additionally require a chemical kinetics mechanism for the source terms, Se and S Yα, an equation of state along with constitutive relations for the viscous shear stress tensor, τ ij, the heat flux vector, Q j, and the mass flux vector of species α, J j,α. The pressure dependent kinetics schemes available in Clemson s code (and the results herein) pertain to hydrogen-oxygen (8 species, 19 steps) and hydrogen-air (12 species, 24 steps), 11 as well as a surrogate kerosene mechanism (10 species, 34 steps) commonly used in the rocket community at large pressure. 12 Clemson s DNS code utilizes the cubic Peng-Robinson equation of state whose resolved form would be: P ( Ψ) RT ( = Ψ) v( Ψ) B m ( Ψ) A m ( Ψ) v( Ψ) 2 + 2v( Ψ)B m ( Ψ), (6) B m ( Ψ) 2 where A m ( Ψ) and B m ( Ψ) are the resolved mixture parameters calculated from a set of mixing rules obtained from the literature. 13 The partial molar enthalpy of species α, H,α (Ψ) is derived from the above equation of state. The shear stress tensor is assumed to take the standard Newtonian form whereas the heat and mass flux vectors are derived from nonequilibrium thermodynamics and fluctuation theory required for high pressure dense fluids. Their forms contain terms proportional to temperature, pressure, and all species mole fractions (ie. Dufour and Soret effects as well as differential and multicomponent diffusion). Additional details of the formulation can be found in Refs. 9, 14, 15 However, the LES formulation above, and the discussion to follow, apply equally to general choices of kinetics, equations of state, as well as constitutive relations. In the above, many LES terms are unclosed and may require modeling. These include the subgrid turbulent stress T ij = ρ(ũ i u j ũ i ũ j ), the subgrid viscous term [τ ij τ ij ( Ψ)], the filtered chemical reaction source terms S e and S Yα, the subgrid pressure [P (Ψ) P ( Ψ)], the subgrid scale pressure flux [P (Ψ)u j P ( Ψ)ũ j ], the subgrid heat flux [Q j (Ψ) Q j ( Ψ)], the subgrid convective energy [ ρẽ t u j ρẽ t ũ j ], the subgrid scale viscous diffusion [u i τ ij ũ i τ ij ( Ψ)], the subgrid molecular enthalpy flux [ N s α=1 H,α(Ψ)J j,α (Ψ) N s α=1 H,α( Ψ)J j,α ( Ψ)], the subgrid scalar flux [ ρ Y α u j ρỹαũ j ], and the subgrid mass flux vectors [J j,α (Ψ) J j,α ( Ψ)]. Of these unclosed terms, the subgrid turbulent stresses and the filtered chemical source terms have received the most attention in the literature. The subgrid stresses are most commonly modeled with either constant or dynamic versions of either the Smagorinsky or the mixed similarity models For transported FDF methods, the chemical source term is in closed form and does not require any direct modeling. The subgrid pressure, the subgrid molecular enthalpy flux, the subgrid molecular shear stress tensor, the subgrid heat flux vector, and the subgrid mass flux vectors are typically neglected in standard LES. However, the Clemson group, 9 among others, have shown that the subgrid pressure, heat flux, and mass fluxes are not necessarily negligible for reacting flows due to their non-linear forms and the fact that reacting flows have larger scalar gradients than non-reacting flows in general. Nevertheless, these terms are not discussed in what follows. α=1 5 of 21

6 III. Turbulent Scalar Manifold Reduction Assumptions Downloaded by Richard Miller on August 28, DOI: / The most fundamental assumptions in general LES approaches to turbulent combustion modeling are centered around the size of the (typically) large dimensional scalar manifold (of size N s, usually on the order of tens to hundreds) that contains the species, energy, and pressure dependent variables and whether or not a smaller attracting manifold exists. Ideally, a turbulent combustion simulation would transport the physical processes of convection, diffusion, and reaction for all of the scalars (contained in the vector Y α, where 1 α N s ) that characterize a given flow. In reality, this task is quite often untenable due to the large computational requirements associated with such a level of description. It is then often necessary to select a reduced set of represented scalars, or linear combination of scalars, (denoted by the vector Y r I, where 1 I M and M N s ) to be transported instead while the remaining unrepresented scalars (denoted by Y u L, where 1 L N s M) are not transported, but rather calculated from a defined mapping function that maps the unrepresented scalars in terms of the represented ones [i.e. YL u = Y L m(y I r) where Y L m is the mapping function]. These reduction arguments suggest that the large N s -dimensional manifold lies on, or at least close to, a much smaller M-dimensional manifold. The reduced manifolds associated with each of the closure models discussed in this review vary quite substantially in their size, complexity, and implementation. In general, LES-Flamelet models assume that the large N s -dimensional manifold can be described by a very low dimensional reduced manifold (typically 1 M 3) derived from one dimensional non-premixed and/or premixed laminar flames at similar thermodynamic conditions to the turbulent flame of interest and stored in a pre-constructed library that can be accessed during the turbulent simulation to obtain the necessary chemical information. Examples of these very Low Dimensional Manifolds (LDMs) generated from one dimensional flames include Flamelet Generated Manifolds (FGM), Flame Prolongation of ILDM (FPI), and Flamelet Progress Variable (FPV) techniques, among others. A detailed review of each of these methods is not provided here as the major assumption associated with all of these techniques is related to the assumed very small size of the reduced manifold, not the generation of the reduced manifold itself. Although LES-LEM and LES-FDF models commonly integrate manifold reduction techniques into their formulations, they do not usually assume such strict limitations as those associated with LES-Flamelet models. In fact, most LEM and FDF modeling approaches do not assume that the N s -dimensional turbulent scalar manifolds can be accessed from laminar flame libraries parameterized by only a few parameters during an LES and therefore typically rely on either directly solving or tabulating the (stiff) chemical kinetics for most of the scalars during the simulation. In this context, LEM and FDF methods are therefore able to offer a reduced description of the overall N s -dimensional manifold that is computationally tractable but not so restrictive as to limit the reduced manifold to sizes of 1 to 3. A recent detailed review of the majority of the various reduced manifolds associated with turbulent combustion is contained in Ref. 7 therefore very little elaboration is provided further here. It is, however, important to discuss the manifold reduction argument itself as it begs several questions including the overall validity of reduced manifolds in general turbulent flames, the physical mechanisms responsible of increasing or decreasing the size of reduced manifolds, and whether or not commonly used parameters offer suitable projections onto reduced manifolds. Recently, Ref. 7 provided an analytical look at reduced manifolds by examining the conservation equation for the departure vector which represents the difference between the true value for the unrepresented quantities and the mapping function [i.e. y L = YL u Y L m(y I r)]. In order for the large N s-dimensional scalar manifold to lie on a LDM, the departure vector ). The analysis in Ref.7 concluded that, due to the curvature I ) and the invariance properties of the reaction manifold, y L cannot be zero for the LDMs associated with turbulent combustion. At best, the true N s -dimensional scalar manifold lies close to a LDM. At this point; however, nothing has been said regarding how close, or not, turbulent combustion lies to a LDM. Furthermore, the reduced manifolds commonly associated with Flamelet, LEM, and FDF models are usually associated with specific assumptions that may or may not be true in the turbulent flow field of interest. Therefore, these a priori generated manifolds do not offer a direct measure of the departure vector to a real turbulent flame therefore a posteriori methods are more appropriate. To this end, several researchers have analyzed so-called Empirical Low Dimensional Manifolds (ELDM) where high fidelity experimental or numerical datasets are post-processed using dimension reduction techniques (most common are Principal Component Analysis [PCA] and Multivariate Adaptive Spline Regression [MARS]) These ELDM s are not limited by any physical assumptions (since they are only based on regression techniques) and the resulting M parameters from these dimension reduction techniques make up must be null for some definition of YL m(y I r properties of YL m(y r 6 of 21

7 a strong attracting M-dimensional reduced manifold. It should also be noted that these PCA and MARS generated manifolds have been shown to be stronger attracting than other common LDM s used in turbulent reacting flows, including flamelet-based manifolds. 7 Figure 1 shows results from DNS of a high pressure H 2 /Air shear flame 10 that further elucidates the idea that common flamelet parameters are not always strong attractors for LDM s of turbulent combustion. The turbulent nature of this flame is depicted in Fig. 1a as pockets of extinction/re-ignition are evident. The corresponding scatter plot in Fig. 1b of temperature as a function of normalized scalar dissipation on the stoichiometric surface shows the low correlation between commonly used flamelet parameters (mixture fraction and scalar dissipation) and the scalar field (temperature in this case) within turbulent manifolds. Because PCA and MARS are known to generate strong attracting M-dimensional reduced manifolds, these ELDM analyses provide a well-founded approach for quantifying the departure vector for each of the scalars lying on the larger N s -dimensional manifold. Yang et al. 35 conducted both PCA and MARS analyses on DNS datasets of a non-premixed temporally evolving 11-species CO/H 2 jet flame and a 22-species lifted Ethylene jet flame. The results showed that, for the CO/H 2 flame, in order for the scalars to maintain an overall departure less than 5% from the original 12-dimensional manifold, 5 and 2 dimensions were required for PCA and MARS, respectively. For the 23-dimensional Ethylene manifold, 9 and 7 dimensions were required for PCA and MARS, respectively. It should also be noted that the chemical source terms (which are unclosed in reduced manifold space and also require modeling) required even higher dimensions to obtain 5% departure for both PCA and MARS analyses. To put these results into perspective, DNS capable flames, even when projected onto LDM s much stronger attracting than those predicted by flamelet parameters, are not well described by 2 or 3 dimensions except in simple configurations. Given the complex nature (highly turbulent, many species, extinction, re-ignition, pollutant formation, etc.) of the vast majority of flames prevalent in propulsion applications, it is highly unlikely that LDM s (on the order of two or three dimensions) parameterized by flamelet variables will be able to accurately reconstruct all of the pertinent physics within these flames. Therefore, the primary fundamental physical assumption associated with all flamelet models is largely inconsistent with the physics describing today s propulsion applications. Further physical assumptions associated with flamelet models, as well as LEM and FDF models are reviewed in detail below. IV. LES-Flamelet Turbulent combustion models based on flamelet concepts remain some of the most widely researched topics in the field. The concept of the flamelet is that locally a one-dimensional laminar flame subject to appropriate initial and boundary conditions can describe a turbulent flame front. These fully-resolved solutions to one-dimensional flames are stored in pre-constructed flamelet libraries usually generated from simple canonical geometries (typically purely non-premixed counterflow opposed laminar jet flames and simplified one-dimensional laminar premixed flames) and are parameterized by a reduced set of M-number of scalars. The turbulent LES then only needs to transport the filtered M-number of parameters (and usually the variance as well). Chemical information is then accessed from the library as needed, using a statistical relationship (usually a beta PDF) between the stored instantaneous and transported filtered values. Peters originated the flamelet concept under the argument of non-premixed flames being much thinner than the Kolmogorov or other flow length scales. 38 It has since been extended well beyond this flame regime and in many modern applications it is assumed that nearly all turbulent regimes (non-premixed, premixed, and partially premixed) can be modeled from flamelet libraries during the LES. Put in terms of the manifold perspective, the flamelet library solutions make up YL m(y I r ); the manifold of unsolved scalars parameterized by the M-number of solved scalars (usually mixture fraction, progress variable[s], and dissipation[s]; typically two to three at most) , The flamelet concept has been extensively studied throughout the literature for purely premixed and purely non-premixed 27, 32, 33, conditions; and recently it has been extended to regime-independent 30, flames. Regime-independent flamelet-based models, which are required to simulate modern propulsive applications, are similar to purely premixed and non-premixed flamelet models in that they typically parameterize the flamelet library by the same set of solved scalars (mixture fraction, progress variable, etc.). However, the generation of the regime-independent flamelet manifold is usually done by either solving the multi-dimensional flamelet equations in reduced parameter space such that a range of laminar flames from the purely non-premixed limit to the purely premixed limit are stored, 49 or solving the purely non-premixed 7 of 21

8 and purely premixed one-dimensional flame equations separately and then blending the solutions based on a regime identifier that measures the degree to which a local state is non-premixed, premixed, or partially 48, 50 premixed. When considering regime-independent flamelet models, the first task carrying physical implications is the specific choice of the M parameters that make up the reduced manifold as these parameters will a) makeup the basis that spans the N s -dimensional scalar space and b) be transported (in filtered form) during the LES with appropriate transport equations that must be derived (and closed). The basis of M-parameters must constitute a well-defined generalized coordinate system for which the N s scalar equations can be transformed into from physical space. 48 The vast majority of regime-independent LES-Flamelet models employ mixture fraction and progress variable(s) as the selected M-parameters. For purely non-premixed combustion, the mixture fraction, φ, can be viewed as a passive scalar (ranging in value from 0 to 1 in pure oxidizer and pure fuel, respectively) making up a flame-attached coordinate system with one direction normal to isoφ surfaces and two directions tangential to the iso-φ surface. 38 This three dimensional mixture fraction space constitutes a well-defined coordinate system; however, the vast majority of LES-Flamelet models only consider the single flame normal coordinate in the transformed N s scalar equations. This is the result of the thin-flame argument suggesting gradients normal to iso-φ surfaces are much greater than those in the tangential directions. Gradients in the tangential directions have been shown to be non-negligible in recirculation regions and flames with substantial curvature and differential diffusion effects. 51 Suffice it to say, many of the flames associated with propulsive applications are not thin and are known to exhibit substantial amounts of recirculation, curvature, and differential diffusion effects. For purely premixed flamelet combustion, progress variables, usually made up of a single or linear combination of the N s -scalar(s), are selected as the dimension reduction parameters. Unlike the mixture fraction, progress variables are not passive scalars and they do not offer a description of the fuel/oxidizer mixed state but rather the overall burnt state. This description leaves the exact definition of progress variables somewhat open-ended and this is reflected by the many different progress variable definitions that have been used throughout the literature. Generally speaking, progress variables should vary monotonically from fresh to burnt gases such that species concentrations remain single-valued functions of the progress variable. It is also desirable to minimize the non-linear nature of the progress variable from fresh to burnt gases as this can cause large interpolation errors within the flamelet library. A recent study from Niu et al. 52 showed that commonly used progress variables do not always satisfy one or both of these constraints, especially when complex chemical kinetics mechanisms are used (e.g., hydrocarbon-based mechanisms). The shortcomings associated with the mixture fraction and progress variables within complex turbulent flames are central to the idea that variables based on laminar flame arguments do not always offer strong attractors for reduced descriptions of turbulent manifolds. Furthermore, for regime-independent flamelet models, Knudsen et al. 48 showed that the statistical dependence between mixture fraction and progress variables is such that they do not form a well-defined coordinate system for all combustion regimes thereby calling into question the validity of some of the solutions generated from flamelet equations solved in mixture fraction and progress variable spaces. The multi-regime approach of Knudsen et al. 50 alleviates the statistical dependence issue by formulating a mixture fraction and statistically independent progress parameter based coordinate system. However, their partially premixed solutions obtained from linear combinations of nonpremixed and premixed flamelet models via a regime identifier are somewhat ad hoc as there are no physically based arguments that suggest partially premixed behavior is simply a linear combination of nonpremixed and premixed behavior. It should be noted that while the majority of flamelet models tend to build laminar flamelet libraries through solving the flamelet equations in reduced variable space, efforts have been made to incorporate flamelet libraries from physical space solutions to certain laminar flames. One recent study further explored this idea in H 2 /O 2 counter-flow diffusion flames by solving (in physical space) a transport system including detailed chemistry, a real gas state equation, real property models, and a conserved passive scalar. 53 Using this technique, the physics associated with finite rate kinetics, real gas effects, fluid strain rate, and differential diffusion are able to be captured naturally. Furthermore, since the reduced manifold variable of interest evolves with this system, a direct mapping function that implicitly includes all of the aforementioned physics also emerges. Although the particular study cited here was applied only to non-premixed combustion, the idea could theoretically be extended to multi-regime environments where mixture fractions and progress variables are transported in physical space amongst various non-premixed, premixed, and partially premixed laminar flames. 8 of 21

9 In summary, a wide array of techniques are available for generating the M-dimensional reduced manifold and corresponding mapping function YL m(y I r ) from flamelet arguments. However, regardless of the implementation method, the very restrictive assumption that the large N s -dimensional manifold can be mapped using only two or three parameters remains. Furthermore, in an actual LES, the instantaneous quantities stored in the flamelet library need be combined with a PDF in order to obtain their filtered values. Several PDF s are used in practice including Delta functions, 32 Beta functions, 42, 54 and Statistically Most-Likely Distribution (SMLD) functions; 55 each of which carry further physical assumptions and restrictions regarding the description of the unrepresented quantities in the actual turbulent flow field. However, the fact that the majority of propulsive flow fields do not likely lie only on very low dimensional manifolds remains, in these authors views, the dominant physical assumption that restricts the LES-Flamelet approach s use for general multi-regime predictive propulsion applications. V. LES-LEM Downloaded by Richard Miller on August 28, DOI: / LES-LEM 56 is based on the assumption that all length and time scales must be fully resolved. In its most rigorous definition this assumption would require that a full DNS be performed, including the resolution of all chemical processes. Because DNS of the applications considered here are computationally intractable, the LEM approach assumes that the small scale scalar features below the LES filter width can be resolved using a one-dimensional (1D) domain at each finite difference grid point, or within each finite volume cell, aligned with the mean scalar gradient. The fourth-dimension to the general LES solver has a 1D length at each grid point that ranges from the Kolmogorov (or Batchelor) scale to the length scale of the filter width. Molecular diffusion and chemical source terms are fully resolved on the 1D domain, whereas turbulent stirring is accounted for via Kerstein s stochastic triplet mapping The LES-LEM approach first departs from the rigorous LES approach by returning to the instantaneous 56, equations for the scalar mass fractions and energy rather than the filtered forms. In the LES-LEM approach the instantaneous velocity is first decomposed in the traditional LES sense as u i = ũ i + u i where, as usual, the contributions are from the resolved (LES mesh) velocity and the subgrid scales, respectively. At this point in the LES-LEM approach, the transport equations for the instantaneous (not filtered) mass fractions are then decomposed as: ρ Y α t ρ Y α t + ρũ j Y α x j = 0, (7) + ρu Y α j = J j,α + S Yα, (8) x j x j where the first equation is meant to represent the large (LES) scale advection and the second equation represents the instantaneous subgrid scale processes. A similar approach is employed for the energy equation. So, LES-LEM involves the solution of a set of modeled transport equations for the filtered continuity, momentum and energy equations coupled with a set of modeled transport equations for the instantaneous scalar mass fractions and temperature. Modeling the scalar transport begins with the 1D assumption which replaces Eq. (8) with: ρ Y α + F stir = J α t s + S Y α, (9) where s is the 1D subgrid refined mesh coordinate and F stir represents Kerstein s triplet mapping used to model the effects of subgrid turbulent stirring. 61 Several assumptions are made at this point. First, the resolved scalar fields only see advection, therefore all molecular diffusion occurs at the small subgrid scales (note that this is counter to most LES in which the subgrid molecular diffusion is commonly neglected). Second, the resolved field advection is only via the filtered velocity even though the scalar is an instantaneous quantity. It is to be noted that in some approaches 62 the effects of the velocity fluctuations are modeled by decomposing the velocity alternatively as u i = ũ i + (u i )R + (u i )S, where the fluctuation has been decomposed into an LES resolved subgrid fluctuation (superscript R) and an unresolved subgrid fluctuation (superscript S). In this case the resolved 2 3 ksgs. However, subgrid fluctuation can be modeled using a transported subgrid kinetic energy: (u i )R = this is somewhat questionable since the subgrid kinetic energy is positive definite whereas velocity fluctuations are by definition both positive and negative. The next assumption in the LES-LEM approach is that the 3 dimensional subgrid scalar transport, diffusion, and reaction described by Eq. (8) can be accurately 9 of 21

10 resolved in a purely one dimensional (1D) domain located at each finite difference grid point or within each finite volume cell. The assumption is justified by assuming that this 1D domain resides along a notional direction aligned in the flame normal or the maximum scalar gradient direction and thus, does not represent any physical Cartesian direction. 56 Implicit in this argument is that there is only a single scalar gradient direction; ie. all scalars are aligned in the same direction. This will be returned to below. Other assumptions implicit to the LES-LEM approach reside within the triplet mapping procedure. The basic argument is that the subgrid turbulent stirring can be modeled by re-mapping the 1D scalar field according to a set of mapping rules which conform to a k 5/3 spectrum. Many assumptions exist and details can be found in Refs. 56, 57, 61 among others. However, for present purposes the primary question is whether or not the inertial k 5/3 scaling is relevant to the diffusive sub-inertial scales resolved by the LEM. These scales make up a large part of the kinetic energy typically residing within the subgrid in academic LES; however, this may not be the case for industrial or other propulsion applications and their associated much higher Reynolds numbers. The next assumption of the model involves the treatment of the resolved field scalar advection on the LES mesh. Rather than treating this in the traditional Eulerian approach, in LES-LEM the advection is treated by a Lagrangian splicing procedure. 56 In practice, portions of each 1D subgrid domain are moved downwind into adjacent cells based on a set of mass conserving rules. Again, the details of the approach are not considered here. However, it is noted that the approach does have a directional bias; ie. different results can be obtained depending upon the order in which the three directions are treated. In practice, the largest mass flux direction is treated first, followed by the second, then the third. Other issues implicit to the LES-LEM approach include the following. Boundary conditions for the 1D subgrid domains are not clear for the small scales. Typically, Neumann boundary conditions are employed without physical justification. The pressure within each computational cell is also be assumed to be uniform (thereby neglecting any possible subgrid pressure effects). Small scale isotropy is assumed which is questionable particularly for high speed flows (although this assumption is common to many LES models). Another issue is that two sets of energy equations are required. One is the typical filtered energy equation necessary for the resolved field equations (continuity, momentum, energy, plus constitutive equations and an equation of state). The second is the instantaneous energy equation decomposed similarly to Eqs. (7) and (9) needed for the chemical source terms. These two sets of equations are not mathematically consistent and the effects of this on the transported energy are not clear. Finally, two LES-LEM physical assumptions are tested using the Clemson database. Figure 2 presents example results showing that assuming that molecular diffusion resides only within the subgrid 1D scales is highly questionable, at least at academic scale LES. The figure shows PDFs of the ratio of the resolved field mass flux vector magnitudes to the exact filtered mass flux vector magnitudes for the H 2 /O 2 shear flame at P 0 = 100atm and Re 0 = 4, 500 with a filter width of 4.7 times that of the initial vorticity thickness and conditioned on the stoichiometric surface. Although not a strict interpretation of a subgrid variable, we can. That is, the exact filtered mass flux vector calculated by the set of scalars Ψ is equal to the resolved field mass flux vector (i.e. that calculated from the set of filtered scalars Ψ available within an actual LES), plus the difference defined as the subgrid mass flux vector. The data in the figure was originally obtained for the purpose of determining whether decompose the filtered mass flux vector as: J j,α (Ψ) = J j,α ( Ψ) + J sgs j,α or not subgrid molecular diffusion is significant and may need modeling. 10 By substitution the plots are PDFs of J j,α (Ψ) J sgs j,α / J j,α(ψ). Therefore, if the subgrid portion of the mass flux vector is completely negligible the PDF would be a delta function located at unity; which is essentially the case in the figure. Therefore, Fig. 2 shows that resolved field molecular diffusion is dominant in this flame at least along the stoichiometric surface, which is inconsistent with the physical assumptions of LES-LEM. Additional PDFs of the same ratio but for different filter widths, Reynolds numbers, and conditioning regions may be found 9, 10 The in Refs. results show that resolved field diffusion is substantial is all cases. Again, this is at least true for academic Reynolds number flames. However, the assumption of negligible resolved field diffusion may well be much better for higher industrial scale Reynolds numbers (except near walls, laminar-turbulent boundaries, and other potential low Reynolds number regions). The second assumption tested is that the entire subgrid scale scalar field evolutions represented by Eq. (9) can be resolved on a 1D domain residing along the direction of the maximum scalar gradient. Again, representative example results shown in Fig. 3 illustrate that there is no single direction along which all scalar gradients reside even in a nonpremixed Kerosene/Air parallel shear flame from Clemson s database at P 0 = 35atm and Re 0 = 2, 500. There are many scalars present including all of the species mass fractions and 10 of 21

11 Figure 2. The PDF of the ratio of the mass flux vector magnitudes as a function of the resolved (filtered) scalar variables that can be calculated within an actual LES to the exact filtered mass flux vector magnitudes obtained from Clemson s DNS of the H 2 /O 2 shear flame at P 0 = 100atm and Re 0 = 4, 500. The ratio of the filter width to the initial vorticity thickness is 4.7 (spherical top hat with diameter equal to 22 times the streamwise DNS grid spacing) and the data are conditioned on the stoichiometric surface. the temperature. So, there is no single particular scalar gradient to assume the 1D domain resides along. However, if correct, the assumption would have all scalars along the same gradient, therefore, any one could be used as the representative test direction. In the case of Fig. 3 the mixture fraction gradient is used as the reference direction. The figure contains scatter plots of the cosine of the angle between the temperature and the O 2 mass fraction gradients with respect to the mixture fraction gradient. If the LES-LEM directional assumption holds true then only values of -1 and +1 would be expected. However, as the figure shows there is no one single direction along which all of the scalar lie even for this nonpremixed flame. Finally, despite the numerous (questionable) assumptions implicit to LES-LEM, the model has produced good results for a variety of conditions, including nonpremixed and premixed flames, sooting flames, twophase flames, as well as flames involving extinction and reignition. 56 Despite its successes, there are numerous fundamental physical assumptions implicit to the approach, many of which are not consistent with combustion regimes associated with common propulsion applications. These make the approach difficult to justify rigorously. VI. LES-FDF LES-FDF methods share the same lack of implicit assumption as LES-LEM that turbulent combustion lies on a LDM. In contrast to the LES-Flamelet and the LES-LEM approach, the LES-FDF methods solve a modeled transport equation for the joint FDF [or filtered mass density function (FMDF) for variable density flows] typically of at least the (represented) scalars needed to close the chemical source terms. These chemical source terms are highly nonlinear and computationally stiff. Various approaches have been implemented to speed the computations including reduced manifold techniques, flamelet libraries, to the In Situ Adaptive Tabulation (ISAT) method, 65, 66 which dynamically generates a similar manifold library which can be used for interpolation purposes later in the simulation. This method has been used in several FDF simulations and dramatically decreases the computation time Other methods of manifold reductions were referenced in Section III. However, these types of approaches to dealing with the chemical source terms are not specific to LES-FDF approaches and will not be discussed further in what follows. Directly related to LES-FDF approaches though is in choosing the proper set of variables to be trans- 11 of 21

12 (a) (b) Downloaded by Richard Miller on August 28, DOI: / Figure 3. Scatter of the instantaneous angles between the gradient of the mixture fraction and the gradient of the (a) temperature and (b) O 2 mass fraction from Clemson s Kerosene/Air DNS at P 0 = 35atm and Re 0 = 2, 500. ported. This FDF or FMDF may include the scalars only (S-FMDF) 70 to close the chemistry, the velocity only (V-FMDF) that can be used to close terms in the LES momentum equation, 71 the scalars plus the velocity components (VS-FMDF) that can additionally be used to close more terms including the subgrid convection, subgrid pressure-work, and subgrid pressure in the energy equation, 72, 73 the velocity plus the scalars, plus a subgrid turbulence frequency (FVS-FMDF), 74 or the most comprehensive form which solves a transport equation for the joint statistics of the energy, pressure, velocity, and scalars (EPVS-FMDF) 75 needed for high Mach number and pressure dependent chemistry. The FDF or FMDF offers a unique form of combustion closure in that knowing the joint FMDF of the scalars the chemical source terms appear in closed form and no direct modeling assumptions are required for these terms. The modeling of unclosed terms is thus shifted from the chemical kinetics to other terms; the majority of which are chosen to reduce to commonly used models in traditional LES as described below. As such, the modeling of these terms is essentially the same as in any form of LES and therefore modeling errors associated with these terms are not specific to the FMDF approach. However, there remain three primary issues relating specifically to the FMDF approach. The first involves accurately modeling the conditional filtered molecular diffusion (or the conditional filtered dissipation although the majority of available models are for the former). The second involves the high dimensionality of the FMDF transport equations and the corresponding choice in solution procedures. Both of these are discussed in detail below following a review of the basic LES-FMDF approach. An additional issue associated with the model is whether or not to solve both the FMDF and the Eulerian LES transport equations, or only the FMDF equations. In the former approach the FMDF is only used to close specific terms in the traditional LES equations. In the latter approach the FMDF provides the resolved field evolution directly. The approach discussed hereinafter is to solve both sets of equations. This both minimizes errors in the FMDF modeling and provides a self-consistency check (i.e. by ensuring that the two sets of equations produce the same resolved field evolution). Now we present the basic FMDF by examining the (relatively simple) S-FMDF transport equation and its modeling. The statistical scalar field φ α is characterized by a filtered mass density function (FMDF), denoted by F L, F L (ψ; x i, t) = + ζ[ψ, φ(x i, t)] = δ[ψ φ(x i, t)] = ρ(x i, t)ζ[ψ, φ(x i, t)]g(x i x i )dx i, (10) σ δ[ψ α φ α (x, t)], (11) where δ is the delta function, ψ the sample space of the scalar array (which are enthalpy and all mass fractions here). The term ζ[ψ, φ(x i, t)] is the fine-grained density, and Eq. (10) implies that the FMDF is the mass-weighted spatially filtered value of the fine grained density. The integral property of the FMDF is α=1 12 of 21

13 such that: + F L (ψ; x i, t)dψ = + ρ(x i, t)g(x i x i )dx i = ρ(x i, t). (12) For further development, the mass-weighted conditional filtered mean of the variable Q(x, t) is defined as: + Q(x i, t) ψ = ρ(x i, t)q(x i, t)ζ[ψ, φ(x i, t)]g(x i x i)dx i. (13) F L (ψ; x i, t) From its properties, it follows that the filtered value of any function of the scalar variables is obtained by integration over the composition space, Q(x i, t) = + ρ(x i, t)q(x i, t)f L (ψ, x i, t)]dψ. (14) Downloaded by Richard Miller on August 28, DOI: / As such, any LES term that is only a function of the variables within the FMDF can be directly calculated; including the chemical source terms. Finally, following the procedures in 76 and, 70 the exact transport equation for the S-FMDF is: F L (ψ; x i, t) t + [ũi ψ]f L (ψ; x i, t) x i = ψ α [< 1 ρ J i,α ψ > F L (ψ, x i, t)] [S α(ψ)f L (ψ, x i, t)/ρ], (15) x i ψ α where the brackets within the first term on the right hand side indicate the standard filter operator for convenience. It is important to note that the above equation is exact, albeit unclosed. Also, due to the high dimensions of the FMDF transport equation a direct solution procedure is not computationally feasible. Even if it were, correct boundary conditions are unclear. Therefore, an alternative solution procedure is necessary. By far the vast majority of LES-FDF is solved using a set of stochastically equivalent notional particles in a Monte Carlo sense. However, an alternative Eulerian approach exists and has recently seen use in solving the LES-FDF equations based on the Direct Quadrature Method of Moments (DQMOM) which may have advantages over the particle approach for high speed flows involving shocks However, as the vast majority of LES-FDF is based on the particle approach only it will be discussed hereinafter. The notional particle approach is based on an ensemble of Monte Carlo particles to represent the FMDF. It is noted that the particles are not Lagrangian fluid particles and therefore only represent the statistics of certain quantities. Note that once a set of models is chosen and incorporated into the original FMDF, the particles form a set of equivalent stochastic equations and transport the identical FMDF as the modeled 5, 77, 82, 83 Eulerian set of equations in the limit of large numbers of particles. The last term on the right hand side of the S-FMDF [Eq. (15)] is due to chemical reaction and is in a closed form. This is the primary advantage of the LES-FDF approach. The unclosed nature of SGS convection and mixing is indicated by the conditional filtered values. The convection term is typically modeled in a manner consistent with conventional LES by decomposing the term as: ũ i ψf L (ψ; x i, t) = ũ i F L + [ũi ψ ũ i ]F L, (16) where the second term on the right hand side denotes the influence of SGS convective flux. This term is typically modeled as: [ũi ψ ũ i ]F L = µ t (F L / ρ) x i. (17) The first Favre moments corresponding to Eqs. (16) and (17) are: u i φ α = ũ i φα + ( u i φ α ũ i φα ), (18) ρ[ u i φ α ũ i φα ] = D t φ α x i, (19) where D t = µt Sc t is the subgrid molecular diffusivity, Sc t is typically assumed constant and equal to 0.7, and unity Lewis number and simple Fickian diffusion have been assumed. Therefore, when integrated to recoup the standard corresponding Eulerian LES transport equation, the above model is the standard gradient 13 of 21