(RANS) employ turbulent transport approximations with an effective turbu-lent viscosity that is by orders of magnitude larger than the molecular viscosity.
In particular if steady state versions of these equations are used, this tends to suppress large scale instabilities, which occur in flows with combustion even more frequently than in nonreacting flows. If those instabilities are to be re-solved in numerical simulations, it is necessary to resort to more advanced, but computationally more expensive methods such as direct numerical simulation (DNS) or large eddy simulation (LES). As noted in the preface, DNS is still out of reach as a method to predict turbulent flows with combustion for practical engineering applications for many years to come.
Large eddy simulation, in contrast, does not intend to numerically resolve all turbulent length scales, but only a fraction of the larger energy-containing scales within the inertial subrange. Modeling is then applied to represent the smaller unresolved scales, which contain only a small fraction of the turbulent kinetic energy. Therefore the computed flows are usually less sensitive to mod-eling assumptions. The distinction between the resolved large scales and the
modeled small scales is made by the grid resolution that can be afforded. The model for the smaller scales is called the subgrid model. In deriving the basic LES equations, the Navier–Stokes equations are spatially filtered with a filter of size, which is of the size of the grid cell (or a multiple thereof) in order to remove the direct effect of the small scale fluctuations (cf. Ghosal and Moin, 1995). These show up indirectly through nonlinear terms in the subgrid-scale stress tensor as scale Reynolds stresses, Leonard stresses, and subgrid-scale cross stresses. The latter two contributions result from the fact that, unlike with the traditional Reynolds averages, a second filtering changes an already fil-tered field. In a similar way, after filtering the equations for nonreacting scalars such as the mixture fraction, one has to model the filtered scalar flux vec-tors that contain subgrid scalar fluxes, Leonard fluxes, and subgrid-scale cross fluxes.
Review papers have been written, for instance, by Lesieur and M´etais (1996) and by Moin (1997). Often the gradient transport assumption employing the Smagorinsky model is introduced for the Reynolds stress tensor and the scalar fluxes. The Smagorinsky model for the subgrid-scale stress tensor takes the form
τ = −2 ¯ρC¯ s2| ¯S| ¯S, (1.168) where ¯S is the filtered rate of strain tensor and Cs is the Smagorinsky coeffi-cient. The Smagorinsky model is a generalization of Prandtl’s mixing length model. It requires Cs to be positive and to be known in advance. It has been found, however, that to obtain satisfactory results, largely different values of Cs, ranging from 0.01 to 0.3, depending on the flow and grid resolution, must be chosen.
As an example for the scalar fluxes, the subgrid-scale heat flux vector in a compressible nonreacting flow can be modeled as (cf. Moin et al., 1991)
¯jq = − ¯ρCs2| ¯S|
Prt
∇ ¯T . (1.169)
Here Prt is a subgrid-scale turbulent Prandtl number. Some of the more re-cent approaches to modeling the additional stresses are discussed by Speziale (1998).
A breakthrough in subgrid modeling is the introduction of a method called dynamic modeling by Germano et al. (1991). In the dynamic subgrid-scale model, a test filter ˆ is introduced in addition to the grid filter . Often the test filter is set to two times the grid filter. A variable Smagorinsky coefficient can then be calculated, which depends on the filtered stresses and fluxes, as well as on those that are resolved at the test filter level. The Smagorinsky coefficient
1.14 Combustion models used in large eddy simulation 59 may thereby take positive or negative values. A positive value implies that energy flows from the resolved to the subgrid scales, which is in agreement with the concept of the cascade hypothesis, while a negative coefficient implies an inverse cascade or “backscatter”. It has been argued by Piomelli et al. (1991) that this back scatter has a physical basis, but with the original formulation of the dynamic model, where up to 30% negative values of Cswere calculated, it leads to undesirable numerical instabilities. Various remedies to resolve this problem have been proposed (cf. Germano et al., 1991, Piomelli and Liu, 1995, and Meneveau et al., 1996). The dynamic model was extended by Moin et al. (1991) to scalar transport where it served to determine the subgrid-scale turbulent Prandtl number.
When finite-difference methods are used in LES of inhomogeneous flows, different numerical discretization errors may inflict serious damage on sim-ulations. Among these are the truncation error due to the representation of derivatives by finite differences and the aliasing error that arises when nonlin-ear terms are represented on a discrete grid. A systematic study of this issue has been undertaken by Kravchenko and Moin (1997) showing that both aliasing and truncation errors of low-order schemes can degrade LES computations. Of these, the aliasing error was found to be the source of the most serious prob-lems, because it interfered with the energy conserving nature of the scheme.
Mathematical and physical constraints of LES have recently been reviewed by Ghosal (1999).
For combustion simulations with LES, in addition to accurate resolution, a reliable subgrid model for scalar fields is of particular importance. Cook and Riley (1994) have proposed a scale similarity assumption for the subgrid variance of the mixture fraction:
Z2= Z2− ¯Z2= cZ(Z2− ˆ¯Z2). (1.170) Here the bar denotes filtering at the grid filter and the hat denotes filtering using the test filter. The hypothesis behind scale similarity is that the largest unresolved scales have a structure similar to the smallest of the resolved scales.
A theoretical estimate for cZis given by Jim´enez et al. (1997), who predict that it depends on the exponent of the scalar spectrum function in the large Reynolds limit. However, Cook (1997) derives a method for calculating the coefficient of scale similarity models that he shows to depend on the grid size of the test filter and the Reynolds number.
An alternative approach assumes that production equals dissipation in the subgrid variance equation. Pierce and Moin (1998a) use this to calculate the constant C in the resulting expression for the subgrid-scale Favre variance Z2
of the mixture fraction,
Z2= C2|∇ ˜Z|2, (1.171) by a dynamic modeling approach. Here|∇ ˜Z| is the gradient of the resolved Favre mean mixture fraction. Comparing this dynamic model with the scale similarity model, Pierce and Moin (1998b) found remarkable differences in the large eddy simulation of a swirling, confined, coaxial jet flame.
Germano et al. (1997) extend the scale similarity model to reacting scalars and postulate that the subgrid variance of a scalar or the co-variance of two different scalars can be calculated from known quantities at the two filter levels.
They find, however, that the similarity constant depends not only on the mesh size but also on the chemical time scale. This is because filtering over thin reaction zones underestimates the unresolved chemical source term. As the grid filter decreases to values of the order of the reaction zone thickness, an increasing part of the reaction rate occurs between the two filter levels. Therefore the similarity constant corresponding to cZin (1.170) decreases with the mesh size (cf. Vervisch and Veynante, 2000).
Certain formulations of subgrid models profitably take guidance from clo-sure procedures that have been successful on the RANS level. Cook and Riley (1994) have proposed a presumed shape beta function pdf formulation, called Large Eddy Probability Density Function (LEPDF), for the mixture fraction. It uses the filtered mean and the subgrid variance taken from (1.170) to achieve closure for nonpremixed combustion. For the reactive scalars they apply the Conserved Scalar Equilibrium Model. Later on, Cook et al. (1997) extend this by using profiles for the reactive scalars from a steady state flamelet model based on a single one-step reaction. This model was found to be reasonably accurate compared to DNS data of homogeneous, isotropic decaying turbu-lence. Cook and Riley (1998) performed further a priori testing (comparison with DNS data) of this model by varying the activation energy of the one-step model. This shows a better agreement with DNS data than models using equi-librium chemistry models or a closure using filtered mean reactive scalars for the reaction rate.
De Bruyn Kops et al. (1998) performed a full LES calculation of the filtered mean and variance of the mixture fraction, determined by using their balance equations, a presumed shape for the scalar dissipation rate, and a subgrid-scale presumed shape pdf. A table was constructed by integrating steady state flamelet profiles. The table is parameterized in terms of the filtered mean and the variance of the mixture fraction and the mean scalar dissipation rate. The model accurately reproduces the spatial average of the filtered species concentration obtained from DNS data.
1.14 Combustion models used in large eddy simulation 61 The large eddy presumed shape beta function LEPDF model by Cook and Riley (1994) has been tested by Jim´enez et al. (1997) together with other assumptions, such as the lognormal pdf for scalar gradients and the scale similarity assumption. They find that presuming a beta function pdf yields excellent agreement with DNS data not only for higher moments of temper-ature in the fast chemistry limit, but even for the pdf of the mixture fraction itself. DesJardin and Frankel (1998) have performed a priori and a posteriori assessments of several subgrid-scale combustion models. They compared LES calculations using a Scale Similarity Filtered Reactive Rate Model (SSFRRM) and a Conserved Scalar Equilibrium Model. They find that the SSFRRM model provides the best agreement with the DNS data. A model based on a transport equation for the subgrid-scale pdf, called Filtered Density Function (FDF), was developed by Colucci et al. (1998) and compared to presumed subgrid pdfs.
Dynamic modeling has been applied by R´eveillon and Vervisch (1997, 1998) to test a closure of the mixing term in the subgrid pdf equation for the mixture fraction. Similarly, Cook and Bushe (2000) use a priori testing of a subgrid-scale model for the scalar dissipation rate.
There are so far only a few large eddy simulations of large scale turbulent nonpremixed flames. Pierce and Moin (1998b) calculate a swirling, confined, coaxial jet flame based on the presumed beta function subgrid LES approach including heat release in the fast chemistry limit and show convincing compar-isons with experimental data. Branley and Jones (2001) compute a hydrogen jet diffusion flame using the Smagorinsky model with Cs= 0.1, a subgrid-scale Prandtl number of 0.7, the presumed shape beta function subgrid pdf model with the variance calculated from (1.171) with C= 0.13, and an equilibrium model for hydrogen–air combustion. Forkel and Janicka (1999) perform a large eddy simulation of the diluted hydrogen–air diffusion flame documented by Tacke et al. (1998). They use the dynamic procedure, clipping negative values of the Smagorinsky constant. Chemical equilibrium profiles are tabulated as a function of the mixture fraction. It is found that inlet conditions must be well defined to describe the flow and the mixture fraction field close to the noz-zle accurately enough. In a recent paper Branley and Jones (1999) calculate a swirling methane flame using dynamic modeling for the velocity and a unity Prandtl number for the scalar flux. The conserved scalar pdf is again obtained from the presumed shape beta function subgrid pdf model with the variance obtained from (1.171) with C= 0.09. The chemistry model is based on a sin-gle flamelet profile computed at a strain rate of 200/s. The results show good qualitative agreement with measurements.
A recent simulation of a turbulent jet diffusion flame by Pitsch and Steiner (2000) based on the Lagrangian Flamelet Model is shown in Figure 1.10. Only
Figure 1.10 (see Color Plate I, Left). Large eddy simulations of a turbulent jet diffusion flame by Pitsch and Steiner (2000). (Reprinted with permission by H. Pitsch and H.
Steiner.)
the conservation equations for mass, momentum, and the mixture fraction were solved in the LES. The subgrid stresses and scalar fluxes are determined using the dynamic model indicating that a constant Schmidt number approximation with a value of Sct= 0.4 is appropriate for LES. The mixture fraction variance is evaluated by (1.171). The mean scalar dissipation rate, including resolved and subgrid contributions, is calculated as proposed by Girimaji and Zhou (1996).
In this study the shape of the scalar dissipation rate conditioned on the mixture fraction has not been presumed but is given by a model based on the assumption of a homogeneous distribution of this quantity in radial direction. Comparison