A methodology termed the ''velocity-scalar filtered density function'' ͑VSFDF͒ is developed and implemented for large eddy simulation ͑LES͒ of turbulent flows. In this methodology, the effects of the unresolved subgrid scales ͑SGS͒ are taken into account by considering the joint probability density function ͑PDF͒ of the velocity and scalar fields. An exact transport equation is derived for the VSFDF in which the effects of the SGS convection and chemical reaction are closed. The unclosed terms in this equation are modeled in a fashion similar to that typically used in Reynolds-averaged simulation procedures. A system of stochastic differential equations ͑SDEs͒ which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô-Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by a finite difference LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VSFDF results are compared with those obtained by the Smagorinsky model, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that the values of both the SGS and the resolved components of all second order moments including the scalar fluxes are predicted well by VSFDF. The sensitivity of the calculations to the model's ͑empirical͒ constants are assessed and it is shown that the magnitudes of these constants are in the same range as those employed in PDF methods.

A methodology termed the "velocity-scalar filtered mass density function" ͑VSFMDF͒ is developed and implemented for large eddy simulation ͑LES͒ of variable-density turbulent reacting flows. This methodology is based on the extension of the previously developed "velocity-scalar filtered density function" method for constant-density flows. In the VSFMDF, the effects of the unresolved subgrid scales ͑SGS͒ are taken into account by considering the joint probability density function of the velocity and scalar fields. An exact transport equation is derived for the VSFMDF in which the effects of SGS convection and chemical reaction are in closed forms. The unclosed terms in this equation are modeled in a fashion similar to that in Reynolds-averaged simulation procedures. A set of stochastic differential equations ͑SDEs͒ are considered which yield statistically equivalent results to the modeled VSFMDF transport equation. The SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô-Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed. In nonreacting flows, it is shown that the VSFMDF results agree well with those obtained by a "conventional" finite-difference LES procedure in which the transport equations corresponding to the filtered quantities are solved directly. The VSFMDF results are also compared with those obtained by the Smagorinsky closure, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that all of the first two moments including the scalar fluxes are predicted well by the VSFMDF. Moreover, the VSFMDF methodology is shown to be able to represent the variable density effects very well. The predictive capabilities of the VSFMDF in reacting flows are further demonstrated by LES of a reacting shear flow. The predictions show favorable agreement with laboratory data, and demonstrate several of the features as observed experimentally.

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A methodology termed "frequency-velocity-scalar filtered mass density function" ͑FVS-FMDF͒ is developed for large eddy simulation ͑LES͒ of turbulent flows. The FVS-FMDF takes account of unresolved subgrid scales ͑SGSs͒ by considering the joint probability density function ͑PDF͒ of the frequency, the velocity, and the scalar fields. An exact transport equation is derived for the FVS-FMDF in which the effects of convection and chemical reaction are in closed forms. The unclosed terms in this equation are modeled in a fashion similar to PDF methods in Reynolds-averaged Navier-Stokes simulations. The FVS-FMDF transport is modeled via a set of stochastic differential equations ͑SDEs͒. The numerical solution procedure is based on a hybrid finite-difference ͑FD͒/Monte Carlo ͑MC͒ method in which the LES filtered transport equations are solved by the FD, and the set of SDEs is solved by a Lagrangian MC procedure. LES of a temporally developing mixing layer is conducted via the FVS-FMDF, and the results are compared with those via the Smagorinsky SGS closure. All these results are also assessed by comparison with those obtained by direct numerical simulation ͑DNS͒. The FVS-FMDF predictions show favorable agreements with DNS data.

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The transport equation of entropy is introduced in large eddy simulation to perform exergy analysis of turbulent combustion systems. The sources of exergy destruction can be evaluated by analyzing entropy generation terms, which appear in unclosed forms in this equation. The closure is based on the filtered density function (FDF) methodology. The primary advantage of FDF is that chemical reaction and its entropy generation effects appear in closed forms. This methodology involves a stochastic model, which is being developed to account for the subgrid scale transport of entropy.

An overview is presented of the recent developments in the application of large eddy simulation (LES) for prediction and analysis of local entropy generation in turbulent reacting flows. A challenging issue in such LES is subgrid-scale (SGS) modeling of filtered entropy generation terms. An effective closure strategy, recently developed, is based on the filtered density function (FDF) methodology with inclusion of entropy variations. This methodology, titled entropy FDF (En-FDF), is the main focus of this article. The En-FDF has been introduced as the joint velocity-scalar-turbulent frequency-entropy FDF and the marginal scalar-entropy FDF. Both formulations contain the chemical reaction and its entropy generation effects in closed forms. The former constitutes the most comprehensive form of the En-FDF and provides closure for all of the unclosed terms in LES transport equations. The latter is the marginal En-FDF and accounts for entropy generation effects, as well as scalar-entropy statistics. The En-FDF methodologies are described, and some of their recent predictions of entropy statistics and entropy generation in turbulent shear flows are presented.

An overview is presented of the state of progress in large eddy simulation of turbulent combustion via the filtered density function (FDF). This includes the formulations based on both the joint velocity-scalar FDF, and the marginal scalar FDF. In the former, the most up-to-date and comprehensive form of the model is presented along with its applications for LES of some relatively simple flows. In the latter, results are presented of the most recent LES of a complex turbulent flame. Both of the models are described in the context of a variable density flow via consideration of the filtered mass density function (FMDF).