Progress in Large Eddy Simulation of High Speed Turbulent Mixing and Reaction

Abstract

The filtered density function (FDF) method is being extended for subgrid scale (SGS) closure as required in large eddy simulation (LES) of high speed turbulent reacting flows. The primary advantage of FDF is that the effects of SGS chemical reactions appear in a closed form. The suitable means of invoking FDF in high speed flows is via consideration of the SGS statistics of the energy, the pressure, the velocity and the scalar fields. This formulation is under way in which modeled stochastic differential equations are being developed to account for the SGS transport of all of these fields. The simplest subset of this model considers the SGS transport of the scalar field. Results are presented of our latest LES of scalar mixing in a high speed shear flow via this method. M odeling and simulation of high speed turbulent reacting flows have been the subject of widespread investigations for several decades now. The state of the practice in simulations of such flows typically solves the Reynolds averaged Navier-Stokes (RANS) equations, expanded to include scalars’ transport. Closure is usually through two-equation turbulence models in conjunction with Boussinesq and gradient-diffusion assumption. Chemical reaction source terms are usually formulated using the law of mass action, and the effects of turbulence fluctuations on reaction rates are either completely ignored or modeled via eddy break up and/or assumed probability density function (PDF) methods. This first generation model has been incorporated in majority of CFD codes worldwide. This technology, however, is severely limited in many respects and the shortcomings are well documented in literature. The physics of high speed combustion is rich with many complexities. From the modeling standpoint, some of the primary issues are the development of accurate descriptors for turbulence, chemistry, compressibility, and turbulence-chemistry interactions. The phenomenon of mixing at both micro- and macro-scales and its role and capability (or lack thereof) to provide a suitable environment for combustion and the subsequent effects of combustion on hydrodynamics, are at the heart of hypersonic physics. From the computational viewpoint, novel strategies are needed to allow affordable simulation of complex flows with state-of-the art physical models. The power of parallel scientific computing now allows inclusion of more complex physical phenomena which in turn translate into greatly improved predictive capabilities. It is now widely accepted that the optimum means of capturing the detailed, unsteady physics of turbulent combustion is via large eddy simulation (LES). 1, 2 The primary issue associated with LES is accurate modeling of the subgrid scale (SGS) quantities. The filtered density function (FDF) methodology 1, 3 has proven particularly effective for this closure. The FDF is the counterpart of the probability density function (PDF) methodology in RANS. 3, 4 The idea of using the PDF method for LES was first suggested by Givi. 5 But it was the formal definition of FDF by Pope 6 which provided the mathematical foundation of LES/FDF. Within the past several years, significant progress has been made in developments and applications of the FDF. In its simplest form, the “assumed” FDF method was suggested by Madnia et al., 7, 8 where all of the drawbacks of this simple approach were highlighted. Similar to PDF methods, there are different ways by which transport of the FDF can be considered. These differ in the flow variables which are being considered, and whether the method is applicable to constant density or variable density flows. The marginal scalar FDF (SFDF) was developed by Colucci et al. 9 This work demonstrated, for the first time, that solution of