A conservative and consistent scalar filtered mass density function method for supersonic flows

Abstract

A novel scalar filtered mass density function (SFMDF) method is developed for high-speed flows, especially for supersonic reactive flows. The total energy is proposed as the energy form for SFMDF, instead of the commonly used enthalpy or sensible enthalpy. Such an energy form is entirely consistent with the one typically used in large eddy simulation (LES) for fully compressible flows, so that the exact/modeled energy equations in both LES and SFMDF are readily identical. Moreover, the total energy can formulate the SFMDF energy transport equation in such a way that the high-speed source term is strictly conservative. Following the conservative formulation, numerically robust conservative schemes are readily available for flows with discontinuities. Tests in one-dimensional Euler equations show that the temperature redundantly obtained based on the total energy (with conservative high-speed source terms) shows better agreement with the analytical result than the one based on the enthalpy. The proposed LES-SFMDF method is further tested in a shock tube interacting with an isotropic turbulent flow, a compressible two-dimensional non-reactive temporally developing mixing layer, and a supersonic three-dimensional reactive temporally developing mixing layer. Results show that SFMDF with the total energy can considerably improve the temperature distribution in both non-reactive and reactive flows. The proposed LES-SFMDF method with the total energy predicts the turbulence–chemistry interaction better than LES-SFMDF with the enthalpy as well as LES with the well-stirred reactor model in supersonic combustion. This conservative and consistent SFMDF method can be readily extended to more sophisticated probability density function methods in high-speed flows.