A filtered Rankine–Hugoniot method for consistent stochastic particle LES-FDF modelling of combustion in high-speed flows

A new model consistency scheme based on the filtered form of the Rankine–Hugoniot (R–H) relation is developed for hybrid Eulerian/Lagrangian Large Eddy Simulation-Filtered Density Function (LES-FDF) methods to model combustion in compressible high-speed flows. The approach particularly pertains to the evolution of the FDF stochastic Lagrangian particles in enthalpy space, accounting for the subgrid effects in the filtered R–H relation to ensure stable solutions and model consistency with the total energy field solved by an Eulerian compressible finite volume scheme. An analytical form of the subgrid term appearing in the filtered R–H relation is derived and it is shown to have a significant effect on the ability to accurately produce the post-shock thermochemical state. Subsequently, two novel numerical schemes are proposed based on either a modelled subgrid scale kinetic energy or a relaxation over a physical timescale. In testing for 1D normal shock cases, the former approach is found to be strongly dependent on the filter width and Mach number, while the latter approach produces consistent results. The relaxation method is further tested for a turbulent 3D shock tube case and a 1D detonation case and found to produce numerically consistent and accurate temperature and species fields. It is shown that the subgrid term is critical for accurately and consistently predicting the temperature fields in both cases and for achieving the correct detonation structure involving a leading shock and autoignition in the second case. Novelty and significance statement LES-FDF methods have not been widely used for high-speed combusting flows with strong shocks. Such flows are common in aerospace combustors and accurate modelling of these is essential for their optimisation, especially with rapid transition to alternative fuels. The first novelty is the development of a filtered Rankine–Hugoniot model to ensure energy consistency between the Eulerian LES and the FDF modelled through Lagrangian particles. The inclusion of a subgrid kinetic energy term in the Lagrangian particle energy equation is shown to be essential for producing consistent solutions and accurate prediction of thermodynamics states through discontinuities. This subgrid term is not closed and, therefore, the second novelty lies in proposing a practical numerical scheme which accounts for it. The consistent model and numerical scheme is validated in canonical shock tube and detonation cases.