Numerical methods for the simulation of shock-induced turbulent mixing have been investigated, focussing on Implicit Large Eddy Simulation. Shock-induced turbulent mixing is of particular importance for many astrophysical phenomena, inertial confinement fusion, and mixing in supersonic combustion. These disciplines are particularly reliant on numerical simulation, as the extreme nature of the flow in question makes gathering accurate experimental data difficult or impossible. A detailed quantitative study of homogeneous decaying turbulence demonstrates that existing state of the art methods represent the growth of turbulent structures and the decay of turbulent kinetic energy to a reasonable degree of accuracy. However, a key observation is that the numerical methods are too dissipative at high wavenumbers (short wavelengths relative to the grid spacing). A theoretical analysis of the dissipation of kinetic energy in low Mach number flows shows that the leading order dissipation rate for Godunov-type schemes is proportional to the speed of sound and the velocity jump across the cell interface squared. This shows that the dissipation of Godunov-type schemes becomes large for low Mach flow features, hence impeding the development of fluid instabilities, and causing overly dissipative turbulent kinetic energy spectra. It is shown that this leading order term can be removed by locally modifying the reconstruction of the velocity components. As the modification is local, it allows the accurate simulation of mixed compressible/incompressible flows without changing the formulation of the governing equations. In principle, the modification is applicable to any finite volume compressible method which includes a reconstruction stage. Extensive numerical tests show great improvements in performance at low Mach compared to the standard scheme, significantly improving turbulent kinetic energy spectra, and giving the correct Mach squared scaling of pressure and density variations down to Mach 10−4. The proposed modification does not significantly affect the shock capturing ability of the numerical scheme. The modified numerical method is validated through simulations of compressible, deep, open cavity flow where excellent results are gained with minimal modelling effort. Simulations of single and multimode Richtmyer-Meshkov instability show that the modification gives equivalent results to the standard scheme at twice the grid resolution in each direction. This is equivalent to sixteen times decrease in computational time for a given quality of results. Finally, simulations of a shock-induced turbulent mixing experiment show excellent qualitative agreement with available experimental data.
展开▼
