Abstract:
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.