The effects of compressibility on Rayleigh-Taylor instability (RTI) are investigated by inspecting the interplay between thermodynamic and hydrodynamic non-equilibrium phenomena (TNE, HNE, respectively) via a discrete Boltzmann model (DBM). Two effec
tive approaches are presented, one tracking the evolution of the emph{local} TNE effects and the other focussing on the evolution of the mean temperature of the fluid, to track the complex interfaces separating the bubble and the spike regions of the flow. It is found that, both the compressibility effects and the emph{global} TNE intensity show opposite trends in the initial and the later stages of the RTI. Compressibility delays the initial stage of RTI and accelerates the later stage. Meanwhile, the TNE characteristics are generally enhanced by the compressibility, especially in the later stage. The global or mean thermodynamic non-equilibrium indicators provide physical criteria to discriminate between the two stages of the RTI.
We present an improved lattice Boltzmann model for high-speed compressible flows. The model is composed of a discrete-velocity model by Kataoka and Tsutahara [Phys. Rev. E textbf{69}, 056702 (2004)] and an appropriate finite-difference scheme combine
d with an additional dissipation term. With the dissipation term parameters in the model can be flexibly chosen so that the von Neumann stability condition is satisfied. The influence of the various model parameters on the numerical stability is analyzed and some reference values of parameter are suggested. The new scheme works for both subsonic and supersonic flows with a Mach number up to 30 (or higher), which is validated by well-known benchmark tests. Simulations on Riemann problems with very high ratios ($1000:1$) of pressure and density also show good accuracy and stability. Successful recovering of regular and double Mach shock reflections shows the potential application of the lattice Boltzmann model to fluid systems where non-equilibrium processes are intrinsic. The new scheme for stability can be easily extended to other lattice Boltzmann models.
