Lattice Boltzmann Methods for thermal flows: continuum limit and applications to compressible Rayleigh-Taylor systems

We compute the continuum thermo-hydrodynamical limit of a new formulation of lattice kinetic equations for thermal compressible flows, recently proposed in [Sbragaglia et al., J. Fluid Mech. 628 299 (2009)]. We show that the hydrodynamical manifold is given by the correct compressible Fourier- Navier-Stokes equations for a perfect fluid. We validate the numerical algorithm by means of exact results for transition to convection in Rayleigh-B\'enard compressible systems and against direct comparison with finite-difference schemes. The method is stable and reliable up to temperature jumps between top and bottom walls of the order of 50% the averaged bulk temperature. We use this method to study Rayleigh-Taylor instability for compressible stratified flows and we determine the growth of the mixing layer at changing Atwood numbers up to At ~ 0.4. We highlight the role played by the adiabatic gradient in stopping the mixing layer growth in presence of high stratification and we quantify the asymmetric growth rate for spikes and bubbles for two dimensional Rayleigh- Taylor systems with resolution up to Lx \times Lz = 1664 \times 4400 and with Rayleigh numbers up to Ra ~ 2 \times 10^10.