Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh–Taylor systems

Abstract

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 FourierNavier-Stokes equations for a perfect fluid. We validate the numerical algorithm by means of exact results for transition to convection in Rayleigh-Bénard 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 RayleighTaylor systems with resolution up to Lx × Lz = 1664 × 4400 and with Rayleigh numbers up to Ra ∼ 2 × 10 10 .

Evaluation of DVFS techniques on modern HPC processors and accelerators for energy‐aware applications

“…At each time step, populations hop from lattice-site to lattice-site and then incoming populations collide among one another, that is, they mix and their values change accordingly. LB models in n dimensions with p populations are labeled as DnQp; we consider a D2Q37 model describing the thermohydrodynamical evolution of a fluid in two dimensions, and enforcing the equation of state of a perfect gas (p = ρT ) [24,25]; this model has been used for large scale simulations of convective turbulence (see e.g., [26,27]). A set of populations (f l (x, t) l = 1 · · · 37), defined at the points of a discrete and regular lattice and each having a given lattice velocity c l , evolve in (discrete) time according to the following equation:…”

Reactive Rayleigh-Taylor systems: Front propagation and non-stationarity

“…-We adopt a numerical scheme based on a recently proposed thermal lattice Boltzmann algorithm [8,9], which is able to reproduce the correct thermohydrodynamics of an ideal gas with good numerical accuracy [10]. To do that, the probability densities f l (x, t) for a particle with velocity c l (belonging to a discrete set, with the index l running over 37 values [9]) at space location x and time t evolve according to the lattice Boltzmann BGK equation [11][12][13] …”

An optimized D2Q37 Lattice Boltzmann code on GP-GPUs

“…In extreme conciseness (full details can be found in [4,5]), one makes contact between this synthetic dynamics and the true dynamics of a compressible gas, starting with a kinetic and thermal description of a system of variable density, velocity and internal energy K , subject to a local body force g g g (gravity); one then is able to show that, after appropriate shift and re-normalization of the velocity and temperature fields, one recovers, through a Taylor expansion in ∆t, the correct thermo-hydrodynamical equations:…”

“…In this section, we introduce the computational methods that we adopt, based on an advanced D2Q37 LB scheme, that correctly reproduces the equation of state of the fluid, regarded as a perfect gas (p = ρT ); full details of the algorithm are given in [4,5].…”