Entropic lattice Boltzmann model for gas dynamics: Theory, boundary conditions, and implementation

Abstract

We present in detail the recently introduced entropic lattice Boltzmann model for compressible flows [N. Frapolli et al., Phys. Rev. E 92, 061301(R) (2015)PLEEE81539-375510.1103/PhysRevE.92.061301]. The model is capable of simulating a wide range of laminar and turbulent flows, from thermal and weakly compressible flows to transonic and supersonic flows. The theory behind the construction of the model is laid out and its thermohydrodynamic limit is discussed. Based on this theory and the hydrodynamic limit thereof, we also construct the boundary conditions necessary for the simulation of solid walls. We present the inlet and outlet boundary conditions as well as no-slip and free-slip boundary conditions. Details necessary for the implementation of the compressible lattice Boltzmann model are also reported. Finally, simulations of compressible flows are presented, including two-dimensional supersonic and transonic flows around a diamond and a NACA airfoil, the simulation of the Schardin problem, and the three-dimensional simulation of the supersonic flow around a conical geometry.

Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

“…where f eq i , g eq i are equilibrium parts computed from (10) and (14) to (17); f (1) i , g (1) i are nonequilibrium parts; and ρ tgt , u tgt , and T tgt are target values which need to be specified. The nonequilibrium parts are obtained based on the Grad's approximation and using the general formula (14) to (17) with the nonequilibrium moments given in Table II [14,30] P (1) αβ…”

“…The number of discrete velocities of the standard lattices is too low to reproduce all the moments required for obtaining the full compressible Navier-Stokes-Fourier (NSF) equations [11]. Increasing the number of discrete velocities and using high-order (multispeed) lattice models is a systematic approach to circumvent these limitations and simulate high-speed compressible flows [12][13][14][15]. However, apart from increased computational cost, a limited temperature range is another restriction of high-order lattices [16].…”

Grid refinement for entropic lattice Boltzmann models

“…Same as for the thermal two-populations ELBM, also for the compressible model [15,26], we employ two populations. However, in the compressible model a multispeed lattice, the DdQ7 d , is used and the second population is needed to change the adiabatic exponent γ ad .…”

“…In the above expressions W i = W i (T ) are the temperature-dependent weights [15,26]. Finally, the relaxation parameter α is computed as the positive root of the entropy condition…”

“…Similar to the f -populations, the different contribution on the relaxation, β 3 , shows up only in the q (1) α tensor; this can be expressed analytically as [26]…”