Bulk and shear viscosities in lattice Boltzmann equations

Lattice Boltzmann equations (LBE) are a useful tool for simulating the incompressible Navier-Stokes equations. However, LBE actually simulate a compressible but usually isothermal fluid at some small but finite Mach number. There has been recent interest in using LBE at larger, but still subsonic, Mach numbers, for which the viscous terms in the resulting momentum equation depart appreciably from those in the compressible Navier-Stokes equations. In particular, the isothermal constraint implies a nonzero “bulk” viscosity in addition to the usual shear viscosity. This difficulty arises at the level of the isothermal continuum Boltzmann equation prior to discretization. A remedy is proposed, and tested in numerical experiments with decaying sound waves. Conversely, an enhanced bulk viscosity is found useful for identifying or suppressing artifacts in under-resolved simulations of supposedly incompressible shear flows.