Narrow Results

A thermohydrodynamic lattice-BGK model for the ideal gas was derived by Alexander et al. in 1993, and generalized by McNamara et al. in the same year. In these works, particular forms for the equilibrium distribution function and the transport coefficients were posited and shown to work, thereby establishing the sufficiency of the model. In this paper, we rederive the model from a minimal set of assumptions, and thereby show that the forms assumed for the shear and bulk viscosities are also necessary, but that the form assumed for the thermal conductivity is not. We derive the most general form allowable for the thermal conductivity, and the concomitant generalization of the equilibrium distribution. In this way, we show that it is possible to achieve variable (albeit density-dependent) Prandtl number even within a single-relaxation-time lattice-BGK model. We accomplish this by demanding analyticity of the third moments and traces of the fourth moments of the equilibrium distribution function. The method of derivation demonstrates that certain undesirable features of the model — such as the unphysical dependence of the viscosity coefficients on temperature — cannot be corrected within the scope of lattice-BGK models with constant relaxation time.

Lattice-Boltzmann models, proposed at the end of the 1980s as the noise-free version of lattice-gas models are based on gas-kinetical-like representation of fluid flow. Their recent modifications, the lattice-BGK models, provide especially simple, effective and stable algorithms for the solution of incompressible flows. The boundary-fitting concept and local grid refinement proposed for the lattice-BGK model conserve the second order accuracy of the original algorithm for flows around complicated geometries in regions of small and moderate Reynolds numbers.

For the simulation of low Mach number reactive flows with significant density changes a modified lattice-BGK model in combination with the conventional convective-diffusion solvers for equations of temperature and species is proposed. Together with boundary-fitting conditions and local grid refinement the scheme enables the accurate consideration of low Mach number combustion in complex geometry as the flows around porous burners or droplets combustion.