Consistent lattice Boltzmann schemes for the Brinkman model of porous flow and infinite Chapman-Enskog expansion

We show that a consistent modeling of porous flows needs at least one free collision relaxation rate to avoid a nonlinear dependency of the numerical errors on the viscosity. This condition is necessary to get the viscosity-independent permeability from the Stokes flow and to parametrize properly (with nondimensional physical numbers) the lattice Boltzmann Brinkman schemes. The two-relaxation-time (TRT) collision operator controls all coefficients of the higher-order corrections in steady solutions with a specific combination of its two collision rates, a possibility lacking for the Bhatnagar-Gross-Krook (BGK)-based single-relaxation-time schemes. The analysis is based on exact recurrence equations of the evolution equation and illustrated for the exact solutions of the Brinkman scheme in simply oriented parallel and diagonal channels. The apparent viscosity coefficient of the TRT Stokes-Brinkman scheme in arbitrary flow is only approximated. The compatibility of one-dimensional arbitrarily rotated flows with the nonlinear (Navier-Stokes) equilibrium is examined. An explicit dependency for all coefficients on the relaxation rates is presented for the infinite steady state Chapman-Enskog expansion.