Chapman–Enskog analysis for finite-volume formulation of lattice Boltzmann equation

• Chapman–Enskog expansion of the finite-volume lattice Boltzmann equation formulation.
• Additional error terms proportional to grid size and time-step in the continuity and momentum conservations.
• An expression for apparent viscosity of FV-LBE on equilateral triangular mesh.
• Numerical validation of the proposed theoretical analysis.

The classical Chapman–Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method [D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262–5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier–Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor–Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.