Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

Abstract

This work concerns with the clean inclusion of the forcing term in the lattice Boltzmann method (LBM) for the modeling of non-uniform body forces in steady hydrodynamics. The study is conducted for the two-relaxation-time (TRT) scheme. Here, we consider a simple, but yet sufficiently generic, flow config- uration driven by a spatially varying body force based on which we derive the analytical solution of the forced LBM-TRT and compare two force strategies set by first- and second-order expansions. This proce- dure exactly establishes the macroscopic system satisfied by each force formulation at discrete level. The obtained theoretical results are further verified in two distinct benchmark channel flow problems. Over- all, this study shows that the spatial discrete effects posed by the LBM modeling of the force term may come in through two sources. The first one is a defect inherent to LBM, arising from the non-local spatial discretization of the forcing term, and given by the discrete Laplacian of the body force. While it corrupts the discrete momentum balance with a non-linear viscosity dependent term in the single-relaxation-time schemes, this inconsistency is avoided with the TRT scheme, through its free-tunable relaxation degree of freedom . The other error source is unique to the use of second-order force expansions, where its non- zero second-order velocity moment interferes with the discrete momentum balance through a spurious first-order derivative term, leading to several forms of inconsistency in the LBM steady solution. For simu- lations under the convective scaling it makes the LBM scheme a numerical representation of a differential system distinct from the physical one, whereas under the diffusive scaling it leads to viscosity-dependent numerical errors, which corrupt the otherwise consistent structure of TRT steady-state solutions. In con- trast, the defects reported herein are absent with the first-order force expansion scheme operated within the scope of the LBM-TRT model for steady hydrodynamics.