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

• Autor(a) principal: Silva, Goncalo
• Data de Publicação: 2020
• Tipo de documento: Artigo
• Idioma: por
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10174/28507; https://doi.org/10.1016/j.compfluid.2020.104537
• topic: Lattice Boltzmann Method; Two-Relaxation-Time Scheme; Analytical Solutions; Discrete Error Analysis; Non-uniform body forces

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.