Linear and non-linear Robin boundary conditions for thermal lattice Boltzmann method: cases of convective and radiative heat transfer at interfaces

• Counter-slip approach is used to derive Robin boundary constraints for thermal LBM.
• Equations are presented for convective heat transfer as a linear Robin constraint.
• Equations are suggested to impose surface radiation as a non-linear Robin condition.
• Analyses show first order of convergence rate for linear constraint.
• Second order of convergence rate is found for non-linear constraint.

Despite the wide applications of the linear and non-linear Robin boundary constraints in thermal simulations, not much works are reported on their implementation in lattice Boltzmann framework. In present work, counter-slip energy approach is employed to derive kinetic level equations, representing two particular cases of Robin boundary conditions; convection and combined convection and surface radiation. Loss of generality is avoided in the study and the terms accounting for boundary movement or viscous dissipation effects are incorporated, too. Utilizing a D2Q9 lattice structure, the derived equations are validated with 1D and 2D analytical solutions for conduction heat transfer problems in a square slab. Results of analysis show a first order rate of convergence for the convective boundary condition, while second order rate is found for combined convection and surface radiation constraint.