A cascaded lattice Boltzmann model for thermal convective flows with local heat sources

A cascaded central moment based lattice Boltzmann (LB) method for solving low Mach number thermal convective flows with source terms in two-dimensions in a double distribution function framework is presented. For the passive temperature field, which satisfies the convection diffusion equation (CDE) with a source term to represent internal/external local heat source, a new cascaded collision kernel is presented. Due to the use of a single conserved variable in the thermal energy equation, the cascaded structure in its collision operator begins from the first order moments and evolves to higher order moments. This is markedly different from the collision operator for the fluid flow equations, constructed in previous work, where the cascaded formulation starts at the second order moments in its collision kernel. A consistent implementation of the spatially and temporally varying source terms in the thermal cascaded LB method representing the heat sources in the CDE that maintains second order accuracy via a variable transformation is discussed. The consistency of the thermal cascaded LB method including a source term for the D2Q9 lattice with the macroscopic convection-diffusion equation is demonstrated by means of a Chapman-Enskog analysis. The new model is tested on a set of benchmark problems such as the thermal Poiseuille flow, thermal Couette flow with either wall injection or including viscous dissipation and natural convection in a square cavity. The validation study shows that the thermal cascaded LB method with source term is in very good agreement with the analytical solutions or numerical results reported for the benchmark problems. In addition, the numerical results show that our new thermal cascaded LB model maintains second order accuracy.