Linearized-Boltzmann-type-equation-based finite difference method for thermal incompressible flow

► We develop a finite difference method for thermal incompressible flows. ► The formulation bases on a linearized Boltzmann-type equation. ► Neither a pressure–density relation nor a small Mach number assumption is required. ► The approach is valid for both liquid and incompressible gas flows. ► A variety of thermal incompressible flows are simulated for scheme validation. This study reports on further development of a finite difference method formulated on the basis of a linearized-Boltzmann-type-equation for thermal incompressible flows with external body force effect. In classical lattice Boltzmann methods, a pressure-density relation, and/or a finite Mach number, no matter how small, are required in the solution of the linearized Boltzmann-type equation, thus generating inherent compressibility error unavoidably. In the present approach, the pressure field is determined by a pressure-correction method to ensure incompressibility, thus the approach is valid for both liquid and incompressible gas flows. A variety of thermal laminar incompressible flows, such as Couette flow, falling thin liquid film flow, fluid flow through porous plates, and two- and three-dimensional natural convection flow are simulated. The results compared extremely well with analytical solutions and other known numerical simulations of the thermal incompressible flows investigated.