Numerical simulation of heat transfer flows by a direct solution of generalized kinetic models

The heat conduction between two infinite parallel plates is numerically simulated based on a kinetic relaxation model for one-dimensional flows. For this purpose, two distribution functions which depend only on longitudinal velocity are introduced to avoid the multiplicity of integrals. The kinetic model equations in terms of newly defined functions are used to investigate one-dimensional heat transfer between two walls of constant temperature ratio. Generalization of the kinetic models allows the correct estimation of the heat flux for arbitrary Prandtl numbers. The steady solutions are compared with the results of the exact Boltzmann equation to validate the possibility of applying the present kinetic models. The temperature jumps near the solid surface are naturally achieved by the difference of the particle distributions from a local equilibrium state. It is shown that the relative temperature jumps at both plate surfaces increases as the Knudsen number of the flow increases. The temperature profiles by means of the generalized kinetic models agree well with those of the exact Boltzmann equation for various Knudsen numbers.