Heat Transfer in Thermal Lattice Boltzmann Equation Method

The evaluation of the boundary heat flux and total heat transfer in the lattice Boltzmann equation (LBE) simulations is investigated. The boundary heat fluxes in the discrete velocity directions of the thermal LBE (TLBE) model are obtained directly from the temperature distribution functions at the lattice nodes. With the rectangular lattice uniformly spaced the effective surface area for the discrete heat flux is the unit spacing distance, thus the heat flux integration becomes simply a summation of all the discrete heat fluxes with constant surface areas. The present method for the evaluation of total heat transfer is very efficient and robust for curved boundaries because it does not require the determination of the normal heat flux on the boundary and the surface area. To validate its applicability and accuracy, several numerical tests with analytical solutions are conducted, including 2-dimensional (2D) steady thermal flow in a channel, 1-D transient heat conduction in an inclined semi-infinite solid, 2-D transient conduction inside a circle, and 3-D steady thermal flow in a circular pipe. For straight boundaries perpendicular to one of the discrete velocity vectors, the total heat transfer is second-order accurate. For curved boundaries only first-order accuracy is obtained for the total heat transfer due to the irregularly distributed lattice fractions cut by the curved boundary.