Consistent 2-D and 3-D thermal boundary conditions with
three different formulations for thermal lattice Boltzmann
simulations are proposed. The unknown energy distribution functions are made functions of known energy distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definition of internal energy density. The proposed thermal boundary conditions are applied to two-dimensional thermal Poiseuille flow, thermal Couette flow, thermal Couette flow
with wall injection, natural convection in a square cavity, and three-dimensional thermal Poiseuille flow in a square duct. Numerical simulations indicate that each formulation is second-order accurate, and maintains accuracy over a wide range of Rayleigh numbers.

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