Boundary conditions for thermal lattice Boltzmann equation method

We propose a thermal boundary condition treatment based on the “bounce-back” idea and interpolation of the distribution functions for both the Dirichlet and Neumann (normal derivative) conditions in the thermal lattice Boltzmann equation (TLBE) method. The coefficients for the distribution functions involved are determined to satisfy the Dirichlet or Neumann condition with second-order accuracy. For the Dirichlet condition there is an adjustable parameter in the treatment and three particular schemes are selected for demonstration, while for the Neumann condition the second-order accurate scheme is unique. When applied to inclined or curved boundaries, the Dirichlet condition treatment can be directly used, while the Neumann condition given in the normal direction of the boundary should be converted into derivative conditions in the discrete velocity directions of the TLBE model. A spatially coupled formula relating the boundary temperature, boundary normal heat flux, and the distribution functions near the boundary is thus derived for the Neumann problems on curved boundaries. The applicability and accuracy of the present boundary treatments are examined with several numerical tests for which analytical solutions are available, including the 2-dimensional (2-D) steady-state channel flow, the 1-D transient heat conduction in an inclined semi-infinite solid, the 2-D steady-state and transient heat conduction inside a circle and the 3-D steady-state circular pipe flow. While the Dirichlet condition treatment leads to second-order accuracy for the temperature field, it only gives a first-order accurate boundary heat flux because of the irregularity of the cuts by the curved boundary with the lattices. With the Neumann condition on the curved boundary, the accuracy for the temperature field obtained is first-order. When the tangential temperature gradient on the boundary is decoupled, second-order convergence of the temperature field can be obtained with Neumann conditions.