A new Neumann boundary condition scheme for the thermal lattice Boltzmann method

In this paper we propose a new scheme for implementing the Neumann boundary condition (BC) with the thermal Lattice Boltzmann Method (LBM). It consists in transforming the wall heat flux into a source term applied only at the boundary nodes, supposing the boundary thermally insulated and performing a polynomial extrapolation for correcting the temperature values at the boundary nodes, but just at the output data. Five problems are considered: one and two-dimensional heat conduction, two-dimensional forced and free convection, all with planar surfaces, and a two-dimensional heat conduction problem with curved walls. The proposed procedure is explored in these problems by the comparison between the numerical results and reference solutions, presenting very accurate results. The performance of the new proposed scheme is compared with three traditional schemes for Neumann BC implementation: the bounce-back rule, the finite difference scheme and finite volume approach. For some tests, our methodology produces more accurate results in comparison to other traditional BC methods, where resides the main advantage of the proposed scheme. •A new scheme for implementing the Neumann boundary condition in LBM is presented•The new scheme achieved better accuracy than traditional schemes in some cases.•Its implementation is straightforward and preserves the second order accuracy of LBM.•The methodology ensures the correct energy conservation at boundaries.