We present a spectral-element discontinuous Galerkin lattice Boltzmann method to solve incompressible natural convection flows based on the Bousinessq approximation. A passive-scalar thermal lattice Boltzmann model is used to resolve flows for variable Prandtl number. In our model, we solve the lattice Boltzmann equation for the velocity field and the advection-diffusion equation for the temperature field, reducing the degrees of freedom compared with the passive-scalar double-distribution model, which requires the solution of an additional set of evolution equations to resolve the temperature field. Our numerical solution is represented by the tensor product basis of the one-dimensional Legendre-Lagrange interpolation polynomials on the Gauss-Lobatto-Legendre quadrature nodes and body-conforming hexahedral elements. Within the discontinuous Galerkin framework, we impose boundary and element-interface conditions weakly through the numerical flux. A fourth-order Runge-Kutta scheme is used for time integration with simple mass matrix inversion due to fully diagonal mass matrices. We studied natural convection fluid flows in a square cavity and a horizontal concentric annulus for Rayleigh numbers in the range of Ra3\~{}5. Validation of our numerical approach is conducted by comparing with finite-difference, finite-volume, multiple-relaxation-time lattice Boltzmann, and spectral-element methods. Compared with other methods, our computational results show good agreement in temperature profiles and Nusselt numbers using relatively coarse resolutions.

}, author = {S. S. Patel and M. S. Min and K. C. Uga and T. Lee} }