A spectral-element discontinuous Galerkin lattice Boltzmann method for simulating natural convection heat transfer in a horizontal concentric annulus

• Spectral element discontinuous Galerkin-lattice Boltzmann method solves thermal flow.
• We achieve high-order of accuracy on complex geometries using a spectral element approximation.
• A discontinuous Galerkin (DG) scheme provides a unique way of implementing LBM bounce-back through flux.
• Optimal accuracy is achieved on non-uniform coarse grids.

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. As a result, we reduce the degrees of freedom when compared with the passive-scalar double-distribution model, which requires the solution of several equations to resolve the temperature field. Our numerical solution is represented by the tensor product basis of the one-dimensional Legendre–Lagrange interpolation polynomials. A high-order discretization is employed on body-conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Within the discontinuous Galerkin framework, we weakly impose boundary and element-interface conditions through the numerical flux. A fourth-order Runge–Kutta scheme is used for time integration with no additional cost for mass matrix inversion due to fully diagonal mass matrices. We study natural convection fluid flows in a square cavity and a horizontal concentric annulus for Rayleigh numbers in the range of Ra = 103–108. We validate our numerical approach by comparing it with finite-difference, finite-volume, multiple-relaxation-time lattice Boltzmann, and spectral-element methods. Our computational results show good agreement in temperature profiles and Nusselt numbers using relatively coarse resolutions.