An internal penalty discontinuous Galerkin method for simulating conjugate heat transfer in a closed cavity

Using the discontinuous Galerkin (DG) method for conjugate heat transfer problems can provide improved accuracy close to the fluid-solid interface, localizing the data exchange process, which may further enhance the convergence and stability of the entire computation. This paper presents a framework for the simulation of conjugate heat transfer problems using DG methods on unstructured grids. Based on an existing DG solver for the incompressible Navier-Stokes equation, the fluid advection-diffusion equation, Boussinesq term, and solid heat equation are introduced using an explicit DG formulation. A Dirichlet-Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux of interface quadrature points in the fluid-solid interface. Formal h and p convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved. The algorithm is then further validated against 3 existing benchmark cases, including a thermally driven cavity, conjugate thermally driven cavity, and a thermally driven cavity with conducting solid, at Rayleigh numbers from 1000 to 100000. The computational effort is documented in detail demonstrating clearly that, for all cases, the highest-order accurate algorithm has several magnitudes lower error than first- or second-order schemes for a given computational effort.