A pressure-based solver for low-Mach number flow using a discontinuous Galerkin method

Over the past two decades, there has been much development in discontinuous Galerkin methods for incompressible flows and for compressible flows with a positive Mach number, but almost no attention has been paid to variable-density flows at low speeds. This paper presents a pressure-based discontinuous Galerkin method for flow in the low-Mach number limit. We use a variable-density pressure correction method, which is simplified by solving for the mass flux instead of the velocity. The fluid properties do not depend significantly on the pressure, but may vary strongly in space and time as a function of the temperature. We pay particular attention to the temporal discretization of the enthalpy equation, and show that the specific enthalpy needs to be ‘offset’ with a constant in order for the temporal finite difference method to be stable. We also show how one can solve for the specific enthalpy from the conservative enthalpy transport equation without needing a predictor step for the density. These findings do not depend on the spatial discretization. A series of manufactured solutions with variable fluid properties demonstrate full second-order temporal accuracy, without iterating the transport equations within a time step. We also simulate a Von Kármán vortex street in the wake of a heated circular cylinder, and show good agreement between our numerical results and experimental data.