Treatment of discontinuous and singular source terms in hyperbolic systems

On many occasions the Navier-Stokes equations are integrated over one or more space directions to remove small-scale features from the solution, such as the size of individual rotor blades in large wind farms, or the micro-pores in flow through porous media. The result of these manipulations is the appearance of additional terms on the right-hand side of the conservation laws. These additional terms may exhibit discontinuous or singular behavior, even if the underlying behavior they represent is regular. Naïve treatment of these additional terms results in non-physical features in the solution or inaccurate predictions of mass, momentum, or energy transfer. This paper proposes a rigorous and straightforward approach for accurately handling source terms with singularities or discontinuities for systems governed by conservation laws, discretized using finite volume schemes with Godunov-type fluxes. We distinguish between volume and surface source terms. Both types can be handled by carefully constructed upwind procedures. This discretization exhibits second-order accuracy, for formally first-order numerical models. The method is validated using the Euler equations with large source terms, with both preconditioned and non-preconditioned flux functions. We also present a novel, elegant method for introducing jump conditions in the solution via singular source terms. This ensures global compatibility relations are met without the need for internal boundaries, as commonly done in actuator disk models. The robustness of the method is demonstrated on a Riemann problem with abrupt area change, matching analytical solutions for 10 different wave patterns. Finally, the new methods are applied to a multi-dimensional, open rotor problem and compared to other results in the literature.