Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains

Abstract

We describe a method for computing time-dependent solutions to the compressible Navier–Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time-varying domain. By writing the Navier–Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration. The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge–Kutta method. For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high-order approximations on complex geometries.