A two-dimensional linearized unsteady Euler scheme for pulse response calculations

An unsteady linearized Euler scheme for use on moving meshes is presented. This is derived from a scheme for the full non-linear unsteady Euler equations. This scheme is based on the Jameson cell-centred scheme, but is time-accurate and includes the necessary terms to account for grid motion. It is assumed that the unsteadiness in the flow and mesh is small. Using this assumption the discrete unsteady Euler equations are linearized about the full, non-linear, steady mean flow. The resulting equations are solved in the work presented here using a dual-time scheme. In the basic scheme no assumptions are made about the form of the perturbations other than that they are small. This permits the direct calculation of non-periodic flows, e.g. pulse responses. Linear pulse responses are a useful tool as they can be used to calculate the flow due to general inputs. The equations that would result from the assumption of harmonic flow are also derived. Results are presented for heave, pitch and ramp test cases and compared to full non-linear Euler results calculated using a dual-time scheme.