An unstructured Shock-Fitting solver for two-dimensional flows

The computation of flows with shocks by means of the shock-fitting technique was proposed by Moretti since the late sixties and used by several researchers in the seventies and eighties. In those years the use of this technique along with numerical schemes based on the quasi-linear equations allowed the accurate computation of flows characterized by strong discontinues with limited computational resources. In the nineties, however, the increased computing power along with the availability of modern general-purpose codes based on the integral conservation equations produced a loss of interest in the shock fitting technique. Despite the widespread use of shock-capturing codes, shock solutions obtained by means of shock-capturing schemes are plagued by a number of problems pertaining to accuracy, stability and, more in general, solution quality. Despite the continuous efforts made over the last 20 years, these shortcomings have not been completely overcome and appear to be particulary severe when unstructured-grids are used. The complex and difficult implementation of the floating shock-fitting technique in structured-grid solvers has largely contributed to discourage the interest in this type of technique. Indeed, the motion of the shocks throughout the flow-field is obtained by means of floating shock gridpoints which can appear and disappear. This requires the implementation of algorithms which are un-natural in a structured grid setting. As a matter of principle, the management of shock points appears to be simpler within an unstructured grid context. Indeed, the addition, motion and cancellation of shock points and the mesh modifications which are necessary in the neighbourhood of the shock front are operations which are less problematic inside an unstructured solver rather than in a structured one. To our knowledge, only a few attempts have been made to incorporate shock-fitting ideas within shock-capturing, unstructured-grid solvers. These include the work by Van Rosendale , Parpia and Parikh, Tr\'epanier and co-workers and, more recently, Hanel and co-workers. In the present work, a new floating shock-fitting technique featuring the explicit computation of shocks by means of the Rankine-Hugoniot relations has been implemented in an unstructured solver based on Roe's fluctuation splitting schemes . This original technique will be presented along with some numerical simulations of two-dimensional flows characterized by a single strong shock and by interacting shocks