Shock detection and discontinuity tracking for unsteady flows

A method is presented for the numerical solution of inviscid flows with discontinuities in quasi-one-dimensional unsteady problems. The numerical technique, belonging to the family of the fitting techniques, is based on the method of characteristics for the calculation of discontinuity points, boundary points and, when required, of grid points close to them. The integration of the remaining grid points, enclosed between the discontinuities, is performed by a finite difference scheme following the “λ formulation”. The time step adopted for the numerical integration of the two different sets of points can be substantially decoupled. A simple and effective criterium for the unsteady shock detection is also proposed. Both the general philosophy and the details of the formulation of the method are illustrated and the efficiency of the numerical procedure is analyzed by means of suitable test cases involving shocks, contact and gradient discontinuities, and their interactions. Moreover, as applicative examples, the results of the simulation of complicated flow transients in convergent/divergent nozzles and in closed-end tubes are presented.