Iterative space-marching method for compressible sub-, trans-, and supersonic flows

A new iterative fully coupled implicit space-marching method is proposed for solving the two-dimensional steady Euler equations for compressible flows at Mach numbers ranging from subsonic to supersonic. A special treatment of the streamwise pressure gradient component permits us to calculate both supersonic flow regions, where the Euler equations are hyperbolic, and subsonic regions, where the equations reveal elliptic properties, To take into account the elliptic effects of subsonic and transonic flows, space-marching sweeps are carried out iteratively, A new parabolic pressure correction procedure is developed to accelerate the convergence rate. This procedure con be applied for subsonic and transonic regimes and is consistent with the characteristic analysis of the Euler equations. At each marching station, a Newton iterative technique is used to solve the nonlinear system of equations in a fully coupled manner. To resolve strong shocks and contact discontinuities as well as smooth flowfields with high accuracy, implicit symmetric second-order total variation diminishing and upwind second-order Richardson schemes are employed to approximate the transverse and streamwise derivatives, respectively. The method is tested on the problem of the flow in a duct with a circular are bump for different Mach number regimes. Numerical calculations show that the method is accurate, is robust, and can efficiently be applied for calculating subsonic, transonic, and supersonic hows without streamwise separation.