Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows

Several new implicit schemes for the solution of the compressible Navier?Stokes equations are presented. These methods are derived from a hierarchy of average-state approximate solutions to the Riemann problem, ranging from the Lax?Friedrichs flux to the exact Riemann-solver flux. In contrast to linearised approximations, these methods will (with certain provisos on the signal velocities) enforce the entropy condition and preserve positivity without the need for additional corrections. The hierarchy also encompasses and explains the origin of many other upwind and centred methods, including the space-time scheme (due to Chang) and the more recent FORCE scheme (due to Toro). Based on an analysis of the above hierarchy, attention is focussed on the development of a new implicit scheme using a positivity-preserving version of Toroet al.'s HLLC scheme, which is the simplest average-state solver capable of exactly preserving isolated shock, contact, and shear waves. Solutions obtained with this method are essentially indistinguishable from those produced with an exact Riemann solver, whilst convergence to the steady state is the most rapid of all the implicit average-stage schemes considered and directly comparable to that of the unmodified Roe scheme. A new two-step implicit method is applied to various test cases, including turbulent flow with shock/boundary-layer interaction. The new time-stepping scheme is composed of two backward Euler steps, but has twice the convergence rate of the backward Euler scheme and alleviates the convergence problems that are often experienced when employing compressive limiter functions.