An implicit fluctuation splitting scheme for compressible flows

The authors are currently involved in a national research project on turbomachinery CFD. One major goal of this project is to develop an accurate, robust and efficient methodology for the three-dimensional compressible Navier-Stokes equations, based on a genuinely multidimensional upwind discretization of the Euler fluxes. In the last years, the first author has concentrated his efforts towards optimizing the efficiency of the solution procedure, using linear Fluctuation Splitting (FS) spacial discretizations of the Euler equations [1]. The method employs an implicit Euler time discretization and evaluates the Jacobian using either approximate analytical formulas (Picard linearisation) or two-point first- order-accurate differences. The resulting large sparse linear system is solved using a preconditioned GMRES strategy. The last three authors have developed and optimized a nonlinear hybrid formulation which, according to the local flow conditions, uses the most suitable FS spacial discretization among the matrix LDA and PSI schemes, which are optimal for subsonic and supersonic flow conditions, respectively, and the matrix N scheme, which is only first-order-accurate but can capture shocks monotonically [2]. Such a hybrid formulation employs a Runge- Kutta time integration procedure with multigrid and has been lately extended to the Reynolds averaged Navier-Stokes equations, using both an algebraic and a two-equation (k —w) turbulence model [3]