Numerical flux functions for Reynolds‐averaged Navier–Stokes and kω turbulence model computations with a line‐preconditioned p‐multigrid discontinuous Galerkin solver

We present an eigen-decomposition of the quasi-linear convective flux formulation of the completely coupled Reynolds-averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p-multigrid algorithm. To this end, we formulate a framework with a backward-Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix-free implementations and memory-lean line-Jacobi preconditioners and compare the effects of some parameter choices. In particular, p-multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.

[1]  D. Wilcox.  Reassessment of the scale-determining equation for advanced turbulence models , 1988 .

[2]  Dimitri J. Mavriplis,et al.  Unstructured-Mesh Discretizations and Solvers for Computational Aerodynamics , 2007 .

[3]  B. V. Leer,et al.  Experiments with implicit upwind methods for the Euler equations , 1985 .

[4]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[5]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[6]  Alessandro Colombo,et al.  Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations , 2012 .

[7]  Ewald Krämer,et al.  A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows , 2008 .

[8]  G Vijayasundaram,et al.  Transonic flow simulations using an upstream centered scheme of Godunov in finite elements , 1986 .

[9]  Stefan Langer,et al.  Investigation and application of point implicit Runge–Kutta methods to inviscid flow problems , 2012 .

[10]  Dimitri J. Mavriplis,et al.  Efficient Solution Techniques for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations on Hybrid Anisotropic Meshes , 2010 .

[11]  P. Tesini,et al.  High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations , 2009 .

[12]  David L. Darmofal,et al.  p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2005 .

[13]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[14]  Ralf Hartmann,et al.  Adjoint-based error estimation and adaptive mesh refinement for the RANS and k-ω turbulence model equations , 2011, J. Comput. Phys..

[15]  Rainald Löhner,et al.  A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids , 2006 .

[16]  Marco Luciano Savini,et al.  Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k–ω turbulence model equations , 2005 .

[17]  Ralf Hartmann,et al.  Discontinuous Galerkin methods for computational aerodynamics — 3D adaptive flow simulation with the DLR PADGE code , 2010 .

[18]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .