A fully coupled Newton-Krylov solver with a one-equation turbulence model
Todd T. Chisholm · TSpace (University of Toronto) · 2007
An efficient Newton-Krylov method is presented for the computation of steady, compressible, high Reynolds number flows about single- and multi-element airfoils using structured grids. A second-order centred-difference method is used to discretize the Navier-Stokes equations, using either scalar or matrix dissipation for stability. Turbulence is modeled following Spalart and Allmaras. The equations are solved with Newton's method, using the Generalized Minimal Residual method to approximately solve the linear system. Incomplete lower-upper preconditioning is used to speed linear system convergence. Full coupling is used between the mean flow and turbulence model equations. Experiments are presented to determine the most efficient, robust approach. Extensive use of a node- and iteration-varying time step stabilizes the Newton method, and allows rapid convergence. In particular, a new time step replaces the M-matrix method of Spalart and Allmaras to ensure positive updates. Equation scaling ensures that the linear solver provides an adequate solution of the linear system. Implementation details of the reverse Cuthill-McKee reordering are examined and optimized, revealing that a downwind node should be chosen as the root node in the reordering. A potential problem with errors in the matrix-free method is addressed by optimizing the perturbation parameter. The solver has been applied to a variety of test cases, from simple inviscid single-element examples to complex, multi-element configurations at high angles-of-attack and Reynolds numbers. It performs very well in all examples, typically converging in around 1000 equivalent function evaluations when transition terms are disabled, and around 2000 when they are enabled. It is also shown to be a good choice for use with numerical airfoil optimization.