University of Toronto
Toronto, Ontario, Canada
During the 1990s, significant progress has been made in algorithms and turbulence models for computational aerodynamics. In this presentation, the results of three recent projects will be reviewed. The projects are:
1. Development and assessment of spatial discretization methods, including numerical dissipation schemes and higher-order methods.
2. Development of a fast Newton-Krylov algorithm for steady flows.
3. Flow computations for a three-element airfoil configuration.
Improved spatial discretization methods: Whether introduced ``artificially'' or through upwinding, numerical dissipation is a necessary evil in solving the compressible Navier-Stokes equations for high-Reynolds-number flows over aerodynamic geometries. The goal is to provide stability, convergence, and oscillation-free solutions while introducing as little error as possible. Numerical dissipation schemes which are quite successful for inviscid flows, such as the popular scalar scheme, can be excessively dissipative in boundary layers when applied to viscous flows. In [1], the convective upstream split pressure scheme with low-Mach-number preconditioning is compared with a matrix artificial dissipation scheme. The comparison will be summarized in this presentation. Consideration is given to the grid density required to achieve a given level of accuracy in computing lift and drag for turbulent flows over airfoils and to the CPU expense. Once errors from numerical dissipation are reduced, the possibility of using a higher-order spatial discretization becomes more attractive. Preliminary results will be shown which demonstrate that a higher-order method can substantially reduce the number of nodes required to achieve a given level of accuracy.
A fast Newton-Krylov algorithm: Two of the fastest approaches for obtaining steady solutions to the spatially discretized Navier-Stokes equations include multigrid and Newton- Krylov methods. The second portion of the presentation will focus on the Newton-Krylov algorithm of Pueyo and Zingg [2]. This algorithm uses the generalized minimum residual method (GMRES) to solve the large system of linear equations arising at each Newton iteration. GMRES is preconditioned using an incomplete lower-upper factorization with some fill. Within this framework, there are a number of parameters and strategies which can be optimized. Results are presented in terms of the CPU time required to achieve convergence, normalized with respect to the cost of a single evaluation of the residual vector, allowing comparison with other algorithms across computing platforms (with some caveats). For a suite of turbulent subsonic and transonic flows over airfoils, the Newton-Krylov algorithm is seen to be extremely fast, achieving a twelve-order residual reduction in the CPU time equivalent to 500 to 1000 residual function evaluations, corresponding to a convergence rate per function evaluation between 0.945 and 0.972 [3].
Flow computations far a three-element airfoil configuration: The final portion of the presentation will cover results obtained using the TORNADO flow solver developed at the University of Toronto Institute for Aerospace Studies [4] for the flow over a three-element airfoil corresponding to a take-off configuration. This test case was the basis for a recent code validation exercise documented in [5]. The results show that excellent predictions of surface pressure, lift, and drag can be obtained over a wide range of angles of attack using either the Menter-SST turbulence model or the Spalart-Allmaras model. Detailed examination of wake and boundary-layer velocity profiles shows that the two models produce comparable accuracy, with the Menter-SST model giving a better prediction of the slat wake at low angles of attack.
References:
[1] Nemec, M., and Zingg, D.W., ``Aerodynamic Computations Using the Convective Up- stream Split Pressure Scheme with Local Preconditioning,'' \forcenl AIAA Paper 98-2444, June 1998.
[2] Pueyo, A., and Zingg, D.W., ``An Efficient Newton-GMRES Solver for Aerodynamic Computations,'' \ AIAA Paper 97-1955, June 1997.
[3] Pueyo, A., and Zingg, D.W., ``Improvements to a Newton-Krylov Solver for Aerodynamic Flows,'' \forcenl AIAA Paper 98-0619, Jan. 1998.
[4] Nelson, T.E., Godin, P., De Rango, S., and Zingg, D.W., ``Flow Computations for a Three-Element Airfoil System,'' submitted to the {\it CASI J.}, March 1998.
[5] Fejtek, I., ``Summary of Code Validation Results for a Multiple Element Airfoil Test Case'', AIAA Paper 97-1932, June 1997.