Multigrid Acceleration for Transonic Aerodynamic Flow Simulations Based on a Hierarchical Formulation

Summary

In [1], the authors used a hierarchical formulation based on a Helmholtz velocity decomposition to simulate transonic flows over airfoils. The potential flow formulation is augmented with entropy and vorticity corrections and the numerical results are compared to standard Euler and Navier-Stokes calculations. For many aerodynamic applications, the corrections are limited to relatively small regions; the flow in the near and far fields is irrotational and isentropic. The entropy and vorticity corrections are governed by convection/diffusion equations while the nonhomogeneous potential equation is of mixed type; elliptic in the subsonic domain and hyperbolic in the supersonic one. The forcing function represents the necessary correction for mass conservation. Upwind schemes are used for the convection terms of the scalar equations of the corrections and for the potential equation in the supersonic region. The formulation can be viewed as an implementation of a viscous/ inviscid interaction procedure which is equivalent to Navier-Stokes equations in the inner field.

In this paper, convergence acceleration techniques are applied for such a formulation using single and multiple grids. For a single grid, optimal relaxation parameters for subsonic potential flow regions and local artificial time steps for the scalar correction equations have been used. A full multigrid technique is implemented to the augmented potential equation. Only three grids: coarse, intermediate and fine meshes are used. Results for both inviscid and viscous transonic flows are presented. It is noticed that on a single grid, the potential and the viscous flow calculations, based on the present formulation, have comparable convergence histories. The limited applications of multigrid result in an order of magnitude saving of the work units for both calculations. More savings should be achievable with more sophisticated multigrid procedures.