REYNOLDS-AVERAGED NAVIER-STOKES EQUATIONS

SUMMARY An agglomeration multigrid strategy is developed and implemented for the solution of three-dimensional steady viscous flows. The method enables convergence acceleration with minimal additional memory overhead and is completely automated in that it can deal with grids of arbitrary construction. The multigrid technique is validated by comparing the delivered convergence rates with those obtained by a previously developed overset-mesh multigrid approach and by demonstrating grid-independent convergence rates for aerodynamic problems on very large grids. Prospects for further increases in multigrid efficiency for high-Reynolds-number viscous flows on highly stretched meshes are discussed. With unstructured mesh techniques having proved their usefulness for two- and three-dimensional steady inviscid flow solutions, the push is now on for realistic and practical three-dimensional unstructured mesh viscous flow solvers. While useful inviscid calculations can be performed in threedimensions using several hundred thousand grid points, the situation is quite different for viscous flow cases. Judging from current viscous block-structured and overset-structured grid calculations, the accurate aerodynamic simulation of isolated aircraft components can be expected to require the use of several million grid points,' whereas accurate computations for entire vehicles can be expected to require tens of millions of grid points.' Thus explicit time stepping is clearly not feasible and the development and implementation of efficient, robust and automated algorithms are essential if unstructured mesh techniques are to be employed for such cases. The task of developing an efficient solver is further complicated by the stiffness associated with the extreme grid stretching which is generally required for resolving high-Reynolds-number flows. Implicit solution techniques have been demonstrated for accelerating convergence to steady state of unstructured grid solvers for both two- and three-dimensional problem^.^" While these methods result in substantial reductions in CPU time for a given solution, they most often greatly increase the amount of required memory. In fact, simply storing the Jacobian of the discrete equations requires two to three times more memory than an explicit scheme. Matrix-free implicit methods provide a low-storage alternative: but the effectiveness of such schemes is limited by the lack of good matrix-free preconditioners.