THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH EXTREMELY FLAT FINITE ELEMENTS IN THE BOUNDARY LAYERS

Viscous and laminar flow simulation are a source of problems at very high Reynolds number (Re) because of boundary layers. Discretization grids extremely thin are needed in perpendicular direction to the boundary layer and that produces a bad, or at least mediocre, conditioning of the linear systems. We propose here, not a new approximation in order to solve the problem, but a new way of resolving the linear system that arises from a finite element discretization of the incompressible and laminar Navier-Stokes equations. It is a simple idea that consists in changing the method of resolution of the linear systems in the boundary layer into a block relaxation method adapted to physics, that imitates the method of resolution of the parabolics Prandtl boundary layer equations, while in the remaining domain a Cholesky or conjugate gradient method is employed. The global linear system is solved by blocks. The first block corresponds to the boundary layer and wake; the second corresponds to the remaining domain. We have obtained better results on a wing profile at Reynolds number 100,000 because this method seems not to propagate the round off errors due to the bad conditioning of the global linear system.