Performances of upwind methods in predicting shear-like flows

Upwind methods are considered as the most appropriate numerical tools for predicting high speed flows. However, disturbing problems may arise in dealing with shear-like flows such as boundary layers. Here, the fluid dynamics is dominated by the diffusion processes and the convective terms, which are estimated through an upwind method, have to play a secondary and almost negligible role. Unfortunately, the presence of density and velocity gradients inside boundary layers introduces, in some upwind method, purely numerical effects. In some methods, the boundary layer region may grow, unphysically, beyond the correct thickness because of a spurious dissipation generated by an incorrect evaluation of the convective terms. In other methods, pressure oscillations may appear inside the boundary layer. Such situations, well known in the scientific community, can be properly analyzed by developing a linearized form of the first order scheme for the Euler conservation laws in a particular problem, with the evaluation of the fluxes made as dictated by the formulation of each upwind method. This analysis provides suggestions and hints useful to understand and predict the behavior of an upwind method in simulating shear-like flows.