Aerodynamic flows around thin airfoils and membranes are very challenging to simulate accurately because of complex flow structures, including geometry-induced separation of the boundary layer, shear layer transition to turbulent behavior, reattachment, relaminarization of the boundary layer, and formation of primary and secondary recirculation bubbles. A physical insight on these structures can be obtained through the numerical Large Eddy Simulation (LES) of the flow around a simpler geometry, the thin flat plate, at shallow incidences of one and three degrees and Reynolds number above 105, which is the focus of this investigation. In order to avoid the numerical instabilities associated with the mesh spreading generally required by such external flow, two solutions have been developed and tested. The first one consists of the traditional sub-grid model used along with a mixed numerical scheme, in which a stable but dissipative part is active only in turbulence-free zones where mesh is highly non-regular, while an unstable but non-dissipative scheme is employed in turbulence- crytical zones, where the mesh is as regular as possible. The second solution, developed and validated in the current investigation, is based on a damping force, aimed to eliminate the smaller scales while preserving as much as possible all other structures. Compared to previous investigations using Reynolds Average (RANS) equations, both solutions provided more accurate and detailed information about the flow, including velocity, pressure and turbulent fluctuations mean profiles, allowing a deeper physical understanding. In particular, the main bubble reattachement lenght was predicted within 5% of the experimental data, while K - W RANS results were found to disagree in more than 20%.