Issues for a mathematical definition of LES∗

Since the early work of [7], Large Eddy Simulation (LES) has become over the years an increasingly popular method, as evidenced by the vast amount of publications on the subject in the literature, and is now considered a tool of choice for simulating three-dimensional incompressible viscous flows at large Reynolds numbers. Heuristically speaking, Large Eddy models are obtained by applying a low-pass filter to the Navier–Stokes equations. The filtered equations are then similar to the original equations but for the presence of the so-called subgrid scale stresses accounting for the influence of the small scales onto the large ones. Assuming that the behavior of the small scales is almost universal, the objective of LES is to model the subgrid scale stresses (the so-called closure problem) and to compute the dynamics of the large scales by using the filtered equations. Although this description of LES is widely accepted, it nevertheless falls short of an unambiguous mathematical theory. Our impression is that LES is at the present time a fuzzy concept. Some authors think of LES as the solution to the filtered equations whereas others think of it as finite-dimensional approximations thereof. Others expect LES to reproduce the statistics of the large scales instead of approximating