Existence of weak solutions for a scale similarity model of the motion of large eddies in turbulent flow. (English) Zbl 1084.35059

Summary: In turbulent flow, the normal procedure has been seeking means \(\overline{u}\) of the fluid velocity \(u\) rather than the velocity itself. In large eddy simulation, we use an averaging operator which allows for the separation of large- and small-length scales in the flow field. The filtered field \(\overline{u}\) denotes the eddies of size \(O(\delta)\) and larger. Applying local spatial averaging operator with averaging radius \(\delta\) to the Navier-Stokes equations gives a new system of equations governing the large scales. However, it has the well-known problem of closure. One approach to the closure problem which arises from averaging the nonlinear term is the use of a scale similarity hypothesis. We consider one such scale similarity model. We prove the existence of weak solutions for the resulting system.


35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76F65 Direct numerical and large eddy simulation of turbulence
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