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The paper presents a finite element error analysis for a projection-based variational multiscale (VMS) method for the incompressible Navier-Stokes equations. In the VMS method, the influence of the unresolved scales onto the resolved small scales is modeled by a Smagorinsky-type turbulent viscosity.

SUMMARYSome variants of a three-scale projection-based finite element variational multiscale (VMS) method are studied for turbulent channel flow computations at Re = 180. Different spaces for the large scales, two eddy viscosity models and two ways of discretizing the projection terms in time are explored. The results obtained with the resolved small scales in the definition of the eddy viscosity are very sensitive to the temporal discretization of the projection terms. The computations were performed on three grids commonly used in turbulent channel flow simulations.

In turbulent flows it is only feasible to simulate large flow structures. Variational multiscale (VMS) methods define these flow structures by projections into appropriate function spaces. This paper presents a finite element VMS method which defines the large scale projection space adaptively. The adaption controls the influence of an eddy viscosity model and it is based on the size of the so-called resolved small scales. The adaptive procedure is described in detail. Numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented. It is shown that the method selects the large scale space in a correct way and that appropriately chosen parameters improve the results compared to the basic method, which uses the same local large scale space in the whole domain and for all times.

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