## A finite element variational multiscale method for the Navier–Stokes equations.(English)Zbl 1073.76054

Summary: This paper presents a variational multiscale method (VMS) for the incompressible Navier–Stokes equations which is defined by a large scale space $$L^{H}$$ for the velocity deformation tensor and a turbulent viscosity $$\nu_T$$. The connection of this method to the standard formulation of a VMS is explained. The conditions on $$L^{H}$$ under which the VMS can be implemented easily and efficiently into an existing finite element code for solving the Navier–Stokes equations are studied. Numerical tests with the Smagorinsky large eddy simulation model for $$\nu_T$$ are presented.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76M30 Variational methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76F65 Direct numerical and large eddy simulation of turbulence 76D05 Navier-Stokes equations for incompressible viscous fluids

### Keywords:

Smagorinsky large eddy simulation model

FEATFLOW; MooNMD
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