Stabilized finite element methods in the inertial range: Monitoring artificial dissipation

In this note we will first discuss finite element discretizations of the incompressible Navier-Stokes equations with some artificial dissipation added in the form of either a turbulence model or some numerical stabilization term. We will then review some results on interior penalty stabilized finite element methods for incompressible flow. These results depend on the regularity of the solution and are typically relevant only when the mesh parameter h is sufficiently small. In the case of high Reynolds number flow in three space dimensions it is unrealistic in most cases of industrial importance to resolve all scales and hence the computation must aim at the resolution only of the large eddies. In this case we propose to monitor the relative artificial dissipation and relate it to the power spectrum in the inertial range as a means of validation of large eddy simulations. Finally a numerical example is given on a two dimensional test case as an illustration to the discussion