Calculation of shear stresses in the Fourier–Galerkin formulation of turbulent channel flows: projection, the Dirichlet filter and conservation

We consider turbulent channel flows computed with DNS and LES employing a Fourier–Galerkin method. We show that the balance of total mean shear stress, a measure of conservation of the numerical method, is satisfied only weakly (i.e., in an integral sense), rather than strongly (i.e., pointwise). A study of filters induced by projectors reveals that a certain filter, the Dirichlet filter, provides a tool for extracting shear stresses that are strongly conservative. Numerical results support the theory and demonstrate that spurious oscillations, present in the unfiltered stresses, are also suppressed.