Spectral analysis of the dissipation of the residual-based variational multiscale method

We analyze the application of the residual-based variational multiscale (VMS) method to the solution of a generic partial differential equation. We prove that the model terms introduced by this method are dissipative under certain restrictions that are likely to be satisfied by spectral approximations of conservation laws. In particular, we demonstrate that the Fourier-spectral approximations of Burgers equation and the incompressible Navier–Stokes equations satisfy these conditions. Hence we show that the residual-based VMS model for these equations is dissipative, even though the form of the model term does not resemble an eddy viscosity. We analyze the scale dependence of this dissipation using fully resolved numerical solutions and conclude that the VMS model accurately captures the scale dependence of the cross-stress terms (u¯u′) and under-predicts the contribution from the Reynolds-stress terms (u′u′)(u′u′). These observations have implications on the use of the residual-based VMS method in spectral approximations of nonlinear conservation laws.