Variational multiscale method based on the Crank-Nicolson extrapolation scheme for the non-stationary Navier-Stokes equations

In this report, a variational multiscale (VMS) method based on the Crank-Nicolson extrapolation scheme of time discretization for the turbulent flow is analysed. The flow is modelled by the fully evolutionary Navier-Stokes problem. This method has two differences compared to the standard VMS method: (i) For the trilinear term, we use the extrapolation skill to linearize the scheme; (ii) for the projection term, we lag it onto the previous time level to simplify the construction of the projection. These modifications make the algorithm more efficient and feasible. An unconditionally stability and an a priori error estimate are given for a case with rather general linear (cellwise constant) viscosity of the turbulent models. Moreover, numerical tests for both linear viscosity and nonlinear Smagorinsky-type viscosity are performed, they confirm the theoretical results and indicate the schemes are effective.