Numerical Analysis of Modular Vms Methods with Nonlinear Eddy Viscosity for the Naiver-stokes Equations

This paper presents a stability, convergence and error analysis for two modular, projection-based variational multiscale (VMS) methods for the incompressible Naiver-Stokes equations. In VMS methods, the influence of the unresolved scales onto the resolved small scales is modeled by a Smagorinsky-type turbulent viscosity acting only on the marginally resolved scales. We analyze a method of inducing a VMS treatment of turbulence in an existing NSE discretization through an additional, uncoupled projection step. For two nonlinear eddy viscosity pparameterizations, we prove an error estimate for this approach. Numerical tests are given that confirm and illustrate the theoretical estimates. One method uses a fully nonlinear step inducing the VMS discretization. The second induces a nonlinear eddy viscosity model with a linear solve of much less cost.