SUPG-YZβ formulation for solving convection-dominated steady linear reaction-convection-diffusion equations

In this computational study, stabilized finite element solutions of convection-dominated steady linear reaction-convection-diffusion equations are examined. Although the standard Galerkin finite element method (GFEM) is one of the most robust, efficient, and reliable methods for many engineering simulations, it suffers from instability issues in solving convection-dominated problems. To this end, this work deals with a stabilized version of the standard GFEM, called the streamline-upwind/Petrov-Galerkin (SUPG) formulation, to overcome the instability issues in solving such problems. The stabilized formulation is further supplemented with YZβ shock-capturing to provide additional stability around sharp gradients. A comprehensive set of test computations is provided to compare the results obtained by using the GFEM, SUPG, and SUPG-YZβ formulations. It is observed that the GFEM solutions involve spurious oscillations for smaller values of the diffusion parameter, as expected. These oscillations are significantly eliminated when the SUPG formulation is employed. It is also seen that the SUPG-YZβ formulation provides better solution profiles near steep gradients, in general.

Error Estimates for the Smagorinsky Turbulence Model: Enhanced Stability Through Scale Separation and Numerical Stabilization

In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $$L^2$$ L 2 -norm for smooth flows in the pre-asymptotic high Reynolds number regime.