On performance of a Pressure-Projection algorithm for incompressible Navier Stokes

A time accurate implicit Galerkin finite element algorithm for the incompressible Navier Stokes equations is assessed for convergence and conservation properties. The algorithm inherits it's philosophy from the group of algorithms known as Pressure Projection methods. The implementation of such a method typically employs numerical linear algebra procedures that are non-Newton, which strongly affects algorithm convergence, hence conservation properties Using sparse matrix methods, a full Newton process is achieved for solving the two-dimensional thermal cavity benchmark problem at large Rayleigh number.

The Newton algorithm performance, fully coupling the velocity, temperature and continuity constraint function degrees of freedom, is clearly superior. Sobolev norms of these variables are used as primary indicators of solution convergence. Time evolution of these norms, along with the norm of the physically motivated pressure. Poission equation solution, provides ample insight into the superior performance and conservation properties compared to the common segregated quasi-Newton procedure. The results of this study generate benchmark quality solutions for Rayleigh numbers ranging from 10^6 to 10^8.