Surveys on Mathematics for Industry a Stabilized Finite Element Formulation for the Reynolds-averaged Navier-stokes Equations

Finite element technologies have been applied to fluid dynamrcs problems for some time now. The early work focused on Galerkin 's method and various attempts to improve its stability while retaining its basic character. In the late 1970s a new approach was introduced, namely, "stabili zed methods," such as SUPG and Galerkin/least-s quares. These methods have been applied to many areas in fluid dynamics including compressible flow, hypersonic flow, chemically reacting flow and lately to turbulent flow. Before discussing turbulence we review (in Sect. 2) the methodology of stabil ­ i zed methods. In Sect. 3 we present the Reynolds-averaged Navier -Stokes (RANS) e quations, the governing e quations for turbulent flow. These e quations are assem ­ bled in Sect. 4 where we obtain a symmetric advective -diffusive system. When the system is written in this form it gives rise to a discrete entropy production statement. In Sect. 5 we discuss the implementation of a particular model and the impact of the methodology on it. Numerical examples are given in Sect. 6 and conclusions drawn in Sect. 7.