Numerical Aspects Of k−epsilon Turbulence Modeling Using a Finite Element Incompressible Navier-Stokes Formulation

Abstract

In this work we present a stabilized equal order finite element formulation of
incompressible Navier-Stokes equations augmented by a κ−_ turbulence model. The aim
of this paper is to evaluate the main numerical difficulties associated with the solution of
this kind of problems, mainly the possitiveness of the mathematical operators involved and
the rate of convergence of the whole system. We propose a particular way to circumvent
these drawbacks using
• .- an extra stabilization term in the transport equations of the turbulence quantities
to avoid undershoots in the κ − _ fields,
• .- a smooth enough cutoff function to avoid negative values of κ − _ fields, and
• .- an almost fully implicit monolithic solution strategy in order to reach good convergence
rates of the whole system.
The incompressible Navier-Stokes equations are spatially discretized by a SUPG-PSPG
technique and temporally solved by a backward Euler scheme.
This work was done as part of our project regarding to the implementation of PETSc-
FEM code (http://minerva.arcride.edu.ar/petscfem/petscfem), a general purpose, multiphysics
library running on Beowulf (Intel processors+Unix/Linux OS) cluster 2 and based
on the MPI message passing library1 and the Parallel Extensible Toolkit for Scientific
Computations (PETSc),3 written in object oriented programming using C++.