On the implementation of the κ-ε turbulence model in incompressible flow solvers based on a finite element discretisation

Turbulence plays an important role in many chemical engineering processes (fluid flow, mass and heat transfer, chemical reactions) which are dominated by convective transport. Since the direct numerical simulation (DNS) of turbulent flows is still prohibitively expensive, eddy viscosity models based on the Reynolds Averaged Navier-Stokes (RANS) equations are commonly employed in CFD codes. One of the most popular ones is the standard k − ε model which has been in use since the 1970s. However, its practical implementation and, especially, the near-wall treatment has always been some somewhat of a mystery. Algorithmic details and employed ‘tricks’ are rarely reported in the literature, so that a novice to this area of CFD research often needs to reinvent the wheel. The numerical implementation of turbulence models involves many algorithmic components all of which may have a decisive influence on the quality of simulation results. In particular, a positivity-preserving discretization of the troublesome convective terms is an important prerequisite for the robustness of the numerical algorithm. This paper presents a detailed numerical study of the k − ε model implemented in the open-source software package FeatFlow (http://www.featflow.de) using algebraic flux correction to enforce the positivity constraint [1, 2]. Emphasis is laid on a new implementation of wall functions, whereby the boundary conditions for k and ε are prescribed in a weak sense. Furthermore, the advantages of Chien’s low-Reynolds number modification are explored. Two representative benchmark problems are used to evaluate the performance of the proposed algorithms in 3D.