Implementation, Realization and an Effective Solver of Two-Equation Turbulence Models

Currently, when the Reynolds-Averaged Navier-Stokes (RANS) equations are solved using turbulence modeling, most often the one-equation model of Spalart and Allmaras is used. Then, it is only necessary to solve the RANS equations in conjunction with a single transport equation for modeling turbulence. For this model, considerable assessment and analysis has been performed, allowing the possibility of a reliable solution method for an eddy viscosity required to compute the Reynolds stresses in the RANS equations. Such evaluation along with analysis has not been achieved to realize similar performance with two-equation models of the k-w type.
The primary objective of this paper is to present and discuss the components of an effective numerical algorithm for solving the RANS equations and the two transport equations of k-w type turbulence models. All the important details of the turbulence model as actually implemented are given, which is sometimes not done in various papers considering such modeling. The viability and effectiveness of this solution algorithm are demonstrated by solving both two-dimensional and three-dimensional aerodynamic flows. In all applications, a linear rate of convergence without oscillations or other evidence of unstable behavior is observed. This behavior is also particularly true when the proposed algorithm is applied to systematically renewed mesh sequences, which is generally not observed with algorithms solving more than one transport equation. Thus, numerical integration errors are systematically reduced, allowing for a significantly more reliable assessment of the effectiveness of the turbulence model. Additionally, in this paper, analysis of the solution algorithm, including linear stability, is also performed for a particular flow problem.