Multi-layer analytic solution for k-ω model equations via a symmetry approach

Abstract

Despite being one of the oldest and most widely-used turbulence models in engineering computational fluid dynamics (CFD), the k-ω model has not been fully understood theoretically because of its high nonlinearity and complex model parameter setting. Here, a multi-layer analytic expression is postulated for two lengths (stress and kinetic energy lengths), yielding an analytic solution for the k-ω model equations in pipe flow. Approximate local balance equations are analyzed to determine the key parameters in the solution, which are shown to be rather close to the empirically-measured values from the numerical solution of the Wilcox k-ω model, and hence the analytic construction is fully validated. The results provide clear evidence that the k-ω model sets in it a multilayer structure, which is similar to but different, in some insignificant details, from the Navier-Stokes (N-S) turbulence. This finding explains why the k-ω model is so popular, especially in computing the near-wall flow. Finally, the analysis is extended to a newly-refined k-ω model called the structural ensemble dynamics (SED) k-ω model, showing that the SED k-ω model has improved the multi-layer structure in the outer flow but preserved the setting of the k-ω model in the inner region.