Multi-layer analytic solution for k-{\omega} model equations via a symmetry approach

Despite being one of the oldest and most widely-used turbulence models in engineering CFD, the k-{\omega} model has not been fully understood theoretically because of its high non-linearity 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-{\omega} model equations in pipe flow. Approximate local balance equations are analyzed to determine key parameters in the solution, which are shown to be rather close to the empirically-measured values from numerical solution of the Wilcox k-{\omega} model, hence the analytic construction is fully validated. Furthermore, the predictions of three critical locations in the model's three transition functions are validated, which enables an in-depth understanding of the parameter setting of the model. These results provide clear evidence that the k-{\omega} model sets in it a multi-layer structure, which is similar to but different, in some insignificant details, from the Navier-Stokes turbulence. This finding explains why the k-{\omega} model is so popular, especially in computing the near-wall flow. Finally, the analysis is extended to a newly-refined k-{\omega} model called SED k-{\omega}, showing that the SED k-{\omega} model has improved the multi-layer structure in the outer flow but preserved the setting of the k-{\omega} model in the inner region.