  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.
