A multi-layer description of Reynolds stresses in canonical wall bounded flows

A complete description of the Reynolds stress tensor is obtained for all three canonical wall turbulence (channel, pipe and turbulent boundary layer - TBL). The result builds on a multi-layer description of length (order) functions and their ratios, including viscous sublayer, buffer layer, meso-layer for the near wall (inner) region, and bulk flow or a central core (absent in TBL) for the outer region. It is shown that the streamwise mean kinetic-energy profile is quantified with high accuracy over the entire flow domain. The model contains only three \textit{Re}-dependent parameters for Reynolds number (\textit{Re}) covering nearly three decades. Furthermore, the inner peak location is predicted to be invariant at y$^{+}=$15, while its magnitude shows notable \textit{Re} and geometry effects, predicted to be .9.2 for high \textit{Re}'s pipe flows. A mechanism is proposed for the emergence of outer peak in pipes, whose magnitude is predicted to scale as .Re$_{\tau }^{0.05}$ beyond a critical Re$_{\tau }$ about 10$^{4}$(). The recently reported logarithmic dependence in the bulk is recovered, but with an alternative explanation. The result is successfully extended to TBL flows by a fractional total stress and an absence of core. Equally accurate descriptions of vertical and spanwise kinetic-energy are also presented for the three flows. The result has been used to modify turbulent engineering models (i.e. k-$\omega $ model) with significant improvement.