Improved engineering models for turbulent wall flows

We propose a new approach, called \textit{structural ensemble dynamics} (SED), involving new concepts to describe the mean quantities in wall-bounded flows, and its application to improving the existing engineering turbulence models, as well as its physical interpretation. First, a revised $k-\omega $ model for pipe flows is obtained, which accurately predicts, for the first time, both mean velocity and (streamwise) kinetic energy $\left\langle {u'u'} \right\rangle $ for a wide range of the Reynolds number (\textit{Re}), validated by Princeton experimental data. In particular, a multiplicative factor is introduced in the dissipation term to model an anomaly in the energy cascade in a meso-layer, predicting the outer peak of $\left\langle {u'u'} \right\rangle $ agreeing with data. Secondly, a new one-equation model is obtained for compressible turbulent boundary layers (CTBL), building on a multi-layer formula of the stress length function and a generalized temperature-velocity relation. The former refines the multi-layer description - viscous sublayer, buffer layer, logarithmic layer and a newly defined bulk zone - while the latter characterizes a parabolic relation between the mean velocity and temperature. DNS data show our predictions to have a 99{\%} accuracy for several Mach numbers \textit{Ma}$=$2.25, 4.5, improving, up to 10{\%}, a previous similar one-equation model (Baldwin {\&} Lomax, 1978). Our results promise notable improvements in engineering models.