Near-asymptotics overlap solutions for transport moments in turbulent channesl and boundary layers

The log profile overlap solutions for turbulent channel have been complemented recently by solutions for the dissipation$^{1}$, $\varepsilon$ and the kinetic energy$^2$, $\langle q^2 \rangle/2$. The dissipation varies as $1/y^+$, while the turbulence kinetic energy varies logarithmically. The Reynolds shear stress is nearly constant. We show from similar arguments that the transport moments, $T =-\langle p v \rangle/\rho - \langle q^2 v \rangle/2 +2 \nu \langle u_i s_{ij} \rangle$, also vary logarithmically. So all the terms in the kinetic energy balance in overlap region, $[\partial T/\partial y - \langle uv \rangle dU/dy - \varepsilon]=0$, vary inversely with $y^+$. Boundary layer results are the power-law equivalents, but indistinguishable. Both are shown to be consistent with recent experimental and DNS data. This presents a problem for the usual eddy viscocity models for this region, $\nu_t \propto \langle q^2 \rangle^2 / \varepsilon$, since both cannot be true. References: 1) Wosnik, M. {\it etal} (2000) JFM 421, 115; 2) Hultmark, M. (2012) JFM 707,575