A problem with near-wall Reynolds stress models

A common mistake in modern turbulence practice is to confuse the quantity $D_{ik} = \nu \langle \partial u_i/\partial x_j~ \partial u_k/\partial x_j \rangle$ with the true dissipation tensor $\varepsilon_{ik} = 2 \nu \langle s_{ij}~ s_{kj} \rangle$, where $s_{ij}$ is the fluctuating strain-rate. The traces $D_{kk}$ and $\varepsilon_{kk}$ are in fact equal ONLY when the turbulent flow is homogeneous. Many flows at high Reynolds number are approximately locally homogeneous; but even when the traces are nearly equal, the component dissipations are not$^{1}$, and they are never the same inside $y^+=30$ of a wall-bounded flow.$^{2}$ It is argued that a major problem with RS-models in wall-bounded flows is the use of $D_{ik}$ in the modeled equations.

\noindent 1) George and Hussein 1991 Locally axisymmetric turbulence, JFM, 223, 1-23.

\noindent 2) Foucaut et al. 2018 SPIV meas. of full dissip. tensor in a turb. b.l. (submitted)