Modified law of the wall leading to turbulent channel flow universal velocity profiles valid down to $Re_{\tau}=395$

Velocity profile modeling is revisited using the results from databases of turbulent channel flow DNS at $Re_{\tau}=u_{\tau}\,h/ \nu= 2000$, $950$, $550$, and $395$. We consider the turbulent region: $y^+ = Re_{\tau}\,\eta$ (with $\eta=y/h$) larger than $70$). A new model for the effective turbulent viscosity, $\nu_t=-\overline{u'v'}/\frac{d\overline{u}}{dy}$, is proposed, that fits well the DNS results all the way to the channel center. The velocity profile is then obtained by integration: it corresponds to a ``modified law of the wall,'' $\frac{1} {\kappa}\left(\log(y^+ + y_0^+) -\eta\right) + C$, with the added classical ``law of the wake,'' $D\,g(\eta)$. The new $- \eta$ term in the modified law of the wall is really required in such still limited Reynolds number channel flows, as an important correction to the usual log term: both terms ``work together,'' as both are multiplied by the same $\frac{1}{\kappa} $ value (recall that $D$ is not related to $\kappa$). Only at the highest Reynolds numbers does this correction become negligible. As to the $y_0^+$ shift in the log term itself (value around 6), something also recently proposed by Spalart et al (Phys. Fluids in press), it too is required as a consequence of the $\nu_t$ near wall behavior. The present velocity profile is quite universal: it fits very well, with the same value of all constants, all $Re_{\tau}$ cases. In particular, the von K\`arm\`an constant is obtained as $\kappa=0.37$: same as Zanoun et al (Phys. Fluids 15 (10):3079, 2003), and close to $0.38$ as Spalart et al.