A simple eddy viscosity formulation for turbulent boundary layers near smooth walls

The aim of this study is to improve the prediction of near-wall mean streamwise velocity profile $U^+$ by using a simple method. The $U^+$ profile is obtained by solving the momentum equation which is written as an ordinary differential equation. An eddy viscosity formulation based on a near-wall turbulent kinetic energy $k^+$ function (R. Absi, Analytical solutions for the modeled $k$-equation, ASME J. Appl. Mech. \textbf{75}, 044501, 2008) and the van Driest mixing length equation (E.R. van Driest, On turbulent flow near a wall, J. Aero. Sci. \textbf{23}, 1007, 1956) is used. The parameters obtained from the $k^+$ profiles are used for the computation of $U^+$ (variables with the superscript of + are those nondimensionalized by the wall friction velocity $u_\tau$ and the kinematic viscosity $\nu$). Comparisons with DNS data of fully-developed turbulent channel flows for $109 < Re_{\tau} < 2003$ show good agreement (where $Re_{\tau}$ denotes the friction Reynolds number defined by $u_\tau$, $\nu$ and the channel half-width $\delta$).