A hybrid RANS closure scheme for the near-wall turbulence

In this study, we propose a parameterization for the eddy viscosity ($\nu_t$) that can be employed in a wall-resolving standard $k$-$\epsilon$ closure model. To this end, we use the equilibrium assumption between the production rate of the turbulent kinetic energy $(P)$ and $\epsilon$ in a wall-bounded turbulent flow. Using this assumption and the linear shear stress distribution, the appropriate velocity scale is $U_S=(\epsilon/S)^{1/2}$ while the corresponding length scale is $L_S=f_\mu \kappa y (1-y/\delta)^{3/4}$, where $\kappa$ is von K\'{a}rm\'{a}n's constant, $f_\mu$ is van Driest's damping function, $y$ represents the vertical distance from the wall and $\delta$ is one half of the channel depth. Consequently, $\nu_t$ results as a product of these two characteristic scales, i.e. $\nu_t=U_SL_S$. {\it `A priori'} tests are performed to assess the validity of the proposed eddy viscosity and the corresponding characteristic scales using the direct numerical simulation (DNS) data of unstratified channel flow. Furthermore, a one-dimensional standard $k$-$\epsilon$ model was developed and `{\it a posteriori}' tests were performed. The comparison of both `{\it a priori}' and `{\it a posteriori}' tests with DNS data show excellent agreement.