Analytical Solutions for the Modeled $k$ Equation

The semitheoretical function of Nezu and Nakagawa (1993, Turbulence in Open-Channel Flows, A. A. Balkema, ed., Rotterdam, The Netherlands) for the turbulent kinetic energy $k$ is valid only where local equilibrium is a good approximation. From an estimation of the difference between the energy production and its dissipation $G−ϵ$⁠, we present in this study an analytical solution for the modeled $k$ equation. Comparisons with direct numerical simulation data of turbulent channel flows show good agreement. A universal function for $k+$ is deducted for $y+<20$⁠.