Analytical Eddy Viscosity Model for Velocity Profiles in the Outer Part of Closed- and Open-Channel Flows

The main equations in analytical modeling of turbulence, used in open-channel flows, are the parabolic profile of the eddy viscosity and the exponentially decreasing turbulent kinetic energy function. However, when using the definition of the eddy viscosity as a product between velocity and length scales and by taking the velocity scale as the root-square of turbulent kinetic energy, we show that the parabolic eddy viscosity profile is incompatible with the turbulent kinetic energy function. Taking into account this shortcoming, we consider an eddy viscosity formulation which is in agreement with the turbulent kinetic energy profile in the equilibrium region. This eddy viscosity is written in a form that allows the calibration of the two $${\text{R}}{{{\text{e}}}_{{{\tau }}}}$$ -dependent parameters which have a linear behavior (where $${\text{R}}{{{\text{e}}}_{{{\tau }}}}$$ is the friction Reynolds number). All results were validated by both direct numerical simulation and experimental data in the same range of friction Reynolds numbers, respectively $$300 < {\text{R}}{{{\text{e}}}_{{{\tau }}}} < 5200$$ for closed-channel flows and $$923 < {\text{R}}{{{\text{e}}}_{{{\tau }}}} < 6139$$ for open-channel flows. Comparisons with the direct numerical simulation data of the eddy viscosity in closed-channel flows for eight different flow conditions show good agreement. Mean streamwise velocities are obtained from solving of the momentum equation. For closed-channel flows, mean velocity profiles show very good agreement. For open-channel flows, results confirm that the use of the parabolic eddy viscosity is unable to improve the velocities while the proposed method shows good agreement. These results show the ability of this analytical eddy viscosity model to predict accurately the velocities in the outer region for both closed- and open-channel flows, without any ad hoc function or parameter.