Reinvestigating the Parabolic-Shaped Eddy Viscosity Profile for Free Surface Flows

Rafik Absi

doi:10.3390/hydrology8030126

Abstract

The flow in rivers is turbulent. The main parameter related to turbulence in rivers is the eddy viscosity, which is used to model a turbulent flow and is involved in the determination of both velocities and sediment concentrations. A well-known and largely used vertical distribution of eddy viscosity in free surface flows (open channels and rivers) is given by the parabolic profile that is based on the logarithmic velocity profile assumption and is valid therefore only in the log-law layer. It was improved thanks to the log-wake law velocity profile. These two eddy viscosities are obtained from velocity profiles, and the main shortcoming of the log-wake profile is the empirical Coles’ parameter. A more rigorous and reliable analytical eddy viscosity model is needed. In this study, we present two analytical eddy viscosity models based on the concepts of velocity and length scales, which are related to the exponentially decreasing turbulent kinetic energy (TKE) function and mixing length, namely, (1) the exponential-type profile of eddy viscosity and (2) an eddy viscosity based on an extension of von Karman’s similarity hypothesis. The eddy viscosity from the second model is Re*-independent, while the eddy viscosity from the first model is Re*-dependent (where Re* is the friction Reynolds number). The proposed analytical models were validated through computation of velocity profiles, obtained from the resolution of the momentum equation and comparisons to experimental data. With an additional correction function related to the damping effect of turbulence near the free surface, both models are similar to the log-wake-modified eddy viscosity profile but with different values of the Coles’ parameter, i.e., Π=0.2 for the first model and Π=0.15 for the second model. These values are similar to those found in open-channel flow experiments. This provides an explanation about the accuracy of these two analytical models in the outer part of free surface flows. For large values of Re* (Re* > 2000), the first model becomes Re* independent, and the two coefficients reach asymptotic values. Finally, the two proposed eddy viscosity models are validated by experimental data of eddy viscosity.

Highlights

Determination of velocity distribution in open-channel flows and rivers is a topic of high interest and is involved in different practical applications [1,2,3,4,5,6,7]
The main interest in laboratory investigations of free surface turbulent flows is related to the experimental conditions that are chosen to be in agreement with the assumptions related to the models
We present two eddy viscosity models based on the concepts of velocity and length scales, which are related, respectively, to the exponentially decreasing turbulent kinetic energy (TKE) function [11] and mixing length, namely, (1) the exponential-type profile of eddy viscosity [35,36] and (2) an eddy viscosity based on an extension of von Karman’s similarity hypothesis [37,38,39]

Summary

Introduction

Determination of velocity distribution in open-channel flows and rivers is a topic of high interest and is involved in different practical applications [1,2,3,4,5,6,7]. The main interest in laboratory investigations of free surface turbulent flows is related to the experimental conditions that are chosen to be in agreement with the assumptions related to the models. Experiments in laboratory flumes allowed analytical models of turbulence to be developed that are in the form of analytical solutions, semi-theoretical or empirical relationships [17,18,19,20]. These models were proposed for mean velocities, turbulent kinetic energy (TKE), mixing length, eddy viscosity with their link to flow parameters (flow depth, friction velocity) [11,18]. Measured data in the central part of rivers are well described by the analytical expressions obtained from laboratory open-channel flows [21]

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