Analytical investigation of the vertical structure of periodic flow

Abstract

Analytical solution to the vertical two-dimensional periodic flow in a rectangular basin is presented in this paper. This solution takes into account the effects of the bottom friction through a slip condition, the horizontal turbulent viscosity, and the vertical turbulent viscosity. These induce a better evaluation of the free surface elevation and velocity.The introduction of a slip condition at the bottom considerably modifies the phase for the free surface elevation and especially the amplitude for the velocity at the bottom layer whereas the introduction of the horizontal turbulent viscosity considerably modifies the phase for the free surface elevation.The model proposed here is more relevant for value on the bottom friction coefficient ranging from 10−4 ms−1 to 10−1 ms−1. Within this range, the slip condition at the bottom is really effective and an ideal value on the bottom friction coefficient for the slip condition is as much as 10−2 ms−1.The analytical solution confirms the existence of a large difference in phase between the bottom stress and the depth-averaged velocity and that this difference increases with the bottom friction. This leads to warn of using the depth-averaged velocity for calculating the bottom stress.