Analytical investigation of the vertical structure of periodic flow

Highlights

• 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 shows that an ideal value on the bottom friction coefficient for the slip condition is around 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.

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.

Introduction

The development of numerical models for two- and three-dimensional flows for applications in lakes, estuaries, and coastal areas have led to a need to verify the simulation results from these complex numerical models, since direct analysis of the results is difficult to use for the assessment of model accuracy, convergence, robustness, and stability. Comparisons with analytical solutions for simplified flow problems play an important role in such assessments.

Analytical solutions to the periodic flow problems were proposed, for example, by Ippen [1], Johns and Odd [2], Lynch and Gray [3], Chen [4], Guillou et al. [5], and Phan and Larson [6]. Ippen [1] developed an analytical solution for one-dimensional (1D) tidal flow describing the propagation of the gravity wave without bottom friction and horizontal turbulent viscosity. Johns and Odd [2] introduced a vertical structure for the tidal flow in the analytical solution. Lynch and Gray [3] and Chen [4] proposed analytical solutions for the 1D tidal flow, including bottom friction but neglecting the horizontal turbulent viscosity. Guillou et al. [5] improved the analytical solution proposed by Johns and Odd [2] and Chen [4] by taking into account the bottom friction effects on the free surface elevation with regard to phase delay and the vertical velocity profile. Recently, Phan and Larson [6] proposed analytical solutions with an extension and generalization of the solutions given by Johns and Odd [2], Lynch and Gray [3], Chen [4], and Guillou et al. [5] by taking into account the effects of the horizontal turbulent viscosity, and the bottom friction through a slip condition.

A slip boundary condition at the bottom is sufficient to impose when modeling loose boundaries occurring in coupled hydrodynamic-sediment problems. Therefore, in order to investigate the effects of the horizontal turbulent viscosity and the slip boundary condition against a no-slip boundary condition at the bottom on the free surface elevation and especially on the velocity near the bottom layer, the analytical solution to the vertical 2D flow simplified from that to the 3D flow proposed by Phan and Larson [6] is used. Also, the solution presented in this paper provides a solution that is simple enough for numerical model testing and evaluation.