Significance of averaging coefficients in stability analysis of shallow wake flows

Abstract

Flows behind obstacles (such as islands) are shallow if the transverse scale of the flow is much larger than water depth. Field, laboratory and numerical data show that the flow pattern in shallow wakes exhibits a complex eddy‐like motion. Experimental and theoretical analyses provide evidence for the presence of two‐dimensional coherent structures in shallow water flows and show that the development of shallow wakes is different from the wakes in deep water due to the following reasons: first, the development of three‐dimensional instabilities is prevented by limited water depth and second, bottom friction acts as a stabilizing mechanism for suppressing the transverse growth of perturbations. Several authors have used the linear and weakly nonlinear stability theory in order to understand when shallow flows become unstable. Two‐dimensional depth‐averaged Saint‐Venant equations are usually used for the analysis. One of the main assumptions in shallow water theory is the independence of the velocity distribution on the vertical coordinate. In many cases, however, this assumption may not be valid. This paper presents an attempt to evaluate the influence of the assumption on the results of linear stability analysis of shallow wake flows with bottom friction. Momentum correction coefficients β ₁ and β ₂ are used in order to take into account the non‐uniformity of the velocity distribution in the vertical direction. Linear stability calculations show that the stability boundary is quite sensitive to the variation of the parameters β ₁ and β ₂. The role of the linear and weakly nonlinear stability analysis on the formation of two‐dimensional coherent structures in shallow water flows is discussed.

First Published Online: 14 Oct 2010

Keyword : momentum correction coefficients, shallow wake flows, stability analysis