Absolute and convective instabilities of plane turbulent wakes in a shallow water layer

Abstract

In shallow turbulent wake flows (typically an island wake), the flow patterns have been found experimentally to depend mainly on a shallow wake parameter, S=c_fD/h in which c_f is a quadratic-law friction coefficient, D is the island diameter and h is water depth. In order to understand the dependence of flow patterns on S, the shallow-water stability equation (a modified Orr–Sommerfeld equation) has been derived from the depth-averaged equations of motion with terms which describe bottom friction. Absolute and convective instabilities have been investigated on the basis of wake velocity profiles with a velocity deficit parameter R. Numerical computations have been carried out for a range of R-values and a stability diagram with two dividing lines was obtained, one defining the boundary between absolute and convective instabilities S_ca, and another defining the transition between convectively unstable and stable wake flow S_cc. The experimental measurements (Chen & Jirka 1995) of return velocities in shallow wakes were used to compute R-values and two critical values, S_A=0.79 and S_C=0.85, were obtained at the intersections with lines S_ca and S_cc. Through comparison with transition values observed experimentally for wakes with unsteady bubble (recirculation zone) and vortex shedding, S_U and S_V respectively, the sequence S_C>S_A> S_U>S_V shows vortex shedding to be the end product of absolute instability. This is analogous to the sequence of critical Reynolds numbers for an unbounded wake of large spanwise extent. Experimental frequency characteristics compare well with theoretical results. The observed values of S_U and S_V for different flow patterns correspond to the velocity profile with R=−0.945, which is located at the end of the wake bubble, and it provides the dominant mode.