ABSOLUTE AND CONVECTIVE INSTABILITIES OF PLANE TURBULENT WAKES IN A SHALLOW-WATER LAYER

In shallow turbulent wake flows (typically an island wake), the flow p atterns have been found experimentally to depend mainly on a shallow w ake parameter, S=c(f)D/h in which c(f) is a quadratic-law friction coe fficient, D is the island diameter and h is water depth. In order to u nderstand the dependence of flow patterns on S, the shallow-water stab ility equation (a modified Orr-Sommerfeld equation) has been derived f rom the depth-averaged equations of motion with terms which describe b ottom friction. Absolute and convective instabilities have been invest igated 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 obtain ed, one defining the boundary between absolute and convective instabil ities S-ca, and another defining the transition between convectively u nstable and stable wake flow S-cc. The experimental measurements (Chen & Jirka 1995) of return velocities in shallow wakes were used to comp ute 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 compar ison with transition values observed experimentally for wakes with uns teady bubble (recirculation zone) and vortex shedding, S-U and S-V res pectively, the sequence SC>SASU>S-V shows vortex shedding to be the en d product of absolute instability. This is analogous to the sequence o f critical Reynolds numbers for an unbounded wake of large spanwise ex tent. Experimental frequency characteristics compare well with theoret ical results. The observed values of S-U and S-V for different flow pa tterns correspond to the velocity profile with R = -0.945, which is lo cated at the end of the wake bubble, and it provides the dominant mode .