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=cfD/h in
which cf 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
Sca, and another defining the transition
between convectively unstable and stable wake
flow Scc. The experimental measurements
(Chen & Jirka 1995) of return velocities in
shallow wakes were used to compute R-values and two critical
values, SA=0.79 and
SC=0.85, were obtained at the intersections with
lines Sca and Scc. Through
comparison with transition values observed experimentally for wakes with unsteady
bubble (recirculation zone) and vortex shedding, SU
and SV respectively, the sequence
SC>SA>
SU>SV
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
SU and SV 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.
Small-scale instabilities of an island wake flow in a rotating shallow-water layer
