Three-Dimensional Flow Patterns in Two-Dimensional Wakes

Abstract

The development of three-dimensional patterns in the wake of two-dimensional objects is examined from the point of view of hydrodynamic stability. It is first shown that for parallel shear flows, which are homogeneous along their span, the time-asymptotic state of the instability is always two-dimensional. Subsequently, the effect of flow inhomogeneities in the spanwise direction is examined. Slow modulations of the time-average flow in the span wise direction, and localized regions of strongly inhomogeneous flow are separately considered. It is shown that the instability modes of an average flow with a slow modulation along the span have a spanwise wavelength equal to twice that of the average flow. Moreover, for the same average flow two instability modes are possible, identical in every respect except from their spanwise structure. Localized inhomogeneities on the other hand can generate through linear resonances inclined vortex filaments in the homogeneous part of the fluid. The theory provides an explanation for the vortex patterns observed in recent flow visualization experiments, and a theoretical justification of the cosine law for the frequency of inclined vortex shedding (Williamson, 1988).