A model of wave dynamics in the far wake of a cylinder

Abstract

Interesting interactions in the far wake behind a cylinder, leading in particular to the appearance of a strong secondary oblique wave, have recently been discovered experimentally by Williamson and Prasad [J. Fluid Mech. 256, 269, 315 (1993)]. They are induced from a very small amount of noise, added to the decaying primary wave. The problem is investigated theoretically with simple amplitude equations, based on temporal instability of the small amplitude waves. The symmetry of the wake flow requires that quadratic interactions arise only among triads of wave numbers involving one varicose and two sinuous waves, or three varicose waves. As the primary wave, corresponding to vortex shedding, is sinuous, the interacting secondary waves must be of opposite parities. In this case, it is found that the sinuous wave will always prevail downstream. The preferential appearance of the secondary oblique wave in the far wake can be reproduced by letting an initially very small varicose parallel wave interact with the primary wave. The secondary oblique wave results from a classical three-wave, quadratic nonlinear interaction between the waves. In addition, our model reproduces the observation that upon increasing the noise amplitude the oblique wave appears sooner (further upstream) in the wake. The occurrence of parallel waves very far downstream, which depends on the frequency relationship between the waves, can be understood by considering the interactions between parallel and oblique secondary waves of both parities. We propose a reasonable scenario explaining the experimental observations of Williamson and Prasad.