A van der Pol equation with parametric forcing is introduced as a model for vortex shedding from a forced cylinder in a uniform flow. The parametric forcing function is developed so that the boundaries of the synchronization region correspond to the boundaries found experimentally by Williamson and Roshko (1988). A scaling constant is also developed so that the response magnification factor corresponds with the lift magnification factor measured by Bishop and Hassan (1964).
During the past several years, investigators have demonstrated that the Ginzburg-Landau equation (A1barede and Monkewitz, 1992) or its close relative, the van der Pol equation (Noack et aI., 1991), arises as the leading order approximation for the vortex shedding instability from a stationary cylinder in a uniform flow. In this paper, we extend the van der Pol equation to the description of the vortex shedding instability from a forced cylinder in a uniform flow. As discussed by Billah, these models fail to provide a good representation to the width and asymmetry of the synchronizatrion region as measured by Williamson and Roshko (1988) as a function of the forcing amplitude.
Equation (15) gives a dual valued curve for the response amplitude R2 as a function of the detuning 4. Based on the general behavior of a forced van der Pol oscillator (Minorsky, 1962, Chapter 18), we anticipate, and our later calculations verify, that only the upper curve, defined by the positive sign in equation (15), corresponds to a stable synchronized solution. Assuming that the dimensionless fluid quantity u and the lift force acting on the cylinder are related in some manner, we have also plotted on Figure 4 the data for the maximum lift amplification factor provided by Bishop and Hassan (1964).