A Nonlinear Oscillator Model For Vortex Shedding From a Forced Cylinder. Part 1: Uniform Flow And Model Parameters

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). VORTEX SHEDDING AND SYNCHRONIZATION Billah, these models fail to provide a good representation to the width and asymmetry of the synchronization region as measured by Williamson and Roshko (1988) as a function of the forcing amplitude. (3) (2) d2u (2 ) du (dY) -+E u -1 -+u=eu! y,r2 r dr where aI' a:z, and r are constants. This "quadratic" form for f, when multiplied by u as in Eq. 2, provides that no even ordered subharmonics (1/2, 1/4, etc.) arise in the solution of Eq. 2 for harmonic forcing. It also provides, via the IdyldrfY term, that the We introduce the dimensionless time r= OJ,t, the dimensionless axial coordinate {;= zlD where D is the diameter of the cylinder, the dimensionless near wake fluid quantity u = 2qlQo, and the dimensionless cylinder displacement y = Y/D. In terms of these dimensionless quantities, Eq. 1 becomes: For harmonic forcing at frequency OJ, that is for y = Asinnr where A is the dimensionless forcing amplitude and n = aiOJs is the dimensionless forcing frequency, a notable feature of the experimentally determined fluid response is the suppression of the natural shedding frequency and the entrainment or synchronization of the response frequency to the forcing frequency. Williamson and Roshko (1988) have conducted an extensive investigation of synchronized vortex shedding from an oscillating cylinder. For the fundamental synchronization defined by n"" 1, they find that the width of the synchronization region increases with increasing A and that the boundaries of the synchronization region are not symmetric about n= 1. They also find a secondary synchronization defined by n"" 1/3 (i.e., a 1/3 subharmonic), but no secondary synchronization defined by n"" 1/2 (a 1/2 subharmonic). Using analytical experimentation, we have developed a form for the coupling function that allows us to replicate closely the synchronization findings of Williamson and Roshko. Specifically, we takefiy, dyldr) as (1) During the past several years, investigators have demonstrated that the Ginzburg-Landau equation (Albarede 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. The model equation is given by: INTRODUCTION where t is time and z is the axial coordinate along the cylinder. Also, q is any near wake fluid quantity associated with the vortex shedding instability, Qo is the amplitude of q for flow over an unforced cylinder, OJs is the shedding frequency, and E is a scaling constant. The cylinder displacement normal to the flow is denoted by Y. The parametric form of the forcing function, qf(Y,dYldt), is based on an observation by Billah (1989) concerning the nature of the feedback between forced cylinder motions and the global vortex shedding instability. For harmonic forcing, Eq. 1 can be solved by approximation techniques for the synchronized shedding response. This solution allows us to formulate the coupling functionfiY,dYldt) so that the boundaries of the synchronization region correspond to the boundaries found experimentally by Williamson and Roshko (1988). The value of E is determined by matching the magnification factor for q with the lift magnification measured by Bishop and Hassan (1964). A number of papers have appeared, over the past 20 years, using the van der Pol equation to model vortex shedding from a forced cylinder. An in-depth review can be found in Billah (1989). In all cases, the forcing function has been assumed to be nonparametric; that is, the right-hand side of Eq. 1 has been taken only as a function of Yand its time derivatives. As discussed by Received March 3, 1995: revised manuscript received by the editors September 20, 1995. The original version (prior to the final revised manuscript) was presented at the Fifth International Offshore and Polar Engineering Conference (lSOPE-95), The Hague, The Netherlands, June 11-16, 1995.