Energy transfer in a vortex induced vibrating tethered cylinder system

Flow induced vibration of a tethered body submerged within a uniform flow represents a fundamental example of fluid-structure interaction. However, little research in this field has been undertaken. This paper presents results from a direct numerical simulation of the flow past a tethered cylinder with mass ratio less than 1. The Navier-Stokes and dynamic equations of motion of the cylinder are solved using a Galerkin spectral element/Fourier method. The fluid forces acting on the cylinder, as well as the tension in the tether are computed and used to determine the resulting cylinder motion. The energy transfer from the fluid to the cylinder is calculated, and two distinct modes of oscillation are observed. INTRODUCTION A simple extension to the problem of a hydro-elastically mounted oscillating cylinder is a cylinder whose motion is confined to an arc by a restraining tether. By considering a wide range of mass ratios, m* (the mass of the cylinder divided by the mass of the displaced fluid), we may allow for both buoyant bodies (m* < 1) and dense bodies (m* > 1), and hence describe a parameter space encompassing a range of practical applications. To date, little progress has been made regarding the fluid-structure interaction of a tethered body, and virtually no work has been published regarding the flow around a tethered cylinder. In this paper, we report on the development of a new numerical code used to simulate the two-dimensional flow field around a tethered cylinder. This code has been validated against preliminary experiments performed in the Monash FLAIR water channel. Most previous work regarding tethered bodies has focused on the free surface interaction with tethered buoys (ShiIgai et al. (1969), and Ogihara (1988)). In each of these studies, the tethered bodies oscillate due to the combined effect of a uniform (or sheared) free stream and free surface waves. This combination induces complicated body motion, which inhibits clear interpretation of the individual forcing mechanisms acting on the body. Only the related studies of Williamson et al. (1997), Govardhan et al. (1997), and Jauvtis et al. (2001) deal purely with the interaction of a uniform flow field and a tethered body (specifically a tethered sphere). Their experimental study was performed by placing spheres of various mass ratios and tether lengths into a water channel at various flow speeds. Sustained, large peak-topeak oscillations in the transverse flow–field direction were noted, with amplitudes in the order of two sphere diameters. Small stream-wise oscillations of the order of about 0.4 sphere diameters were also observed. Their results were independent of sphere mass ratio or tether length if they were plotted against the reduced velocity ( * n U U f D = , where fn is the natural frequency of the tethered sphere system). From this information, four distinct modes of shedding were observed corresponding to local maxima in the peak-to-peak transverse oscillation of the sphere (see figure 4b in Govardhan et al. (1997). Two of the observed modes were present for low reduced velocity experiments (u* < 25 corresponding to m* < 1). It may be anticipated that the flow around a circular cylinder will follow similar phenomenon. This investigation presents a new numerical algorithm for simulating the flow past a two-dimensional tethered body, and relates the solutions obtained to the vortex-induced oscillation of bluff bodies. A cylinder mass ratio, m* = 0.833 was simulated. It was found that the oscillation frequency remains close to the vortex shedding frequency of the flow as in Angrilli et al. (1974) and Govardhan et al. (2000). The energy input from the fluid to the cylinder over one oscillation cycle was calculated and used to establish two separate modes of vortex induced oscillation. PROBLEM FORMULATION The coordinate system and geometry of the problem are shown in Figure 1. The fluid forces acting on the tethered body are composed of Drag (FD), Lift (FL), and Buoyancy (B) terms. A restoring tension force (T) in the tether is also present. The problem is fully described in two dimensions by the coupled system of the incompressible Navier-Stokes equations (equation 1a and 1b) and the equations of motion describing the body acceleration in response to calculated fluid forces (equations 2a and 2b).