| A general framework based on the extended Hamilton's principle for external viscous flows is presented. The indicated method is shown to yield the correct governing equations when applied to the problem of vortex-induced oscillations of an elastically-mounted circular cylinder embedded in a uniform flow. The rigid cylinder has either one or two degrees-of-freedom, depending on the particular application considered. Assuming nominally two-dimensional flow, and taking the continuity equation as a given constraint, the Navier-Stokes equations for an incompressible viscous fluid, and one or two linear structural oscillators are derived. The oscillators are appropriately coupled to the fluid, and to each other in the two degree-of-freedom case, via the hydrodynamic forcing. The framework is believed to be amiable to more complicated problems, such as the vortex-induced vibrations of a flexible cylinder, although this aspect is not explored in this work. The true strength of the framework lies in that it represents a physically sound basis from which reduced-order models for fluid-structure interaction problems can be obtained. To illustrate this, a wake-body model for the flow past a circular cylinder with only a transverse degree-of-freedom is derived from the general variational equation. By wake-body model, it is meant a coupled system of two ordinary differential equations: the nonlinear equation representing the transverse motion of a representative near-wake fluid mass (the wake-oscillator), and the linear equation representing the transverse motion of the cylinder. The reduction in complexity is presented as a hierarchical process, in which the validity of each step is corroborated by physical reasoning and results from the literature. |
