Numerical Flow Structure Interaction for Cylinders Undergoing Vortex-Induced Vibrations
Authors: Schulz K. , Kallinderis Y. 
Published: October 2008

Abstract
A numerical pressure correction algorithm is implemented for solution of the incompressible Navier-Stokes equations using deformable hybrid grids and a finite volume discretization. The pressure correction method is loosely coupled with an elastic body structural response to obtain vortex-induced vibration solutions both at low and moderately high Reynolds numbers. Maximum cylinder displacements are compared to available experimental data for both cases and frequency ratios between the vortex shedding frequency and natural structural frequency are compared for the lower Reynolds number results. Additional examples of two-dimensional cylinder motion a seafloor VIV configuration and multiple cylinders undergoing VIV are presented illustrating the flexibility of the flow-structure algorithm for arbitrary motions and geometries.

Introduction
Incompressible fluids are encountered in a variety of engineering problems including offshore marine sciences, low-speed aerodynamics and a wide class of civil and industrial engineering applications. Of particular interest is how incompressible fluids interact with the structures they surround and in turn, how these structures respond to the external fluid forces. In some instances this fluid-structure coupling can result in a resonant structural condition yielding severe structural displacements which can lead to failure of the structure. One such instance of flow-structure coupling arises with the prediction and suppression of vortex-induced vibrations (VIV) which is a critical concern to the offshore oil industry. particularly with the design of offshore pipelines risers, and spars. These structures are typically cylindrical in shape and are an essential part of any offshore oil exploration or production. Modeling of the structural aspects of these elements has reached a substantial degree of maturity but the understanding and prediction or VIV is still a perplexing issue. Although typical amplitudes of vibration for risers undergoing VIV arc small the risers can still fail as a result of the persistent high frequency dynamic stresses. Most all of the current models used to predict VIV response characteristics are derived from a vast database of experimental results with a large scatter of predicted responses being observed [1]. However, surprisingly little work has been done to apply the governing incompressible Navier-Stokes equations to numerically obtain flow structure solutions about cylindrical members. Numerical methods for solving the Navier-Stokes equations have matured substantially in recent years and the present work is aimed at coupling an established Navier-Stokes method with a rigid body structural response to offer an additional avenue for confronting the difficulties of vortex-induced vibrations.

Governing Equations and Numerical Method
The governing incompressible Navier-Stokes equations are now presented in non-dimensional limn. with relevant primitive variables normalized by a characteristic length (D) and the free-stream velocity (U?). Terms in the equations are rearranged into in viscid. viscous and source terms. The source term arises for cases involving moving bodies which require non-stationary computational grids. The non-dimensional Navier-Stokes equations in two dimensions can be written in Cartesian coordinates in differential form as:(mathematical equation)(available in full paper) where U is the state vector. F1 and GI are the inviscid flux vectors, Fv and Gv are the viscous flux vectors and S is the moving grid source term.