Analysis of Wave Plus Current-Induced Forces on Cylinders
Authors: Sarpkaya Turgut , Dalton Charles 
Published: October 2008

ABSTRACT
The numerical experiments, carried out through the use of the vorticity-stream function equations and their finite difference form on wave plus current flow are described. A third-order in time, second-order in space, three-level predictor-corrector finite-difference scheme has been used. The results have revealed, for the first time, the existence of a very interesting wake comprised of three rows of vortex pairs at certain Keulegan-Carpenter numbers and relative current ratios. The existence of such a street has been vindicated by physical experiments.

INTRODUCTION
The in-line oscillations of a cylinder in a current or the translation of a cylinder in an oscillating flow or the. combined wave-current flow about a stationary cylinder has been the subject of intense interest in recent years (see, e.g., Sarpkaya & Storm1) in connection with the understanding of the behavior of hotwire anemometers and the fluid loading of structures subjected to currents, gusts and other types of unsteady flows. The main common thread in all these is that the vortex shedding and, hence, the drag and inertia coefficients, are profoundly affected by the current relative to their no-current values. The elucidation of the resulting vortex motion has been strongly aided by the ability to perform numerical experiments in the range of computer-realizable Reynolds numbers. The purpose of this paper is not yet another discussion of the force-transfer coefficients but rather an exploration of the relationship between the shedding of vortices and the primary governing parameters (available in full paper) for a flow assumed to be two-dimensional. In the indllstrially-significant-rallge of the goV'erning parameters (K, Vr, Re or ?), the existing numerical codes (with or without ad hoc turbulence models) are not in a position to predict the force-transfer coefficients, even for a simple circular cylinder. No three-dimensional finite-difference solution of the Navier-Stokes equations for the flow past a cylinder of finite aspect ratio (with no slip condition on the cylinder and on the bounding end walls) exists.
A finite difference analysis of the Navier-Stokes equations for a sinusoidally-oscillating ambient flow about a circular cylinder at Keulegan-Carpenter numbers K = 5 (Re = 1000) and K = 7 (Re = 700) has been attempted by Baba & Miyata2, assuming a physically unrealistic symmetric wake in both simulations. Their results have shown that the calculations can be carried out only for short times (less than two cycles of flow oscillation) with a non-super computer. Murashige, Hinatsu and Kinoshita3 have used a similar method to analyze three cases (K = 5, 7, and 10) at higher Reynolds numbers around 104. The flow was perturbed by artificial means to trigger asymmetry. At K = 10, a transverse vortex street appeared, in agreement with experimental observations. The numerical simulation of steady flow past a circular cylinder undergoing in-line and/or transverse oscillations through the use of two dimensional unsteady Navier-Stokes equations was undertaken by Lecointe and Piquet4 for relatively small amplitudes (A/D = 0.13).