This paper presents the results of a numerical calculation of the drag coefficient on a circular cylinder in planar oscillatory flow at low Keulegan-Carpenter numbers. This calculation represents the classical hydrodynamic damping problem. The two dimensional time-dependent stream-function/vorticity equations for an incompressible fluid are solved by a finite-difference procedure. The drag coefficient values are determined through an integration of the pressure and shear stress around the surface of the cylinder. The results show that the Stokes-Wang result is duplicated at ? =1035 (? = d2/vT, where d is the cylinder diamter, T is the period of the flow oscillation, and v is the kinematic viscosity of the fluid). For values of ? as high as 130,000, the calculated drag coefficients agree closely with experimental values.
The hydrodynamic damping of moving bodies is important in several areas of offshore engineering. Small amplitude, high frequency oscillations can occur in situations involving marine riser vibrations, the oscillations of the tethers on compliant structures, etc. The prediction of fatigue and the analysis of the dynamic behavior of a moving structure require an understanding of hydrodynamic damping. The accurate prediction of damping is then an important factor in describing platform motions and, therefore, in accomplishing platform design. Our analysis considers damping due to viscous effects on small bodies only; radiation damping will not be considered in this study.
Understanding of the fluid motion generated by a body undergoing sinusoidal oscillations is the first step in determining the hydrodynamic damping. The flow field can be characterized by two of the following parameters: KC =Umax T/d =2?a/d, Re =Umax d/v and ? =Re/KC =d2/uT, where Umax is the maximum oscillatory velocity, T is the period of oscillation, d is the diameter, u is the kinematic viscosity, and a is the amplitude of oscillation. In the above definition, KC is the Keulegan-Carpenter number, Re is the maximum velocity Reynolds number and ? is the frequency parameter. For small KC (KC < <l) and large ? (?> > 1), the drag coefficient can be predicted by the Stokes-Wang law, (Mathematic equation)(available in full paper)
The Stokes-Wang drag coefficient is developed from the Navier-Stokes equations in the absence of the convective terms which were taken to be negligible, an assumption which is applicable only for low values of KC. When KC increases, the convective terms increase in magnitude and the Stokes-Wang law no longer adequately represents the solution. The higher KC problem has no analytical solution; therefore, the force description is based on experimentally determined coefficients. The conventional force description is given by the Morison equation which typically represents the force on a fixed cylinder due to an oscillating fluid. The Morison equation decomposes the instantaneous force into a linear sum the drag and inertia forces.