the body are critically dependent upon the strength (circulation) and location of the discrete vortices. 

 These parameters are in turn dependent upon the starting conditions (separation point, magnitude of 

 the vorticity, number of vortices, etc.) and upon the evolution of the vortices. 



It is possible to predict the major features of the complex nonlinear interaction that characterizes 

 vortex-excited oscillations, but application of the discrete vortex method (DVM) is thus far limited to 

 the case of a rigid cylinder. Although Sarpkaya and Shoaff (E12, E13) have made considerable progress 

 in limiting the number of empirical parameters that must be used with the DVM and have achieved 

 good agreement with rigid cylinder experiments, the prediction of the vortex-excited response of flexi- 

 ble bluff bodies such as cables and marine pilings will require the formidable step to a three- 

 dimensional code. Moreover, the effects of yaw angle, shear, and roughness will no doubt be 

 extremely difficult to incorporate into the boundary layer calculations, i.e. starting conditions, without 

 resort to further empiricisms. 



E.4 Numerical Models. Few, if any, numerical models for calculating the flow around bluff bodies 

 have been developed for practical applications. This is primarily due to the difficulties that are encoun- 

 tered in achieving the required small grid sizes and time steps as the Reynolds number is increased to a 

 practical value (say Re == 200 at a minimum). The numerical solution of the Navier-Stokes equations 

 of motion for flow past a cylinder are highly sensitive to the grid size, especially near the cylinder, and 

 to the time step size. 



A numerical solution of the Navier-Stokes equations for the flow around a circular cylinder was 

 obtained recently by Hurlbut, Spaulding and White (El 7, El 8). Three cases were considered: a 

 cylinder vibrating in a still fluid, a cylinder vibrating in line with an incident uniform flow, and a 

 cylinder vibrating normal to an incident uniform flow. The maximum Reynolds number achieved was 

 Re = 100. The solution of the governing equations in the presence of the oscillating cylinder was 

 achieved by transforming the equations for the computational grid system to a noninertial (accelerating) 

 system. Good quantitative agreement was achieved with experimental data for the steady drag and 

 unsteady lift forces on a vibrating cylinder at Re = 80. The computer model developed by Hurlbut, 



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