E.3 Discrete Vortex Methods. The discrete vortex method (DVM) is a potential flow representation 

 of the separated shear layers and vortex wake of a body from which flow separation has taken place. 

 The DVM approach has been applied recently to the case of a freely oscillating circular cylinder by 

 Sarpkaya and Shoaff (E12,E13). The method has only come into general usage with the advent of 

 modern high speed computers, and applications of the method have dealt primarily with modeling the 

 flow separation from two-dimensional bodies with fixed separation points as described by Maull (El 4) 

 and Clements and Maull (E15). Stansby (E16) has applied the DVM approach to the periodic flow past 

 a circular cylinder in order to model wave/structure interactions. The history of the method and the 

 mathematical details of it are discussed by Sarpkaya and Shoaff" (E12) and by Clements and Maull 

 (E15). 



A problem with implementing the DVM approach has been the number of empirical parameters 

 that must be employed to minimize undesirable features. These features include instability of the vor- 

 tex sheets, the need for extreme accuracy of the flow field in the vicinity of the separation point, and 

 the difficulty in tracking the evolution of a random distribution of discrete vortices. Sarpkaya and 

 Shoaff appear to have minimized these problems by the introduction of a boundary layer calculation on 

 the cylinder to precisely determine both the location of the separation point and the amount of shed 

 vorticity and by a method of rediscretization for the evolving vortex sheets. The evolution of the vor- 

 tex wake behind a vibrating circular cylinder is shown in Fig. E2. The arrow in each step of the figure 

 represents the incident flow relative to the cylinder. By application of Blasius' theorem to the flow the 

 time-dependent and steady fluid forces can be calculated. The form of the equation employed by Sarp- 

 kaya and Shoaff" is 



N 



{z„ - Zq) - {Z„ - Zq)" 



— /zq X f"" ~ ■"'^'o- (E.3.1) 



Here zq denotes the (complex) position of the center of the cylinder and the (■) notation denotes 



differentiation with respect to time. The circulation and location of the nib. discrete vortex in the wake 



are given by r„ and z„, respectively, and the second term in the brackets represents the image point of 



the «th discrete vortex within the cylinder. It is readily seen from equation (E.3.1) that the forces on 



169 



