^■iegel and Delmonte 



For deep water the horizontal component of water particle velocity is 

 approximately u = ( 7rH/T) cos Zirt/l Bit y = 0. An average of u 

 can be used to represent V ; i. e. , V ~ u ss ttH/ZT , where 



u is the "average" horizontal component of water particle velocity 



due to a train of waves of height H and period T. For at least one 

 pair of eddies to have time to form it can be argued that it is neces- 

 sary for T > l/f w 2 DT/ v H N g , if N « 0. 2, H > 10 D/tt . 



Keulegan and Carpenter (1958) studied both experimentally and 

 theoretically the problem of the forces exerted on a horizontal circu- 

 lar cylinder by an oscillating flow. In their experimental work the os- 

 cillations were of the standing water wave type, created by oscillating 

 a tank of water. The cylinder was placed with its center in the node 

 of the standing wave so that the water motion was simply back and 

 forth in a horizontal plane. The axis of the cylinder was normal to the 

 direction of flow (i. e. , parallel to the wave front), and about half way 

 between the water surface and the bottom. They found that C-q (and 

 C>r) depended upon u max T/D, (the Keulegan-Carpenter number 

 Nj^o) , where u = u max cos 2 7rt/T. They observed that when Nj^c 

 was relatively small no eddy formed, that a single eddy formed when 

 Nj^q was about 15, and that numerous eddies formed for large values 

 of the parameter. It is useful to note that this leads to a conclusion 

 similar to the one above. For example, if one used the deep water 

 wave equation for u max = 7rH/T, then u max T/D > ttH/D > 15 , 

 and H > 15D/7T for one eddy to form. 



It appears from the work described above that a high Reynolds 

 number oscillating flow can exist which is quite different from that 

 which occurs in high Reynolds number steady rectilinear flow, unless 

 the wave heights are larger than the diameter of the circular cylinder. 

 Even then, owing to the reversing nature of the flow, the "wake" dur- 

 ing one portion of the cycle becomes the approaching flow during an- 

 other portion of the cycle. It is likely that Nj^q ^ s ®£ greater signi- 

 ficance in correlating Cjy and Cj^ with flow conditions than is N-^ 

 (Wiegel, 1964, p. 259), and that the ratio H/D should be held cons- 

 tant to correlate model and prototype results, or at least should be 

 the appropriate value to indicate the prototype and model flows are in 

 the same "eddy regime" (see Paape and Breusers, 1967, for similar 

 results for a circular cylinder and for a flat plate oscillating in water). 



In studying forces exerted by waves on circular cylinders one 

 usually uses the equation developed by Morison, O'Brien, Johnson 

 and Schaaf (1950). For a cylinder with its axial normal to the direc- 

 tion of wave advance the horizontal component of force per unit length 

 of cylinder is given by 



764 



