Figure 24 shows an effect of R on the peak water around the rear 

 half of a circular pile, but R has a more fundamental influence. In 

 general, R characterizes the flow regime at a circular pile, as clearly 

 shown in measured drag forces on circular cylinders in unidirectional 

 flow. From first principles, the drag force, fp, per unit length of 

 a cylinder may be expressed as 



f^ = 0.5C^pau|u| 



(6) 



where p is fluid density, u is flow velocity, and Cd is drag co- 

 efficient. Roshko (1961] and Schlichting (1968) discussed the distinct 

 unidirectional flow regimes revealed in measured values of Cp at 

 various R (Fig. 36). A subcritical range with C^ == 1.2 is separated 

 from a transcritical range with Cp == 0.7 by a supercritical range of 

 varying but lower C^ at Reynolds numbers between 2-10^ and 3.5-10 . 

 The C£) measured in wave flow is about the same as that indicated in 

 Figure 36, although the data are scanty (Fig. 7-62 in U.S. Army, Corps 

 of Engineers, Coastal Engineering Research Center, 1977) . 



Subcrilicol Range 



Recommended for Design in Wove Flow (US Army, Corps of Engineers, 

 Coostol Engneermg Reseorch Center, 19771^ 



lis B s. sfi' 1 1 



Wind Tunnel Meosuremenli \ 

 (RoshKo,l96l) ^\ 



10* lO' 



Reynolds Number, R = 2Uo/i/ 



10* 



10' 



Figure 36. Flow regimes indicated by measured drag coefficient at a 

 circular cylinder. 



Roshko (1961) describes the supercritical flow range as characterized 

 by a laminar separation bubble. Proceeding around the pile, the follow- 

 ing phenomena occur: laminar flow separation, transition to turbulent 

 flow, flow reattachment, and finally turbulent flow separation. The 

 laboratory test waves were well ordered, whereas natural waves have a 

 somewhat turbulent character, likely to be important to the flow separa- 

 tion process in this supercritical range. Also, the test piles were 

 smooth, whereas realistic models would have a somewhat rough face, also 

 likely to be important beyond the subcritical range. These other model- 

 ing inaccuracies would be important if these tests had entered into 

 higher Reynolds number flows. 



62 



