The largest negative lift forces do not occur at clearances where the 

 choking effect is absent (corresponding to k = 1 and cj) = 90°) . Rather, 

 the largest values of the effective negative coefficient of lift correspond 

 to values of c}) = 45° and k = 0.75. Interestingly, when k = 1 and (j) = 90° 

 where the positive lift forces have decreased to zero and the choking 

 effect does not develop, the maximum effective negative coefficient of 

 lift is approximately 4.5, the same magnitude as the potential flow solu- 

 tion for the positive coefficient of lift for zero bottom clearance. 

 However, as the bottom clearance is increased further, k and cJ) remain 

 at 1 and 90°, respectively, while the effective negative coefficient of 

 lift decreases to zero (with the diminishing lift forces). 



The significance of these results is easily seen by following these 

 relationships for a given pipe and wave as the pipeline is raised from the 

 bottom, and k goes from to 1. In the interval from k = to 1/2, the 

 magnitude of the maximum upward lift forces remains the same, which is 

 approximately equal to the potential flow solution for a cylinder in 

 contact with the bottom (Cl = 4.5). However, at the same time, the nega- 

 tive lift forces increase continuously, reaching a magnitude equal to the 

 positive lift forces at k = 1/2 (CL(l-k) = CL(k) = 4.5). Simultaneously, 

 there is a shift in the positions of both the maximum positive and nega- 

 tive lift forces, since <^ increases from 0° to 30°. 



In the interval k = 1/2 to 1 , the maximum positive lift forces 

 continuously decrease to zero. At the same time, the maximum negative 

 lift forces increase to reach a maximum value at k = 0.75 (where CL(k) = 

 6. or 7.), and then decrease back to a maximum corresponding to CL(k) = 

 4.5 at k = 1. The point of maximum negative lift corresponds to $ = 45°, 

 the midpoint of the phase shift cycle. 



The phase shift of the maximum lift forces is only half as much in the 

 interval k = to 1/2 (where c}) goes from 0° to 30°) as in the interval 

 k = 1/2 to 1 (where ^ goes from 30° to 90°). 



At k = 1, only negative lift forces exist, and these go to zero as 

 the bottom clearance is increased further. 



All of the above interrelationships between cJ), k, C^, CL(l-k) , and 

 CL(k) were the same for all pipe diameters tested, regardless of the angle 

 of orientation (provided that Cj^ was calculated considering only the com- 

 ponent of the horizontal velocity perpendicular to the pipeline axis) . 

 Thus, for the range of conditions tested, these interrelationships were 

 independent of the scale and configuration of the pipeline. Also, there 

 is no mention of the wave conditions, which indicates the interrelation- 

 ships are independent of the wave conditions as well. 



The relationships between the parameters, C^, (j), and k, defining the 

 lift force equation are useful, since if either cJ) or k is known, the 

 other two parameters can be determined. All that is needed is a rela- 

 tionship between (^ or k and the wave and pipeline conditions. 



86 



