5. Interrelationships Between Cl, <i> , and k. 



(|) and k were defined as varying from 0° to 90° and to 1 , respec- 

 tively, with increasing clearance. (}> = 0° and k = correspond to the 

 case of a pipeline in contact witii the bottom (no clearance), while the 

 maximum values of (}> = 90° and k = 1 correspond to the case of a large 

 enough clearance so that the choking phenomenon does not occur at any time 

 throughout the wave cycle. Since a simultaneous increase of both parameters 

 was noted in the data for increasing clearance between the two limiting 

 cases, it was suspected that a direct relationship may exist between (p and 

 k. Such a relationship was found, as shown in Figure 39. The same rela- 

 tionship held for all three pipe diameters tested, regardless of the ori- 

 entation angle, indicating that the relationship was independent of these 

 two factors, and was thus valid for any pipeline configuration in which 

 the lift effect was present. 



In this plot and the ones that follow, the data for orientation angles 

 from 0° to 30° were plotted for each pipe diameter, without differentiating 

 the data corresponding to each angle. The relationships shown were found 

 to be valid regardless of the angle of orientation, provided the data were 

 handled as discussed above (using the component of the horizontal velocity 

 perpendicular to the pipeline axis) . The data corresponding to each pipe 

 diameter are distinguished by using different plot symbols. The same 

 relationships hold for orientation angles of 45°, but these data were not 

 plotted in order to minimize scatter so that differences between the pipe 

 diameters could be detected more easily. In general, the same relation- 

 ships held for orientation angles up to 60°. But in some cases, the lift 

 effect was negligible at high orientation angles, so the values of the 

 associated parameters (Cl, <t>, and k) were less accurate. Thus, plotting 

 all of the data corresponding to the larger orientation angles would intro- 

 duce additional scatter, obscuring the valid relationships which were 

 consistent when the lift forces were significant. 



A relationship was found between the coefficient of lift, Cl, and the 

 parameters, (j) and k (Figs. 40 and 41). Cl appears to be better correlated 

 with k than with (J). Note that for minimum values of k and (f), corresponding 

 to the case of a pipeline in contact with the bottom, the value of Cl is 

 approximately 4.5. This value is of interest, since it agrees with the 

 potential flow solution (Cl = 4.495) for the value of the coefficient of 

 lift for a circular cylinder in contact with a plane wall, subject to an 

 inviscid steady flow (Yamamoto, Nath, and Slotta, 1973). 



Maximum values of Cl occur at approximately k = 1/2, corresponding to 

 maximum lift forces that are equal in both the upward and downward direc- 

 tions. The average value of the coefficient of lift at this point is about 

 9.0, with values extending up to about 10.5. These maximum values of Cl 

 are attained at approximately ^ = 25° to 30° in the ({) versus Cl plot. 



Since the coefficient of lift, Cl, defines the combined magnitude of 

 both the positive and negative lifts, it can be separated into two parts: 

 (a) the part defining the magnitude of the positive lift, CL(l-k) , and 



77 



