90° from the wave crest and increased while continuously shifting posi- 

 tions to reach a maximum negative value at a position 180° from the 

 wave crest (Fig. 7, a). 



Since a sinusoidal function of twice the frequency of the wave 

 (sin 20 or cos 26) can be expressed as cos^O, using the appropriate trigo- 

 nometric relationships, and since the lift force is a function of the hori- 

 zontal velocity squared (u^ax cos 6)^, using linear wave theory, the lift 

 force equation was expressed as F^^ = 1/2 C. p A u ^ [cos^ (0 - cj)) - k] . 



However, it is the symmetrical properties of this equation and linear 

 wave theory that allow this expression to work so well. When higher 

 order wave theories are applied to this relationship, problems due to 

 nonsymmetry are encountered. This is easily seen by graphically compar- 

 ing the transition from positive to negative lift forces with increasing 

 bottom clearance with this lift model, using both linear and higher order 

 theories . 



The horizontal component of the water particle velocity for both 

 Stokes' third-order waves and linear waves is shown in Figure 8, along 

 with the corresponding lift forces on a pipeline for the two extreme 

 cases of: (a) a pipeline on the bottom with no clearance, and (b) a 

 pipeline with a large enough bottom clearance so that the choking phe- 

 nomenon does not occur. By gradually shifting the linear theory lift 

 force curve for case (a) (no bottom clearance) to the right 90° from 

 the wave crest, while simultaneously lowering it so that the forces 

 become negative, the lift force curve for case (b) is obtained (com- 

 pare Figs. 7 and 8). This same transformation of trie wave force record 

 was observed with increasing bottom clearance in the experimental data. 



However, if this procedure is repeated with the Stokes' third-order 

 lift force record, the correct force record for case (b) is not obtained 

 (compare Figs. 7 and 8). In reality, rather than a mere shift of the 

 force record downward and to the right with increasing bottom clearance, 

 a simultaneous transformation of the shape of the lift force record 

 would also occur for highly nonlinear waves. This gradual transforma- 

 tion of the shape occurring simultaneously with the shift would provide 

 a continuous change in the lift force record with increasing clearance 

 between the two limiting cases (a) and (b) (Fig. 8) . 



However, the lift force phenomenon is not a direct function of the 

 instantaneous water particle velocity acting at the center of the pipe 

 section if the pipeline was absent. Rather, it is a complicated function 

 of the asymmetrical distorted flow pattern and accelerating velocity 

 field acting on the pipeline, which in turn causes the choking phenomenon 

 to occur, with the resulting change in the relative differences in the 

 flow velocities and corresponding pressure distribution over the top and 

 bottom of the pipeline. Boundary layer flow through the bottom constric- 

 tion, the formation of a turbulent jet and associated eddies, and a cyclic 

 change in the location of the stagnation point with the accelerating 

 velocity field further complicate matters. In addition, the eddies and 



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