clearance constriction, more of the flow must be diverted over the top 

 of the pipe section, resulting in a downward shift in the stagnation 

 point, as well as an increase in the flow velocities and associated 

 pressure drop over the top side of the pipeline. 



The induced changes in the flow pattern, velocities, and associated 

 pressure distribution over the pipe section due to choking through the 

 bottom clearance constriction result in an upward lift force, rather 

 than the downward lift force predicted by potential flow theory. 



3. Thus, for an oscillatory wave-induced flow, the lift force acts 

 downward in those parts of the wave cycle where the horizontal water 

 particle velocities are not high enough to produce choking through the 

 bottom clearance. In this case, the unrestricted flow is faster through 

 the bottom clearance constriction than over the top of the pipe section, 

 so the corresponding pressure distribution results in a negative lift 

 toward the bottom boundary. 



However, in those parts of the wave cycle where the horizontal 

 velocities are sufficient to induce choking through the bottom clearance 

 constriction, the lift force acts in an upward direction. 



4. For a given pipe diameter and wave condition, as the bottom 

 clearance is increased, higher velocities are necessary to produce the 

 choking effect. Thus, the negative lift force can reach a greater 

 magnitude and occur later into the wave cycle before the choking condi- 

 tion is induced. 



Correspondingly, the positive lift that occurs only after the 

 choking condition develops is limited to a smaller part of the wave 

 cycle, and the maximum magnitude of these forces decreases with 

 increasing clearance. In addition, since there is a small timelag 

 involved in the development of the choking phenomenon and the transition 

 from negative to positive lift, the maximum positive lift occurs later 

 into the wave cycle, although its magnitude is diminishing. 



5. All major features of the wave-induced lift force phenomenon 

 can be described adequately by a modified lift force equation, Fl = 

 1/2 Cl p A Ujj^g^jj^^ [cos^ (G - ())) - k], where cj) represents a phase shift 

 in the position of the maximum positive (upward) lift force relative 

 to the point of maximum horizontal velocity at the center of the wave 

 crest, and k represents the proportion of the total lift force cycle 

 that acts in the negative (downward) direction. The values of 4> and 



k vary from 0° and 0, respectively, for the case of a pipeline touching 

 the bottom, and increase with increasing clearance (for a given pipe- 

 line and wave condition) to maximum values of 90° and 1, respectively, 

 when the pipeline is far enough from the bottom so that the choking condi- 

 tion does not develop. (}> = 0° and k = correspond to lift forces that 

 are positive throughout the wave cycle, with maximums occurring at the 

 points of maximum horizontal velocity under the wave crests and troughs. 

 (j) = 90° and k = 1 correspond to negative lift forces throughout the wave 



