Wave Induoed Forces and Motions of Tubular Structures 



justification for this simplification. 



Maruo [ 1954] and Havelock [ 1954] have discussed the forces 

 on submerged bodies which are caused by waves of small amplitude 

 in an ideal fluid. Maruo deals with the two-dimensional problem of 

 a horizontal cylinder completely submerged below the surface of an 

 inviscid fluid, and gives some results which can be conipared directly 

 with Eq. (7). In particular, he shows that, for the case of a deeply 

 submerged cylinder, the "exact" total force is equal to twice the 

 Froude-Krylov force, Eq. (7), and that this corresponds to a value of 

 the added mass coefficient equal to that of the cylinder in an infinite 

 fluid combined with the wave motion at the centerline of the cylinder. 

 If the cylinder is near the free surface, the force differs from this 

 deeply submerged value by an amount dependent upon the depth of 

 submergence and the wave length. The error, however, is small 

 if both the depth of submergence and the wave length are greater than 

 several cylinder diameters. Similarly, C. M, Lee [1970] has 

 analyzed the problem of an oscillating cylinder submerged beneath 

 the free surface, and has shown that the added mass coefficient for 

 forced motion approaches the infinite fluid value within a small error 

 if the depth of submergence is nnore than two times the cylinder 

 diameter and the length of generated waves is more than about five 

 times the cylinder diameter. For our present purposes , these 

 results imply that we may assume a constant value of the added 

 mass coefficient in Eq. (7), since in the majority of practical situ- 

 ations the cylindrical members will be sufficiently deeply submerged 

 and of sufficiently small diameter compared to wave lengths of 

 interest to fulfill the above conditions. Thus the first and last terms 

 in Eq. (7) can be expected to give a good approximation to the non- 

 dissipative parts of the force on the individual member considered 

 here. 



The drag or velocity dependent force acting on an oscillating 

 body under a train of waves is associated with two phenomena: (1) the 

 dissipation of energy in surface waves which are generated as a 

 result of motion of the body, and (2) the viscous effects which are 

 felt both as tangential forces on the surface of the body, and as a 

 deviation of the pressure distribution from its ideal fluid value. This 

 latter effect, which is associated with the formation of a wake and 

 vortices downstream of the body, will cause the added mass coefficient 

 to differ from its ideal fluid value as well. The drag force associated 

 with free surface wave effects decays to zero with increasing depth 

 of submergence at the same rate that the added mass coefficient 

 approaches the infinite fluid value. Therefore wave damping is of 

 little significance to the configurations being considered here. The 

 drag forces associated with viscosity are generally of much greater 

 importance, and also less clearly defined. The usual method of 

 approximating these forces, Wiegel [ 1964] , is to assume that they 

 behave in a manner similar to the drag on a body immersed in a flow 

 of constant velocity. In such case, the drag force is expressed as a 

 quadratic function of velocity and the drag coefficient is found to be a 



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