singular Pevtuvbation Problems in Ship Hydrodynamias 



the nineteenth century for solving free- streamline problems » But It 

 Introduces a new difficulty: It cannot be used to treat free stream- 

 lines which are affected by gravity, which means that only Infinite - 

 Froude-number problems can be treated directly. This leads to a 

 further great difficulty, which Is discussed In some detail In 

 Section 5.1. 



In Section 5.3, a brief discussion Is presented of the problem 

 studied by Salvesen [ 1969] . It contains two aspects of Interest: It 

 is a case In which the free-surface conditions can be linearized 

 because of the depth of the moving body, and I have already commented 

 in the Introduction that there are very Interesting fundamental ques- 

 tions Involved In such procedures. Also, It presents a clear example 

 of the classical phenomenon discussed In the section on multiple scale 

 expansions: The wave length obtained In the first approximation must 

 be modified In subsequent approximations , or the solution becomes 

 unbounded at Infinity -- where we know perfectly well that the waves 

 are bounded In amplitude. 



Finally, Section 5.4 describes two recent attempts to approach 

 the problem of extremely low- speed motion. The difficulty Is basically 

 this: In the usual linearization, we assume that all velocity components 

 (at least In the vicinity of the free surface) are much smaller than the 

 forward speed -- which becomes nonsense If we subsequently decide 

 to let U, the forward speed, approach zero. What is needed Is a 

 perturbation scheme In which somehow the small parameter Is pro- 

 portional to U. Then It Is certainly permissible to allow U to 

 approach zero. Section 5.41 shows a very straightforward procedure 

 for doing this; however, it leads to a sequence of Newmann problems, 

 and so the wave nature of the fluid motion is lost. In Section 5.42, 

 an alternative method is discussed. It is an application of the multi- 

 scale expansion procedure to which Section 1.3 was devoted. 



5.1, Gravity Effects in Planing 



Before we try to treat this problem properly, let us consider 

 briefly a well-known approach to the 2-D planing problem and deter- 

 mine why it is not completely satisfactory. In the middle 1930's, 

 A. E. Green wrote several papers on the subject, and the essence of 

 his approach is well-presented by Milne-Thomson [ 1968] . A flat 

 plate is located with its trailing edge at the origin of coordinates, as 

 shown in Fig. (5-1). There is an incident stream with speed U 

 coming from the left, and, at infinity upstream, there is a free sur- 

 face at y = h. The effects of gravity are neglected. The fluid is 

 assumed to leave the trailing edge smoothly (a Kutta condition), and 

 a jet of fluid is deflected forward and upward by the plate. In the 

 absence of gravity, the jet never comes down to trouble us again. 

 In the figure, A marks the leading edge of the plate and C marks 

 the stagnation point. 



The physical plane shown in Fig. (5-1) is also the complex 



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