Submerged Two-Dimensional Bodies 



Accuracy of the Mathematical Representation of the Body 



We assumed in the mathematical formulation that the body could be repre- 

 sented by its singularities in a uniform flow without accounting for the free sur- 

 face. The boundary condition on the cylinder wall is by this assumption correct 

 only to the first order. To be correct to the second order, we should have mod- 

 ified these singularities such that the same body would be generated in a fluid 

 with a linearized free surface. This is one of the critical assumptions in this 

 work, and it was therefore felt that an investigation of its accuracy should be 

 included. 



A computer program was written which could trace the streamlines around 

 the singularity distribution including the linearized free surface. This is a 

 rather time consuming computation, especially because we do not know a priori 

 the value of the stream function at the stagnation points. The streamlines were 

 therefore only traced for two speeds, V = 4 ft /sec and v = 6 ft/sec, both with 

 the same submergence b = 1.25 ft. Each case took about 10 minutes on the IBM 

 7090 computer. The results are shown in Fig. 7. 



ABOUT -j^ INCH 



SPEED = 4 ft/sec AND SUBMERGENCE - 1.25 ft 



SPEED = 6 ft/sec AND SUBMERGENCE =1.25 ft 



U I FT SCALE 



Fig. 7 - Effect of a linearized free 

 surface on the flow around assumed 

 singularities 



It is seen clearly that when the linearized free-surface effect is included, 

 the body is no longer closed, and that the forward and aft stagnation points are 

 on different streamlines. This effect was first pointed out by Tuck (4). The 

 author is of the opinion, however, that for lower speeds the given singularity 

 distribution does represent the body fairly well and can be used with reasonable 

 accuracy for a second-order theory. For larger speeds it is obvious that the 



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