Theory of Unsteady Propeller Forces 



REPLY TO DISCUSSION 



Ryusuke Yamazaki i 



I would like to express my gratitude to Dr. Pien, Dr. Morgan, and Mr. Bavin 

 for their useful discussions and valuable comments. I have intended in this paper 

 to find a theoretical clue to calculate the unsteady propeller forces including the 

 propulsion performance of a ship on the basis of my present knowledge, and so 

 the theory in it is not a completed closed system mainly because of an imperfec- 

 tion in the treatment of the viscous flow which holds vorticity. I agree with Dr. 

 Morgan's opinion about the interaction of the ship hull and the propeller. I have 

 formed a plan as follows. By using the method developed in this paper, we can 

 calculate the stream line, the flow velocity, and the pressure near the hull sur- 

 face, and then, applying the two-dimensional boundary layer theory to the flow 

 along each stream line, we can obtain the frictional resistance of the hull, the 

 separation point of the boundary layer, and the wake velocity behind the hull and 

 can correct numerically the equations of the boundary conditions on the hull, pro- 

 peller, and rudder and the balance of the forces acting on these three parts. By 

 repeating the procedure the precise values of the unsteady propeller forces are 

 expected to be obtained. 



On the other hand, the flow state of the three-dimensional boundary layer 

 surrounding such a three-dimensional body as a ship is generally different from 

 that obtained by means of the above-mentioned two-dimensional process, even 

 in the case of steady condition. Therefore, as described in this paper, to solve 

 the problem of fluid flow near the hull, especially viscous flow, we must apply 

 the three-dimensional turbulent boundary layer and turbulent wake theory, which 

 is not yet completed. For example, at present we cannot calculate numerically 

 the exact values of the pressure in a laminar flow with vorticity for high Reynolds 

 numbers. In the future I want to study further the viscous flow near the hull, 

 which contains the boundary layer, its separation point, and the wake. 



In reply to Dr. Pien and Mr. Bavin, I plan to carry on the numerical calcula- 

 tion of the bearing and surface forces for the typical ship simultaneously. How- 

 ever, it seems very difficult, because the velocity components induced by the 

 velocity potential contain infinite series of improper integrals of Bessel func- 

 tions. In the examples of this paper the influence of the blade area ratio on the 

 bearing forces is rather small compared with the results obtained by Krohn and 

 Miller, as Mr. Bavin said, and I consider the reason to be the differences of the 

 wake distributions and the geometrical shapes of propeller blades except for the 

 blade area ratio. 



On the basis of the unsteady lifting surface theory with higher order terms, 

 the general theory to calculate hydrodynamic performance characteristics of a 

 heavily loaded propeller working unsteadily in a nonuniform flow was developed 

 in the beginning of Ref. 6, in which exact expressions were obtained for calculat- 

 ing the spatial and temporal distortion of the geometrical shape of the free vortex 

 sheet with the fluctuating strength from a regular helical surface with a constant 

 pitch. However, in the numerical examples of this paper and Ref. 6, I calculated 



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