Yamazaki 



have not yet been obtained for the case where the velocity field around the ship 

 satisfies the boundary conditions on the hull, rudder, and propeller simultane- 

 ously. 



In my previous paper (8) a general hydrodynamical theory was developed for 

 propulsion of a ship with a single propeller and a single rudder in which the mu- 

 tual interactions among the hull, rudder, and propeller are generally taken into 

 account. We will now develop this theory further for the computation of the un- 

 steady propeller forces. At first we shall employ the following assumptions: 



1. The field of fluid flow around the ship can be represented by an incom- 

 pressible inviscid irrotational flow superposed linearly on a flow caused by 

 viscosity such as the viscous wake behind the ship hull. 



2. A discontinuous flow such as a flow containing a free vortex or cavity is 

 not produced at the ship hull. 



3. No cavitation is produced at the propeller and the rudder. 



Then the ship speed can be assumed to be kept almost constant, because the 

 inertia of the ship is very large and the periods of vibratory forces generated by 

 the rotating propeller are very small. Similarly, the angular velocity of the pro- 

 peller is assumed to be almost constant. Further, since the influence of the free 

 water surface on the flow field around the ship can be generally represented by 

 the velocity potentials due to appropriate distributions of singular points in the 

 upper half space of unlimited still water (8) , we may safely treat the problem of 

 the ship moving on the surface of still water by replacing it with that of a sub- 

 marine moving in unlimited water modified by superposing a flow field due to 

 the added singular points. Thus, we can define clearly the unsteady propeller 

 forces including both the surface and the bearing forces. Therefore, we will 

 consider the submarine with a single propeller and a single rudder being moved 

 straight with a constant speed by rotating the propeller with a constant angular 

 velocity in unlimited still water as the simplest example and then derive a mathe- 

 matical expression for the unsteady propeller forces in this case. Finally nu- 

 merical examples will be presented to determine the effects of skew and chord 

 length of the propeller on the bearing forces under a given nonuniform flow. 



FUNDAMENTAL THEORY 



Consider a submarine with a single propeller and a single rudder being 

 moved straight with a constant velocity by rotating the propeller with a constant 

 angular velocity in unlimited still water , for which the hydrodynamical theory 

 presented in the previous paper (8) can be applied. We will use the term ship 

 instead of submarine in the following for convenience. 



At first we define a rectangular coordinate system - xyz fixed in space 

 and a cylindrical coordinate system - xr9 by the relations 



X = X , y = r cos 6 , z = r sin , (1) 



18 



