A General Theory for Marine Propellers 



evaluation. However, if the modification is too drastic, the solution obtained 

 may no longer be related to the problem to be solved. 



The main difficulty in attempting to functionally carry out the integration 

 involved in a propeller theory is the cosine factor in the integrand. In the past, 

 efforts have been made to find an approximating expression for either the whole 

 integrand or the distance factor in the denominator of the integrand, but no sat- 

 isfactory expression has been obtained, because the range of integration extends 

 to negative infinity. On the other hand, it is well known that a cosine function 

 within a small range of the argument can be accurately approximated by a 

 second-degree polynomial. Hence by dividing the range of integration into 

 steps and replacing the cosine factor of the integrand by an appropriate second- 

 degree polynomial the integration can be carried out functionally within each 

 step. This approach greatly reduces the required computing time compared to 

 the usual tedious numerical integration. Based on this numerical technique a 

 very efficient computing program can be developed. Such a development is now 

 under way, and some preliminary results are included in this paper. 



BASIC CONCEPT . 



When a propeller blade advances through a fluid, a pressure field is moving 

 with the blade. As a result, an unsteady motion is created throughout the fluid. 

 As was stated in the Introduction, the acceleration of a fluid particle of any un- 

 steady flow consists of two parts: 



3?=^(^'>- ■:.:.. ..... (« 



where q is a velocity vector. The first part, Bq/Bt, is due to the time rate of 

 change of velocity in a fixed space. The second part, (qV)q, is due to the move- 

 ment of the fluid particle. 



For an inviscid fluid, in the absence of an external force field, we have the 

 equations of motion 



^f^-^- e.:-^ .:-.^.:: , . ■ (3) 



where Vp is the gradient of the pressure field p, and pf is the fluid density. 

 Combining Eqs. (1) and (3) we write 



Bq 1 



3^=-^-Vp- (qV)q 



or 



^=-.^Vp-,(i, = ).,.f-. (4) 



where <^ is a vorticity vector. 



89 



