Pien and Strom-Tejsen 



vortex- propeller theory is chosen as a point of departure, it is difficult to take 

 contraction of the slipstream into account. Such an attempt has been made in 

 the hope of developing a theory for heavily loaded propellers, but so far it has 

 not been successful. Propeller theory based on vortex representation has been 

 extended to cover unsteady operation, e.g., Refs. 11 through 18. Despite many 

 simplifications having been made in such unsteady propeller theories, the nu- 

 merical analysis is still so complicated that hours of computing time on a high- 

 speed computer would be needed. Therefore, it is not practical to generalize 

 such a theory any further. 



To overcome some of the difficulties inherent in a vortex- propeller theory, 

 theories based on the concept of acceleration potential have been developed re- 

 cently, e.g., Refs. 1 and 19 through 21. The starting point is the Euler equations 

 of motion; the concept states that the pressure gradient at any point divided by 

 the fluid density gives the acceleration of the fluid element at that point. Strictly 

 speaking, the pressure field of a fluid is not a potential function. Hence, such 

 propeller theories are viewed as linearized theories and are applicable only to 

 lightly loaded propellers. Unless the acceleration potential itself is modified, 

 there is not much room for generalization. 



It seems imperative to take a fresh look at the problem before attempting 

 to develop a general lifting- surf ace theory for marine propellers. When a pro- 

 peller blade is moving through a fluid, the fluid motion in a fixed space is un- 

 steady. Various fluid particles experience certain accelerations. The accelera- 

 tion of a moving fluid particle dq/dt consists of two parts: 



^=^<,V),. (1) 



The first part, Bq/3t, is due to the unsteadiness of the velocity field or the ac- 

 celeration at a fixed space. The second part (qV) q is due to the motion of the 

 fluid particle. It is more convenient if we consider these two parts separately. 

 As a matter of fact the time integration of the acceleration at a fixed space plus 

 the initial velocity field yields the velocity field at any time: 



t 



I 



3Ldt . (2) 



3t 



Then the second part of the acceleration of a moving fluid particle can be ob- 

 tained from the velocity field at that time. The fluid- particle acceleration due 

 to the movement of the particle can be left out in the calculation of the velocity 

 field at any time. Therefore it is logical to define a new acceleration potential, 

 whose gradient will yield an acceleration field in a fixed space. The general 

 theory of marine propellers described herein is based on this concept. 



The amount of numerical work involved in any propeller theory is always a 

 practical concern. A propeller theory is useful only if the computer time re- 

 quired is within a reasonable limit. The kernel function involved in a propeller 

 theory is quite complicated, and the usual practice is to evaluate it numerically. 

 In some cases, modification of the kernel function is made for convenience in 



