Pien and Strom-Tejsen 



The tangential force component k^ has an order of magnitude of q^ because ^ is 

 the difference in q on both sides of the surface. In general, a tangential force 

 cannot produce a normal velocity component on the tangent plane. However, if 

 the lifting surface has some curvature in it, as it usually does, the tangential 

 component of the generalized force k^ would have some effect on the lift distri- 

 bution but the effect is also of the third order. Therefore, consistent with a sec- 

 ond order accuracy, it is only necessary to consider the normal component of the 

 singularity distribution k,, on the lifting surface. This conclusion is significant 

 because it is on the basis of this conclusion that the detailed theory has been de- 

 veloped. Of course, a higher than second order result can be obtained by taking 

 all the neglected small quantities into account with an iterative procedure. 



In practice we encounter two different types of problems: performance pre- 

 dictions of a given propeller geometry and design for a specified load require- 

 ment. In a performance prediction problem, we choose a proper expression for 

 k^ with a number of relevant parameters. Then o can be expressed in terms of 

 these parameters. The acceleration at a field point P is -va>. The time integra- 

 tion of -vo gives the velocity q at P. By choosing P to be on the surface pres- 

 ently occupied by the lifting surface, we obtain q on the lifting surface in terms 

 of these parameters. These parameters are determined by the boundary condi- 

 tion on the lifting surface. Subsequently the lift distribution F is calculated from 

 Equation (All). In a design problem an iterative procedure is necessary since 

 the orientation of the lifting surface is not known. It has been found that the con- 

 vergence is very rapid in such iterations. 



Any of the existing propeller theories based on a linearized acceleration 

 potential can be applied only to lightly loaded propellers since the accuracy of 

 the computed induced velocity on a lifting surface suffers from two possible 

 sources of error. These are (1) the linearization of the governing equation, the 

 equation of motion; (2) the assumption that the lift distribution is perpendicular 

 to the velocity of the lifting surface rather than perpendicular to the relative 

 velocity between the fluid and the lifting surface. 



In the case of a propeller theory based on vorticity distribution, effort can 

 be made to have the bound vorticity properly oriented in space. However, a long 

 iterative procedure is necessary in a performance-prediction problem even in a 

 steady case because the geometry of the slipstream is not known. It is extremely 

 difficult to use such a theory to analyze an unsteady propeller problem. 



It is felt that the theory developed here not only has the advantage of better 

 accuracy but also has its simplicity in its application, especially to unsteady 

 propeller problems. 



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