Panel Discussion 



which are exact for polunomials of degree n- l viz., n. They correspond to 

 Gaussian integration, giving a great improvement in accuracy. For unsteady 

 motions a special treatment for the infinite wake is needed. The improvement 

 in integration time as compared to a conventional method (e.g., Truckenbrodt) 

 is about a factor of 10. 



Coming back to the problem of applicability, viz., the alternative of using 

 exact theories in design or faster approximation methods, which are checked or 

 corrected by lifting- surface theory. 



Pien (NSRDC, Washington, D.C.) remarks that we can predict rpm and thrust 

 quite accurately. The main problem is to predict loading over the propeller 

 blade with relation to the cavitation problem and secondly to predict vibratory 

 force accurately. The question is the accuracy of the theory. 



Theories have two purposes understanding the physical problem, this is 

 already reached in history, but in order to reach quantitative predictions we can- 

 not modify the problem too much in order to reduce computing time, we have to 

 come as close to the problem as is possible. 



The main point is geometry of the slipstream, we have very nonuniform in- 

 flow and the free vorticity has to follow this flow. This is a drawback of the 

 vortex representation of the propellor. Going back to acceleration theory we 

 either know the loading or assume the loading and go back to the history of the 

 blade and bypass the helical sheet. 



We have to reach the stage of high loading and nonuniform flow, which seems 

 difficult on present computers. 



Timman remarks that for this purpose two ways are open much more compli- 

 cated calculations or simplified models which simulate special features. In his 

 opinion the formulation of linearized lifting- surface theory contains. 



Pien's time history: Going from the acceleration potential to the velocity 

 potential requires an integration over the wake, which is essentially the same as 

 in integration over the time history, since the free vortices in the wake carried 

 along with the flow with the strength they have when they were generated. 



For results as accurate in the nonlinear case as in the linear case it is nec- 

 essary to put on more effort, but is it important to include some effects and 

 leaving out others. It would be of interest to know whether it is contemplated to 

 work on lifting surface theory with cavitation. 



Weissinger asks whether in Dr. Pien's method the calculation of the shape 

 of the wake vortices would give regular helices. There are some linearization 

 assumptions in the theory, and it could be that the improvement is essentially an 

 improvement in computing time. 



Pien remarks that his theory bypasses the free vorticity and only calculates 

 dq/dt at the time the propellor blade passes. 



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