Pien and Strom-Tejsen 



REPLY TO DISCUSSION 



Pao C. Pien and J. Strom-Tejsen 



We would like to thank Dr. Wu for his penetrating comments. In introducing 

 the exact acceleration potential, we discussed a simple case of a lifting surface 

 moving forward with a constant velocity V . Our objective was to obtain the per- 

 turbation velocity caused by the action of the lifting surface in a coordinate frame 

 F, fixed to the lifting surface. In our basic concept we used another reference 

 frame F', in which an exact acceleration potential could be found as the two 

 frames approached each other. Then the time integration of the negative gradient 

 of the acceleration potential could yield the perturbation velocity in frame F' . 



When frame F* coincides with frame F, the velocity field in frame F differs 

 from that in frame F' by the known relative velocity between F and F' . Dr. Wu 

 has commented that if the flow is steady in frame F, p/pf + q^/2 is a constant, 

 and there is no possible singularity in that frame. This is in accordance with .our 

 definition of the exact acceleration potential $. However it does not imply that 

 there is no possible singularity distribution for d) when referred to other frames. 

 To facilitate our discussion we introduce the pressure equation with respect to a 

 moving frame F' as given by L. M. Milne-Thomson in "Theoretical Hydrodynam- 

 ics" as follows: 



where q^ is the magnitude of the fluid velocity in frame F' and u is the transla- 

 tion speed of frame F' . If there is no force field 0, we may write 



By taking the gradient on both sides of the equation we obtain 



P , 1 _ 2N ^"Jr 



which shows that in any reference frame, the sum of the pressure divided by the 

 fluid density and half of the velocity squared gives an exact acceleration poten- 

 tial. This is the foundation of our approach. The body frame F is a particular 

 case in which the acceleration potential is a constant. It simply means that we 

 should not choose it as our reference frame for an acceleration potential. If a 

 frame fixed in space far ahead of the lifting surface is chosen as our reference 

 frame, q^ becomes the perturbation velocity w. Therefore p/Pf + w^/2 is an 

 exact acceleration potential in that frame. It is up to us to choose whichever 

 reference frame is the most convenient one to use. 



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