Recent Progress in the Calculation of Potential Flows 



2TTCr(p) 



1 



n r (p,q) 



(q) dS = -n(p)-V„ 



(5) 



where the unit outward normal vector has been written n (p) to show explicitly 

 its dependence on location. The onset flow velocity V^ may or may not vary 

 with position. Equation (5) is seen to be a Fredholm integral equation of the 

 second kind, over the body surface S. Once this equation is solved for the source 

 density distribution cr, the potential cp may be evaluated from Eq. (4) and the dis- 

 turbance velocity components from the derivatives of Eq. (4) in the three- 

 coordinate directions. 



This method of solution is very general. The body surface s is not re- 

 quired to be slender, analytically defined, or even simply-connected; that is, 

 there may be several bodies, as in the example of Fig. 1. The only restriction 

 is that s must have a continuous normal vector n(p), which means that the 

 method cannot be guaranteed to give correct results for bodies with corners. In 

 practice, this difficulty can be avoided by rounding off any corners with a small 

 radius. Trial calculations show that the method does however give correct re- 

 sults for convex corners, but there may or may not be significant errors near 

 unrounded concave corners. The onset flow V„ is not restricted to being a uni- 

 form stream. It may be any flow consistent with the assumption that the pertur- 

 bation velocity field due to the body is irrotational. This is satisfied if the onset 

 flow has a constant vorticity — a uniform shear, for example — since it can be 

 shown that the perturbation velocity is irrotational. 



The efficiency of the method is that only the body surface itself needs to be 

 considered, not the entire exterior flow-field. Thus the dimensionality of the 

 problem is reduced by one: from three to two in three-dimensional problems, 

 and from two to one in axisymmetric and two-dimensional problems; for in 

 these cases the double integral of Eq. (5) can be reduced to a single integral by 

 performing one integration analytically. The area of interest is also shifted 

 from the infinite to the finite. 



General Description of the Method of Solution 



The central problem of the present method of flow calculation is the numer- 

 ical solution in Eq. (5). The integral equation is replaced by a set of linear alge- 

 braic equations in the following way. 



The body surface is approximated by a large number of surface elements, 

 each of which is small in comparison to the characteristic dimensions of the body. 

 Over each surface element the value of the surface source density is assumed 

 constant. That assumption reduces the problem of determining the continuous 

 source density function a to the problem of determining a finite number of values 

 of cT, one for each of the surface elements. The contribution of each element to 

 the integral in Eq. (5) can now be obtained by taking the constant but unknown 

 value of cr on that element out of the integral and then performing the indicated 

 integration of known geometrical quantities over the element. Requiring Eq. (5) 

 to hold at one point of the approximate body surface, i.e., requiring the normal 

 velocity to vanish (or to take on a prescribed value) at one point, gives a linear 



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