Recent Progress in the Calculation of Potential Flows 



source density is then constant over each frustum only in the axial direction and 

 varies in a prescribed manner circumferentially. This is the situation, for ex- 

 ample, when there is flow over a body of revolution at angle of attack. As Ref. 1 

 shows, this problem may be solved without resort to fully three-dimensional 

 techniques. 



For two-dimensional and axisymmetric bodies, the edges of adjacent ele- 

 ments are coincident, but this is not necessarily so for three-dimensional 

 bodies. In general, a three-dimensional body cannot be approximated by plane 

 quadrilaterals in such a way that the edges of adjacent elements are coincident. 

 Any errors due to these gaps are of a higher order than, and negligible with 

 respect to, the errors due to the approximation of the body by plane elements 

 in the first place. Nevertheless, the fact that small gaps exist between the 

 elements is sometimes disturbing to people hearing about the method for the 

 first time. It should be kept in mind that the elements are simply devices for 

 finding the surface source distribution and that the polyhedral body shown in 

 Fig. 3 has no direct physical significance, in the sense that the flow eventually 

 calculated is not that about the polyhedral body. Even if the edges of adjacent 

 elements are coincident (as, for example, can be arranged for any body of 

 revolution), the normal velocity is zero at only one point of each element and 

 there is flow through the remainder of the element. Also, the computed veloci- 

 ties will be infinite on the edges of the elements whether these are coincident or 

 not, as long as there is a break in slope or in source density. The unimportance 

 of the gaps has been further demonstrated by calculating axisymmetric bodies 

 with element distributions that had coincident edges and then recalculating with 

 slight gaps. The two types of element distributions gave essentially identical 

 results. The gaps between the elements can be eliminated by the use of plane 

 triangular elements. This procedure, however, results in no increase in ac- 

 curacy — in fact may cause a loss of accuracy — and so greatly complicates the 

 input to the digital computer program as to impair its usefulness as a design 

 tool. On many bodies of technical importance such as ships, wings, and hydro- 

 foils, approximation of the shape by quadrilaterals is much more natural than 

 approximation by triangles. However, the triangle is merely a special case of 

 a quadrilateral, and the present method can, in fact, handle triangular approxi- 

 mation, if that is desirable. 



This method of geometric representation has been used without modification 

 as a basis for analyzing complicated shapes in hypersonic flow. Figure 4 is 

 taken from some of this work to give an impression of the accuracy of the method. 

 The figure was made by an SC4020 plotter (3). 



Induced Velocities 



On each element one point is selected at which velocities and pressure are 

 to be evaluated. For two-dimensional and axisymmetric bodies, the point selec- 

 ted is the midpoint of the line segment that is the trace of the element in the 

 plane of the profile curve, i.e., the average of two successive points that were 

 used to define the profile curve (Fig. 3). This is the obvious choice for two- 

 dimensional bodies and probably a reasonable selection for axisymmetric 



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