Recent Progress in the Calculation of Potential Flows 



Assume for the moment that the surface source density on the jth element 

 has a constant value of unity. Denote by V. j the vector velocity at the control 

 point of the ith element that is induced by a unit source density on the yth 

 element. The formulas for the induced velocity V^j are the basis of the pres- 

 ent method of flow calculation. They are obtained by integrating the formulas 

 for the velocity induced by a point source over the element in question, and thus 

 depend on the geometry of the element and the location of the point where the 

 velocity is being evaluated. Since there is no restriction on the location of the 

 control point of the ith element with respect to the >th element, the formulas 

 for V- J are those for the velocity induced by an element at an arbitrary point 

 in space. The dependence of the induced velocity on the geometry of the ele- 

 ment means that there are three completely distinct sets of formulas forV^j , 

 corresponding to the three different types of elements that are needed. Differ- 

 ent kinds are used according to whether the bodies are two-dimensional, axi- 

 symmetric, or three-dimensional. The axisymmetric situation is further sub- 

 divided into the case where the flow is also axisymmetric (i.e., the source 

 density is independent of circumferential location), and the case where the flow 

 is not axisymmetric but is due to a uniform stream perpendicular to the axis of 

 symmetry of the body (i.e., the source density varies with circumferential loca- 

 tion in a known way). The induced- velocity formulas (1) are rather lengthy and 

 will not be given explicitly here. A brief discussion of their nature follows. 



In two-dimensional and three-dimensional cases the elements are those of 

 a plane, and the integration over an element may be performed analytically to 

 obtain explicit expressions for Vij in terms of logarithms and inverse tangents. 

 (Obviously, the two-dimensional formulas can be obtained as limiting cases of 

 the three-dimensional formulas, but this is not a computationally efficient pro- 

 cedure.) In three-dimensional cases, so many elements are required to approx- 

 imate adequately the body surface, that the use of the rather complicated induced- 

 velocity formulas obtained by direct integration is quite time-consuming. Ac- 

 cordingly, these formulas are used only when the control point of the ith ele- 

 ment is within a few element dimensions of the jth element. For points farther 

 away, approximate formulas based on a multiple expansion are used. If the point 

 in question is farther from the centroid of the element than four times the max- 

 imum dimension of the element, the actual quadrilateral source element may be 

 replaced by a point source of the same total strength located at its centroid, 

 with no loss in the overall accuracy of the method and with a very large saving 

 in computation time. In both two-dimensional and three-dimensional cases, the 

 computation is not significantly complicated by the condition i = j; i.e., the 

 velocity induced by an element at its own control point is calculated without un- 

 due difficulty, because the integration is analytic. This velocity has a magnitude 

 of 2tt and a direction normal to the element [see the discussion preceding Eq. 5)J. 



For axisymmetric bodies the surface element is a frustum of a cone, and 

 the integration over the element of the velocity induced by a point source cannot 

 be performed analytically. First, the integration in the circumferential direction 

 is accomplished, to give the velocity induced by a ring source, which is expressed 

 in terms of the complete elliptic integrals. The resulting expressions are then 

 integrated numerically over the line segment that is the trace of the element in 

 the plane of the profile curve, as shown in Fig. 3. The number of coordinates 

 used in the numerical-integration scheme decreases with increasing distance of 

 the control point from the element in question. Thus, a saving in computation 



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