Smith 



time is effected with no loss of overall accuracy. The case of i = j , i.e., the 

 calculation of the velocity induced by an element at its own control point, re- 

 quires special handling for axisymmetric elements. Here the result cannot be 

 predicted in advance, as it can be for the two-dimensional and three-dimensional 

 elements, because of the complicated nature of the ring- source formulas. The 

 procedure is described in detail in Ref. 1. Basically, it consists of a series ex- 

 pansion of the integrand about the singularity at the control point. In the case 

 of axisymmetric flow, the induced velocities have two components, one parallel 

 to the axis of symmetry of the body and one radially outward from or inward to 

 this axis. In the case of flow due to a uniform stream perpendicular to the axis 

 of symmetry, the circumferential variation of the surface source density gives 

 rise to an additional circumferential component of induced velocities. 



The Set of Linear Equations for the Values of Surface Source Density 



A complete set of N^ induced-velocities V^j is computed, to give the veloci- 

 ties induced by all elements at each other's control points. (It will be recalled 

 that N denotes the total number of elements used to approximate the body sur- 

 face.) In this calculation a constant unit-value of source density is assumed on 

 each element. The quantity 



A,. = n,.\,. , (6) 



obtained by taking the dot product of V- j with the unit normal vector n^ of the 

 ith element, is thus the normal velocity induced at the control point of the ith 

 element by a unit source density on the ith element. Multiplying A^j by the 

 constant but unknown value of o-^ of the source density on the jth element then 

 gives the actual normal velocity at the control point of the ith element due to 

 the ith element. This is the contribution of the ith element to the integral of 

 Eq. (5), where that equation is being required to hold at the control point of the 

 ith element. Summing the normal velocities due to all elements at the control 

 point of the ith element, setting the result equal to the negative of the normal 

 component of the onset flow at that point, and repeating the process for the con- 

 trol points of all elements will give a set of linear algebraic equations for the 

 values of the source density on the elements. Specifically 



L ^j^j =-"i-'^- ' i = 1, 2, ...,N, (7) 



j = i 



where Vco. is the onset flow evaluated at the control point of the ith element. 

 The set of equations in Eq. (7) is the approximation of Eq. (5). 



The set of linear equations in Eq. (7) is solved by an elimination procedure, 

 the method of successive orthogonalization, if the order N is less than 275. 

 This number of elements is sufficient for good accuracy in most two-dimensional 

 and axisymmetric cases. ForN greater than 275, the capacity of the computer 

 does not permit solution by direct elimination, and an iterative procedure must 

 be used. In practice, this means iterative solutions are used for three- 

 dimensional bodies and elimination for two-dimensional and axisymmetric bodies. 

 Many conventional matrix-iteration techniques are not efficient in this case, 



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