Recent Progress in the Calculation of Potential Flows 



because the matrix Aij is neither symmetric nor sparse. In fact, none of the 

 terms of A^j need iteration by zero in general. The matrix does, however, 

 have a dominant main diagonal. It will be recalled that the diagonal terms A^ ^ 

 are exactly in in two-dimensional and three-dimensional cases. They are 

 fairly close to this value in axisymmetric cases. To a first approximation, the 

 sum of all diagonal terms equals the sum of all off -diagonal terms, and thus on 

 the average each diagonal term equals the sum of the other term in its row. For 

 convex bodies all terms are positive, and thus similar statements hold for the 

 absolute values of the terms. Because of the dominance of the main diagonal, 

 the Gauss-Seidel iterative procedure has been found to be quite effective in the 

 solution of the set of equations in Eq. (7). It has converged in all cases. 

 Usually, convergence is quite rapid, although for certain extreme types of 

 bodies this may not be true. Unfavorable cases typically require as many as 

 200 iterations, but normal cases converge in about 16 iterations or, more pre- 

 cisely, 4 iterations per decimal place in a. Methods of accelerating the con- 

 vergence have been studied and found effective, but have not been incorporated 

 into the method. A great deal more information about convergence rates is 

 given in Ref. (1). 



Calculation of Velocities ;, ^ -:•- , .:_..;;:; ■ r; : .-: j ; sl-:?^ ;. 



Once the set of equations in Eq. (7) has been solved, the velocities at the 

 control points of the elements are calculated from 



^i = L ^ij^j ^ ^- • i = 1' 2, ...,N. ■ (8) 



j = i 



Potentials can also be calculated, using similar types of formulas, if desired. 

 The pressure coefficient Cp is computed by means of Bernoulli's formula. For 

 unsteady flow, it is 



f = P(t)-i|V|S^^. •, (9) 



p ^ d t 



where P(t) is independent of position in the field. For steady flow, Eq. (9) leads 

 to the well-known formula for pressure coefficient. 



c = P^~ P- = 1 - fXi] . (10) 



Velocities and pressures at points off the body are calculated from Eqs. (8) and 

 (10) after sets of V^j appropriate to the points in question have been calculated. 

 This method is well suited to the simultaneous calculation of several onset flows 

 at once, since the induced velocities V; j do not depend on the onset flow as long 

 as the basic form of the source density is not affected, which is always true for 

 two-dimensional and three-dimensional cases. This feature has been found 



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