Recent Progress in the Calculation of Potential Flows 



39> 

 9r 



-113,535.4 



Fig. 17 - External boundary condition approximates 

 that obtained from hypersonic flow theory 



1. cp(0.8) = 1, (P(l) = 



2. qp(0.8)=0,(p(l)=l 



2 



3. qp(0.8) = 0, CP(1)= PjCcos 6) - 1/2 (3 cos 6-1) 



4. cp (0.8) = 0, qp(l)= P4(cos 8) - 1/8 (35 cos'^ 9 - 30 cos^5+3) . 



Here e is the angular coordinate measured from the symmetry axis 

 as shown in the figure. The Legendre polynomials Pj and P4 are de- 

 fined above. Two element numbers were used. In the smaller case 

 each element on each sphere had a 3° angular extent. For an exterior 

 problenn this is certainly adequate. However, the elenaent length on 

 the outer sphere is longer than one -fourth the 0.2 distance between 

 the spheres. In the larger case the elements were simple halved to 

 give a 1-1/2° angular spacing and an element length of about one - 

 eighth the distance between the spheres. Calculations were com- 

 pared with analytic solutions for the values of the radial (nornnal) 

 derivative of the potential on the sphere surfaces. The maximum 

 errors in the calculated derivative are shown in Table 3 as per- 

 cents of the maximum value of the derivative on the surface for each 

 case. The accuracy is quite good even for the smaller (large element) 

 case. For all boundary conditions halving the element size halves 

 the error, so accuracy is linear in element number. Computing time 

 is quadratic in element number, and thus it is advisable to use small- 

 point number cases. 



359 



