has more than five intervals, 2-deg intervals are sufficient; finer intervals, which 

 increase costs significantly, appear to be unnecessary. 



To check whether there were enough collocation points, we chose 10 points and 

 created the foil shape by plotting streamlines. The results have been compared with 

 the case of four collocation points given in Figures 7 and 8. The simple approxi- 

 mation obtained using Equation (39) is shown to express amazingly close agreement 

 with the plotted streamlines. 



In Equation (28) V„ is the local mean speed, which is not known without exami- 

 nation, so the value from the lifting line theory has been substituted for it. How- 

 ever, a more accurate formulation would be Equation (53) (Appendix A) instead of 

 Equation (28). 



-S-k+aU^Z- C47) 



V /^V\)V\V/V V 

 V V 



The computer results for these two cases were almost the same. The present program 



used Equation (47), although slightly more computer time was needed. When G/V is 



s 



quite large, it may improve the solution. 



To check whether the solution satisfies the boundary conditions well, the left- 

 hand and right-hand sides of Equation (47) were plotted in Figures 9-12; and the 

 radial components of velocities on the blades were plotted, Figures 13 and 14. These 

 calculations were made with and without the hub boundary conditions being satisfied 

 when the degrees of p and x in the double polynomials in Equations (31) through (33) 

 were taken as 3 in one case and as 5 in the other case. During this process, we 

 noticed an interesting phenomenon: an instability occurred in the numerical value 

 of radial velocity for the solution which did not satisfy the hub boundary condition. 

 That is, if the hub boundary condition was not specified, a slight change of parame- 

 ters, such as cavity length, number of intervals, or the degree of polynomials, 

 produced large changes in radial velocities. Yet, the numerical values of the thrust 

 and torque coefficients or the pitch distribution did not change too much. This may 

 be because the linear boundary conditions on a cavity or foil do not include any 

 constraint on the radial velocity. In the present problem, the only constraint on 



26 



