Interior field (x < x ) 



(7) 

 (8) 



B = > mI m A \K' m A (9) 



One recognizes that the induction factors depend on 3 variables, viz., on the ratio 

 x /x, the number of blades z and the pitch angle /?,. For an application, it is impor- 

 tant to note that these factors are independent of the circulation and are solely deter- 

 mined by the geometry of the vortex system. Numerical calculations of the factors 

 have been carried out for z = 3 to 5 [6]. 



It is now possible to solve a fairly general propeller problem, viz., to determine 

 the velocity components which are induced at any radius x of a bound vortex line if 

 the bound circulation is an arbitrarily given function compatible with the end condi- 

 tions at both the hub and the tip. This problem requires an integration of (1). This 

 integration has two difficulties, firstly that the integral (1) is an improper integral and 

 secondly that the induction factors depend on the unknown induced velocity compo- 

 nents by way of the pitch angle /?,-. The first difficulty is removed by expanding both 

 the given circulation function and the induction factors into Fourier series. It is then 

 possible to determine the principal value of the integral [3, 6]. The second difficulty 

 requires successive approximations starting in the first step with both w a and w t equal 

 zero to determine a first approximation for /?;. The convergence of this process is very 

 rapid for functions G (x) which are met in practical applications. Having determined 

 the components of the induced velocity the force components which correspond to 

 G (x) follow immediately from the law of Kutta-Joukowsky. 



It should be mentioned that the analysis of a free-running propeller with mini- 

 mum kinetic energy within the slipstream for a given thrust arises as a special case of 

 the afore written general equations. In this case the vortex sheets are of a true helical 

 shape corresponding to a rule by Betz [7]. From this rule the relation 



(w a /v) + (\ i /x)(w t /v) = w*/v (10) 



follows, within which relation the quantities Ai and w* are independent of the radius. 

 Introducing expressions (1), an integro-differential equation for the bound circulation 

 G (x) of the free-running optimum propeller is obtained: 



11 dG / \i \ dxo w* 



— ( i a + -it = 2- (11) 



! x n dx \ x ) X — .To v 



This equation permits a numerical calculation of the Goldstein factor which plays an 

 important role in the application of propeller theory. From Stoke's law, this factor is 

 defined by the following relation: 



k= (zG) / [ 2x- ] (12) 



V 



'/(' 



From both (11) and ( 1 ) the function « is obtained in a way which is independent of 

 Goldstein's former analysis carried out on a basis of Betz's displacement theorem [8]. 



157 



