Lehman and Kaplan 



,025 1 



.020- 



,015 



^ u2wR2 



,010- 



,005 • 



0- 



total 



vorticity 



thickness 



V- 



— r 

 .6 



.8 



1.0 



1.2 



— r 

 1.4 



Fig. 44 - Axial force as a function of the advance ratio for a 

 four -bladed propeller: /3 = 0.05, N = 4, tQ/2c = 0.10 



as the total axial force, for a number of conditions. Typical results are given in 

 Fig. 47, where it is shown that the dipole term is the predominant term, and the 

 same effect is true for the higher harmonics of the axial force. Thus it is es- 

 sential to include this particular induced singularity effect to obtain the major 

 component of the total axial force caused by a rotating propeller. 



The theoretical expressions show that the forces decay exponentially with 

 the distance between the propeller and the appendage, with this variation of the 

 form e'^'^^^o/Re ^ so that there is a faster decay with distance for the higher har- 

 monics and more blades. The exponential variation with distance holds for both 

 the terms due to the propeller vorticity and propeller thickness, so that the total 

 also varies in this manner. The variation with distance between the propeller 

 and appendage is shown by Figs. 48 and 49 for both the axial and transverse 

 force blade-rate components for the particular case where J = 1.0. The varia- 

 tion for a larger number of blades, and for the higher harmonics, is in accord- 

 ance with the exponential form indicated above. Since there is no dependence on 

 the advance ratio J in the expression for variation with distance, the theory im- 

 plies that the same decay rate will occur for all advance ratios. 



Another theoretical result is the proportionality of the axial force to the ap- 

 pendage thickness. The computations illustrated in the figures are carried out 

 for only one value of appendage thickness, and the axial force due to any other 



220 



