A General Theory for Marine Propellers 



Let us assume that an harmonic analysis of the circumferential wake veloc- 

 ity variation has been done; for the gth harmonic we may write 



Uo(x,r,0) = Uo(x,r)g cos q0 + iuo(x,r)^ sin q<; 

 Vo(x,r,0) = Vo(x,r)3 cos qcjy + iVo(x,r)^ sin qc; 

 Wq (x, r ,0) = Wq (x, r )3 cos q<?^ + iw^Cx, r )^ sin qg 



(80a) 

 (80b) 

 (80c) 



where Uq(x,t)^, VQ(x,r)g, and w^(x,r)^ are the amplitudes of the real or cosine 

 parts and Uo(x,r)jj, vo(x,r)j^, and vjq(x,t)^ are the imaginary or sine parts of the 

 three wake velocity components respectively. 



For this gth-harmonic wake velocity a qth-harmonic load distribution is in- 

 duced over the blade to assure that the fluid velocity relative to the blade will 

 always be tangent to the cambered surface. Figure 3 shows an elementary cam- 

 bered surface at P' (x' , r,0' ). The relative fluid velocity there must be tangent 

 to the cambered surface. Let us first ignore the radial fluid velocity component. 

 Then we must have 



tan = 



Uo(x 



and 



tan V' 



Vo(x 



Uo(x 



, r ) cos qcf) + u(x' , r,0' ), 



, r ) cos 



+ V ( X , r 



), 



r), sin q4> + u(x',r,(j^'). 



^o('''''')b ^i" "^"^ + v(x',r,0')^^ 



(81a) 



(81b) 



Equation (81a) states that the real part of the resultant fluid velocity must be in 

 the direction of P'D and Eq. (81b) states the same fact for the imaginary part. 

 The propeller blade velocity does not enter these equations, since the propeller 

 rotates with a constant angular velocity n and advances with a constant velocity 

 V. There is no harmonic content in n and V, except for q equal to zero. For q 

 equal to zero we have 



tan 



"o(x'' '•)a + u(x'. r.0') - V 

 VoC'^'-Oa + v(x',r,0') - rn 



(81c) 



The induced fluid velocity at P'(x' , r,<?^' ) due to a load distribution over the 

 cambered surface can be approximated by the induced velocity at P(x, r,0) on the 

 chord due to the same loud distribution along chord lines rather than mean lines. 

 Since the distance PP' is very small, the wake velocity at P' can also be approxi- 

 mated by the wake velocity at P. Then we have for q ^ 



tan i// 



UQ(x,r)g cos q0 + u(x,r,r6). 

 Vq(x,t) cos q0 + v(x,r,0). 



(82a) 



115 



