Pien and Strom-Tejsen 



Based on these considerations, it is logical to first develop a general theory 

 for marine propellers in inviscid fluid. Since the induced velocity due to blade 

 loading is independent of that due to blade thickness, a propeller with zero blade 

 thickness will be considered here as a first step. 



DEVELOPMENT OF THE THEORY 



Derivation of Kernel Functions 



It is assumed that the propeller blade thickness is zero and that the fluid is 

 inviscid. The steady case of a propeller operating in open water is considered 

 first; the propeller operating in the behind condition is discussed. 



Figure 2 defines a cylindrical coordinate system. The position of a moving 

 blade at time t equal to zero is also shown. The essential part of our problem 

 is to calculate the induced velocity at any point P(x, r,0) due to the action of the 

 blade. It is convenient in our discussion to introduce three equations: 



u(x,r,0) = \ [l(^,p,0) K^(x,t,4>;^,p,0) dpd0 , (19a) 



v(x,r,0) = r|L(if,p,0) KJx,r,cP;^,p,0) dpdd , (19b) 



w(x,r,0) = j JL(^,p,0) K^(x,r,0;^,p,0) dpd0 , (19c) 



where u(x,r,0), v(x,r,0), and w(x,r,0) are respectively the axial, tangential, 

 and radial induced velocity components at P (x, r,^), L(i,p,0) is the blade load 

 distribution, and Kj(x,r,0; ^,p,0') represents the contribution to the axial compo- 

 nent of the induced velocity at P(x, r,0) of a unit blade loading at Q(^,p,i9), etc. 



Our first objective is to derive expressions for the kernel functions Kj, K^, 

 and K^. In the case of zero blade thickness the blade loading L(cf,p,i9) can be 

 represented by the induced pressure jump Ap^ as discussed in the previous 

 chapter, and the field value of the acceleration potential $ at P (x, r.c/)) due to a 

 unit pressure dipole at QC^,/?,^) is 



where n' is the normal of the blade surface at Q(^,p,0) and R is the distance 

 from point Q(^,p,6') to P(x,r,0) . 



96 



