Cox 



17. Morgan, W.B., and Wrench, J.W., "Some Computational Aspects of Pro- 

 peller Design," Methods in Computational Physics, Vol. 4, Academic 

 Press, 1965 



18. Pien, P.C., and Strom-Tejsen, J., "A General Marine Propeller Theory," 

 Seventh ONR Symposium on Naval Hydrodynamics, 1968 



APPENDIX A 

 FORMULATION OF THE PROBLEM [15] 



Consider a propeller blade with no 

 translation velocity, rotating with angu- 

 lar velocity oj in a fluid whose free- 

 stream velocity is u(r), see Fig. Al. 

 Perturbation velocities u*, u*, u^ 

 are defined such that ° 



V* = U + U*, V* = u*, v*^ -- u*^, 



while for upstream, i.e., x* — . -», 



< r* < cOj 



V* = U, V* = V* = . 



X r e^ 



The linearized equations of motion 

 with respect to fixed axes are 



3u* 3u* ^ p* 



Fig. Al - The co- 

 ordinate system 



where u(u*, u*, u^^) is the perturbation velocity vector and p* the perturba- 

 tion pressure [9,10,11]. A perturbation velocity potential <p* exists such that 



u^ = V 



hence, taking the divergence for both sides of Eq. (Al), it can be shown that an 

 acceleration potential p* = p* /p exists. Provided that du/dr is a second- 

 order term, Eq. (Al) can be written as 



— + u — 



^t 3x* 



(A2) 



which possesses a solution 



950 



