A General Theory for Marine Propellers 



22. Lamb, H., "Hydrodynamics," Sixth Edition, Dover, New York, p. 60 ^, ,: 



23. Abott, I.R., and von Doenhoff, A.E., "Theory of Wing Sections," Dover, 

 New York, 1958, p. 79 



24. Watkins, C.E., Woolston, D.S., and Cunningham, H.J., "A Systematic Kernel 

 Function Procedure for Determining Aerodynamic Forces on Oscillating or 

 Steady Finite Wings at Subsonic Speeds," NACA Report R-48, 1959 



Appendix 

 SUPPLEMENTAL DISCUSSIONS ON THE BASIC CONCEPT SECTION 



During informal discussions with Prof. J. Weissinger, a point was raised as 

 to the general applicability of our preliminary conclusion that the term q ^ ^ in 

 Eq. (4) was zero and could be omitted in the subsequent mathematical develop- 

 ment of the basic concept for the marine propeller problem. After a closer ex- 

 amination, we found that our conclusion q < ^ is zero in the wake was apparently 

 an error. In this appendix we would like to rectify our error by reasoning that 

 we can assume as a simplification that q x ^ is zero and offering the following 

 discussion to supplement our reasoning and to bridge the gap in the formulation. 



We shall begin with the general equation of motion for an inviscid fluid par- 

 ticle under the influence of an external force field, 



-,^.. ...I:; ^'^'.'u. ....... ....... (^1) 



where q is the velocity vector, Ap is the gradient of the pressure field p,pf is 

 the fluid density, and F is the external force per unit mass. The left-hand term 

 represents the acceleration of the fluid particle which may be expressed in two 

 parts, namely, a term representing the local acceleration at a fixed space and a 

 convective term due to the movement of the fluid particle as follows: 



^--P-. (qV) q . ' ' (1) 



Combining Equations (Al) and (1) we obtain • • ' >■'■"' ^ ' ' ^ 



1^= - J- Vp - (qV) q + F . (A2) 



at y { 



Since . ,: 



(qV) q = -| Vq2 - q X ^ , 



127 



