24. R. A. dimming, et al, "Highly Skewed Propellers," 



Trans . SNAME, Vol 80, 1972. 



25. T. T. Huang, et al, "Stem Boundary-Layer Flow on 



Axisymmetric Bodies," Twelfth Symposium on 

 Naval Hydrodynamics, National Academy of 

 Sciences, Wash. D.C., 1978. 



26. Damon E. Cummings, "Numerical Prediction of 



Propeller Characteristics," Journal of Ship Research, 

 Vol 17, No. 1, March, 1973". 



27. Nancy Groves, "An Integral Prediction Method for 



Three-Dimensional Turbulent Boundary Layers on 

 Rotating Blades." Paper presented at "Propellers 

 '8 1 Symposium," SNAME, May, 1 98 1 . 



-{ 



I r d* U 



c d\l/\ W 



— + ot 1 — 



D dxJV 



For this cross product to be zero, the slope of the streamline 



dip 

 "dT, 



_c_ W_ 

 D V 



(124) 



i l r u n 



— VI + N„ 2 a — 



2 > R n V V 



APPENDIX - STREAMLINE COORDINATE SYSTEM 



It is often convenient to have an orthogonal coordi- 

 nate system on the surface of the blade. In particular, for 

 performing boundary-layer computations, an orthogonal 

 coordinate system with one variable along the streamlines 

 reduces the number of terms in the governing equations. 

 To determine the differential equation of the streamline 

 path, lei 



*(x r ) 



(120) 



For lines along the surface which are normal to the 

 streamlines, let 



k. (x R ) 



(125) 



be the chordwise position as a function of radius. Then a 

 vector on the blade surface tangent to this line is 



ds ((£ (x R ), x R ) 



be the radius of the streamlines as a function of the chord- 

 wise coordinate x„. Then 



s*(xj = s(x„, iMxJ) 



(121) 



(126) 



is the position vector of the streamlines on the blade surface. 

 Hence a tangent to the streamline is 



-* 



= D 



— e , + I a e j + 



c/x R = t// \ 3x R/x, 

 N!Xe 



R 



D- 



D 



I \ d<p 



d77 



di// 



(122) 



C d K. 1 I r „ 



— e, + ae, +-V1 +N R 2 ?. 



D ' dx D ' 2 K 



The condition to be satisfied is that _t be perpendicular to 

 the velocity vector, or 



(127) 



d^ 



./ '■> dii 



2 K o dx, 



For this tangent vector to be parallel to the velocity vector 

 on the surface, the vector cross product, t ^ X q, must 

 be zero. Hence, for the velocity on the blade surface given 

 by 



U 



= — e, + 

 V "' 



the cross product is 



(123) 



dx 





u w „ 



X {— e, + — e 



lv -' V - 



U / c d k \ i , W 



(128) 



Thus the slope of lines on the surface which are normal to 

 the streamline is: 



1 I T- TV U 



— VI + N„ 2 — + a — 



2 " R„ V V 



o V 



S. R. 

 D V 



(129) 



One now has differential equations to determine an orthog- 

 onal network over the blade surface. The differential arc 

 length along the streamlines is 



ds = | dsj = h, dx 



3s 



d 



(130) 



(131) 



21 



