concepts, only a few papers consider the marine propeller. In fact, a 

 majority (over 80 percent) of the work in this area is associated directly 

 with the aircraft industry. However, although the particular applications 

 are quite different, the results of the aircraft tip vortex alleviation 

 work can be applied, to varying degrees, to the marine propeller. The 

 limits of applicability and the disparities in the literature will be high- 

 lighted in later sections of this report. 



REPRESENTATION OF TIP VORTEX ROLLUP 

 The earlier attempts — Lamb and Prandtl — to represent the complex vortex 

 rollup phenomenon generally consisted of a simplified, two-dimensional, 

 inviscid theory, where a vortex sheet emanates from the trailing edge of 

 a lifting wing and rolls up, in the form of a spiral, under the action of 

 its self-induced velocity field. The initial strength of the vortex sheet 

 is determined by the spanwise load distribution of the wing. This over- 

 simplified model failed to correctly predict the sizes and strengths of 

 the observed vortices. As more experimental data emerged, later models 



became more realistic and elaborate; for example, these models began to 



1* 

 incorporate both the viscous effects governed by the wing-tip boundary 



2 

 layer and an observed axial velocity in the vortex core, which basically 



introduced a three dimensionality to the models. A recent vortex core 

 representation, shown in Figure 2, includes four distinct regions: (1) 

 a viscous inner region, (2) a smoothed-out spiral where the velocity dis- 

 tribution is essentially inviscid, (3) a tightly wound spiral, and (4) an 



external region containing the unrolled portion of the vortex sheet. 



1 3 



Some results from this theory are compared with experiment in Figure 



3, which shows the variation of the vortex core axial velocity w_ with 



Reynolds number R . The observed disagreement is not totally unexpected 



since the theory is confined to laminar flow, which renders comparison with 



high-Reynolds-number, turbulent-flow, wind-tunnel experiments somewhat 



uncertain. In addition, these models are limited to very simple wing 



4 

 planform and loading distribution. Recently, numerical techniques have 



*A complete listing of references is given on page 59. 



4 



