Foa 



the pressure exchange phase, and with or without a subsequent mixing phase. 

 Similar but generalized treatments were developed in Refs. 5, 7, and 8. These 

 analyses considered situations in which the coning angle was zero and the width 

 of the interaction space was constant and very small compared to its mean ra- 

 dius, so that the flows could again be treated as two-dimensional in the interac- 

 tion region. Heat transfer and mixing effects were neglected. Under these con- 

 ditions, the effect of the interaction could again be described by a velocity vector 

 diagram like that of Fig. 4 and the thrust augmentation ratio could be calculated 

 as 



(^Id+M ^2d) COS ^^ - (1 + m) ^0 



The analysis of Ref. 8, which was limited to the case of an interaction duct 

 of constant cross -sectional area, revealed the necessity of velocity nonuniformi- 

 ties at the merger station — a necessity reflecting the implictly assumed irrota- 

 tionality of both flows in the deflection phase. Thus, the absolute velocities of 

 the secondary fluid particles have different orientations at the merger station, 

 although the overall transverse momentum of this flow is, of course, still zero 

 at this station in the absolute frame. The results of this analysis, for static op- 

 eration and for two values of the secondary-to-primary density ratio, are pre- 

 sented in Figs. 9 and 10. Here, as in subsequent performance charts, A^ and A^ 

 denote the cross -sectional areas of the primary and of the secondary flow, re- 

 spectively, at the merger station. These figures show that an increase of the 

 secondary -to -primary density ratio has a very favorable effect on the thrust 

 augmentation obtainable with the bladeless propeller (whereas it has the oppo- 

 site effect on the performance of the ejector). It should be noted that, because 

 of the specified absence of mixing, the bladeless propeller considered in this 

 analysis does not, for ^^. = 0° , reduce to an ejector: instead, it is reduced to 

 a totally ineffective device, for which ,^ = 1.0 throughout. 



An elegant generalization of this analysis was developed by Hohenemser in 

 Ref. 5, through the introduction of an "equivalent spin angle," defined as /3j ^' = 

 tan" ' (y/u.'). Use of this parameter in lieu of the actual spin angle made it pos- 

 sible to bypass the axial momentum equation, and hence to bypass consideration 

 of the shape of the interaction duct. Except for cases involving very large den- 

 sity differences between the two fluids, equivalent and actual spin angles were 

 found to differ relatively little from one another. In fact, the results of the two- 

 dimensional analyses of Refs. 5 and 8 are practically identical. 



A further refinement of the theory was introduced by Hohenemser, also in 

 Ref. 5, through a strip approach similar to the strip concept of propeller theory. 

 Here, the primary jet is assumed to be thin in comparison with the width of the 

 interaction space and is further assumed to penetrate this space at a coning 

 angle small enough to make it permissible to neglect primary velocity compo- 

 nents normal to the direction of the secondary flow. The interaction is then 

 treated as an infinite succession of infinitesimal steps, in each of which the pri- 

 mary jet and the elemental layer of secondary flow which it penetrates deflect 

 each other to a common direction. Coriolis and centripetal accelerations are 

 again neglected, as if these secondary flow layers were plane rather than annu- 

 lar. The resulting axial velocity distribution at the shroud exit is nonuniform, 



1360 



