Malavard 

 1. On the blade surface, the zero normal velocity condition for radius r is 



A' + ^' i*(<f.T) , (13) 



where <t> = <P'/'^'R^, ^ = Vq/ojE, ^ = r/R, and i* is respectively the slope of the 

 upper or lower surface of the blade at a given reduced radius, <f , and the curvi- 

 linear abscissa along a chord, t, n the normal directed towards the fluid. 



The tangential velocities are connected to the pressure by 



'^-^ = — =^Cp(^--) . (14) 



where Cp(^,T) = (p~ - p'^)/(pco^R^/2) is the local pressure coefficient. 



This expression can be integrated with respect to r, which brings us to a 

 condition similar to that of paragraph 3B in the Hydrodynamic Problem section 

 of this paper. As in that paragraph, there are here two problems: 



Direct problem: i (^,t) is given, which is equivalent to giving the form of 

 the blade, or — 



Inverse problem: Cp(^,T) = c (^) g(??,T-Tj) is given. 



2. On the trailing edge the Joukowski condition is conveyed by 







3. The pressure continuity in the wake is conveyed, according to the 

 linearized Bernouilli equation, by a potential difference S0(,f) = 0+ - 0^, which 

 depends only on <? , between the two sides of the helicoidal free vortex sheet. 



4. At far downstream, the potential presents, as in the lifting line theory, 

 the helicoidal symmetry of p-order. The blowing of the propeller implies the 

 existence of induced velocities in the axial and tangential directions. If, then, 

 the reproduction of the field is limited at infinity by a surface perpendicular to 

 the axis, conditions on the normal derivative to this surface must be respected. 



According to the conditions in Eqs. (13) and (14), it is evident that the func- 

 tion 4^ is defined in the present c ase by c onditions resembling those in para- 

 graph 3B, except for the factor \f>J~~^~T^, which is taken into account here. It 

 is easy to see that the equilibrium condition of a cavity with constant pressure 

 P is given by 



30 

 Br 



398 



(15) 



