Weissinger and Maass 



steady radial wash v may be expressed accordingly as the superposition of the 

 individual contributions or 



"^ = ^q "^ ^7 "*" ^7' + ^r' "'' "^r' ■*" f^7 "^ ^y' + V~ + Vp + Vp + vj] . (6.1) 



The expression in the bracket turns out to be zero. 

 Putting 



7 = 70 + '>'i ' '>'o = VgQ(x) , 7j = V[A(x) sin<;6 + B(x) cos 0] , (6.2) 



one finds easily that q and y^ are the distributions of the ducted propeller at 

 zero incidence, which account for the effects of thickness and camber. So we 

 are left with the condition 



+ v„ 



1 



-Va sin 4) - Vp. . (6.3) 



The left-hand side, corresponding to the model of ring-airfoil theory, can be 

 written as v sin Tj A + v cos Ti B by means of the operator Tj . 



Since only the first harmonic is present in either the case of the ring air- 

 foil at incidence or that of the propeller at incidence, the distribution r will 

 turn out to vary sinusoidally with fit such that, from Biot-Savart integration, is 

 foimd 



Vp, 3 V[F(x) sin0 + G(x) cos 0] . (6.4) 



The functions F(x) and G(x) involve complicated double integrals over the am- 

 plitude and phase angle of r(r) which must be determined by solving the 

 unsteady -propeller problem explicitly. 



K F(x) and G(x) are known, Eq. (6.3) splits into the two equations 



T^ A = -a - F , Tj B = -G , (6.5) 



which can be interpreted as the equation for a ring airfoil at incidence with a 

 modified ^-dependent camber and can be solved by the numerical methods de- 

 scribed in Sees. 1 and 2. 



The distribution r(r) is determined in the following manner. At each pro- 

 peller blade the unsteady axial velocity component due to y^ and the unsteady 

 tangential component due to both the incidence cross flow and the duct trailing 

 vortices y[ are calculated. The component of this velocity vector taken per- 

 pendicular to the effective free stream composed of v and fir gives the unsteady 

 downwash at the propeller blade. This downwash has a sinusoidal distribution 

 over the blade chord. Now, at a representative radius r = tq (e.g., ro^= 0.7 Rp) 

 the solution of Kemp (41) for the sinusoidal gust problem gives a value r(ro) 

 that depends on a phase angle and an amplitude factor, both of which can be ex- 

 pressed by Bessel and Hankel functions of the reduced frequency nd/(2\/v2Tn27^) 

 (d = propeller chord length). The application of Kemp's two-dimensional theory 

 implies that the blade interference can be neglected, i.e., that the blade number 

 N is small. 



1244 



