Weissinger and Maass 



CO 



7 = c^ cot — + / c sin nS , (4.18) 





and introducing the coefficient vectors - 



{q} = -fqo' ^r ■■•} . "fc} = {cq, Cj, •••} , 

 the solution can be expressed as 



{c} = ^^ Ct [0]{q} , (4.19) 



with a matrix [0] which depends only on \ and which has been tabulated in (153) 

 as a 7x7 matrix for k = 0.25, 0.5, 0.75. For small \. the matrix is close to the 

 unit matrix. Interpolation for k may be possible. The vector q has also been 

 tabulated for several values of k, Xp c, and /x(0.9 </x< i). Similarly, the con- 

 tinuous part u~ = ±7/2 +u~ of the axial velocity u^ induced by y at the duct is 

 expressed by a Fourier series whose coefficients can be calculated as a vector 

 [s] {c} with a tabulated matrix [S] . Then, the total axial velocity at the duct and 

 the pressure coefficient can easily be calculated by means of Legendre functions. 



For /i = 1, the ring vortex distributions y^ on the duct is given by y^ = 

 7+7^ with 7^-0 for -c/2 < X < Xp and y^ = const, t o for Xp < x < c/2. 

 Since the induced velocities must be continuous inside the duct, the jump of 7^ 

 at x = Xp must be cancelled by a jump of y equal to -74,. Therefore, the Birn- 

 baum series (4.18) of y converges very slowly, corresponding to a slow conver- 

 gence of the Fourier series (4.7) for the discontinuous function Qj ^^(a)- There- 

 fore, for fj- = 1 the procedure requires a large number of Fourier coefficients 

 q^, and a tabulation to n = 12 as given in (153) may not be sufficient. Probably, 

 as indicated above, the methods of the first part of Sec. 3. should be preferred. 



Of course, slow convergence will also occur if the tip clearance is very 

 small, i.e., if /x is close to unity. In this case the logarithmic singularity of 

 vy^ [or Qi/2(o^)] is replaced by a sharp peak at x = Xp. The peak is still pres- 

 ent for fairly large tip clearance, as can be seen in Fig. 18 (j = 0, cjp = 0.1, 

 \ = 0.5, /x = 0.9), at X = Xp = for a cylindrical duct. The corresponding vor- 

 tex distribution is shown in Fig. 19, together with the distribution for the pro- 

 peller located at Xp = -0.25c. At x = Xp, an indication of the jump occurring 

 for M = 1 can be observed. The corresponding pressure distributions are 

 shown in Fig. 20. The distributions on the outer surface are practically inde- 

 pendent of the axial propeller position. The inner duct surface pressure decays 

 almost to zero immediately behind the propeller plane. The duct-to-propeller 

 thrust ratios cx^/cx for the two cases are 0.683 and 0.706 for Xp = -0. 25c and 

 Xp = 0, respectively. Even in this case of large tip cleraance this ratio is al- 

 most independent of Xp; for /x = 1 it is totally independent. 



In (153), the influence of an improved determination of the pitch is also in- 

 vestigated. The new pitch j j = Ji(x) is defined by j 1 = J + u where u is the 

 mean axial velocity on the slipstream surface induced by y and y,.. Then, from 

 (4.13), an improved slipstream distribution y^^ ^ is obtained and the induced 

 radial velocity v~< ^ ) is introduced in the right-hand side of the integral 



1236 



