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Published online by Cambridge University Press: 28 July 2016
The Townend Ring has been the subject of a large amount of experimental research and its general characteristics are now well understood. The following paper is the first theoretical treatment of the ring which has appeared. It was originally presented as a thesis to the University of Delft by Capt. G. Otten and published in Holland in November, 1932. The author has made the following abbreviated translation of it into English in collaboration with Dr. H. C. H. Townend, who also suggested a few modifications.
* Engineer Officer, Royal Dutch Air Force, East Indies.
† Published in R. & M. 1267, and also in “Aircraft Engineering,” April, 1930, and the Journal of the Royal Aeronautical Society, October, 1930.
* In later calculations (p. 897) a more accurate expression for v is desirable, but as the inclusion of terms of the second order presents great difficulties we have written from (6a)
Q = -2hu1=2hf/ρ(V-u'+u)
from which v is deduced in the usual manner. This method may not be quite satisfactory, as only part of the second order terms are used, but for small values of a, however, they seem to be the more important ones.
* As the motion is two-dimensional, the induced drag is zero, so that the frictional drag coefficient alone appears.
* This formula differs from that derived from (9) 
, only by the term [l/4]k2 in the denominator, which can be regarded as a quantity of the third order.
* Dimensions published in Erge. Aerod. Vers. Göttingen III., 1927, p. 28.
† loc. cit.
† Hydro und Aeromechanik, Zweiter Band, Berlin, 1931, p. 143.
* Compare Whittaker, E. T. and Watson, G. N., Modern Analysis (Cambridge, 1920), p. 251.Google Scholar It should be remembered that k '2 = 1 - k 2.
* The derivation is due to Prof. J. M. Burgers, of Delft.
* Journal of the Royal Aeronautical Society, October, 1930, p. 820.
† In this Chapter, airscrew interference is presumed to be absent.
* The remaxks in this paragraph relate, strictly, to engines not mounted in a body, but the argument is probably applicable to engine-body combinations also.
† loc. cit.
* loc. cit.
† loc. cit. (Journal, p. 825).
‡ loc. cit. (Journal, p. 828).
§ loc. cit. (Journal, p. 828).
* Fifth Congres Internationale de l’Aviation (The Hague, 1930), pp. 568 and 573.
* See Glauert, H., M.A., “The Elements of Aerofoil and Airscrew Theory,” Cambridge, 1930, p. 162.Google Scholar Here the horseshoe vortices are bent.