Turbulent Flow in Pipes and Channels. 337 



curvature over the range of velocities below the " critical 

 velocity/' there exists an indication of a tendency for the 

 series of points to show more than one interval ofdiscontinuous 

 slope ; the " critical " flow is not very sharply marked, but 

 a value is assigned corresponding to a value of the "critical 

 constant'" in the neighbourhood of K = 2080. Distributions 

 of velocities over the cross-section are studied in detail for 

 one of the pipes by means of a Pitot tube ; the results 

 indicate a gradual transition of shape from the "stream- 

 line ".to the "turbulent" distribution, and a close inspection 

 of these curves, even at the lower velocities, shows a tendency 

 for the curves to assume forms similar to those obtained by 

 the writer for two-dimensional flow of air. The observations 

 of Kohlrausch are not sufficiently numerous, nor the resolving* 

 power of the Pitot tube method of measuring velocities 

 sufficiently high, to reveal in their full detail the type of 

 distributions observed by the use of the linear hot-wire 

 anemometer. 



Perhaps the most remarkable observations on the subject 

 are those of Dowling 42 , which show an indication of an 

 approach to the results of the writer. The criterion of a 

 change in the flow distribution was obtained by an ionization 

 method. The most marked change was obtained for a mean 

 velocity giving values of K averaging 2480 for air and 2500 

 for carbon dioxide ; in the case of long thin tubes the values 

 of K were somewhat increased to 3120 and 3080 respectively, 

 and the anomalous effect of metallic tubes was also noted. 

 The most significant result of these observations was the 

 discovery of a second well-marked critical velocity con- 

 siderably below that from which Reynolds's Criterion is 

 established ; in fact, many of the curves obtained by 

 Dowling resemble in their general appearance those obtained 

 by the writer, connecting the velocity at a certain point and 

 the pressure-gradient in presenting a succession of well- 

 marked points of discontinuous slope considerably earlier 

 than that which is generally supposed to mark the commence- 



42 Dowling, J. J ., " Steady and Turbulent Motion in Gases," Roy. 

 Dub. Soc. Proc. vol xiii. p. 375 (1912). In these experiments, air 

 ionized over a short portion of the tube by a radium source, was 

 examined for electrical conductivity by the usual electrometer method 

 at a distant point. The rate of recombination depends on the nature of 

 the velocity-distribution curve in the tube between the ionizing source 

 and the exploring electrode : the electrometer leak is plotted against the 

 mean velocity and the resulting curve gives a remarkably sensitive 

 indication of sudden changes in the nature of the flow. 



Phil. Mag. S. 6. Vol. 31. No. 184. April 1916. 2 A 



