583 



in wliicli formula V represents tlie mean velocity (i.e. the volume 

 of fluid which in unit of time flows through a section of the tube, 

 divided by the area of that section), d the diameter of the tube, 

 and Q the density of the fluid, then C is called the coefiicient of 

 the resistance, and appears to be a function of the characteristic 



number introduced by Reynolds: R= — ^ ((it is the coefiicient of 



ft 



viscosity of the fluid). The value of C for different cases is given 

 in textbooks; as an example may be mentioned: 



a. for rough walled tubes C is approximately independent of R; 

 however, it is a function of the roughness; 



h. for very smooth tubes of circular diameter: 



C= 0,1582 /?-"'). 



The greater part of the theoretical investigations on the turbulent 

 motion treat the problem: how does it originate?') An explanation 

 of the increase of resistance which accompanies the appearance of 

 the turbulent state of flow has been given by Reynolds and Lorentz'). 

 More than once it has been remarked that this problem is one of 

 statistical nature *). The resistance experienced by the fluid and 

 indicated by our measuring apparatus is a mean value. It is possible 

 that such a mean value may be calculated sufiliciently approximate 

 without an exact knowledge of the fluctuating and never exactly 

 returning relative motions. 



In the following lines a preliminary attempt is made to determine 

 the value of the resistance and to explain the quadratic law. In the 

 first part (paragraphs 2 and 3) two equations given by Reynolds 

 and Lorentz are discussed and put into such a form that immediately 

 appears what quantities are wanted in order to calculate the resistance. 

 In the second pai't (paragraphs 4 and 5) a simple idealized "model" 

 of the turbulent flow is constructed which allows these quantities 

 to be determined. 



Instead of the flow through a tube or channel a more simple 



1) Gomp. fi. R. VON Mises. Elemente der technischen Hydromechanik I (1914) 

 p. 57 and H. Blasius, Mitt, iibei- Forschungsaibeiten, herausgeg vom V, D. I., 

 Heft 131 (1913). 



') Gf. F. NoETHER, ZS. fur angew. Math. u. Mechanik 1, p. 125. 1921. 



') 0. Reynolds, Scientific Papers IT, p. 575 — 577; 

 H. A. Lorentz, I.e. p. 66—71. 



*) Among others by Th. von Karman at a lecture at the 'Versammlung der 

 Mathematiker und Physiker" in Jena 1921; comp. a remark in the ZS. fur angew. 

 Math. Li. Mechanik 1, p. 2.50, l'.i21. 



