NO. 4 WIND TUNNEL EXPERIMENTS IN AERODYNAMICS 1/ 



The properties of the fluid which determine its mechanical behavior 

 are its density p, its viscosity /x, and its compressibility. Instead of 

 the viscosity, it is generally more convenient to use the kinematic 



viscositv V— ^ which will do equally well when p is given. And simi- 

 P 



larly, the speed C of sound waves in the fluid is fixed by the density 

 and compressibility so that, conversely, C together with p fixes the 

 compressibility. The properties of the fluid which concern us may 

 therefore be specified by stating the values of the density p, the kine- 

 matic viscosity v, and the acoustic speed C in the fluid. 



We have now enumerated the quantities on which the force R may 

 be supposed to depend, and if nothing has been overlooked there 

 must be a complete relation connecting R with the other quantities. 

 We may state the fact that such a relation subsists by writing the 

 equation 



/ {R,S,D,r',r", ,p,v,C)=o, (i) 



and our first task is to obtain from general principles any information 

 we can about the form of this unknown function /, which wilj enable 

 us to restrict the amount of experimentation required to finish the 

 work of finding the form of the equation. 



Application of the Principle of Dimensional Homogeneity 



By the well known " principle of dimensional homogeneity," all the 

 terms of a complete physical equation must have the same dimen- 

 sions, and this fact enable's us to simplify equation (i). Let n repre- 

 sent a dimensionless product of the form 



Ti = R''S^D''p'vC^, (2) 



the numerical exponents a, jS, y, etc., being such as to satisfy the 

 dimensional equation 



[7?'^5-^Z)V"^Cf] = [i] (3) 



when the known dimensions oi R, s, D, p, v, and C are inserted. Then 

 it may readily be shown : * ist, that since three fundamental units are 

 needed as the basis of an absolute system for measuring the six kinds 

 of quantity, R, S, D, p, v, and C, the number of possible independent 

 expressions of the form (2) is 6 — 3 or 3; and 2d, that if these expres- 

 sions are denoted by IT^, n,, Ilg, any correct equation involving the 

 quantities which appear in equation (i) and no others, must neces- 



* Physical Review (2), 4, p. 345, October, 1914. 



