IV-28 



Consider a fluid which is maintained by some means or other in an 



unstable state, i.e., a state which results in the occurrence of turbulence. Suppose 



that an external source of energy provides a power P per unit mass of the medium. 



m-l^ 1 l^ 



The dimensions of P are power per unit mass = —3 — • — = -3~. We suspect the 



t m t 



existence of a spectrum of the velocity of the turbulent motion which is pretty uni- 

 versal, i.e. , a spectrum which, at least for a substantial portion of the frequency, 

 is independent of the details of the energy source and independent of the detailed 

 properties of the medium, but subject only to the condition that turbulence is main- 

 tained. The resulting spectrum E^(k) can, if it is to be universal, depend only on 

 the variables k and P: 



E = E (P, k) (IV- 65) 



1 -t? 



The dimensions of k are j. The dimensions of the spectrum E^ are -g- , as may 



be seen from the definition of a spectrum as in (IV- 64), with the realization that 



for the velocity spectrum the mean square velocity fluctuations < u^ > must be 



used instead of < u^ >. There is clearly no dimensionless combination of P and k, 



and we are, therefore, looking for a combination of P and k which is dimensionally 



consistent with E : 

 u 



E^~P™k" or p-~^^7H- <IV-66) 



It follows at once that m - 2/3 and n = -5/3, and in the range of interest, the 

 spectrum must be of the form: 



2/3 -5/3 



E ~P k (IV- 67) 



u 



The above is the Kolmogoroff law of turbulence, stating that for ranges in which 

 a universal law of turbulence is applicable, the spectrum must decrease as the 5/3 

 power of the wave number. Clearly, the above law can only be expected to be valid 

 for values of the wave number below the viscous cutoff. In other words, for tur- 

 bulent eddies so small that the viscosity of the medium becomes important, the 

 spectrum will in general also depend on the viscosity, and the above dimensional 

 argument will no longer be applicable . Suppose that the eddy size for which this 

 occurs is 'E.yisc' ^^ shall call this the inner scale of the turbulence and would 

 expect the Kolmogoroff spectrum for the velocity to be valid for wave numbers less 



2n 



than — . 



"Wise 



artbur a.lLittlcJitf. 



S-7001-0307 



