C,8 • SPECTRAL THEORY OF ISOTROPIC TURBULENCE 



intimately connected with each other by simple mathematical transfor- 

 mations. Physically speaking, however, the two methods of description 

 put different emphasis on the different aspects of the same phenomena. 

 The spectral theory is often found to give a clearer description of the 

 basic mechanism of turbulence. 



Spectral analysis has long been used for the study of electromagnetic 

 waves, such as the radiation of heat and Hght. It was first introduced into 

 the study of turbulence by Taylor [19]. Taylor made spectrum measure- 

 ments, behind a grid in a wind tunnel, of the velocity fluctuation as regis- 

 tered by a hot wire fixed in the wind tunnel. This is a fluctuation in time. 

 But Taylor assumed^ that "the sequence of changes in u at the fixed 

 point are simply due to the passage of an unchanging pattern of turbu- 

 lent motion over the point." The variation is then essentially the same 

 as that in space, and the spectrum he observed corresponds to a one- 

 dimensional Fourier analysis of the field of turbulence in the direction of 

 the wind. 



The field of turbulence in the wind tunnel is obviously not homogene- 

 ous in the direction of the wind. However, in developing the theory, we 

 shall consider a homogeneous field and its Fourier analysis. In isotropic 

 turbulence, the analysis would be the same in all directions, provided we 

 are always dealing with the component of velocity in the direction chosen 

 for the analysis. The transverse component in general has a different 

 spectrum whether the turbulence is isotropic or not. 



In the case of turbulent motion, we may formulate the Fourier trans- 

 form relations between the power spectrum and the correlation function 

 as follows. If ^Fi(K)dK is the amount of kinetic energy per unit mass, 

 associated with the longitudinal component of the velocity, and lying in 

 the range of wave numbers (k, k -\- dK), then Fi{k) is related to the longi- 

 tudinal correlation function /(r) by the pair of Fourier transform relations : 



u^fir) = / Fi(k) cos KrdK 



r. (8-1) 



Fi{k) = -^=- I /(r) cos Krdr 



It is clear from the first formula in Eq. 8-1 that 



u2 = IJ Fi{K)dic (8-2) 



recapitulating the original physical interpretation of Fi{k). To clarify our 

 concepts, a derivation of Eq. 8-1 will be given in the next section. 



^ A theoretical analysis justifying Taylor's assumption was given by Lin [20]. A 

 thorough experimental investigation, including measurements of velocity correlations 

 involving both time and space separations, was made by Favre, Gaviglio, and Dumas 

 (see [21] and the references quoted). 



<211 ) 



