C,13 • SMALL SCALE STRUCTURE OF TURBULENCE 



provided the integral involved is convergent. If, in addition, h{r, t) van- 

 ishes sufficiently rapidly at infinity so that 



Hm r'h = (12-8) 



r — > 00 



the relation (Eq. 12-5) is obtained. 



It must be noted that there is no a priori reason^ for the convergence 

 of the integrals (Eq. 12-3) and the vahdity of Eq. 12-8. As a matter of 

 fact, recent investigations of Batchelor and Proudman [31] show that 

 even if /(r) is exponentially small at infinity at an initial instant, because 

 of the influence of the long range pressure forces, one can only be sure 

 that it will be no larger than 0(r^^) when r is large, although the possi- 

 bility of an exponentially small behavior is by no means excluded. We 

 are therefore only assured of the leading term in Eq. 12-4 and the existence 

 of the Loitsiansky 'parameter, 



J = u' r f{r)r'dr (12-9) 



However, the constancy of J depends on the relation (Eq. 12-8), which is 

 shown to be not generally true by the analysis of Batchelor and Proud- 

 man. On the other hand, for low Reynolds numbers based on the turbu- 

 lence level u, the term on the right-hand side of Eq. 12-7 becomes negli- 

 gible, and the Loitsiansky parameter is indeed approximately constant. 

 (Cf. Art. 14 and 15 for the part dealing with the final period of decay.) 



From a physical point of view, any prediction of the behavior of the 

 largest eddies must be regarded with some reserve, since it is expected to 

 be dependent on the experimental apparatus. If the general scale of turbu- 

 lence is much smaller than the dimensions of the experimental apparatus, 

 it would appear that this complication may be avoided by a proper 

 interpretation of the above results. The integrals (Eq. 12-3) may, for ex- 

 ample, be considered as extending over a distance much larger than the 

 scale of turbulence but still much smaller than the scale of the apparatus. 



Generalization of the above discussions to the anisotropic case has 

 been made by Batchelor [25]. The earlier conclusions are again modified 

 by the work of Batchelor and Proudman [31]. In fact, in the anisotropic 

 case, the correlation tensor Rij is shown to be in general of the order of 

 r~^, so that even the existence of a Loitsiansky parameter is in doubt. 



C,13. Small Scale Structure of Turbulence. Kolmogoroff's 

 Theory. We now turn to consider the small scale structure of turbu- 

 lence. Here the formal relations analogous to Eq. 12-1, 12-2, 12-3, and 

 12-4 are obtained by expanding cos {kt) into a power series in the first 

 equation in Eq. 8-1 : 



u^fir) = IJ Fi(k) cos {Kr)dK (13-1) 



« Cf. Birkhoff [30]. 



< 221 ) 



