C,13 • SMALL SCALE STRUCTURE OF TURBULENCE 



pensated, or even over-compensated, by the stretching of the vortex 

 tubes, due to the diffusive nature of turbulent motion. (Hence one may 

 expect more stretching of the vortex tubes than compression.) Taylor 

 calculated the relative magnitudes of the various quantities by deter- 

 mining /o' and /j", and he found that all the three terms in Eq. 13-6 are 

 of the same order of magnitude for his experiments. Such measurements 

 were more accurately made later by Batchelor and Townsend [33] and by 

 Stewart [11]. 



As noted before (Art. 8), in many experiments the dissipation of 

 energy is practically all associated with the high frequency components 

 which contain a neghgible amount of energy. Combining this fact with 

 the mechanism of vortex-stretching just discussed, one can form a reason- 

 able picture of the process of turbulent motion. There are the large energy- 

 containing eddies which contribute very Httle to the viscous dissipation 

 directly. By their own diffusive motion, small eddies are formed, i.e. the 

 kinetic energy of turbulent motion goes down to smaller scales. It is at 

 these small scales that viscous forces become most effective and the pre- 

 dominant part of the energy dissipation occurs. Thus one forms the pic- 

 ture of an energy reservoir in the large eddies, and a dissipation process 

 in the small eddies which may be presumed to depend very little on the 

 structure of the large eddies except to the extent of the amount of energy 

 supplied to them. This forms the physical basis of Kolmogoroff's theory 

 of locally isotropic turbulence [34]. 



Before we go on with the discussion of his theory, it should be empha- 

 sized that the picture is correct only when the diffusive mechanism is 

 strong; i.e. when the inertial forces are large compared with the viscous 

 forces. In other words, the Reynolds number of the turbulent motion 

 must be relatively large. This is well illustrated by the detailed calcu- 

 lations made by Taylor and Green [35] on a model of isotropic turbu- 

 lence. ^^ Indeed, they found that for very low Reynolds numbers of turbu- 

 lence, defined by 



V 



the stretching mechanism is not strong enough, so that the magnitude of 

 the vorticity decreases steadily. On the other hand, if the motion starts 

 out at a fairly high R\, the mean square vorticity (and hence also the 

 rate of energy dissipation) first increases to several times its original value 

 due to the stretching mechanism. The kinetic energy of the motion, how- 

 ever, decreases steadily. Eventually, it becomes very low, and the stretch- 

 ing process is so weakened that the vorticity of the motion also decreases 

 steadily. 



Kolmogoroff's theory. In line with the above ideas, Kolmogoroff pos- 

 tulates that, at large Reynolds numbers of turbulent motion, the local 



" See also Goldstein [36]. 



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