Some Progress in Turbulence Theory 



4.0 



3.5 



Nu 



3.0 



2.5- 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.1 



t 



Fig. IZ - Conaparison of direct-interaction and computer- 

 experiment results for the evolution of Nusselt number at infi- 

 nite Prandtl number, Rayleigh number = 3000. The solid curve 

 is the nunaerical experiment; the dashed curve, the direct- 

 interaction approximation; the triangles are the quasilinear 

 approximation; and the dot-dash curve is the quasinormal ap- 

 proximation. The squares show points from another numerical 

 experiment, with 124 horizontal wave vectors, and give a meas- 

 ure of the statistical error at the time of evolution, when that 

 error was found to be maximum. (See Ref. 24 for further de- 

 tail and for normalization.) 



the near future. This should permit meaningful comparison with laboratory as 

 well as computer experiment. 



LAGRANGIAN- HISTORY DIRECT-INTERACTION APPROXIMATION 



We noted in the section on Isotropic Turbulence Decay that the direct- 

 interaction equations for isotropic turbulence failed to give the Kolmogorov k"^^^ 

 inertial-range spectrum, yielding instead a k"^' ^ spectrum. The trouble here can 

 be traced back to a deep-lying cause: the use of Eulerian description. The princi- 

 pal idea behind Kolmogorov' s theory is that eddies of large size convect eddies of 



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