Some Progress in Turbulence Theory 



.4 



1.2 



1.6 



2.0 



k/k. 



Fig. 10 - The direct-interaction and LHDI re- 

 sults of Fig. 9 compared with Stewart and 

 Townsend measurements of v.'^'p^ (k) (Ref. 15) 



of infinite Prandti number, a rather inhospitable limit in which to test a turbu- 

 lence theory because laboratory experiments have found strong evidence of a 

 tendency toward ordered motion in this limit. Extension of this work to lower 

 Prandti numbers is in progress. 



Figures 12 and 13 illustrate the results for convection in a horizontally in- 

 finite layer of fluid contained between slippery, infinitely conducting boundaries, 

 at a Rayleigh number of 3000. In both the computer experiment and the direct- 

 interaction equations, only the gravest three Fourier modes of temperature 

 fluctuation in the vertical z -direction were retained in combination with all 

 horizontal wave vectors whose x and y components fell within a chosen octave. 

 Cyclic boundaries were taken in the horizontal such that the numerical experi- 

 ment involved usually 76 distinct wave-vector projections in the horizontal, and 

 averages were performed over an ensemble of ten realizations. The initial con- 

 ditions were zero velocity everywhere, with Gaussian- distributed temperature 

 fluctuation. The direct-interaction equations were integrated with an initial tem- 

 perature fluctuation spectrum corresponding to the distribution used in the nu- 

 merical experiment. 



Figure 12 shows the evolution of Nusselt number (the ratio of mean heat 

 transfer across the layer to what the transfer would be without convection) ac- 

 cording to the numerical experiment and the direct- interaction equations. Also 

 shown are the results of two other statistical approximations, the quasilinear 



801 



