Some Progress in Turbulence Theory 



t u(0)/L(0) 



Fig. 8 - Evolution ofskewness for a variety of initial con- 

 ditions. Curve 4 is for the run shown in Figs. 3-5; curve 

 10, for the run shown in Figs. 6 and 7. (See Ref. 12 for 

 full details.) 



More definitive and detailed tests of the direct- interaction solutions for 

 isotropic turbulence decay at moderate Reynolds numbers probably must come 

 from computer simulation of the flows. Such computer experiments now seem 

 feasible at the R^ values cited above, and it is to be hoped that they will be 

 carried out in the near future. 



In the absence of these computer experiments at the present time, it is of 

 interest to report a test of direct- interaction results against computer simula- 

 tion for a simpler dynamical problem with the same kind of nonlinear ity, viz., 

 the interaction of a small, discrete set of shear waves [17]. The equations of 

 motion here are obtained by writing the Navier-Stokes equation in Fourier form 

 and then deleting all terms that refer to wave numbers outside a small set. The 

 computer experiment is performed by integrating the equations of motion for an 

 ensemble of initial conditions, and then averaging. Figure 11 shows a typical 

 result for the interaction of a set of three shear waves confined to two dimen- 

 sions. In this run, all of the energy was initially contained in two of the waves. 

 Curves 1, 3, and 5 show the evolution of the energy in the three waves accord- 

 ing to the computer experiment. Curves 2, 4, and 6 show the corresponding 

 evolution according to the direct- interaction equations. 



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