Some Progress in Turbulence Theory 



simply omitted from the dynamical equations, statistical weights are altered, 

 and the faithfulness of the solution destroyed. Moreover, there is no way to 

 interpolate between selected high k modes, because of the jagged variation of 

 amplitude from mode to mode, which, although unpredictable, incorporates 

 elaborate dynamically created statistical correlations. 



Suppose, on the other hand, we deal with statistical equations: the direct- 

 interaction approximation and a sequence of (presumed valid) higher approxi- 

 mations built upon it after the fashion of that described in the previous section. 

 Here the smoothness of the statistical functions does permit interpolation at 

 high wave numbers. If these equations are solved by Monte -Carlo evaluation of 

 multiple k -space integrals, we are, in effect, sampling just a few of the high k 

 modes, and interpolating — which is what we wanted to do and could not with the 

 computer simulation. If only the direct- interaction equations, without higher 

 corrections, are solved, a semiquantitative solution (errors in spectrum on the 

 order of 10%- 20%) emerges with great computational economy. As higher cor- 

 rections are admitted, a more accurate solution is obtained (if the proposals of 

 the previous section are valid), at the expense of evaluating more elaborate 

 multiple k -space integrals. This means that the Monte-Carlo sampling involves 

 longer sample chains of interacting Fourier modes. 



If a prediction of only the energy spectrum is desired, the sequence of sta- 

 tistical approximations appears to offer greater economy at moderate Reynolds 

 numbers. Suppose, however, that a more elaborate statistical function were 

 desired — say, the joint- probability distribution of the velocity at three points, 

 or the flatness factor of some probability distribution. A formulation of the 

 statistical equations to yield such functions with acceptable accuracy could be 

 expected to be very elaborate and to require, ultimately, the sampling of very 

 long chains of interacting Fourier modes in very complicated equations. Here 

 it would likely be more economical to work with direct simulation, where all 

 sets of Fourier-mode interactions are explicitly calculated from the outset. 



ACKNOWLEDGMENT 



This work was supported by the Office of Naval Research under Contract 

 N00014-67-C-0284. I am grateful to Dr. J. R. Herring for permission to repro- 

 duce Figs. 12 and 13. • - - 



REFERENCES 



1. Orszag, S.A., "Numerical Methods for the Simulation of Turbulence," Proc. 

 Int. Symp. on High-Speed Computing in Fluid Dynamics, Monterey, Cali- 

 fornia, 1968 



2. Kraichnan, R.H., "Invariance Principles and Approximation in Turbulence 

 Dynamics," in Dynamics of Fluids and Plasmas, ed. S.I. Pai, New York: 

 Academic Press, 1966 



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