Some Progress in Turbulence Theory 



the error of the prediction for the long-time eddy-diffusivity /<(») coming in the 

 first diagonal Pade approximation. Extension of the Pade technique to isotropic 

 turbulence dynamics and turbulent flow in pipes and channels is underway. The 

 outlook is favorable, but with insufficient work done to make any definite 

 statements. 



NUMERICAL INTEGRATION OF THE .... 



DIRECT-INTERACTION EQUATIONS 



We noted earlier that although statistical equations like the direct-interaction 

 equations are much more complicated than the original Navier- Stokes equation, 

 they can be less troublesome to solve numerically, because the solutions of the 

 statistical equations, being averages, are smoother and more stable than the ve- 

 locity fields of individual flow realizations. . r. ,. 



To illustrate, consider the isotropic turbulence decay discussed in the sec- 

 tion on Isotropic Turbulence Decay. Recent numerical techniques, using the 

 Cooley-Tukey fast Fourier transform, make it possible to integrate a flow repre- 

 sented by 32 X 32 X 32 Fourier modes (i.e., 32 values for each component of wave 

 vector), by direct solution of the Navier- Stokes equation in a computation time of 

 about one minute per time step on an IBM 390/95 computer [l]. This is suffi- 

 cient Fourier resolution to describe fairly well the energy- containing and 

 dissipation- range wave numbers in isotropic turbulence decay at R;^ - 20. In 

 order to follow the evolution for a time equivalent to the evolution times of the 

 direct- interaction solutions of section of this paper just mentioned, the order of 

 100 time steps would be needed, giving a total computation time per realization 

 of the order of an hour. An ensemble of perhaps 10 members would probably 

 give acceptable statistics at the higher wave numbers where the number of 

 similar modes is large. 



In contrast, the direct-interaction solutions illustrated in the above- 

 mentioned section require less than a minute per run on the same computer. 

 This time is based on a numerical scheme for the direct-interaction equations 

 which involves logarithmic steps in wave number and linear steps in time [12]. 

 The favorable properties of the statistical functions are exploited several ways 

 in this scheme. First, the spatial symmetries, homogeneity and isotropy, have 

 already been used in writing Eqs. (12) - (16), since the unknown functions are 

 scalar functions of scalar wave number. The use of logarithmic steps in wave 

 number (about 20 to cover the entire k range) is possible because of the smooth 

 dependence of u (k ; t, t') and G(k; t, t') on k. Finally, a time step about five 

 times larger than permissible for the straight computer simulation is permissi- 

 ble because of the high stability of the direct-interaction equations. 



It should be noted that the isotropy and homogeneity properties are no help 

 in the computer experiment, except possibly in reducing the number of realiza- 

 tions needed to get good statistics. It still is necessary to follow the complex 

 vector amplitude of each of the 32 x 32 x 32 Fourier modes. On the other hand, 

 if no use of symmetry and smoothness had been made in integrating the direct- 

 interaction equations, the machine time for the latter would have been stagger- 

 ing. In this case, a tensor function of two vector arguments, of the form 



809 



