Kraichnan 



u. .(k,k', t, t') would replace u(k; t, t'). If the direct- interaction equations 

 were then integrated using point-to-point integration on the 32 x 32 x 32 grid, 

 and the same time step as for the computer experiment, the number of multipli- 

 cations required would be billions of times greater than for integration of the 

 Navier- Stokes equation for a realization. 



The saving in computation time over direct simulation offered by the direct- 

 interaction equations rises with Reynolds number. Consider a steady- state 

 turbulent flow. Here logarithmic steps in both wave number and frequency can 

 be used with the direct-interaction equations, with the result that computation 

 time rises only as the logarithm of Reynolds number. With direct simulation, 

 on the other hand, the number of Fourier modes must go up as the cube of ratio- 

 of-dissipation wave number to energy- containing wave number, while the per- 

 missible time step gets smaller. 



Further marked reductions in the integration time for the direct- interaction 

 equations can be achieved through the use of more economical representations 

 of the statistical functions than by (linear or nonlinear) grids in k . The integra- 

 tions illustrated in the section on Isotropic Turbulence Decay used about twenty 

 logarithmic steps in k. However, the resulting functions are so smooth that 

 they could be represented adequately by two or three coefficients in an aptly 

 chosen representation by orthogonal functions (e.g., Laguerre functions). Sav- 

 ings of this kind become of increasing value in nonisotropic problems, such as 

 turbulent flow in a pipe, where the loss of symmetry raises the dimensionality 

 of the final statistical equations. 



Increased dimensionality also makes the Monte-Carlo evaluation of multiple 

 integrals attractive for reducing computation time. The use of Monte-Carlo 

 methods was what made feasible the evaluation of the higher- order corrections 

 to the direct- interaction dispersion equations, discussed in the previous section. 

 The use of representation by well-chosen orthogonal functions, coupled with 

 Monte-Carlo evaluation of integrals, makes integration of the direct- interaction 

 equations for simple shear flows [38], such as a flow in an infinite pipe or chan- 

 nel, appear practicable with presently available computers. Work toward this 

 end is in progress. 



The use of Monte-Carlo methods for evaluating the statistical equations is 

 of theoretical as well as of practical interest, and leads to a point of view in 

 which direct simulation and representation by statistical approximation appear 

 as intimately related complements. Let us again take isotropic turbulence as 

 an example. Suppose an initial spectrum is prescribed, with initial, multivari- 

 ate, Gaussian statistical distribution. Suppose first that the initial R;*^ is low 

 enough that only a few modes need be retained in the computer simulation. Inte- 

 gration time per realization is then small, and solution for a sizable ensemble 

 of realizations is feasible. For R^ -- 40, however, the order of 10^ to 10^ 

 Fourier modes must be retained, and the computation task becomes onerous. 

 Apart from the cost of computer time, the direct simulation seems fundamentally 

 inefficient at this stage. Most of the effort goes into handling the large number 

 of high k modes, whose behavior is statistically redundant. We would like to 

 handle only a few of these modes, which then would typify the others. This is not 

 possible with direct simulation. If a sizable fraction of the high k modes are 



810 



