Kraichnan 



There are two general ways in which predictions of turbulence averages 

 may be attempted with the Navier-Stokes equation as a starting point. The 

 more straightforward is direct computer simulation of a turbulent flow, or 

 ensemble of flows, followed by averaging. This involves numerical integration 

 of the Navier-Stokes equation forward in time for each flow, individually. The 

 second way is to seek dynamical equations for the statistical averages them- 

 selves. These latter equations then are solved numerically. There are ad- 

 vantages and disadvantages to each approach, and the two methods should be 

 regarded as complements rather than competitors. Direct integration of the 

 Navier-Stokes equation offers pictures of individual flow structures which are 

 impossible to recover from statistical equations. On the other hand, statistical 

 theory can exhibit more clearly some of the physics of the transport processes 

 that characterize turbulence, and, equally important, can result in less demand- 

 ing numerical computations. Direct computer simulation of turbulent flows has 

 been limited, in the past, to two-dimensional flows, because of computer limi- 

 tations. However, continuing advances in the speed and capacity of computers, 

 together with numerical techniques like the Cooley-Tukey fast Fourier trans- 

 form, appear to be changing this situation rapidly. As we shall discuss later 

 in this paper, useful three-dimensional calculations already seem to be fea- 

 sible [1]. 



The present paper reports on a family of statistical approximations, the 

 "direct- interaction" approximation and its relatives, which the author and his 

 colleagues have explored during the past ten years. The unique feature of the 

 direct- interaction approximation is that it is an exact description of a model 

 dynamical system, in addition to being an approximation to turbulence dynamics. 

 The model system has the same expression for energy as the Navier-Stokes 

 system, together with other common properties. This assures important in- 

 ternal consistency properties of the approximation. 



The direct- interaction approximation and its variants have been applied 

 to isotropic turbulence decay, turbulent dispersion, convection of scalar con- 

 taminants by turbulence, random solutions of Burgers' equation, hydromagnetic 

 turbulence, turbulence in a Vlasov plasma, and buoyant turbulent convection. In 

 the sections which follow, the nature of the approximation is reviewed, and 

 brief reports are given on some of these applications, including comparisons 

 of the results with laboratory and computer experiments. The paper concludes 

 with discussions of error estimates, higher approximations which reduce errors 

 systematically, and, finally, the numerical techniques called for by the direct- 

 interaction equations. 



THE NATURE OF DYNAMICAL EQUATIONS 

 FOR STATISTICAL QUANTITIES 



The simplest statistical field variables are the mean velocity vi (x, t ) = 

 <v (x, t)> and the velocity covariance tensor V-^- (x, t ; x', t') =<Ui (x, t)uj(x', t')>, 

 where v^ (x, t) is the velocity field in an individual realization of the flow, < > 

 denotes the average over an ensemble of realizations, and u. (x, t ) = v^ (x, t ) - 

 V. (x, t ) is the departure of the velocity field, in a realization, from the ensemble 

 mean. We use ensemble averages here and in what follows because they are 

 always applicable, while space or time averages are appropriate only if the flow 

 exhibits statistical homogeneity or stationarity. 



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