Some Progress in Turbulence Theory 



The dynamical equations obeyed by v and U differ fundamentally from the 

 underlying Navier Stokes equation for v (x, t). The latter is a vector differential 

 equation which can be integrated forward in time once the initial values v(x, 0) 

 and the boundary conditions are specified. In contrast, there is no closed-form 

 dynamical equation which can be integrated to generate v and U from their 

 initial values. The equations for v and u can be formulated in several ways 

 [2,3], but, always, an infinite sequence or series of some sort arises. In one 

 formulation, there is an infinite set of coupled equations which determines 

 simultaneously v,U , and the infinite set of higher moments<u- (x, t)uj (x, t')u x 

 (x", t") >5 • • . . In a second formulation, v and U are expanded in an infinite 

 perturbation series, with a characteristic Reynolds number of the flow serving 

 as the expansion parameter. A third formulation yields integrodifferential 

 equations for v and U which contain, within them, infinite series in v and U 

 themselves. Finally, the infinite set of equations for v , U , and higher-order 

 moments can be replaced by an equivalent, single, functional equation for the 

 probability-distribution characteristic functional [4,5]. All these ways of formu- 

 lating the statistical equations require as input information the initial values of 

 all moments of all orders. This infinite set of initial values replaces, in the 

 statistical description, the initial values of the velocity field in all the individual 

 members of the ensemble. The dynamical coupling of moments of different 

 order comes from the nonlinearity of the Navier-Stokes equation; that is, from 

 the advection term (v*V)v. 



In order to compute v and u from the statistical equations, it is necessary 

 to find an approximating algorithm which replaces the infinite dynamical equa- 

 tions by something integrable in a finite number of operations. Grave difficul- 

 ties have been encountered here, because in all the formulations the infinite 

 sequences or series are divergent [2]. At the time of writing (December 1968), 

 no known scheme guarantees converging approximations to the correct values 

 of V and U. Here is a sharp contrast to the original Navier-Stokes equation, 

 which, with reasonable assurance, can be integrated with any desired accuracy, 

 in any given realization, by taking a sufficiently fine grid in space and time. 



In view of the preceding paragraph, what valid motivation is there for pur- 

 suing statistical turbulence theory instead of simply integrating the Navier- 

 Stokes equation for representative turbulent flows? There are two principal 

 reasons. First, equations for the statistical quantities themselves, even if 

 approximate and mathematically difficult in their final usable form, can exhibit 

 the important physics of turbulence more clearly than the Navier Stokes equa- 

 tion. Statistical equations can provide a bridge to intuitive ideas about turbu- 

 lence, such as eddy viscosity and mixing lengths. Second, approximate statisti- 

 cal equations, required as an end result of moderately accurate numerical 

 predictions for averages of interest, demand, in some cases at least, enor- 

 mously less computation time than integration of the Navier-Stokes equation 

 for representative flows. This is because the statistical functions v and u are 

 smooth functions of their arguments and can be specified adequately by fewer 

 numbers than the jaggedly varying velocity field of a typical realization. This 

 is particularly true when there are statistical symmetries, such as isotropy or 

 stationarity. Statistical equations can also be much more stable for machine 

 computation than the Navier-Stokes equation. 



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