SOME PROGRESS IN TURBULENCE THEORY 



Robert H. Kraichnan 

 Dublin, New Hampshire , . 



■ .\. ABSTRACT 



A review is made of efforts to deduce the quantitative properties of 

 turbulence from the Navier -Stokes equation, without the introduction of 

 mixing lengths or other arbitrary paranaeters, by means of the direct- 

 interaction family of statistical approximations. The direct-interaction 

 results for turbulent dispersion, isotropic turbulence, and Boussinesq 

 turbulent convection are connpared with laboratory and computer ex- 

 periments. The nature of expansions for statistical functions is dis- 

 cussed, together with techniques for the estimation of errors and the 

 systematic refinement of the direct-interaction predictions. Finally, 

 the numerical methods appropriate to the direct-interaction equations 

 are discussed. 



INTRODUCTION 



Turbulence is a contradictory phenomenon which simultaneously exhibits 

 both order and disorder. The combination has proved hard to analyze and pre- 

 dict successfully. The random aspect of turbulence arises from the rich va- 

 riety of instabilities accessible to high-Reynolds-number flows. When a laminar 

 flow breaks down into turbulence, so many different degrees of freedom can be 

 excited that the detailed, point-to-point, causal dependence of the resulting tur- 

 bulent velocity field on the initial disturbance field is impossible to unravel. 

 The turbulent velocity field is intricate and irregular in appearance. 



Measurements on a variety of sustained turbulent flows show that the proba- 

 bility distribution of the velocity measured as a function of time at any point 

 within the fully turbulent region is close to a normal distribution. However, the 

 joint probability distribution of the velocity at two or more points is significantly 

 nonnormal. A snapshot of the velocity field at any instant would show an abun- 

 dance of well-defined local features: filaments and streets of high vorticity 

 separated by relatively quiescent regions. 



The intricacy, irregularity, and nonreproducibility of individual turbulent 

 flows make it natural to use a statistical description in which averages over 

 space, time, or an ensemble of realizations of the flow are sought. In contrast 

 to the instability of an individual turbulent flow, laboratory results suggest that 

 appropriate statistical averages are relatively stable and reproducible. The 

 transport properties of turbulent flows are naturally described by averages. 



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