SCALING OF CORRELATION-FUNCTIONS OF VELOCITY-GRADIENTS IN HYDRODYNAMIC TURBULENCE

As is demonstrated in Refs. 2 and 3 in the limit of infinitely large R eynolds numbers, the correlation functions of the velocity predicted b y Kolmogorov's 1941 theory (K41) are actually solutions of diagrammati c equations. Here we demonstrate that correlation functions of the vel ocity derivatives, del(alpha)upsilon(beta), should possess scaling exp onents which have no relation to the K41 dimensional estimates. This p henomenon is referred to as anomalous scaling. This result is proved i n diagrammatic terms: We have extracted a series of logarithmically di verging diagrams, whose summation leads to the renarmalization of the normal K41 dimensions. For a description of the scaling of various fun ctions of del(alpha)upsilon(beta), an infinite set of primary fields O (n) with independent scaling exponents DELTA(n) can be introduced. Sym metry reasons enable us to predict relations between the scaling of di fferent correlation functions. We also formulate restrictions imposed on the structure of the correlation functions due to the incompressibi lity condition. We also propose some tests which make it possible to c heck experimentally the conformal symmetry of the turbulent correlatio n functions. Further, we demonstrate that the anomalous scaling behavi or should reveal itself in the asymptotic behavior of the correlation functions of the velocity differences. We propose a method to obtain t he anomalous exponents from the experiment.