Relation of Fourth-Order to Second-Order Moments in Stationary Isotropic Turbulence

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Abstract
An investigation is made of the hypothesis that the fourth-order moments of the two-time velocity amplitude distribution in stationary, isotropic, incompressible turbulence are related to second-order moments as in a normal distribution. It is concluded that this hypothesis is inconsistent with the equations of motion, and the inconsistency is exhibited as a gross violation of energy conservation in the inertial range. The origin of the inconsistency is discussed. The arguments developed are used to demonstrate inconsistencies in Chandrasekhar's recent theory of turbulence. Qualitative considerations are presented with regard to the consistency of the hypothesis that fourth-order moments of the simultaneous amplitude distribution are related to second-order moments as in a normal distribution. The general validity of this restricted hypothesis is questioned also.

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