On third-order mixed moments

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Abstract
It is shown that, for third-order mixed moments of two quantities (of zero mean), four sets of orthogonal axes exist (rotated 12π from each other) in which both moments involving odd powers of one of the variables vanish. The mixed moments in an arbitrary coordinate system are characterized by two real, positive numbers, and the angle from the principal axes. If one of the eigenvalues is larger than the other, the moments involving odd powers of the same variable will always have the same sign. Such distributions, termed “simple” have third-order mixed moments obeying certain inequalities. This is applied to the mixed moments of the temperature and velocity gradients in turbulent flows. Where these have been measured, the distributions are simple; where incomplete measurements exist, they satisfy the subsidiary inequalities derived for simple distributions. If the distribution of such quantities is always simple, a rather tight bound is provided for the temperature gradient production term.