= Longitudinal velocity-gradient skewness
{title2=$\langle(\partial_xu_x)^3\rangle=-\frac2{35}\langle\omega_i\omega_jS_{ij}\rangle$}
For isotropic <incompressible flow>, directional averaging gives $\langle(\partial_xu_x)^3\rangle=(8/105)\langle\operatorname{tr}S^3\rangle=-(2/35)\langle\omega_i\omega_jS_{ij}\rangle$. Positive mean <vortex stretching> therefore implies negative longitudinal derivative <skewness>. <Statistical homogeneity> alone does not determine the sign in a fixed direction.
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