Longitudinal velocity-gradient skewness ($⟨({\partial }_x{u}_x{)}^3⟩=-\frac{2}{35}⟨{\omega }_i{\omega }_j{S}_{ij}⟩$)

For isotropic incompressible flow, directional averaging gives $⟨({\partial }_x{u}_x{)}^3⟩=(8/105)⟨\mathrm{tr}{S}^3⟩=-(2/35)⟨{\omega }_i{\omega }_j{S}_{ij}⟩$. Positive mean vortex stretching therefore implies negative longitudinal derivative skewness. Statistical homogeneity alone does not determine the sign in a fixed direction.