= Betchov relation
{c}
{title2=$\langle\omega_i\omega_jS_{ij}\rangle=-4\langle abc\rangle$}
For differentiable homogeneous <incompressible flow>, $\langle\operatorname{tr}A^3\rangle=0$. One divergence representation is $\operatorname{tr}A^3=\partial_i[u_j(\partial_j u_k)(\partial_k u_i)-u_i\operatorname{tr}A^2/2]$. Splitting the <velocity gradient tensor> into strain and rotation gives $\langle\omega_i\omega_jS_{ij}\rangle=-4\langle abc\rangle$. It connects mean <vortex stretching> to the cubic strain statistic, without requiring isotropy.
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