More than eighty years ago, Kolmogorov and Obukhov made a breakthrough in turbulence theory by establishing the relation for the three-point correlation function of the velocity field in turbulent flow.
\[ \langle v_\alpha(\vec{r}_0)\,v_\beta(\vec{r}_0)\,v_\gamma(\vec{r}+\vec{r}_0) \rangle = \frac{\mathcal E }{(d-1)(d+2) V} \left( \delta_{\alpha \gamma} r_{\beta} + \delta_{\beta \gamma} r_{\alpha} - \frac{2}{d}\delta_{\alpha \beta} r_{\gamma} \right); \]
\[ \langle v_\alpha(\vec{r}_0)\,v_\beta(\vec{r}_0)\,\omega_\nu(\vec{r}+\vec{r}_0) \rangle = e_{\mu\gamma\nu}\,\partial_\mu(\vec{r})\,\langle v_\alpha(\vec{r}_0)\,v_\beta(\vec{r}_0)\,v_\nu(\vec{r}+\vec{r}_0) \rangle \propto e_{\mu\gamma\nu}\Bigl(\delta_{\alpha\gamma}\delta_{\beta\nu} + \delta_{\alpha\nu}\delta_{\beta\gamma} - \frac{2}{d}\delta_{\alpha\beta}\delta_{\mu\gamma}\Bigr) = 0; \]
The last relation follows from the symmetry of the expression in the brackets with respect to the exchange of the indices \( \gamma \leftrightarrow \mu \).
This relation indicates no constraints on the rotational part concerning triple vorticity correlations and sheds no light on the scale invariance of turbulence theory.
Moreover, the linear term in coordinates does not have any support in the Fourier spectrum. In Fourier space, this term becomes linear combinations of gradients of delta function \( \vec{\nabla}\delta^3(\vec{k}) \), rather than some constant flow through the whole spectrum. Such a constant flow would require an \( |x|\log|x| \) term rather than a linear term in coordinates. This linear term is an example of the harmonic terms added to the BS integral for the velocity field.