How to derive the Karman-Howarth-Monin relation for anisotropic turbulence?

Question

I find the derivation of the Karman-Howarth-Monin relation in the book Turbulence from Frisch (1995) a bit to short. Can someone point me to a more detailed derivation of that relation, if possible in a publicly available source (lecture notes, thesis, arXiv paper)? Or can someone explain to me how Frisch obtains the equation that follows the sentence on page 78:

Starting from the Navier-Stokes equation (6.6), we obtain $$\begin{align} \partial_t \frac{1}{2}\langle v_i v'_i \rangle =& - \frac{1}{2} \partial_j \langle v_i v_j v'_i\rangle - \frac{1}{2} \partial'_j \langle v'_i v'_j v_i\rangle \\ & - \frac{1}{2} \langle v'_i \partial_i p \rangle - \frac{1}{2} \langle v_i \partial'_i p' \rangle\\ &+ \frac{1}{2} \langle v'_i f_i \rangle + \frac{1}{2} \langle v_i f'_i \rangle \\ & + \frac{1}{2} \nu \left( \partial_{jj} + \partial'_{jj} \right) \langle v_iv'_i\rangle. \qquad (6.11) \end{align} $$

Answer 1

Let me here just derive the equation (6.11) that follows the sentence, you mention. The Navier-Stokes equation (6.6a) reads

$$\partial_t v_i + v_j\partial_j v_i = -\partial_i p + f_i + \nu~\partial_j\partial_j v_i.$$

The incompressibility condition (6.6b) reads

$$\partial_jv_j=0. $$

Hence we have in the unprimed and the primed point that

$$\partial_t v_i = -\partial_j(v_j v_i) -\partial_i p + f_i + \nu~\partial_j\partial_j v_i,$$

and

$$\partial_t v'_i =-\partial'_j(v'_j v'_i) -\partial'_i p' + f'_i + \nu~\partial'_j\partial'_j v'_i,$$

respectively. Therefore, averaging yields

$$\begin{align} \partial_t \langle v_i v'_i \rangle &= \langle v'_i \partial_t v_i \rangle + \langle v_i \partial_t v'_i \rangle \\ &= - \langle v'_i\partial_j(v_j v_i) \rangle - \langle v_i \partial_j'(v'_j v'_i) \rangle \\ & -\langle v'_i \partial_i p \rangle -\langle v_i \partial'_i p' \rangle\\ & +\langle v'_i f_i \rangle +\langle v_i f'_i \rangle \\ & +\nu \langle v'_i\partial_j\partial_jv_i\rangle + \nu \langle v_i\partial'_j\partial'_j v'_i\rangle \\ &= -\partial_j \langle v_i v_j v'_i\rangle -\partial_j' \langle v'_i v'_j v_i\rangle \\ & -\langle v'_i \partial_i p \rangle -\langle v_i \partial'_i p' \rangle\\ & +\langle v'_i f_i \rangle +\langle v_i f'_i \rangle \\ & +\nu\left( \partial_j\partial_j + \partial'_j\partial'_j\right)\langle v_iv'_i\rangle, \end{align} $$

where we have used that averaging and differentiation commute, and also that primed velocities are independent of unprimed derivatives, and vice-versa.

The full Karman-Howarth-Monin relation is derived by Marc Brachet on p.8-9 here, essentially following the book by Uriel Frisch (1995).