How to derive the Karman-Howarth-Monin relation for anisotropic turbulence? 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? 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
$$\begin{align} 
\partial_t \frac{1}{2}\langle v_i v'_i \rangle =& 
- \frac{1}{2} \partial_j \langle v_i v_j v'_i\rangle 
- \frac{1}{2} \partial'_j \langle v'_i v'_j v_i\rangle \\
& - \frac{1}{2} \langle v'_i \partial_i p \rangle 
- \frac{1}{2} \langle v_i \partial'_i p' \rangle\\
&+ \frac{1}{2} \langle v'_i f_i \rangle 
+ \frac{1}{2} \langle v_i f'_i \rangle \\
& + \frac{1}{2} \nu \left( \partial_{jj} + \partial'_{jj} \right) \langle v_iv'_i\rangle.
\tag{6.11} \end{align} $$

 A: 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 Ref. 1 essentially following Ref. 2.
References:

*

*Marc Brachet, A Primer in Classical Turbulence Theory; p. 8-9.


*Uriel Frisch, Turbulence: The Legacy of A. N. Kolmogorov, 1995; p. 79.
