7
$\begingroup$

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 $$\begin{align} \partial_t \frac{1}{2}\langle v_i v'_i \rangle =& - \frac{1}{2} \partial_j \langle v_i v_j v'_i\rangle - \frac{1}{2} \partial'_j \langle v'_i v'_j v_i\rangle \\ & - \frac{1}{2} \langle v'_i \partial_i p \rangle - \frac{1}{2} \langle v_i \partial'_i p' \rangle\\ &+ \frac{1}{2} \langle v'_i f_i \rangle + \frac{1}{2} \langle v_i f'_i \rangle \\ & + \frac{1}{2} \nu \left( \partial_{jj} + \partial'_{jj} \right) \langle v_iv'_i\rangle. \tag{6.11} \end{align} $$

$\endgroup$
1
  • 3
    $\begingroup$ Why don't you exhibit the lines that follow and define the terms? There is the possibility that someone knows the math but not the name "Karman-Howarth-Monin relation", and the expanded form would increase you pool of potential answers. $\endgroup$ May 29, 2011 at 16:27

1 Answer 1

8
$\begingroup$

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:

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

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

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.