The intermittency of the Eulerian velocity increment is a characteristic of the downward energy cascade in 3D turbulence. In the inertial range and using Kolmogorov second similarity hypothesis, it is easy to express the ELVSF as a function of the ETVSF by a linear expression (4/3 factor). If intermittency effects are included, is it possible to derive a similar relation that links these two quantities together? Thank you.
Hi Abdalllah,
If the flow is locally isotropic then you can compute the transverse structure function from the longitudinal one (see e.g. Monin & Yaglom 2007)
D_{nn} = D_{ll} + \frac{r}{2} \frac{ \partial{D_{ll}} } { \partial{r} } (1)
(I hope you are familiar with Latex).
For the third-order structure function, it can be shown that
D_{lnn} = \frac{1}{6} \frac{ r \partial{D_{lll}} } { \partial{r} } (2)
As far as I am aware, a similar relation for higher-order structure functions has not been unambiguously derived (see the discussion in http://dx.doi.org/10.1017/jfm.2014.222 around page 400). These two expressions are purely kinematics relations, and they are valid independently of any intermittency effects. As you may know, the 4/3 factor between Dnn and Dll in the inertial range is obtained by considering the flow locally isotropic together with the K41 relation in the inertial range Dll \propro r^{2/3}.
Incorporating intermittency effects, i.e. considering Dll \propro r^{zeta}, then one obtains from Eq. (1), Dnn = (1+zeta/2)Dll. Therefore, if local isotropy holds, the scaling exponent for longitudinal and transverse structure function should be the same. In other words, the intermittency correction is the same for either Dll or Dnn. However, for higher order structure functions (>3), as I said before there is no simple relation between longitudinal and transverse structure functions. Therefore, the difference between the intermittency corrections for transverse and longitudinal structure functions remains an open question...
Regards...
I did a mistake, the discussion regarding isotropic relations for higher order structure function is in http://dx.doi.org/10.1017/jfm.2013.11.
Sorry
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