I'm looking for existing works which give a spectral expression (in terms of spectra, transfers etc...) of the skewness of the velocity derivative transverse to the mean flow, in shear flows (dU1/dx3 = S) , i.e. there is one non-zero component of the mean velocity field U1 = S x3.
The definition in term of "physical" derivatives is:
/3/2
Spectral expressions exist in the homogeneous isotropic framework, but I cannot find anything in shear flows.
Thank you for your interest.
My question is clearly written: I'm looking for the spectral expression of the velocity derivative skewness in shear flows. The difficulty being that shear flows are anisotropic, and I'm not sure that such spectral expressions can be derived similarly to the isotropic case.
Maybe the missing point is that I'm not doing DNS, but spectral modelling of second-order moments. Consequently, I don't have access easily to
Thanks for this link. I will investigate further.
The key point in the determination of the spectral skewness in shear flows is purely mathematical, and relies on the general expression of a nth order tensor in a homogeneous flow (with n even and n=6 at maximum) that depends on the velocity ui and its derivatives.
Such expressions have been partially derived by Pope (2000, Turbulent Flows) for homogeneous isotropic turbulence up to 6th order tensor. And by Batchelor as well (Theory of Homogeneous Turbulence) up to the 3rd order for skew-isotropy. The calculation is quite straightforward as it involves only sums of kronecker delta-functions that can be grouped together using symmetries.
Is it possible to give the general shape of a nth order tensor in a homogeneous shear flow (that involves the velocity and its derivatives) ? I'm not sure this is possible when there are not strong symmetries...
Hi Antoine--
You can get a little help starting this by looking at Robertson's (1940) "Invariant theory of isotropic turbulence", or Matthaeus+Smith's 1981 anisotropic extension of such.
Thanks Mark, this is exactly the kind of general decomposition I'm looking for in the Homogeneous Anisotropic Turbulence framework.
The article by Matthaeus is of great interest. Nevertheless, it only addresses the case of second-order tensors... But this is a great start for me and maybe I'll be able to extend their results to higher orders.
Antoine, I think you might have find something similar to this kind of framework in the second volume of the book:
A.S. Monin and A.M. Yaglom Statistical fluid mechanics : mechanics of turbulence
Unfortunately I have it only in Russian so here is a link to a proper english name of it
Thanks Andrey for your advice.
Fortunately, I have this book in the library on my lab.
Indeed, there are many details, and I am glad to see that they performed the same (lengthy) calculations I made for the isotropic case. There are also indications regarding the expressions of correlations in homogeneous flows without any particular symmetries. I have to investigate it more to see if the extension to shear flows is possible.
Just to keep you informed: I have found interesting details in a Journal of Fluid Mechanics by W. K. George (1991), in his theory of local axisymmetry
(http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=397700)
try a new method
Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...
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