[jennap223]

Hi I have a general equation about the first, second and third moments of turbulence. like u'^2, w'^2; u'w', u'w'v', etc.

Can someone please explain what all these moments mean literally (in words)? I cannot find any good documentation on their literal meanings which I'm sure are dependent on the flow at hand, but I'd like a general description.

Thanks so much!

[davidwilcox]

the first moment is the mean, U_bar. The second moment is the variance u^2 and the third moment is skewness (u^3). u^2 for example is your Reynolds normal stress which comes about when you do a Reynolds average on the Navier-stokes equations (the convective/advective terms). The skewness part, or the triple products is seen in the turbulent diffusion term. The third moment depends only on the lack of symmetry of say a probability density function P(u) which means if P(u) is symmetric about its origin, u^3 is 0.

OKAY, now in English. why are these terms important. If you look at the transport of the turbulence (can either be reynolds stresses or turbulent kinetic energy), you want to know which of those terms contribute to the evolution/eduction of these 'creatures'. if you look at the production term for a 2d flow for an example, production =uv dU/dy. So uv and dU/dy contribute to the generation of turbulence. In short, you want to see the contributing mechanism of a turbulent flow field be it from production (second moment terms coupled with velocity gradients), turbulent diffusion terms(derivative of terms like uvw, uuu,etc), etc.

As for literature, i would highly recommend Paul Durbin's Statistical theory and Modeling of turbulent flows. Also anything by G.I. Taylor ( this guy prides himself in explaining things in a very simple manner visavis GK Bachelor)

[FMDenaro]

Originally Posted by jennap223

Hi I have a general equation about the first, second and third moments of turbulence. like u'^2, w'^2; u'w', u'w'v', etc.

Can someone please explain what all these moments mean literally (in words)? I cannot find any good documentation on their literal meanings which I'm sure are dependent on the flow at hand, but I'd like a general description.

Thanks so much!

Just to add some comments:

the fluctuations u',v',w' can have different meaning depending on the mean variable they refer to. For example in the statistical averaging formulation (for example RANS) they are the residual from the steady mean variable <u>. However, in LES we continue to denote u',v',w' but they refers as to the filtered variable.
The quadratic terms are also the counterpart in the kinetic energy contained by the residuals.
I can suggest reading basic textbooks on turbulence, from the old book https://mitpress.mit.edu/books/first-course-turbulence up to the more recent of https://pope.mae.cornell.edu/TurbulentFlows.html

[jennap223]

Thanks for both of your replies, examples and sources.
They are a good start, but I'll have to check out the mentioned resources as well.