[rdemyan]

It seems like most of the derivations in turbulence theory that involve eddies assume that the eddy is circular. So the eddy turnover time (t) is given by,



where is the eddy diameter and is the eddy characteristic velocity.

But what if I have an elliptical eddy? Can I use an equivalent diameter for the ellipse where the equivalent diameter is determined by setting the area of the ellipse equal to the area of a circle. So,



where a is the radius of the major axis of the ellipse and b is the radius of the minor axis of the ellipse.

Therefore,



So for an elliptical eddy is it reasonable to use an equivalent diameter as derived above?

[LuckyTran]

Not sure where this theory is headed. You set the circle equal to the ellipse and then what? Real eddies don't just revolve in place but are carried by a background flow while undergoing simultaneous stretching so that real eddies are neither circular, nor elliptical.

Btw the prevailing treatment of microscales for example already permits eddies of different lengthscales in each direction: L11, L22, L33, and even L12, L13, L23! It is simply not true that most derivations assume they are circular. Most derivations don't assume any shape at all.

[rdemyan]

Originally Posted by LuckyTran

Not sure where this theory is headed. You set the circle equal to the ellipse and then what? Real eddies don't just revolve in place but are carried by a background flow while undergoing simultaneous stretching so that real eddies are neither circular, nor elliptical.

Btw the prevailing treatment of microscales for example already permits eddies of different lengthscales in each direction: L11, L22, L33, and even L12, L13, L23! It is simply not true that most derivations assume they are circular. Most derivations don't assume any shape at all.

Okay. But the order of magnitude expression for the energy dissipation rate does seem to assume one length scale, namely, . I would assume since is labeled as a diameter, the eddy is circular in shape for this order of magnitude expression.

[LuckyTran]

1,2,3,4,5,6,7,8,9 are all formally of the same order of magnitude and does not imply that 1 is equal to 9. Unless your eddy is a needle, then all their dimensions are the same order of magnitude. Where are we going? If you want to consider eddies of arbitrary dynamics, then do so.

[FMDenaro]

Originally Posted by rdemyan

Okay. But the order of magnitude expression for the energy dissipation rate does seem to assume one length scale, namely, . I would assume since is labeled as a diameter, the eddy is circular in shape for this order of magnitude expression.

Don't forget that in the energy dissipation range is assumed the isotropy behaviour. You are mixing the large anisotropic eddies with the smallest isotropic scales.
No one assumes that large eddies have a circular section.

[rdemyan]

Originally Posted by FMDenaro

Don't forget that in the energy dissipation range is assumed the isotropy behaviour. You are mixing the large anisotropic eddies with the smallest isotropic scales.
No one assumes that large eddies have a circular section.

Thanks, that is helpful.