We saw in the turbulence class that the right region of the Lumley diagram which is called the axisymmetric in cigar, includes flows such as boundary layer of jet flow. Are there any similarities between the 2 flows?
The Lumley triangle is a way of characterizing the Reynolds stress tensor (indirectly via its anisotropy).
The "right side" is one big Reynolds stress and two small (and equal) ones. Hence the 'cigar'. Which is a shape with one long dimension and two-small dimensions.
The "left side" is two big (and equal) Reynolds stresses and one small one.
Sometimes called pancake shaped.
The top boundary is call Two-component (or 2C) turbulence. One of the Reynolds stresses (or two of them) is zero there (just like turbulence very near a wall).
When two different flows are at the same location on the Lumley triangle invariant map it means surprisingly little. Just that the ratio of the anisotropy levels is the same.
You can get the same anisotropy levels and have completely different looking eddies making those anisotropy levels.
In a boundary layer one of the Reynolds stress (the one normal to the wall) is severely damped (by pressure responses bouncing off the wall) - so you get two big and one small.
In an axisymmetric jet you have strong mean flow gradients in the two
tangential to the axis directions - causing large production of turbulence for those two directions of the Reynolds stress.
So ... in the B.L. one stress damped so two big and one small Reynolds stress.
In the axi-jet, two stresses get energy from the mean flow, so two big and one small. So both have the same anisotropy - for totally different reasons.
Sorry. Lumley diagram says very little really. It is a trick borrowed by Lumley (like much of his stuff) from Non-Newtonian fluid mechanics. Don't over-read its implications.
Dear Abdallah.
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You can clearly indicate what you call "Lumley diagram"?
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Maybe I know him under another name!
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Thank you Rogerio. i think it is called also the Anisotropy Invariant Map. The variations of the second invariant of the anisotropy tensor with respect to the third invariant.
Regards!
Dear
I am currently used the Lumley diagram in the context of rotating flows. For a given geometry and set of control parameters, you can be on one branch or the other depending on the spatial location. You are on the 2C limit close to a wall and for a high Reynolds number, you tend often to the isotropic limit. Other cases are quite rare.
I am not sure if one can classify flows in this diagram in the general case.
thank you Sébastien for your answer. Please I would like to know if there is any physical explanation of the sign of the third invariant? (The abscissa of the Lumely diagram) Thank you!
Abdallad.
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At the moment I have not the time to give a correct answer, but I suggest you take a read at: Introduction to Continuum Mechanics Lai, Rubin and Krempl. Perhaps there has the answer. It is rigorous but readable (based in the physical sense).
What you call Lumley Diagram is well known as "Lumley Triangle". If you search the term "Lumley triangle" you'll find a lot of resources explaining the origin and the meaning of each invariant of the Reynold's stresses and how Lumley constructed and explained the triangle.
The Lumley triangle is a way of characterizing the Reynolds stress tensor (indirectly via its anisotropy).
The "right side" is one big Reynolds stress and two small (and equal) ones. Hence the 'cigar'. Which is a shape with one long dimension and two-small dimensions.
The "left side" is two big (and equal) Reynolds stresses and one small one.
Sometimes called pancake shaped.
The top boundary is call Two-component (or 2C) turbulence. One of the Reynolds stresses (or two of them) is zero there (just like turbulence very near a wall).
When two different flows are at the same location on the Lumley triangle invariant map it means surprisingly little. Just that the ratio of the anisotropy levels is the same.
You can get the same anisotropy levels and have completely different looking eddies making those anisotropy levels.
In a boundary layer one of the Reynolds stress (the one normal to the wall) is severely damped (by pressure responses bouncing off the wall) - so you get two big and one small.
In an axisymmetric jet you have strong mean flow gradients in the two
tangential to the axis directions - causing large production of turbulence for those two directions of the Reynolds stress.
So ... in the B.L. one stress damped so two big and one small Reynolds stress.
In the axi-jet, two stresses get energy from the mean flow, so two big and one small. So both have the same anisotropy - for totally different reasons.
Sorry. Lumley diagram says very little really. It is a trick borrowed by Lumley (like much of his stuff) from Non-Newtonian fluid mechanics. Don't over-read its implications.
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