Recently the focus in turbulence is shifted to the study of shape evolution. Staring from regular shapes at the Kolmogorov dissipative scale, one wants to characterize the geometrical evolution (size, area, eigenvalues of the moment of inertia tensor, etc.) with respect to time. The evolution of higher-order correlations such as tetrahedral shapes in turbulence provides an important information on velocity gradient. I would like to know whether there are analytical / numerical predictions regarding the evolution law of a triangle area formed by three neighboring fluid particles in 2D incompressible HIT. Here what I obtained from a numerical simulation performed during 1000 times steps in the tracking algorithm averaged over 1000 particles. The mean area is plotted versus time. Thank you!
In 2D the incompressibility means area conservation (like in 3D it's volume conservation), so how can it grow?
than you Alex for the answer. However I think that is not the case here since we want to track three particles individually without considering a area / volume conservation. I think that it gives a sort of linear evolution for the large times but I didn't know if there are evidences in the literature about that!
Thank you Aria for the links. I am looking for references which deal with shape evolution in 2D turbulence, i.e. triangle area / eigen values of the moment of inertia tensor, etc.. Herewith enclosed the mean-squared-area evolution versus time of a triangle formed by three fluid particles obtained from a incompressible velocity field generated by a Gaussian white noise stream function. I found a power-like evolution (as it is clearly presented in the log-log plot). I don't know if there are some results in the literature about that!
Dear Abdallah Daddi Moussa Ider
Try to see work of V. Klyatskin for statistical topography.
Good Luck
Those papers should be helpful:
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.5324
http://journals.aps.org/pre/abstract/10.1103/PhysRevE.64.056303
For 3D and tetrades, see
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.064504
I do not understand what does "we want to track three particles individually" mean. if 3 particles are tracked "individually", i.e. their identity is not preserved, then there is no "area". If the original marker is a triangle, i.e. three connected particles, then this is a "control surface" and the area, i.e. mass has to be conserved. Am I missing something?
If it has a turbulent cascade and the triangle is larger than the lower cut-off scale, small vortices will provide a flux of fluid through the sides of the triangle; see the refs above.
it says: "Staring from regular shapes at the Kolmogorov dissipative scale"
His graph also starts with a plateau and starts growing rapidly only later (when the triangle is larger)
You probably want to read
Celani A, Vergassola M.
Statistical geometry in scalar turbulence.
Phys Rev Lett. 2001 Jan 15;86(3):424-7.
wrt to area of the triangle, as pointed to by Jaan Kalda :
Strictly speaking an initially triangular piece of fluid will remain triangular and preserve its area only if the velocity field is affine. The velocity field felt by a triplet of particles is approximately affine (by a Taylor expansion) if the inter-particle distance is below Kolmogorov's scale. Beyond that scale the triangle formed by the three particles is not a material object any more and its area is not expected to remain constant.
Thank you Thomas for your explanations as well as for the paper.
best
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