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Tetrahedron deformation and alignment of perceived vorticity and strain in a turbulent flow
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10.1063/1.4795547
/content/aip/journal/pof2/25/3/10.1063/1.4795547
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/3/10.1063/1.4795547
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Deformation of tetrahedra: alignment of the principal axes of the tetrahedra with the eigenvectors of the strain. The ensemble averages ⟨[ () · (0)]⟩ as a function of time for (a) = 1 (longest axis), (b) = 2 (intermediate axis), and (c) = 3 (shortest axis of the tetrahedron). As expected, the th axis of the tetrahedron, (), aligns perfectly at very short times with the th eigenvector of the strain, (0). After a time of order ∼ , the alignment relaxes and the axes of the tetrahedron do not show any particular alignment with any of the eigenvectors of the strain. The DNS data correspond to = 170, whereas the experiments to = 350.

Image of FIG. 2.
FIG. 2.

Flattening of tetrahedra. (a) PDF of at several times, / = 0.07, 0.13, and 0.21, for different initial tetrahedron sizes ( from /16 to /2) in both DNS ( = 170) and experiment ( = 350). The observed peak of the PDF decreases when time increases. At = 0.21 , a non-zero PDF at ≈ 0 is observed. The probability that ⩽ 10 is shown in (b) for several values of , as a function of the normalized time / . A steep growth of the probability ( ⩽ 10) is seen for /4. The growth of the probability does not depend much on the scale over the inertial range: from = /16 ≈ 20η in DNS and = /17 ≈ 50η in experiments to = /2.

Image of FIG. 3.
FIG. 3.

Scale dependence of the alignment of vorticity and strain (instantaneous statistics): The PDFs of | · | from both DNS (panels (a), (c), and (e)) and experiments (panels (b), (d), and (f)) for tetrahedra with different sizes from dissipative ( = 0 in DNS) to integral (/2) scales. Panels (a) and (b) are for = 1; (c) and (d) for = 2; and (e) and (f) for = 3. The lack of alignment of vorticity with is nearly independent of scale. The PDF of | · | sharply peaks at 1 at very small scales, in particular for the true velocity gradient tensor ( = 0). The effect weakens as the scale increases. Similarly, the maximum at 0 of the PDF of | · | at very small scales becomes milder as scale increases. The DNS data correspond to = 170, whereas the experiments to = 350.

Image of FIG. 4.
FIG. 4.

The mean values ⟨( (0) · (0))⟩. The open symbols are DNS data ( = 170) and the filled symbols are experimental results ( = 350).

Image of FIG. 5.
FIG. 5.

Dependence of the intermediate eigenvalue of strain on scale . (a) PDF of β, as defined in Eq. (13) , for several values of . (b) The average value of β as a function of scale, in which the dashed line indicates ⟨β⟩ for the true velocity gradient as → 0. The DNS data correspond to = 170, whereas the experiments to = 350.

Image of FIG. 6.
FIG. 6.

Alignment between () and (0). Ensemble averages ⟨[ (0) · ()]⟩ for = 1 (a), = 2 (b), and = 3 (c) as a function of the time delay , normalized by . The initial tetrahedron size, , are all in the inertial range. The evolution of the alignment of () with (0) suggests a self-similarity: The curves in (a) all superpose once is expressed in units of . The alignment of () with (0) and (0) shows a stronger dependence on , consistent with the scale dependence of the alignment properties of with and (Fig. 3 ). The DNS data correspond to = 170, whereas the experiments to = 350.

Image of FIG. 7.
FIG. 7.

Evolution of the angle between (0) and (). The PDF of | (0) · ()| is shown for times between 0 ⩽ / ⩽ 0.2, and at three values of : = /2 in DNS ( = 170) (a)-(c), /9 and /17 in experiment ( = 350) (d)-(f), and = /16 in DNS (g)-(i). Panels (a), (d), and (g) are for = 1. Consistent with previous results, the evolution of these PDFs is essentially self-similar for in the inertial range. Panels (b), (e), and (h) are for = 2. The PDFs hardly change over the range of times shown; however, they depend significantly on the value of . Panels (c), (f), and (i) are for = 3. During the period of time shown, the PDFs evolve to peak at 0, i.e., () tends to become perpendicular to (0). There is also a moderate dependence on scale .

Image of FIG. 8.
FIG. 8.

Alignment of () with (0) weighted by vorticity. The averages ⟨[ (0) · ω()]⟩/⟨ω()⟩ are shown as a function of / for different values of in the inertial range. The DNS data correspond to = 170, and the experiments to = 350.

Image of FIG. 9.
FIG. 9.

Alignment of () with (0) weighted by strain. The averages ⟨λ(0)[ (0) · ()]⟩/⟨ω(0)⟩ are shown as a function of / for different values of in the inertial range. The DNS data correspond to = 170, and the experiments to = 350.

Image of FIG. 10.
FIG. 10.

Alignment of () with (0) weighted by both strain and vorticity. The averages ⟨λ(0)[ (0) · ω()]⟩/⟨ω()⟩⟨ω(0)⟩ are shown as a function of / for different values of in the inertial range. The DNS data correspond to = 170, and the experiments to = 350.

Image of FIG. 11.
FIG. 11.

Evolution of ⟨β⟩, the mean of the normalized intermediate eigenvalue of strain. With time rescaled with , the curves corresponding to different values of collapse very well. The DNS data correspond to = 170, and the experiments to = 350.

Image of FIG. 12.
FIG. 12.

Alignment between () and (0) for the true velocity gradient tensor , obtained from DNS ( = 170). The averages ⟨[ (0) · ()]⟩ for = 1, 2, and 3, are indicated by crosses, downwards pointing, and right pointing triangles. The time is normalized by the Kolmogorov time scale, τ = (ν/ɛ). The observed evolutions of the alignment of () and (0) are qualitatively very similar to what is shown in Fig. 6 for tetrahedra with size in the inertial range.

Image of FIG. 13.
FIG. 13.

Alignment between () and (0) measured with isotropic tetrahedra of fixed shapes and size , obtained from DNS at = 170. The averages ⟨[ (0) · ()]⟩ for = 1 (a), 2 (b), and 3 (c), as a function of / , are shown for several values of , all in the inertial range. The tendency of () to align with (0), and to become perpendicular to (0), are much weaker than with tetrahedra freely advected with the flow (compare with Fig. 6 ), or with the true velocity gradient tensor (compare with Fig. 12 ).

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/content/aip/journal/pof2/25/3/10.1063/1.4795547
2013-03-26
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Tetrahedron deformation and alignment of perceived vorticity and strain in a turbulent flow
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/3/10.1063/1.4795547
10.1063/1.4795547
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