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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

### Abstract

We describe the structure and dynamics of turbulence by the scale-dependent perceived velocity gradient tensor as supported by following four tracers, i.e., fluid particles, that initially form a regular tetrahedron. We report results from experiments in a von Kármán swirling water flow and from numerical simulations of the incompressible Navier-Stokes equation. We analyze the statistics and the dynamics of the perceived rate of strain tensor and vorticity for initially regular tetrahedron of size r 0 from the dissipative to the integral scale. Just as for the true velocity gradient, at any instant, the perceived vorticity is also preferentially aligned with the intermediate eigenvector of the perceived rate of strain. However, in the perceived rate of strain eigenframe fixed at a given time t = 0, the perceived vorticity evolves in time such as to align with the strongest eigendirection at t = 0. This also applies to the true velocity gradient. The experimental data at the higher Reynolds number suggests the existence of a self-similar regime in the inertial range. In particular, the dynamics of alignment of the perceived vorticity and strain can be rescaled by t 0, the turbulence time scale of the flow when the scale r 0 is in the inertial range. For smaller Reynolds numbers we found the dynamics to be scale dependent.

© 2013 American Institute of Physics

Received 21 April 2012
Accepted 27 February 2013
Published online 26 March 2013

Acknowledgments: We are very thankful to B. Lüthi, B. Shraiman, E. Siggia, and A. Tsinober for stimulating discussions, and to L. Chevillard and C. Meneveau for communicating to us Ref. 43 and for freely sharing their insight. A.P. thanks IDRIS for providing the computation resources and the ANR for financial support through Contract No. TEC2. H.X. is grateful to the Deutsche Forschungsgemeinschaft for support through Grant No. XU 91/3-1. We thank the Max Planck Society for support. This research was conducted in part at the Kavli Institute for Theoretical Physics at Santa Barbara, CA (supported by the U.S. National Science Foundation under Grant No. NSF PHY05-51164), the Kavli Institute for Theoretical Physics China at Beijing (supported by the Project of Knowledge Innovation Program (PKIP) of the Chinese Academy of Sciences under Grant No. KJCX2.YW.W10) and received additional support from the the European COST Action MP0806.

Article outline:

I. INTRODUCTION

II. DEFINITION OF THE PERCEIVED VELOCITY GRADIENT TENSOR **M**

III. EXPERIMENTS AND NUMERICAL SIMULATIONS

A. Experiments

B. Direct numerical simulations

IV. DEFORMATION OF THE TETRAHEDRA

A. Alignment of tetrahedron geometry with the eigenvectors of the rate of strain

B. Flattening of the tetrahedra

V. PROPERTIES OF **M**: SCALE DEPENDENCE (INSTANTANEOUS STATISTICS)

A. Alignment between vorticity and strain

B. The intermediate eigenvalues of the rate of strain

VI. PROPERTIES OF M: DYNAMICS (TEMPORAL CORRELATIONS)

A. Averaged alignment of the direction of vorticity with the eigenvectors of the rate of strain

B. Elementary modeling considerations

C. Alignment of vorticity with the eigenvectors of the rate of strain: Conditional statistics

D. Evolution of the intermediate eigenvalue of the rate of strain

VII. DISCUSSION AND CONCLUSIONS

A. Comparison between dissipative and inertial scale dynamics

B. Fixed shape tetrahedra

C. Summary

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2013-03-26

2014-04-24

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