TY - JOUR

T1 - The area rule for circulation in three-dimensional turbulence

AU - Iyer, Kartik P.

AU - Bharadwaj, Sachin S.

AU - Sreenivasan, Katepalli R.

N1 - Funding Information:
ACKNOWLEDGMENTS. We are grateful to A. A. Migdal for continuing and stimulating discussions and P. K. Yeung for his sustained collaboration on the DNS data. We thank A. M. Polyakov particularly for his insight on the area rule and G. L. Eyink and E. D. Siggia for uplifting remarks over time. This work is partially supported by the NSF, via Grant ACI-1640771 at the Georgia Institute of Technology. The computations were performed using supercomputing resources provided through the Extreme Science and Engineering Discovery Environment (XSEDE) consortium (which is funded by NSF) at the Texas Advanced Computing Center at the University of Texas (Austin) and the Blue Waters Project at the National Center for Supercomputing Applications at the University of Illinois (Urbana–Champaign).
Publisher Copyright:
© 2021 National Academy of Sciences. All rights reserved.

PY - 2021/10/26

Y1 - 2021/10/26

N2 - An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called area rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the area rule, for both planar and nonplanar loops, using high-resolution data from direct numerical simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is farther from unity and when the moment order increases. The differences do not exceed about 5% for any condition examined here. The aspect ratio dependence observed for the second-order moment is indistinguishable from results for the Gaussian random field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller (< 2%) but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence.

AB - An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called area rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the area rule, for both planar and nonplanar loops, using high-resolution data from direct numerical simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is farther from unity and when the moment order increases. The differences do not exceed about 5% for any condition examined here. The aspect ratio dependence observed for the second-order moment is indistinguishable from results for the Gaussian random field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller (< 2%) but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence.

KW - Direct numerical simulation

KW - Isotropic turbulence

KW - Velocity circulation

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U2 - 10.1073/pnas.2114679118

DO - 10.1073/pnas.2114679118

M3 - Article

C2 - 34663734

AN - SCOPUS:85117418956

VL - 118

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 43

M1 - e2114679118

ER -