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Article

Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models

1
Institute of Geophysics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
2
Federal Research Center for Information and Computational Technologies, Russian Academy of Sciences, Lavrentjev ave. 6, 630090 Novosibirsk, Russia
3
Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany
4
Center for Computational Engineering, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(9), 1482; https://doi.org/10.3390/sym12091482
Received: 12 August 2020 / Revised: 2 September 2020 / Accepted: 4 September 2020 / Published: 9 September 2020
(This article belongs to the Special Issue Symmetry in Fluid Flow)
This paper addresses the problem of the existence of conformal invariance in a class of hydrodynamic models. For this we analyse an underlying transport equation for the one-point probability density function, subject to zero-scalar constraint. We account for the presence of non-zero viscosity and large-scale friction. It is shown analytically, that zero-scalar characteristics of this equation are invariant under conformal transformations in the presence of large-scale friction. However, the non-zero molecular diffusivity breaks the conformal group (CG). This connects our study with previous observations where CG invariance of zero-vorticity isolines of the 2D Navier–Stokes equation was analysed numerically and confirmed only for large scales in the inverse energy cascade. In this paper, an example of CG is analysed and possible interpretations of the analytical results are discussed. View Full-Text
Keywords: conformal symmetry; Lundgren–Monin–Novikov equations; method of characteristics conformal symmetry; Lundgren–Monin–Novikov equations; method of characteristics
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MDPI and ACS Style

Wacławczyk, M.; Grebenev, V.N.; Oberlack, M. Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models. Symmetry 2020, 12, 1482. https://doi.org/10.3390/sym12091482

AMA Style

Wacławczyk M, Grebenev VN, Oberlack M. Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models. Symmetry. 2020; 12(9):1482. https://doi.org/10.3390/sym12091482

Chicago/Turabian Style

Wacławczyk, Marta, Vladimir N. Grebenev, and Martin Oberlack. 2020. "Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models" Symmetry 12, no. 9: 1482. https://doi.org/10.3390/sym12091482

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