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On the Lie symmetries of characteristic function hierarchy in compressible turbulence

Part of: Basic methods in fluid mechanics Turbulence

Published online by Cambridge University Press:  11 April 2022

D.S. PRATURI Open the ORCID record for D.S. PRATURI [Opens in a new window] ,
D. PLÜMACHER  and
M. OBERLACK
D.S. PRATURI
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, 64287 Darmstadt, Germany emails: dspraturi@gmail.com; pluemacher@fdy.tu-darmstadt.de
D. PLÜMACHER
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, 64287 Darmstadt, Germany emails: dspraturi@gmail.com; pluemacher@fdy.tu-darmstadt.de
M. OBERLACK
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, 64287 Darmstadt, Germany emails: dspraturi@gmail.com; pluemacher@fdy.tu-darmstadt.de Center for Computational Engineering, TU Darmstadt, 64293 Darmstadt, Germany email: oberlack@fdy.tu-darmstadt.de
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Abstract

We compute the Lie symmetries of characteristic function (CF) hierarchy of compressible turbulence, ignoring the effects of viscosity and heat conductivity. In the probability density function (PDF) hierarchy, a typical non-local nature is observed, which is naturally eliminated in the CF hierarchy. We observe that the CF hierarchy retains all the symmetries satisfied by compressible Euler equations. Broadly speaking, four types of symmetries can be discerned in the CF hierarchy: (i) symmetries corresponding to coordinate system invariance, (ii) scaling/dilation groups, (iii) projective groups and (iv) statistical symmetries, where the latter define measures of intermittency and non-gaussianity. As the multi-point CFs need to satisfy additional constraints such as the reduction condition, the projective symmetries are only valid for monatomic gases, that is, the specific heat ratio, $\gamma = 5/3$. The linearity of the CF hierarchy results in the statistical symmetries due to the superposition principle. For all of the symmetries, the global transformations of the CF and various key compressible statistics are also presented.

Keywords

Characteristic function hierarchycompressible turbulencesymmetry analysis

MSC classification

Primary: 76M60: Symmetry analysis, Lie group and algebra methods
Secondary: 76F50: Compressibility effects
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Special Issue 5: Symmetries and Differential Equations , October 2023 , pp. 913 - 935
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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