Lie Algebra of the Symmetries of the Multi-Point Equations in Statistical Turbulence Theory

Rosteck, Andreas; Oberlack, Martin

doi:10.1142/s1402925111001404

Cited by 19 publications

(4 citation statements)

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“…General expressions valid for all statistical moments require a more advanced notation and are given in the work of Rosteck. 23 The seminal finding by Oberlack and Rosteck 23,26 is that additional symmetries are present in the averaged system, which we call statistical symmetries. Note that this term may mislead one into asserting that the symmetries themselves are in some way statistical, which is of course not the case-instead, their name stems from the fact that they are specific to a statistical description of turbulence.…”

Section: One-and Two-point Turbulence Descriptionmentioning

confidence: 99%

“…Oberlack and Rosteck 25,26 conducted a symmetry analysis based on the instantaneous system, i.e., the averaged Navier-Stokes system (4) and ( 17) extended by ( 19) and an infinite hierarchy of higherorder equations, but the results can easily be expressed in any other of the presented formulations. All symmetries of the unaveraged Navier-Stokes system, which we refer to as classical symmetries from here on, are preserved in the averaged system.…”

Section: One-and Two-point Turbulence Descriptionmentioning

confidence: 99%

“…( 1) and ( 2) and, therefore, exhibits all classical symmetries given by Eqs. ( 20)- (26). This greatly simplifies the modeling procedure.…”

Section: B Failure Of Classical Approachmentioning

confidence: 99%

“…In Secs. IV B-IV E, we take the invariant modeling approach a step further by (i) incorporating the statistical symmetries ( 41)-( 44) in addition to the symmetries of the Euler and Navier-Stokes equations, i.e., ( 20)- (26), and (ii) generating equations using a formal method instead of a heuristic trial-and-error approach.…”

Section: A Introductory Example: Constructing the Euler Equations Fro...mentioning

confidence: 99%

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Symmetries and turbulence modeling

2020

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This work applies new insights into turbulent statistics gained by Lie symmetry analysis to the closure problem of turbulence. Founded in the mathematics of partial differential equations, Lie symmetries have helped advances in many fields of modern physics. The main reason for this is their ability to encode important physical principles that are implicitly expressed by governing equations. Newly discovered symmetries of the multi-point correlation equations describing turbulent motion have been shown to encode two central effects of turbulent statistics: intermittency and non-Gaussianity. Moreover, these symmetries play a pivotal role in obtaining turbulent scaling laws such as the logarithmic law of the wall. Evidently, correctly preserving these symmetry properties in a turbulence model would render it capable of accurately predicting important effects of turbulent statistics and turbulent scaling. As these symmetry constraints have so far not been taken into account when devising turbulence models, we present a completely new modeling framework that can yield models fulfilling these conditions. In order to accomplish this, it turns out to be helpful, if not necessary, to introduce an entirely new symmetry-based modeling strategy that allows systematically constructing equations based on symmetry constraints imposed on them. From these considerations, it can be shown that in order to create meaningful turbulence models that fulfill these constraints, it is necessary to introduce a new velocity and pressure field. A possible skeleton of model equations for second moment closure is presented.

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Section: One-and Two-point Turbulence Descriptionmentioning

confidence: 99%

Section: One-and Two-point Turbulence Descriptionmentioning

confidence: 99%

“…( 1) and ( 2) and, therefore, exhibits all classical symmetries given by Eqs. ( 20)- (26). This greatly simplifies the modeling procedure.…”

Section: B Failure Of Classical Approachmentioning

confidence: 99%

Section: A Introductory Example: Constructing the Euler Equations Fro...mentioning

confidence: 99%

See 2 more Smart Citations