The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

Grebenev, V. N.; Grishkov, A.; Oberlack, Martin

doi:10.1155/2013/469654

Cited by 2 publications

(4 citation statements)

References 14 publications

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“…the quantity c is the so-called central charge, and its value in general depends on the particular theory under consideration. The proof of this assertion repeats our result given in [14]. We cannot calculate the value of the central charge c without further specification of the statistics of the zero-vorticity characteristic X 1 (s) i.e.…”

Section: Lie Algebra Of the Symmetry Transformation Of The Characteri...supporting

confidence: 86%

“…Therein, conditional averages were calculated from DNS data and Lagrangian particles were tracked in these conditional fields determining characteristics of the first equation from the LMN hierarchy for temperature. In our case, conditional velocities u(x, t)| {ω, x} and v(x, t)| {ω, x} could be calculated from DNS data based on relations (14) and (15). Resulting particle tracks should also be the SLE κ curves, as in the case of zerovorticity lines investigated in [4].…”

Section: Invariance Of the Separation And Coincidence Conditionsmentioning

confidence: 99%

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Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence

Grebenev

Wacławczyk

Oberlack

2019

J. Phys. A: Math. Theor.

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It was clearly validated experimentally in Bernard et al (2006 Nat. Phys. 2 124–8) that the zero-vorticity isolines in 2D turbulence belong to the class of conformal invariant (Schram–Löwner evolution) curves with . The diffusion coefficient classifies the conformally invariant random curves. With this motivation, we performed a Lie group analysis in Grebenev et al (2017 Phys. A: Math. Theor. 50 5502–44) of the first of the inviscid Lundgren–Monin–Novikov (LMN) equations for 2D vorticity fields. This equation describes the evolution of the 1-point probability density function (PDF) . We proved that the conformal group (CG) is not admitted by the 1-point PDF equation itself, however it is permitted under the condition . The main focus of the present work is to prove explicitly the CG invariance of the zero-vorticity Lagrangian path, which is the characteristic of the inviscid LMN hierarchy truncated to the first equation. We also show the CG invariance of the separation and coincidence properties of the PDFs. Besides the derivation of the CG invariance of the zero-vorticity Lagrangian path, we demonstrate that the infinitesimal operator admitted by the characteristic equations forms a Lie algebra which is the Witt algebra, whose central extension represents exactly the Virasoro algebra. The numerical value of the central charge c occurring here could not be calculated. Other mathematical tools need to be involved to link this analytical study with the previous analyses by Bernard et al, who report the value c = 0, which corresponds to for the .

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Section: Lie Algebra Of the Symmetry Transformation Of The Characteri...supporting

confidence: 86%

Section: Invariance Of the Separation And Coincidence Conditionsmentioning

confidence: 99%

Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence

Grebenev

Wacławczyk

Oberlack

2019

J. Phys. A: Math. Theor.

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“…Notice that the quantities c and c coincide. We presented the asymptotic behavior of the transversal correlation function B NN as | r| → ∞ and this is in K 3 with dl 2 D 2 (t), see [6] for more details.…”

Section: Asymptotic Expansion Of the Transversal Correlation Functionmentioning

confidence: 99%

“…The aim of this review is to present both geometric and isometry properties of the 2-point velocity correlation tensor field B i j (x , x,t) for the case of homogeneous isotropic turbulence which were obtained in [3]- [6] and demonstrates their impact in turbulence. The review is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

Grebenev¹,

Oberlack²

2021

JNMP

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The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field B i j (x, x ,t) and isometries of the correlation space K 3 equipped with a (pseudo-) Riemannian metrics ds 2 (t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds 2 (t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (M t , ds 2 (t)), M t ⊂ K 3 . Here the metric ds 2 (t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds 2 (t) in K 3 and present symmetries (or isometric motions in K 3 ) of the metric ds 2 (t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds 2 (t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λ g is a second-order differential invariant and show how λ g can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.

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The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

Cited by 2 publications

References 14 publications

Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence

Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence

Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

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