Second-order invariants of the inviscid Lundgren-Monin-Novikov equations for 2d vorticity fields

Author(s)

Grebenev, V. N. ^[1] ; Grichkov, A. N. ^{[2, 3]} ; Oberlack, M. ^[4] ; Waclawczyk, M. ^[5]

Affiliation

^[1] Fed Res Ctr Informat & Computat Technol, Lavrentjev Ave 6, Novosibirsk 630090 - Russia

^[2] Univ Sao Paulo, Inst Math & Stat, Rua Matao 1010, BR-66281 Sao Paulo - Brazil

^[3] Omsk am FM Dostoevsky State Univ, Litskevich 1 Omsk, Omsk 644053 - Russia

^[4] Tech Univ Darmstadt, Dept Mech Engn, Chair Fluid Dynam, Otto Berndt Str 2, D-64287 Darmstadt - Germany

^[5] Univ Warsaw, Fac Phys, Inst Geophys, Pasteura 7, PL-02093 Warsaw - Poland

• Document type: Journal article
• Source: ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK; v. 72, n. 3 JUN 2021.
• Web of Science Citations: 0

Abstract

In Grebenev, Wacawczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33), the conformal invariance (CI) of the characteristic X1(t) (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the 1-point PDF f1(x1,omega 1,t), x1 is an element of D1 subset of R2) of the inviscid Lundgren-Monin-Novikov (LMN) equations for 2d vorticity fields was derived. The infinitesimal operator admitted by the characteristics equation generates an infinite-dimensional Lie pseudo-group G which conformally acts on D1. We define the conformal invariant differential form ds2=f1.<mml:mfenced close={''}){''} open={''}({''}>dX11</mml:msubsup>2+dX12</mml:msubsup>2</mml:mfenced> along the characteristic <mml:mfenced close={''}|{''}>X1(t)</mml:mfenced>omega 1=0 together with the simple action functional F(X1,ds2). We demonstrate that GY, which is a subgroup of the group G restricted on the variables x1 and f1, gives rise to a symmetry transformations of F(X1,ds2). With this, we calculate the second-order universal differential invariant J2Y</mml:msubsup> (or the multiscale representation of the invariants) of <mml:msub>GY under the action on the zero-vorticity characteristics. We show that F(<mml:msub>X1,ds2) is a scalar invariant and generates all differential invariants, which look like the quantities of different scales, from J2Y by the operators of invariant differentiation. It gives insight into the geometry of a flow domain nearby point <mml:msub>x1 in the sense of Cartan. (AU)

• FAPESP's process: 18/21330-2 - Minimal set of differential invariants of an extended loop group arising in fluid flows
• Grantee: Alexandre Grichkov
• Support Opportunities: Research Grants - Visiting Researcher Grant - International