Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

Cheviakov, Alexei F.; Oberlack, Martin

doi:10.1017/jfm.2014.611

Cited by 31 publications

(28 citation statements)

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“…in Cheviakov's potential vorticity type equation (4.30). Rosenhaus and Shankar (2016) develop Noether's second theorem for quasi-Noether systems, and describe the conservation laws obtained by Cheviakov (2014) and Cheviakov and Oberlack (2014).…”

Section: The Gauge Symmetrymentioning

confidence: 99%

On magnetohydrodynamic gauge field theory

Webb

Anco

2017

J. Phys. A: Math. Theor.

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Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963). It is shown how the polarization vector P in Calkin's approach naturally arises from the Lagrange multiplier constraint equation for Faraday's equation for the magnetic induction B, or alternatively from the magnetic vector potential form of Faraday's equation. Gauss's equation, (divergence of B is zero) is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether's theorem coupled with the gauge symmetries is used to derive the conservation laws for (a) magnetic helicity, (b) cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws associated with curl and divergence equations which applies to Faraday's equation and Gauss's equation. The magnetic helicity conservation law is due to a gauge symmetry in MHD and not due to a fluid relabelling symmetry. The analysis is carried out for the general case of a nonbarotropic gas in which the gas pressure and internal energy density depend on both the entropy S and the gas density ρ. The cross helicity and fluid helicity conservation laws in the non-barotropic case are nonlocal conservation laws that reduce to local conservation laws for the case of a barotropic gas. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed. It is shown that the gauge symmetry functionals in the work of Henyey (1982) satisfy the Casimir determining equations.PACS numbers: 95.30. Qd,47.35.Tv,52.30.Cv,45.20.Jj,96.60.j,96.60.Vg P·dS = P x dy ∧ dz + P y dz ∧ dx + P z dx ∧ dy,( 1.5) is the polarization 2-form. In the usual MHD, non-relativistic limit ∇·P = 0 (i..e. ρ c = 0), and (1.2) simplifies to:

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Section: The Gauge Symmetrymentioning

confidence: 99%

On magnetohydrodynamic gauge field theory

Webb

Anco

2017

J. Phys. A: Math. Theor.

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“…Examples in Sections 4 and 5 involve less than two hundred determining equations; similar numbers of determining equations arise, for example, in the study of point symmetries of magnetohydrodynamics equilibrium equations [35]. We note that both GeM software and the Maple rifsimp routine can efficiently processes substantially more complicated cases; for example, one calculations involving 31,918 and 58,273 determining equations were successfully completed in a related application of local conservation law computations in fluid dynamics models [36].…”

Section: Discussionmentioning

confidence: 89%

“…36) etc., as appropriate, to exclude the dependence of transformation components of the arbitrary elements on variables they do not depend on, as well as the dependence of transformation components of the variables of the system on inappropriate arbitrary elements. For example, must not explicitly depend on the variables x, t. Therefore the restrictions on the component θ are given by In order to simplify computations, additional restrictions can be introduced at this stage, for example,…”

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confidence: 99%

Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models

Cheviakov

2017

Computer Physics Communications

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An efficient systematic procedure is provided for symbolic computation of Lie groups of equivalence transformations and generalized equivalence transformations of systems of differential equations that contain arbitrary elements (arbitrary functions and/or arbitrary constant parameters), using the software package GeM for Maple. Application of equivalence transformations to the reduction of the number of arbitrary elements in a given system of equations is discussed, and several examples are considered. First computational example of a generalized equivalence transformation where the transformation of the dependent variable involves the arbitrary constitutive function is presented.As a detailed physical example, a three-parameter family of nonlinear wave equations describing finite anti-plane shear displacements of an incompressible hyperelasic fiber-reinforced medium is considered. Equivalence transformations are computed and employed to radically simplify the model for an arbitrary fiber direction, invertibly reducing the model to a simple form that corresponds to a special fiber direction, and involves no arbitrary elements.The presented computation algorithm is applicable to wide classes of systems of differential equations containing arbitrary elements.

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“…This framework is based on the idea that SGS models should be consistent with the fundamental physical and mathematical properties of the Navier-Stokes equations and the turbulent stresses. Specifically, consistency of SGS models with the symmetries (Speziale 1985;Oberlack 1997Oberlack , 2002Razafindralandy et al 2007) and conservation laws (Cheviakov and Oberlack 2014) of the Navier-Stokes equations, and the dissipation properties (Vreman 2004;Razafindralandy et al 2007;Nicoud et al 2011;Verstappen 2011), realizability (Vreman et al 1994b) and near-wall scaling behavior (Chapman and Kuhn 1986) of the turbulent stresses is desired. As explained in Section 3.2, the general class of SGS models of Eq.…”

Section: Defining the Model Coefficientsmentioning

confidence: 99%

Nonlinear Subgrid-Scale Models for Large-Eddy Simulation of Rotating Turbulent Flows

Silvis

Verstappen

2019

Direct and Large-Eddy Simulation XI

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Rotating turbulent flows form a challenging test case for large-eddy simulation (LES). We, therefore, propose and validate a new subgrid-scale (SGS) model for such flows. The proposed SGS model consists of a dissipative eddy viscosity term as well as a nondissipative term that is nonlinear in the rate-of-strain and rate-of-rotation tensors. The two corresponding model coefficients are a function of the vortex stretching magnitude. Therefore, the model is consistent with many physical and mathematical properties of the Navier-Stokes equations and turbulent stresses, and is easy to implement. We determine the two model constants using a nondynamic procedure that takes into account the interaction between the model terms. Using detailed direct numerical simulations (DNSs) and LESs of rotating decaying turbulence and spanwise-rotating plane-channel flow, we reveal that the two model terms respectively account for dissipation and backscatter of energy, and that the nonlinear term improves predictions of the Reynolds stress anisotropy near solid walls. We also show that the new SGS model provides good predictions of rotating decaying turbulence and leads to outstanding predictions of spanwise-rotating plane-channel flow over a large range of rotation rates for both fine and coarse grid resolutions. Moreover, the new nonlinear model performs as well as the dynamic Smagorinsky and scaled anisotropic minimum-dissipation models in LESs of rotating decaying turbulence and outperforms these models in LESs of spanwise-rotating plane-channel flow, without requiring (dynamic) adaptation or near-wall damping of the model constants.

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Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

Cited by 31 publications

References 25 publications

On magnetohydrodynamic gauge field theory

On magnetohydrodynamic gauge field theory

Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models

Nonlinear Subgrid-Scale Models for Large-Eddy Simulation of Rotating Turbulent Flows

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