General principles of the motion of fluids

Euler, Leonhard

doi:10.1016/j.physd.2008.02.023

Cited by 18 publications

(12 citation statements)

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“…qγ (−µ * , Γ * ) = γe * + µ * hi µi − µ * , where e * is given by equations (26) and (27). The energy E−µ * (γ ) of these states is given by equation (32).…”

Section: Computation Of Energies and Circulations Of Canonical Solumentioning

confidence: 99%

Solvable Phase Diagrams and Ensemble Inequivalence for Two-Dimensional and Geophysical Turbulent Flows

Venaille

Bouchet

2011

J Stat Phys

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Using explicit analytical computations, generic occurrence of inequivalence between two or more statistical ensembles is obtained for a large class of equilibrium states of two-dimensional and geophysical turbulent flows. The occurrence of statistical ensemble inequivalence is shown to be related to previously observed phase transitions in the equilibrium flow topology. We find in these turbulent flow equilibria, two mechanisms for the appearance of ensemble equivalences, that were not observed in any physical systems before. These mechanisms are associated respectively with second-order azeotropy (simultaneous appearance of two second-order phase transitions), and with bicritical points (bifurcation from a first-order to two second-order phase transition lines). The important roles of domain geometry, of topography, and of a screening length scale (the Rossby radius of deformation) are discussed. It is found that decreasing the screening length scale (making interactions more local) surprisingly widens the range of parameters associated with ensemble inequivalence. These results are then generalized to a larger class of models, and applied to a complete description of an academic model for inertial oceanic circulation, the Fofonoff flow.

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“…qγ (−µ * , Γ * ) = γe * + µ * hi µi − µ * , where e * is given by equations (26) and (27). The energy E−µ * (γ ) of these states is given by equation (32).…”

Section: Computation Of Energies and Circulations Of Canonical Solumentioning

confidence: 99%

Solvable Phase Diagrams and Ensemble Inequivalence for Two-Dimensional and Geophysical Turbulent Flows

Venaille

Bouchet

2011

J Stat Phys

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“…This is well known both to mathematicians and to more practical scientists, such as hydraulic and aeronautical engineers. Euler [42] wrote, "If it is not permitted to us to penetrate to a complete knowledge concerning the motion of fluids, it is not to mechanics nor to the insufficiency of the known principles of motion that we must attribute the cause. It is analysis itself which abandons us here since all the theory of the motion of fluids has been reduced to the solution of analytic formulas".…”

Section: Introductionmentioning

confidence: 99%

Steady water waves

Strauss

2010

Bull. Amer. Math. Soc.

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“…The dependence on the square of the velocity gradient and the square of the cell size is noteworthy and is not justified in [56]. Note that the FSNS momentum equation (15) derived in section 4 predicts both the form and the magnitude of the artificial viscosity. Artificial viscosity methods remain the principal Lagrangian methodology for flows with shocks.…”

Section: Discrete Regularizationmentioning

confidence: 97%

“…Fluids in motion. The Euler equations [15] and Navier-Stokes equations [53] were originally derived on the basis of classical mechanics and experiment. Here, we briefly describe those two sets of evolution equations and their relationship in the more modern context of the kinetic theory of gases [31].…”

Section: Introductionmentioning

confidence: 99%

Conservation laws in discrete geometry

Margolin¹,

Baty²

2019

Journal of Geometric Mechanics

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The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. We describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.

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General principles of the motion of fluids

Cited by 18 publications

References 1 publication

Solvable Phase Diagrams and Ensemble Inequivalence for Two-Dimensional and Geophysical Turbulent Flows

Solvable Phase Diagrams and Ensemble Inequivalence for Two-Dimensional and Geophysical Turbulent Flows

Steady water waves

Conservation laws in discrete geometry

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