Existence of knotted vortex structures in stationary solutions of the Euler equations

2018 ◽  
pp. 133-153
Author(s):  
Alberto Enciso ◽  
Daniel Peralta-Salas
Keyword(s):  
Euler Equations ◽  
Stationary Solutions ◽  
Vortex Structures

Stationary solutions to Euler equations with spherical symmetry

2005 ◽  
Vol 61 (1-2) ◽  
pp. 261-267
Author(s):  
Yanhong Liu ◽  
Changjiang Zhu
Keyword(s):  
Euler Equations ◽  
Spherical Symmetry ◽  
Stationary Solutions

scholarly journals Self-intersecting Interfaces for Stationary Solutions of the Two-Fluid Euler Equations

Annals of PDE ◽  
2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Diego Córdoba ◽  
Alberto Enciso ◽  
Nastasia Grubic
Keyword(s):  
Euler Equations ◽  
Stationary Solutions ◽  
Two Fluid

Selected topics on the topology of ideal fluid flows

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630012 ◽  
Author(s):  
Daniel Peralta-Salas
Keyword(s):  
Euler Equations ◽  
Topological Structure ◽  
Fluid Flows ◽  
Stationary Solutions ◽  
Knot Invariants ◽  
Vortex Lines ◽  
The Subject ◽  
Complicated Topology ◽  
Incompressible Fluid Flows ◽  
Main Ideas

This is a survey of certain geometric aspects of inviscid and incompressible fluid flows, which are described by the solutions to the Euler equations. We will review Arnold’s theorem on the topological structure of stationary fluids in compact manifolds, and Moffatt’s theorem on the topological interpretation of helicity in terms of knot invariants. The recent realization theorem by Enciso and Peralta-Salas of vortex lines of arbitrarily complicated topology for stationary solutions to the Euler equations will also be introduced. The aim of this paper is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics of the subject known as Topological Fluid Mechanics. This is the set of lecture notes, the author gave at the XXIV International Fall Workshop on Geometry and Physics held in Zaragoza (Spain) during September 2015.

scholarly journals On the Semi-Analytical Solutions in Hydrodynamics of Ideal Fluid Flows Governed by Large-Scale Coherent Structures of Spiral-Type

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
Sergey V. Ershkov ◽  
Alla Rachinskaya ◽  
Evgenii Yu. Prosviryakov ◽  
Roman V. Shamin
Keyword(s):  
Euler Equations ◽  
Coherent Structures ◽  
Large Scale ◽  
Stationary Solutions ◽  
Incompressible Fluids ◽  
Bottom Surface ◽  
Stationary Flows ◽  
Long Time ◽  
Partial Type ◽  
Spiral Type

We have presented here a clearly formulated algorithm or semi-analytical solving procedure for obtaining or tracing approximate hydrodynamical fields of flows (and thus, videlicet, their trajectories) for ideal incompressible fluids governed by external large-scale coherent structures of spiral-type, which can be recognized as special invariant at symmetry reduction. Examples of such structures are widely presented in nature in “wind-water-coastline” interactions during a long-time period. Our suggested mathematical approach has obvious practical meaning as tracing process of formation of the paths or trajectories for material flows of fallout descending near ocean coastlines which are forming its geometry or bottom surface of the ocean. In our presentation, we explore (as first approximation) the case of non-stationary flows of Euler equations for incompressible fluids, which should conserve the Bernoulli-function as being invariant for the aforementioned system. The current research assumes approximated solution (with numerical findings), which stems from presenting the Euler equations in a special form with a partial type of approximated components of vortex field in a fluid. Conditions and restrictions for the existence of the 2D and 3D non-stationary solutions of the aforementioned type have been formulated as well.

scholarly journals A note on the stationary Euler equations of hydrodynamics

2015 ◽  
Vol 37 (2) ◽  
pp. 454-480 ◽  
Author(s):  
K. CIELIEBAK ◽  
E. VOLKOV
Keyword(s):  
Vector Field ◽  
Euler Equations ◽  
Hamiltonian Structure ◽  
Stationary Solutions ◽  
Torus Bundle ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Real Analyticity ◽  
Bernoulli Function ◽  
Real Analytic

This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.

A new model of roll waves: comparison with Brock’s experiments

2012 ◽  
Vol 698 ◽  
pp. 374-405 ◽  
Author(s):  
G. L. Richard ◽  
S. L. Gavrilyuk
Keyword(s):  
Euler Equations ◽  
Large Scale ◽  
Stationary Solutions ◽  
Small Scale ◽  
Roll Waves ◽  
Averaged Equations ◽  
Isentropic Euler Equations ◽  
The Waves ◽  
Shock Thickness

AbstractWe derive a mathematical model of shear flows of shallow water down an inclined plane. The non-dissipative part of the model is obtained by averaging the incompressible Euler equations over the fluid depth. The averaged equations are simplified in the case of weakly sheared flows. They are reminiscent of the compressible non-isentropic Euler equations where the flow enstrophy plays the role of entropy. Two types of enstrophies are distinguished: a small-scale enstrophy generated near the wall, and a large-scale enstrophy corresponding to the flow in the roller region near the free surface. The dissipation is then added in accordance with basic physical principles. The model is hyperbolic, the corresponding ‘sound velocity’ depends on the flow enstrophies. Periodic stationary solutions to this model describing roll waves were obtained. The solutions are in good agreement with the experimental profiles of roll waves measured in Brock’s experiments. In particular, the height of the vertical front of the waves, the shock thickness and the wave amplitude are well captured by the model.

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