Further Remarks on the Luo–Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

Sperone, Gianmarco

doi:10.1007/s00332-017-9363-8

Cited by 3 publications

(4 citation statements)

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“…Swirl of opposite signs from above and below the midplane is thus transported towards z = 0, resulting in intense radial vorticity ω r = −∂ z u θ on the critical ring. This has been described clearly by LH [2][3][4], who show very strong evidence that the flow continues to develop a singularity in a nearly (but not exactly [5,6]), self-similar way. The final state of the present simulations shown in figure 2 is representative of the flow as it approaches the singularity.…”

Section: Basics Of the Singularity Mechanismmentioning

confidence: 74%

A fluid mechanic’s analysis of the teacup singularity

Barkley

2020

Proc. R. Soc. A.

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The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.

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Section: Basics Of the Singularity Mechanismmentioning

confidence: 74%

A fluid mechanic’s analysis of the teacup singularity

Barkley

2020

Proc. R. Soc. A.

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“…Therefore the integrands h(ξ) fundamentally cannot collapse because the (very weak) tails at large ξ cannot. This lack of exact self-similarity for this flow is well known[4,5].…”

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confidence: 61%

“…1. As time evolves, the axial flow along the cylinder wall advects the swirl towards the critical ring, where a singularity develops in a nearly, but not exactly, self-similar way [2][3][4][5].…”

Section: Overview Of the Singularitymentioning

confidence: 99%

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A fluid mechanic's analysis of the teacup singularity

Barkley

2019

Preprint

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The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container, periodic in the axial coordinate. The flow is axisymmetric with swirl. The simulations reproduce and corroborate aspects of the seminal work of Luo and Hou [Luo G, Hou T (2014) Potentially singular solutions of the 3D axisymmetric Euler equations. Proc Natl Acad Sci USA 111:12968-12973]. Contrary to standard practice, the analysis here focuses on the pressure and not the vorticity. Linearity of the pressure Poisson equation is exploited to decompose the pressure field into a superposition of independent contributions arising from the meridional flow and from the swirl. The swirl pressure is further decomposed into axial mean and fluctuating components, where the fluctuating component itself has distinct contributions maintaining incompressibility and confining the flow. The key pressure field driving singularity formation is shown to be that confining the fluid within the cylinder walls. This pressure is directly connected to the Luo and Hou model via the Hilbert transformation.

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Imperfect O(2) symmetry in counter-rotating split-cylinder flow

Gutiérrez-Castillo

Lopez

2022

Physics of Fluids

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The effect of a small imperfection in the counter-rotating split-cylinder flow is studied numerically. The defect is characterized by a small parameter ϵ, corresponding to the difference in the magnitude of rotations in each half of the cylinder. With the two half cylinders not rotating exactly in counter rotation, the O(2) symmetry of the exact counter-rotating case (invariance to azimuthal rotations as well as to an involution consisting of reflections about the mid-plane composed of reflections about any meridional plane) is weakly broken. This small defect results in relevant variations in the flow. For slow rotations (characterized by a small Reynolds number), the system remains axisymmetric with the imperfection only breaking the reflection symmetry about the cylinder half-height. At larger Reynolds numbers, in the absence of the imperfection, axisymmetry is broken resulting in steady states with azimuthal wavenumber m. When axisymmetry is broken in the presence of the imperfection, a background rotation is introduced. Depending on the case and the level of imperfection, either rotating waves or slow-fast dynamics with mean background rotations are found instead. The interaction between azimuthal wavenumbers m = 2 and 3 plays a crucial role in the flow. The flow is analyzed in detail, varying ϵ from a very small value of 0.01%, typical of a natural imperfection in an experimental setup, to higher values corresponding to forced symmetry breaking. The ramifications of the imperfection on various solution states found in the exact counter-rotating case for a fixed aspect ratio are investigated.

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Further Remarks on the Luo–Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

Cited by 3 publications

References 6 publications

A fluid mechanic’s analysis of the teacup singularity

A fluid mechanic’s analysis of the teacup singularity

A fluid mechanic's analysis of the teacup singularity

Imperfect O(2) symmetry in counter-rotating split-cylinder flow

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