On the Axisymmetric Steady Incompressible Beltrami Flows

Bělík, Pavel; Su, Xueqing; Dokken, Douglas P.; Scholz, Kurt; Shvartsman, Mikhail M.

doi:10.4236/ojfd.2020.103014

Cited by 6 publications

(10 citation statements)

References 34 publications

(48 reference statements)

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“…Gromeka's developments in the theory of helical flows have been frequently used by many authors from Russia [21][22][23][24][25][26][27] and other countries [28][29][30]. Because Gromeka's works were written in Russian and published mainly in the Scientific Notes of Kazan University or the Bulletin of the Kazan Mathematical Society, they are not well known abroad.…”

Section: Laminar Flowmentioning

confidence: 99%

Ten Years of Passion: I.S. Gromeka’s Contribution to Science

Urbanowicz,

Tijsseling

2024

Fluids

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The work and life of Ippolit Stepanovich Gromeka is reviewed. Gromeka authored a classical set of eleven papers on fluid dynamics in just ten years before a tragic illness ended his life. Sadly, he is not well known to the western scientific community because all his publications were written in Russian. He is one of the three authors who independently derived an analytical solution for accelerating laminar pipe flow. He was the first to eliminate the contradiction between the theories of Young and Laplace on capillary phenomena. He initiated the theoretical basis of helical (Beltrami) flow, and he studied the movement of cyclones and anticyclones seventeen years before Zermelo (whose work is considered as pioneering). He is also the first to analyse wave propagation in liquid-filled hoses, thereby including fluid–structure interaction.

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Section: Laminar Flowmentioning

confidence: 99%

Ten Years of Passion: I.S. Gromeka’s Contribution to Science

Urbanowicz,

Tijsseling

2024

Fluids

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“…The velocity solution is implicit in the recent work of Bělík et al. (2020). Solution (7.1) is the superposition of a radially oscillating velocity field and a rigid motion .…”

Section: Vortices In Cylindrical Geometrymentioning

confidence: 99%

Multipolar spherical and cylindrical vortices

Viúdez

2022

J. Fluid Mech.

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Multipolar spherical solutions to the three-dimensional steady vorticity equation are provided. These solutions are based on the separation of radial and angular contributions in terms of the spherical Bessel functions and vector spherical harmonics, respectively. In this set of multipolar vortex solutions, the Hicks–Moffatt swirling vortex is categorized as a vortex of degree ${\ell }=1$ and therefore as a vortex dipole. This swirling vortex is the three-dimensional dipole in spherical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry. The three-dimensional dipole solution admits two linearly superposable solutions. The first one is a Trkalian flow and the second one is a cylindrical solid-body rotation with swirl. The higher ${\ell }>1$ multipolar vortices found are either vanishing-helicity vortices or Trkalian flow vortices. The multipolar Trkalian flows admit two circular polarizations given by the sign of the wavenumber $k$ . It is also found that piecewise vortex solutions, consisting of interior rotational and exterior potential flow domains, satisfying velocity continuity conditions at the vortex boundary, are possible in the general multipolar Trkalian spherical vortex. A particular polarized dipole solution in three-dimensional cylindrical geometry, consisting as well in the superposition of a Trkalian flow and a rigid motion, is also analysed. This swirling vortex may be interpreted as the three-dimensional dipole in cylindrical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry.

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Section: Introductionmentioning

confidence: 99%

“…Steady axisymmetric inviscid Beltramian solutions in bounded or unbounded space have been found by focusing on the so-called Bragg-Hawthorne equation [17,22,23]. How the structure of the fundamental modes admitted by the linear equation varies by depending on the coordinate system chosen for solving the equation has been revealed in [22,24]. Non-axisymmetric solutions with vanishing helicity, i.e., inner products of velocity and vorticity, as well as the vortices with a constant proportionality coefficient, c hereafter, are presented in [24].…”

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

Takahashi

2023

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This paper is aimed at eliciting consistency conditions for the existence of unsteady incompressible axisymmetric swirling viscous Beltrami vortices and explicitly constructing solutions that obey the conditions as well as the Navier–Stokes equations. By Beltrami flow, it is meant that vorticity, i.e., the curl of velocity, is proportional to velocity at any local point in space and time. The consistency conditions are derived for the proportionality coefficient, the velocity field and external force. The coefficient, whose dimension is of [length−1], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances.

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On the Axisymmetric Steady Incompressible Beltrami Flows

Cited by 6 publications

References 34 publications

Ten Years of Passion: I.S. Gromeka’s Contribution to Science

Ten Years of Passion: I.S. Gromeka’s Contribution to Science

Multipolar spherical and cylindrical vortices

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

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