Finite-thickness effect on speed of a counter-rotating vortex pair at high Reynolds numbers

Habibah, Ummu; Nakagawa, Hironori; Fukumoto, Yasuhide

doi:10.1088/1873-7005/aaa5c8

Cited by 4 publications

(20 citation statements)

References 31 publications

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“…1 should be in good agreement with the simulated dipole velocities (Delbende and Rossi, 2009;Habibah et al, 2018), but instead the dipole propagation velocities are consistently twice as large. Recent work (Habibah et al, 2018) expresses the solution to the Navier Stokes equation in form of a power series in the aspect ratio. To first order the propagation velocity is given by our Equation 1, and a correction to this only appears in the fifth order of the aspect ratio.…”

Section: Dipole Propagation Velocitymentioning

confidence: 51%

“…Originally a is increasing with time and depends on viscosity. Equation 11 is a particular solution to the Navier-Stokes equations (Habibah et al, 2018), and is known to show good agreement with experimental data (Leweke et al, 2016). The vortex shape described by Eq.…”

Section: Flow Separation and Vortex Formationmentioning

confidence: 80%

“…Here b is the distance between the vortex centers, and Γ is the magnitude of the circulation in each of the two vortices, assuming they are of equal strength. Equation 1 is valid as long as the distance between the two vortices is large compared to their core radius (Yehoshua and Seifert, 2013;Delbende and Rossi, 2009;Habibah et al, 2018). Habibah et al (2018) show that a correction to the velocity given by Eq.…”

mentioning

confidence: 99%

“…Equation 1 is valid as long as the distance between the two vortices is large compared to their core radius (Yehoshua and Seifert, 2013;Delbende and Rossi, 2009;Habibah et al, 2018). Habibah et al (2018) show that a correction to the velocity given by Eq. 1 occurs in the 5th order of a/b where a is the core radius of the vortices.…”

mentioning

confidence: 99%

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Flow separation, dipole formation and water exchange through tidal strait

Nøst¹,

Børve²

2021

Preprint

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Abstract. We investigate the formation and evolution of dipole vortices and their contribution to water exchange through idealized tidal straits. Self-propagating dipoles are important for transporting and exchanging water properties through straits and inlets in coastal regions. In order to obtain a robust data-set to evaluate flow separation, dipole formation and evolution and the effect on water exchange, we conduct 164 numerical simulations, varying the width and length of the straits as well as the tidal forcing. We show that dipoles are formed and start propagating at the time of flow separation, and their vorticity originates in the velocity front formed by the separation. We find that the dipole propagation velocity is proportional to the tidal velocity amplitude, and twice as large as the dipole velocity derived for a dipole consisting of two point vortices. We analyse the processes creating a net water exchange through the straits and derive a kinematic model dependent on dimensionless parameters representing strait length, dipole travel distance and dipole size. The net tracer transport resulting from the kinematic model agrees closely with the numerical simulations and provide understanding of the processes controlling net water exchange.

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Section: Dipole Propagation Velocitymentioning

confidence: 51%

Section: Flow Separation and Vortex Formationmentioning

confidence: 80%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Flow separation, dipole formation and water exchange through tidal strait

Nøst¹,

Børve²

2021

Preprint

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“…Nevertheless, our claim is that this establishes a clear route to a possible finite-time Euler singularity. We note that Habibah et al (2018) have recently developed an asymptotic theory for vortexpair propagation in powers of the parameterˆ = δ/s, and have shown that the speed of propagation of the pair is given by the beautifully simple formula U = Γ/4πs +ˆ 2 Q 2 /2s 3 , where Q 2 is the strength of the quadrupole at O(ˆ 2 ) associated with the elliptical deformation of each vortex; in other words, the speed is slightly increased as a result of this deformation. They show streamlines and vorticity contours forˆ = 0.3, which, even for this relatively low value, exhibit a slight flattening of each vortex on the side near to the central plane.…”

Section: An Euler Singularitymentioning

confidence: 91%

Towards a finite-time singularity of the Navier–Stokes equations Part 1. Derivation and analysis of dynamical system

Moffatt

Kimura

2018

J. Fluid Mech.

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The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ is studied, starting from a finite-energy configuration of two vortex rings of circulation $\pm \unicode[STIX]{x1D6E4}$ and radius $R$, symmetrically placed on two planes at angles $\pm \unicode[STIX]{x1D6FC}$ to a plane of symmetry $x=0$. The minimum separation of the vortices, $2s$, and the scale of the core cross-section, $\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities $\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$ is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$ at the tipping points and by $s(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$ is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when $\unicode[STIX]{x1D705}s$ and $\unicode[STIX]{x1D6FF}/\!s$ become of order unity. The dynamical system admits ‘partial Leray scaling’ of just $s$ and $\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of $s,\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter $\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.

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Vorticity and circulation decay in the viscous Lamb dipole

Krasny

2021

Fluid Dyn. Res.

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The Lamb dipole is a steady translating structure in 2D ideal fluid flow with opposite-sign vorticity of compact support in a circular disk. Previous studies have shown that when viscosity is present, the resulting viscous Lamb dipole develops a head-tail structure in which the head expands in size, while a tail of low amplitude vorticity is left behind as the head moves forward; in addition, the maximum vorticity and total circulation on each side of the dipole decay in time. Here we examine these decay properties by comparing numerical solutions of the Navier–Stokes equation (NSE) and diffusion equation (DE) in the Reynolds number range 125 ⩽ Re ⩽ 1000 using the inviscid Lamb dipole as initial condition; this enables us to compare the combined effects of convection and diffusion in the NSE with the sole effect of diffusion in the DE. The results show that for a given Re, the vortex core size, shape, and maximum vorticity are nearly the same for the NSE and DE, indicating that convection has little effect on these properties. Nonetheless, compared to the DE, convection in the NSE inhibits circulation decay at low Re, while it enhances circulation decay at high Re, and the lateral separation of the vortex cores is a critical factor in this transition.

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Finite-thickness effect on speed of a counter-rotating vortex pair at high Reynolds numbers

Cited by 4 publications

References 31 publications

Flow separation, dipole formation and water exchange through tidal strait

Flow separation, dipole formation and water exchange through tidal strait

Towards a finite-time singularity of the Navier–Stokes equations Part 1. Derivation and analysis of dynamical system

Vorticity and circulation decay in the viscous Lamb dipole

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