Lagrangian theory for 3D vortex sheets with axial or helical symmetry

Caflisch, Russel E.; Li, Xiao Fan

doi:10.1080/00411459208203798

Cited by 31 publications

(27 citation statements)

References 8 publications

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“…Furthermore, it is also different in terms of the vortex sheet strength. According to Caflisch and Li, 14 the vortex sheet strength vector in the vortex line, (␣,t), is defined in the cylindrical coordinates by…”

Section: A Singularity Formation In the Axisymmetric Vortex Sheetmentioning

confidence: 99%

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Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet

Sakajo

2002

Physics of Fluids

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We consider an axisymmetric, swirling vortex sheet in an inviscid and incompressible flow. Caflisch et al. pointed out that the vortex sheet acquired a singularity in finite time, but the property of the singularity was not revealed. In the present paper we show convincing numerical evidences of the singularity formation by applying the same numerical methods as what was used in the study of a two-dimensional ͑2D͒ vortex sheet. We find that the radial and axial components of the axisymmetric vortex sheet behave like the 2D singularity that has been observed in many vortex-sheet motions, while the azimuthal component of the sheet behaves differently. Furthermore, the singularity appears along the vortex line and the first derivative of the vortex sheet strength forms a cusp, while the known singularities are associated with the curvature along curves perpendicular to the vortex lines and the sheet strength has a cusp.

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Section: A Singularity Formation In the Axisymmetric Vortex Sheetmentioning

confidence: 99%

“…The vortex line of circulation ⌫ 1 on the axis of symmetry never moves due to the symmetry. The vortex sheet evolves according to the following integrodifferential equations ͑see Caflisch et al 13,14 for the details͒:…”

Section: ͑2͒mentioning

confidence: 99%

Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet

Sakajo

2002

Physics of Fluids

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“…The limiting values of the velocities can be found with the Plemelj formulas; see [6] for a discussion. These limiting values are…”

Section: The Formulation Of the Problemmentioning

confidence: 99%

Well-posedness of two-phase Darcy flow in 3D

Ambrose

2007

Quart. Appl. Math.

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Abstract.We prove the well-posedness, locally in time, of the motion of two fluids flowing according to Darcy's law, separated by a sharp interface in the absence of surface tension. We first reformulate the problem using favorable variables and coordinates. This results in a quasilinear parabolic system. Energy estimates are performed, and these estimates imply that the motion is well-posed for a short time with data in a Sobolev space, as long as a condition is satisfied. This condition essentially says that the more viscous fluid must displace the less viscous fluid. It should be true that small solutions exist for all time; however, this question is not addressed in the present work.

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“…The simulations are based on the Lagrangian formulation of vortex sheet motion in three-dimensional flow [44][45][46]. For the intended application to vortex rings we view the sheet as a parametrized surface x( , θ, t) composed of closed material lines, where measures circulation across the lines and θ is a 2π-periodic parameter along the lines.…”

Section: Lagrangian Formulationmentioning

confidence: 99%

A Particle Method and Adaptive Treecode for Vortex Sheet Motion in Three-Dimensional Flow

Lindsay¹,

Krasny²

2001

Journal of Computational Physics

142

180

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A particle method is presented for computing vortex sheet motion in three-dimensional flow. The particles representing the sheet are advected by a regularized BiotSavart integral in which the exact singular kernel is replaced by the RosenheadMoore kernel. New particles are inserted to maintain resolution as the sheet rolls up. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor approximation in Cartesian coordinates, and the necessary Taylor coefficients are computed by a recurrence relation. The adaptive features include a divide-andconquer evaluation strategy, nonuniform rectangular clusters, variable-order approximation, and a run-time choice between Taylor approximation and direct summation. Tests are performed to document the treecode's accuracy and efficiency. The method is applied to simulate the roll-up of a circular-disk vortex sheet into a vortex ring. Two examples are presented, azimuthal waves on a vortex ring and the merger of two vortex rings.

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Lagrangian theory for 3D vortex sheets with axial or helical symmetry

Cited by 31 publications

References 8 publications

Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet

Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet

Well-posedness of two-phase Darcy flow in 3D

A Particle Method and Adaptive Treecode for Vortex Sheet Motion in Three-Dimensional Flow

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