Discretization of a vortex sheet, with an example of roll-up

Chorin, Alexandre J.; Bernard, Peter S.

doi:10.1016/0021-9991(73)90045-4

Cited by 203 publications

(128 citation statements)

References 2 publications

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“…The approach taken is a refinement of the ideas expressed by Chorin & Bernard (1973). The key element in our computational approach is to desingularize the Cauchy principal value integral which defines the velocity of a Lagrangian point on the vortex sheet.…”

Section: Introductionmentioning

confidence: 99%

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Computation of vortex sheet roll-up in the Trefftz plane

1987

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Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptotic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices' deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighbouring vortices.

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Section: Introductionmentioning

confidence: 99%

“…Another approach was taken by Chorin & Bernard (1973) who replaced the point vortex ' singular velocity field with a bounded velocity field in a small neighbourhood of the singularity. Kuwahara & Takami (1973) smoothed the point-vortex velocity field by letting it evolve in time according to the linear diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Computation of vortex sheet roll-up in the Trefftz plane

1987

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“…In two-dimensional flow, the point vortex method replaces a continuous vortex sheet by a set of discrete point vortices [6,7], but the method fails to converge past a finite critical time when a curvature singularity forms in the underlying sheet [8][9][10][11]. One way to proceed is to regularize the problem by applying the vortex-blob method, and this approach captures the spiral roll-up of the vortex sheet past the critical time [12][13][14]. Aside from the issue of singularity formation, another difficulty arises because perturbations introduced by machine roundoff error are amplified by Kelvin-Helmholtz instability, leading to the rapid loss of computational accuracy [7,10].…”

Section: Introductionmentioning

confidence: 99%

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

Lindsay¹,

Krasny²

2001

Journal of Computational Physics

142

171

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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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“…The vortex blob method convolves the singular kernel in the Biot-Savart law with a smooth function. Here we use a regularization which introduces a smoothing parameter δ into the denominator of the integrand [4][5][6][7]. This is numerically convenient, but the true physical regularization is by viscosity.…”

Section: Introductionmentioning

confidence: 99%

Comparison of regularizations of vortex sheet motion

Nitsche

Taylor

Krasny

2003

Computational Fluid and Solid Mechanics 2003

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This paper presents a numerical comparison of two regularizations of the Euler equations, namely, the vortex blob regularization and regularization by physical viscosity. The initial condition is a flat vortex sheet whose vorticity has been smoothed by convolution. The sheet rolls up into a vortex pair. We compute the frequency of oscillations in the core vorticity and scale it using results for a semi-infinite vortex sheet. The computations indicate that as the smoothing parameter δ and the viscosity ν decrease, the scaled frequency in the vortex blob solutions differs from the one in the Navier-Stokes solutions.

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Discretization of a vortex sheet, with an example of roll-up

Cited by 203 publications

References 2 publications

Computation of vortex sheet roll-up in the Trefftz plane

Computation of vortex sheet roll-up in the Trefftz plane

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

Comparison of regularizations of vortex sheet motion

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