On the energy of elliptical vortices

Vanneste, Jacques; Young, W. R.

doi:10.1063/1.3474703

Cited by 10 publications

(8 citation statements)

References 18 publications

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“…w ) cos(2θ), where A 2 ≈ 0.9 for the voltage configuration described above. This yields a flow field v s = ǫ (yx + xŷ) with strain magnitude between predicted and measured strain is not presently understood, the m = 1 calibration is consistent with direct measuremet of strain using passive advection of vorticity where ǫ/ω ≫ ǫ c /ω [20]. Thus, the calibrated strain is used for the data presented here.…”

supporting

confidence: 58%

Evolution of a Vortex in a Strain Flow

et al. 2016

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Experiments and vortex-in-cell simulations are used to study an initially axisymmetric, spatially distributed vortex subject to an externally imposed strain flow. The experiments use a magnetized pure electron plasma to model an inviscid two-dimensional fluid. The results are compared to a theory assuming an elliptical region of constant vorticity. For relatively flat vorticity profiles, the dynamics and stability threshold are in close quantitative agreement with the theory. Physics beyond the constant-vorticity model, such as vortex stripping, is investigated by studying the behavior of nonflat vorticity profiles.

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confidence: 58%

Evolution of a Vortex in a Strain Flow

et al. 2016

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“…The factor of one half arises due to the fact that tr {I} = 2. which we can readily verify by decomposing the rotation matrix as R(θ) = cos θI + sin θJ. Then R(θ)KR T (θ) = (cos θI + sin θJ) K (cos θI − sin θJ) = cos 2θK + sin 2θL (14) R(θ)LR T (θ) = (cos θI + sin θJ) L (cos θI − sin θJ) = cos 2θL − sin 2θK (15) as we see at once through the application of the multiplication rules of Equation (11). The first of these has been used in forming the rotated strain matrix in Equation (7).…”

Section: A Matrix Basismentioning

confidence: 90%

“…where we note the appearance of an extra additive constant with a value of 1 2 ψ o in the disk-averaged version compared with the ring average. This distinction will be important for interpreting the so-called "excess" domain-integrated kinetic energy of an elliptical two-dimensional vortex such as the Kida vortex, which is based on the value of the stream function; see e.g., [14].…”

Section: Physical Properties Of An Elliptical Disk Of Fluidmentioning

confidence: 99%

“…This solution was introduced by Kida [1] as an extension of the classical unforced Kirchhoff vortex, given a Hamiltonian formulation by Neu [2], then generalized to time-dependent forcing fields by Ide and Wiggins [3]. The Kida vortex has been of considerable interest as a means of understanding such phenomena as instability mechanisms [4][5][6][7][8][9], chaotic advection [10,11], the interaction of diffusion and advection [12], vortex-vortex interactions in shear [13], vortex energetics [14] and vortex interactions with boundaries [15]. Elliptical vortices also play a central role as a basis ingredient in ambitious attempts to approximate the dynamics of more complex or realistic flows [16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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Kinematics of a Fluid Ellipse in a Linear Flow

Lilly

2018

Fluids

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A four-parameter kinematic model for the position of a fluid parcel in a time-varying ellipse is introduced. For any ellipse advected by an arbitrary linear two-dimensional flow, the rates of change of the ellipse parameters are uniquely determined by the four parameters of the velocity gradient matrix, and vice versa. This result, termed ellipse/flow equivalence, provides a stronger version of the well-known result that a linear velocity field maps an ellipse into another ellipse. Moreover, ellipse/flow equivalence is shown to be a manifestation of Stokes' theorem. This is done by deriving a matrix-valued extension of the classical Stokes' theorem that involves a spatial integral over the velocity gradient tensor, thus accounting for the two strain terms in addition to the divergence and vorticity. General expressions for various physical properties of an elliptical ring of fluid are also derived. The ellipse kinetic energy is found to be composed of three portions, associated respectively with the circulation, the rate of change of the moment of inertia, and the variance of parcel angular velocity around the ellipse. A particular innovation is the use of four matrices, termed the IJKL basis, that greatly facilitate the required calculations.

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“…This solution was introduced by Kida [1] as an extension of the classical unforced Kirchhoff vortex, given a Hamiltonian formulation by Neu [2], then generalized to time-dependent forcing fields by Ide and Wiggins [3]. The Kida vortex has been of considerable interest as a means of understanding such phenomena as instability mechanisms [4,5,6,7,8,9], chaotic advection [10,11], the interaction of diffusion and advection [12], vortex-vortex interactions in shear [13], vortex energetics [14] and vortex interactions with boundaries [15]. Elliptical vortices also play a central role as a basis ingredient in ambitious attempts to approximate the dynamics of more complex or realistic flows [16,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

Kinematics of a Fluid Ellipse in a Linear Flow

Lilly¹

2018

Preprint

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A four-parameter kinematic model for the position of a fluid parcel in a time-varying 1 ellipse is introduced. For any ellipse advected by an arbitrary linear two-dimensional flow, the rates 2 of change of the ellipse parameters are uniquely determined by the four parameters of the velocity 3 gradient matrix, and vice-versa. This result, termed ellipse/flow equivalence, provides a stronger 4 version of the well-known result that a linear velocity field maps an ellipse into another ellipse. 5Moreover, ellipse/flow equivalence is shown to be a manifestation of Stokes' theorem. This is done 6 by deriving a matrix-valued relationship, called the geometric Stokes' theorem, that involves a spatial 7 integral over the velocity gradient tensor, thus accounting for the two strain terms in addition to 8 the divergence and vorticity. General expressions for various physical properties of an elliptical 9 ring of fluid are also derived. The ellipse kinetic energy is found to be composed of three portions, 10 associated respectively with the circulation, the rate of change of the moment of inertia, and the 11 variance of parcel angular velocity around the ellipse. A particular innovation is the use of four 12 matrices, termed the IJKL basis, that greatly facilitate the required calculations.

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On the energy of elliptical vortices

Cited by 10 publications

References 18 publications

Evolution of a Vortex in a Strain Flow

Evolution of a Vortex in a Strain Flow

Kinematics of a Fluid Ellipse in a Linear Flow

Kinematics of a Fluid Ellipse in a Linear Flow

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