Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.1014 ◽

2019 ◽

Vol 863 ◽

pp. 513-544 ◽

Cited By ~ 3

Author(s):

Jinhua Hao ◽

Shiying Xiong ◽

Yue Yang

Keyword(s):

Lorentz Force ◽

Diffusion Term ◽

Surface Field ◽

Vortex Lines ◽

Body Forces ◽

Vorticity Flux ◽

Virtual Velocity

We demonstrate that, if a globally smooth virtual circulation-preserving velocity exists, Kelvin’s and Helmholtz’s theorems can be extended to some non-ideal flows which are viscous, baroclinic or with non-conservative body forces. Then we track vortex surfaces frozen in the virtual velocity in the non-ideal flows, based on the evolution of a vortex-surface field (VSF). For a flow with a viscous-like diffusion term normal to the vorticity, we obtain an explicit virtual velocity to accurately track vortex surfaces in time. This modified flow is dissipative but prohibits reconnection of vortex lines. If a globally smooth virtual velocity does not exist, an approximate virtual velocity can still facilitate the tracking of vortex surfaces in non-ideal flows. In a magnetohydrodynamic Taylor–Green flow, we find that the conservation of vorticity flux is significantly improved in the VSF evolution convected by the approximate virtual velocity instead of the physical velocity, and the spurious vortex deformation induced by the Lorentz force is eliminated.

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Topological vortex dynamics in axisymmetric viscous flows

Journal of Fluid Mechanics ◽

10.1017/s0022112094003435 ◽

1994 ◽

Vol 260 ◽

pp. 57-80 ◽

Cited By ~ 13

Author(s):

Mogens V. Melander ◽

Fazle Hussain

Keyword(s):

Phase Portrait ◽

Critical Points ◽

Numerical Solutions ◽

Stokes Equations ◽

Saddle Points ◽

Time History ◽

Vorticity Field ◽

Vortex Lines ◽

Short Period ◽

Virtual Velocity

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

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The degree of knottedness of tangled vortex lines

Journal of Fluid Mechanics ◽

10.1017/s0022112069000991 ◽

1969 ◽

Vol 35 (1) ◽

pp. 117-129 ◽

Cited By ~ 891

Author(s):

H. K. Moffatt

Keyword(s):

Velocity Field ◽

Inviscid Flow ◽

Steady Flows ◽

Vortex Filaments ◽

Vorticity Distribution ◽

Vortex Lines ◽

Body Forces ◽

Infinite Extent

Letu(x)be the velocity field in a fluid of infinite extent due to a vorticity distributionw(x)which is zero except in two closed vortex filaments of strengthsK1,K2. It is first shown that the integral\[ I=\int{\bf u}.{\boldmath \omega}\,dV \]is equal to αK1K2where α is an integer representing the degree of linkage of the two filaments; α = 0 if they are unlinked, ± 1 if they are singly linked. The invariance ofIfor a continuous localized vorticity distribution is then established for barotropic inviscid flow under conservative body forces. The result is interpreted in terms of the conservation of linkages of vortex lines which move with the fluid.Some examples of steady flows for whichI± 0 are briefly described; in particular, attention is drawn to a family of spherical vortices with swirl (which is closely analogous to a known family of solutions of the equations of magnetostatics); the vortex lines of these flows are both knotted and linked.Two related magnetohydrodynamic invariants discovered by Woltjer (1958a, b) are discussed in ±5.

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Vortex reconnection in the late transition in channel flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.492 ◽

2016 ◽

Vol 802 ◽

Cited By ~ 18

Author(s):

Yaomin Zhao ◽

Yue Yang ◽

Shiyi Chen

Keyword(s):

Channel Flow ◽

Fluid Motion ◽

Wall Friction ◽

Sudden Increase ◽

Mean Flow ◽

Vortex Sheets ◽

Vortex Reconnection ◽

Vortex Lines ◽

Symmetric Plane ◽

Vorticity Flux

Vortex reconnection, as the topological change of vortex lines or surfaces, is a critical process in transitional flows, but is challenging to accurately characterize, particularly in shear flows. We apply the vortex-surface field (VSF), whose isosurface is the vortex surface consisting of vortex lines, to study vortex reconnection in the Klebanoff-type temporal transition in channel flow. The VSF evolution can capture the reconnection of the hairpin-like vortical structures evolving from the initial vortex sheets in opposite halves of the channel. The incipient vortex reconnection is characterized by the vanishing minimum distance between a pair of vortex surfaces and the reduction of vorticity flux through the region enclosed by the wall and the VSF isoline of the channel half-height on the spanwise symmetric plane. We find that the surge of the wall-friction coefficient begins at the identified reconnection time. From the Biot–Savart law, the rapid reconnection of vortex lines can induce a velocity opposed to the mean flow, which partially blocks the flow near the central region and generally accelerates the near-wall fluid motion in the flow with constant mass flux. Therefore, the vortex reconnection appears to play an important role in the sudden increase of wall friction in transitional channel flows.

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Geometry Effects on Free Surface Vorticity Flux

Journal of Fluids Engineering ◽

10.1115/1.2823523 ◽

1999 ◽

Vol 121 (3) ◽

pp. 678-683 ◽

Cited By ~ 4

Author(s):

Bill Peck ◽

Lorenz Sigurdson

Keyword(s):

Free Surface ◽

Convenient Method ◽

Gaussian Curvature ◽

Vortex Lines ◽

Geometry Effects ◽

Vorticity Flux

Effects of geometry on the flux of vorticity from a free surface are discussed. Special attention is paid to situations where curvature-dependent contributions to the vorticity flux can be neglected. The geometry of vortex lines embedded in the surface is discussed in this context. These results show that vortex lines can be straight and geometry-induced vorticity flux is produced; conversely vortex lines can be curved and no geometry-induced vorticity flux is produced. A convenient method for assessing vorticity flux from a steady surface based on Gaussian curvature is derived.

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