The collapse of an axi-symmetric, swirling vortex sheet (original) (raw)

On the ‘δ-equations’ for vortex sheet evolution

Journal of Fluid Mechanics, 1993

We use a set of equations, sometimes referred to as the ‘δ-equations’, to approximate the two-dimensional inviscid motion of an initially circular vortex sheet released from rest in a cross-flow. We present numerical solutions of these equations for the case with δ2 = 0 (for which the equations are exact) and for δ2 > 0. For small values of the smoothing parameter δ a spectral filter must be used to eliminate spurious instabilities due to round-off error. Two singularities appear simultaneously in the vortex sheet when δ2 = 0 at a critical time tc After tc the solutions do not converge as the computational mesh is refined. With δ2 > 0, converged solutions were found for all values of δ2 when t < tc, and for all but the two smallest values of δ2 used when t > tc. Our results show that when δ2 > 0 the vortex sheet deforms into two doubly branched spirals some time after tc The limiting solution as δ→0 clearly exists and equals the δ = 0 solution when t < tc. For t…

A model for the global structure of self-similar vortex sheet roll-up

2013

Most vortices are born from the roll-up of a shear layer. The roll-up is traditionally modeled as the self-similar winding of an infinite spiral connected to a thin shear layer: a vortex sheet. We demonstrate a composite vortex sheet-point vortex model to quantify the global structure of two archetypal cases of self-similar roll-up. These cases are Kaden’s single spiral solution of the wing-tip vortex and Pullin’s double spiral solution of the nonlinear impulse response of the Kelvin-Helmholtz instability. The model consists in replacing the spiral with a point vortex of equal vortical intensity and accounting for the mutual interaction of the point vortex and the untouched sheet. We show that the model itself has an attractive self-similar solution which compares well with numerical experiments.

On the connection between thin vortex layers and vortex sheets

Journal of Fluid Mechanics, 1990

The equations for the two-dimensional motion of a layer of uniform vorticity in an incompressible, inviscid fluid are examined in the limit of small thickness. Under the right circumstances, the limit is a vortex sheet whose strength is the vorticity multiplied by the local thickness of the layer. However, vortex sheets can develop singularities in finite time, and their subsequent nature is an open question. Vortex layers, on the other hand, have motions for all time, though they may develop singularities on their boundaries. Fortunately, a material curve within the layer does exist for all time. Under certain assumptions, its limiting motion is again the vortex sheet, and thus its behaviour may indicate the nature and possible existence of the vortex sheet after the singularity time. Similar asymptotic results are obtained also for the limiting behaviour of the centre curve as defined by Moore (1978). By examining the behaviour of a sequence of layers, some physical understanding of the formation of the curvature singularity for a vortex sheet is gained. A strain flow, induced partly by the periodic extension of the sheet, causes vorticity to be advected to a certain point on the sheet rapidly enough to form the singularity. A vortex layer, however, simply bulges outwards as a consequence of incompressibility and subsequently forms a core with trailing arms that wrap around it. The evidence indicates that no singularities form on the boundary curves of the layer. Beyond the singularity time of the vortex sheet, the limiting behaviour of the vortex layers is non-uniform. Away from the vortex core, the layers converge to a smooth curve which has the appearance of a doubly branched spiral. While the circulation around the core vanishes, approximations to the vortex sheet strength become unbounded, indicating a complex, local structure whose precise nature remains undetermined.

Singular Solutions and Ill-Posedness for the Evolution of Vortex Sheets

SIAM Journal on Mathematical Analysis, 1989

The evolution of a planar vortex sheet is described by the Birkhott-Rott equation. Duchon and Robert [C.R. Acad. Sci. Paris, 302 (1986), pp. 183-186], [Comm. Partial Ditterential Equations, 13 (1988), pp. 1265-1295 have constructed exact solutions of this equation that are analytic for all < 0 but have a possible singularity in the curvature of the sheet at 0. This shows that smooth initial data for a vortex sheet can lead to singularity formation at a finite time, in agreement with the results of numerical computation [We present an independent construction of these solutions and use these results to infer that the vortex sheet problem is ill-posed in Sobolev class Hn with n > 3/2.

Geometry and dynamics of vortex sheets in 3 dimension

Theoretical and Applied Mechanics, 2002

We consider the properties and dynamics of vortex sheets from a geometrical, coordinate-free, perspective. Distribution-valued forms (de Rham currents) are used to represent the fluid velocity and vorticity due to the vortex sheets. The smooth velocities on either side of the sheets are solved in terms of the sheet strengths using the language of double forms. The classical results regarding the continuity of the sheet normal component of the velocity and the conservation of vorticity are exposed in this setting. The formalism is then applied to the case of the self-induced velocity of an isolated vortex sheet. We develop a simplified expression for the sheet velocity in terms of representative curves. Its relevance to the classical Localized Induction Approximation (LIA) to vortex filament dynamics is discussed.

Nonlinear vortex development in rotating flows

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008

We present the results of a combined experimental and numerical investigation into steady secondary vortex flows confined between two concentric right circular cylinders. When the flow is driven by the symmetric rotation of both end walls and the inner cylinder, toroidal vortex structures arise through the creation of stagnation points (in the meridional plane) at the inner bounding cylinder or on the mid-plane of symmetry. A detailed description of the flow regimes is presented, suggesting that a cascade of such vortices can be created. Experimental results are reported, which visualize some of the new states and confirm the prediction that they are stable to (mid-plane) symmetry-breaking perturbations. We also present some brief results for the flows driven by the rotation of a single end wall. Vortex structures may also be observed at low Reynolds numbers in this geometry. We show that standard flow visualization methods lead to some interesting non-axisymmetric particle paths in...

Simulation of vortex sheet roll-up by vortex methods

Journal of Computational Physics, 1989

The vortex sheet roll-up characteristic of large amplitude Kelvin-Helmholtz instability is simulated by vortex-in-cell methods and a vortex blob method. Both methods regularize the problem and prevent the irregular motion found in many previous simulations. The regularization properties of the methods are compared and discussed.

Azimuthal instability of a vortex ring computed by a vortex sheet panel method

Fluid Dynamics Research, 2009

A Lagrangian panel method is presented for vortex sheet motion in threedimensional (3D) flow. The sheet is represented by a set of quadrilateral panels having a tree structure. The panels have active particles that carry circulation and passive particles used for adaptive refinement. The Biot-Savart kernel is regularized and the velocity is evaluated by a treecode. The method is applied to compute the azimuthal instability of a vortex ring, starting from a perturbed circular disc vortex sheet initial condition. Details of the core dynamics are clarified by tracking material lines on the sheet surface. Results are presented showing the following sequence of events: spiral roll-up of the sheet into a ring, wavy deformation of the ring axis, first collapse of the vortex core in each wavelength, second collapse of the vortex core out of phase with the first collapse, formation of loops wrapped around the core and radial ejection of ringlets. The collapse of the vortex core is correlated with converging axial flow.

Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability

Journal of Fluid Mechanics, 1982

The instabilit,y of an initially flat vortex sheet to a sinusoidal perturbation of the vorticity is studied by means of high-order Taylor series in time t. All finite-amplitude corrections are retained at each order in t. Our analysis indicates that the sheet develops a curvature singularity at t = t, < 00. The variation oft, with the amplitude a of the perturbation vorticity is in good agreement with the asymptotic results of Moore. When a is O(I), the Fourier coefficient of order n decays slightly faster than predicted by Moore. Extensions of the present prototype of Kelvin-Helmholtz instability to other layered flows, such as Rayleigh-Taylor instability, are indicated.