An infrared method for viscous fluids. I. Point vortices and vortex sheets in two dimensions

1985 ◽  
Vol 28 (11) ◽  
pp. 3220 ◽  
Author(s):  
H. M. Fried
Keyword(s):  
Two Dimensions ◽  
Point Vortices ◽  
Viscous Fluids ◽  
Vortex Sheets ◽  
Infrared Method

Evolution and breakdown of a vortex street in two dimensions

1981 ◽  
Vol 109 ◽  
pp. 435-463 ◽  
Author(s):  
Hassan Aref ◽  
Eric D. Siggia
Keyword(s):  
Opposite Sign ◽  
Initial Perturbation ◽  
Subsequent Development ◽  
Two Dimensions ◽  
Point Vortices ◽  
Vortex Street ◽  
Turbulent Shear ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Vortex Sheets

The initial-value problem defined by two parallel vortex sheets of opposite sign is studied. Strictly two-dimensional, incompressible, nearly inviscid dynamics is assumed throughout. The roll-up of the sheets into a vortex street is simulated numerically using 4096 point vortices. Much longer runs than in previous work are performed, and it is found that only for a finite range of values of the ratio, h/λ, of sheet separation to perturbation wavelength, does a long-lived vortex street emerge. For h/λ [gsim ] 0·6 a pairing transition within each row intervenes. For h/λ [lsim ] 0·3 we find oscillatory modes.Using up to 16384 point vortices, we also study the breakdown of the metastable street to a two-dimensional, turbulent shear flow. The vortex blobs that made up the street may merge with others of the same sign after the breakdown, but otherwise persist throughout the turbulent regime. Neither their disintegration nor amalgamation with vortices of opposite sign was observed. Using dimensional arguments we derive the relevant scaling theory, and show that it applies to a flow started from two random vortex sheets. The resulting turbulence is not self-similar. For the turbulent flow that follows from the breakdown of a regular vortex street two length scales with different power-law growth in time appear to be necessary. The important differences in the asymptotic structure of flows initialized from random and regular sheets leads us to question the idea of universality. The influence of the symmetry of the initial perturbation on the subsequent development is also considered.

scholarly journals Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 943
Author(s):  
Henryk Kudela
Keyword(s):  
Differential Equations ◽  
Point Vortex ◽  
Optimization Procedure ◽  
Vortex Sheet ◽  
Point Vortices ◽  
Vortex Sheets ◽  
Impermeable Barrier ◽  
Initial Location ◽  
Passive Tracers ◽  
Initial Iterate

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

A versatile taxonomy of low-dimensional vortex models for unsteady aerodynamics

2018 ◽  
Vol 858 ◽  
pp. 917-948 ◽  
Author(s):  
Darwin Darakananda ◽  
Jeff D. Eldredge
Keyword(s):  
Large Scale ◽  
Bluff Body ◽  
Point Vortex ◽  
Unsteady Aerodynamics ◽  
Shear Layers ◽  
Point Vortices ◽  
The Body ◽  
Vortex Sheets ◽  
Proposed Model ◽  
Low Dimensional

Inviscid vortex models have been demonstrated to capture the essential physics of massively separated flows past aerodynamic surfaces, but they become computationally expensive as coherent vortex structures are formed and the wake is developed. In this work, we present a two-dimensional vortex model in which vortex sheets represent shear layers that separate from sharp edges of the body and point vortices represent the rolled-up cores of these shear layers and the other coherent vortices in the wake. We develop a circulation transfer procedure that enables each vortex sheet to feed its circulation into a point vortex instead of rolling up. This procedure reduces the number of computational elements required to capture the dynamics of vortex formation while eliminating the spurious force that manifests when transferring circulation between vortex elements. By tuning the rate at which the vortex sheets are siphoned into the point vortices, we can adjust the balance between the model’s dimensionality and dynamical richness, enabling it to span the entire taxonomy of inviscid vortex models. This hybrid model can capture the development and subsequent shedding of the starting vortices with insignificant wall-clock time and remain sufficiently low-dimensional to simulate long-time-horizon events such as periodic bluff-body shedding. We demonstrate the viability of the method by modelling the impulsive translation of a wing at various fixed angles of attack, pitch-up manoeuvres that linearly increase the angle of attack from $0^{\circ }$ to $90^{\circ }$, and oscillatory pitching and heaving. We show that the proposed model correctly predicts the dynamics of large-scale vortical structures in the flow by comparing the distributions of vorticity and force responses from results of the proposed model with a model using only vortex sheets and, in some cases, high-fidelity viscous simulation.

scholarly journals The two-dimensional slow motion of viscous fluids

1922 ◽  
Vol 100 (705) ◽  
pp. 394-413 ◽  
Keyword(s):  
Differential Equation ◽  
Viscous Fluid ◽  
Slow Motion ◽  
Two Dimensions ◽  
Imperial College ◽  
Viscous Fluids ◽  
Financial Assistance ◽  
Two Dimensional ◽  
Flat Plates ◽  
Slow Motions

The present paper is a contribution to the treatment of problems which require a solution of the differential equation▽ 4 ψ = 0. Amongst such problems are to be found not only the very slow motions of a viscous fluid in two dimensions, but also the flexure of thin flat plates. The prosecution of the investigation has been made possible by the support of the Department of Scientific and Industrial Research, which has provided financial assistance to enable two of us to devote the whole of our time to the research, and our thanks are offered to the Department for its assistance. We also desire to acknowledge the facilities afforded by the Governing Body of the Imperial College of Science and Technology in placing a room at our disposal in the Department of Aeronautics.

Inviscid Interaction of Trailing Vortex Sheets Approximated by Point Vortices

1978 ◽  
Vol 15 (12) ◽  
pp. 857-859 ◽  
Author(s):  
Pradeep Raj ◽  
J. D. Iversen
Keyword(s):  
Point Vortices ◽  
Vortex Sheets

scholarly journals On stagnation points and streamline topology in vortex flows

1998 ◽  
Vol 370 ◽  
pp. 1-27 ◽  
Author(s):  
HASSAN AREF ◽  
MORTEN BRØNS
Keyword(s):  
General Solution ◽  
Stagnation Point ◽  
Plane Curve ◽  
Two Dimensions ◽  
Point Vortices ◽  
Vortex Flows ◽  
Stagnation Points

The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N−1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

scholarly journals Stationary states of identical point vortices and vortex foam on the sphere

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120622 ◽  
Author(s):  
Kevin A. O'Neil
Keyword(s):  
Initial Conditions ◽  
Point Vortex ◽  
Intermediate Energy ◽  
Point Vortices ◽  
Stationary States ◽  
Vortex Sheets ◽  
Identical Point ◽  
Equilibrium Configurations ◽  
Stationary Vortex ◽  
Point Vortex Equilibria

Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary ‘vortex foam’ on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included.

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