Long Time Computation of Two-Dimensional Vortex Sheet by Point Vortex Method

2003 ◽  
Vol 72 (8) ◽  
pp. 1968-1976 ◽  
Author(s):  
Sun-Chul Kim ◽  
June-Yub Lee ◽  
Sung-Ik Sohn
Keyword(s):  
Point Vortex ◽  
Vortex Method ◽  
Vortex Sheet ◽  
Two Dimensional ◽  
Long Time

scholarly journals Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

2020 ◽  
Author(s):  
Klas Modin ◽  
Milo Viviani
Keyword(s):  
Euler Equations ◽  
Vortex Dynamics ◽  
Point Vortex ◽  
Point Vortices ◽  
Symplectic Reduction ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Unified Framework ◽  
Long Time ◽  
2D Euler Equations

Abstract Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for $$N=2$$ N = 2 , 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

scholarly journals Singularity formation during Rayleigh–Taylor instability

1993 ◽  
Vol 252 ◽  
pp. 51-78 ◽  
Author(s):  
Gregory Baker ◽  
Russel E. Caflisch ◽  
Michael Siegel
Keyword(s):  
Real Axis ◽  
Point Vortex ◽  
Vortex Method ◽  
Vortex Sheet ◽  
Taylor Instability ◽  
Approximate Theory ◽  
Fluid Interface ◽  
Helmholtz Instability ◽  
The Real ◽  
Rayleigh Taylor Instability

During the motion of a fluid interface undergoing Rayleigh-Taylor instability, vorticity is generated on the interface baronclinically. This vorticity is then subject to Kelvin-Helmholtz instability. For the related problem of evolution of a nearly flat vortex sheet without density stratification (and with viscosity and surface tension neglected), Kelvin-Helmholtz instability has been shown to lead to development of curvature singularities in the sheet. In this paper, a simple approximate theory is developed for Rayleigh-Taylor instability as a generalization of Moore's approximation for vortex sheets. For the approximate theory, a family of exact solutions is found for which singularities develop on the fluid interface. The resulting predictions for the time and type of the singularity are directly verified by numerical computation of the full equations. These computations are performed using a point vortex method, and singularities for the numerical solution are detected using a form fit for the Fourier components at high wavenumber. Excellent agreement between the theoretical predictions and the numerical results is demonstrated for small to medium values of the Atwood number A, i.e. for A between 0 and approximately 0.9. For A near 1, however, the singularities actually slow down when close to the real axis. In particular, for A = 1, the numerical evidence suggests that the singularities do not reach the real axis in finite time.

scholarly journals Convergence of the point vortex method for 2-D vortex sheet

2000 ◽  
Vol 70 (234) ◽  
pp. 595-607 ◽  
Author(s):  
Jian-Guo Liu ◽  
Zhouping Xin
Keyword(s):  
Point Vortex ◽  
Vortex Method ◽  
Vortex Sheet

On the roll-up of a trailing vortex sheet in the very near field

2003 ◽  
Vol 217 (5) ◽  
pp. 263-269 ◽  
Author(s):  
A L Heyes ◽  
S J Hubbard ◽  
A J Marquis ◽  
D A Smith
Keyword(s):  
Vortex Core ◽  
Near Field ◽  
Angle Of Attack ◽  
Point Vortex ◽  
Vortex Sheet ◽  
Measured Data ◽  
Tip Vortex ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Naca 0012

This paper addresses a discrepancy found between the rate of roll-up of a trailing vortex sheet calculated from point vortex simulations and that from measured data. Measurements of the wake behind a rectangular planform NACA 0012 section wing at 7.5° angle of attack show that some 50 per cent of the circulation in the wake is already present in the vortex core or “rolled-up region” at the trailing edge of the wing, and that there is no increase in the circulation contained within this region within one chord length downstream of the trailing edge. This conflicts with two-dimensional point vortex simulations of sheet roll-up which predict no initial core at the trailing edge and a constantly increasing value of circulation in the vortex in the downstream direction. A modification to include the effect of the tip vortex in the simulation is proposed and is shown to represent the behaviour of the vortex sheet in the very near field accurately.

scholarly journals An Efficient Two-Dimensional Vortex Method with Long Time Accuracy

1996 ◽  
Vol 33 (4) ◽  
pp. 1425-1450 ◽  
Author(s):  
Ibrahim Bless Ranero ◽  
Tomás Chacón Rebollo
Keyword(s):  
Vortex Method ◽  
Two Dimensional ◽  
Long Time ◽  
Time Accuracy

The life-cycle of tripoles in two-dimensional incompressible flows

1994 ◽  
Vol 267 ◽  
pp. 53-82 ◽  
Author(s):  
Xavier Carton ◽  
Bernard Legras
Keyword(s):  
Incompressible Flows ◽  
Mutual Influence ◽  
Bottom Topography ◽  
Point Vortex ◽  
Finite Amplitude ◽  
Transformation Process ◽  
Two Dimensional ◽  
The Core ◽  
Elliptical Model ◽  
Long Time

The mechanisms of coherent tripole formation from unstable shielded circular vortices are analysed in the context of two-dimensional incompressible flows. Three stages are identified during the transformation process: the linear growth of the initial normal mode perturbation, its nonlinear amplification and the finite-amplitude saturation under the tripolar form. We give a geometrical discussion of the mutual influence of the core vortex and of the satellites generated from the shield. The role of the angular momentum in determining the finite amplitude saturation is demonstrated using a simple elliptical model of the core vortex associated with two point-vortex satellites. The long-time asymmetric breaking of the tripole into a dipole and a monopole is shown to be driven by the erosion of the core vortex by stripping and diffusion. Finally the influence of bottom topography on tripole formation is considered, providing a rich phenomenology when the height of the topography is varied.

Impulsively started planar actuator surfaces in high-Reynolds-number steady flow

2013 ◽  
Vol 733 ◽  
pp. 302-324 ◽  
Author(s):  
P. B. Johnson ◽  
A. Wojcik ◽  
K. R. Drake ◽  
I. Eames
Keyword(s):  
Point Vortex ◽  
Vortex Method ◽  
Resistance Coefficient ◽  
Flow Past ◽  
Static Water ◽  
Long Time ◽  
Tidal Turbine ◽  
Pressure Discontinuity ◽  
High Reynolds ◽  
Weak Resistance

AbstractThe characteristics of unbounded flow past an impulsively started planar energy extracting device, such as a wind or tidal turbine, are studied theoretically, numerically and experimentally. The initial thrust on an impulsively started device, which can be more than double the steady thrust, is an important consideration for design and safe operation. The energy sink is modelled here as an ‘actuator surface’ which imposes a uniform pressure discontinuity in the fluid proportional to the square of the fluid speed normal to the surface, the fluid density, and a dimensionless resistance coefficient. The flow past the actuator is studied theoretically for the case of weak resistance using an unsteady model which recovers steady linear momentum theory in the limit of long time. For the case of strong resistance the flow is studied numerically using the point vortex method. Experimental measurements of thrust on a mesh towed through static water are compared to the numerical results and show good agreement. The thrust on an impulsively started device is estimated, for a typical installation, to fall to within 10 % of the steady value within ∼1 min. The numerical model is also used to simulate the gradual startup of a device, yielding estimates of the time constant necessary in a control system in order to reduce peak thrusts in practice.

scholarly journals A Fast Two-Dimensional Numerical Method for the Wake Simulation of a Vertical Axis Wind Turbine

Energies ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 49
Author(s):  
Zheng Yuan ◽  
Jin Jiang ◽  
Jun Zang ◽  
Qihu Sheng ◽  
Ke Sun ◽  
...  
Keyword(s):  
Velocity Distribution ◽  
Numerical Method ◽  
Three Dimensional ◽  
Particle Method ◽  
Vortex Method ◽  
Vertical Axis ◽  
Two Dimensional ◽  
Vertical Axis Wind Turbine ◽  
Wake Effect ◽  
Array Design

In the array design of the vertical axis wind turbines (VAWT), the wake effect of the upstream VAWT on the downstream VAWT needs to be considered. In order to simulate the velocity distribution of a VAWT wake rapidly, a new two-dimensional numerical method is proposed, which can make the array design easier and faster. In this new approach, the finite vortex method and vortex particle method are combined to simulate the generation and evolution of the vortex, respectively, the fast multipole method (FMM) is used to accelerate the calculation. Based on a characteristic of the VAWT wake, that is, the velocity distribution can be fitted into a power-law function, a new correction model is introduced to correct the three-dimensional effect of the VAWT wake. Finally, the simulation results can be approximated to the published experimental results in the first-order. As a new numerical method to simulate the complex VAWT wake, this paper proves the feasibility of the method and makes a preliminary validation. This method is not used to simulate the complex three-dimensional turbulent evolution but to simulate the velocity distribution quickly and relatively accurately, which meets the requirement for rapid simulation in the preliminary array design.

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