8/3/2019 L. A. Barba and A. Leonard- Numerical studies of relaxing two-dimensional vortices with non-axisymmetric states

http://slidepdf.com/reader/full/l-a-barba-and-a-leonard-numerical-studies-of-relaxing-two-dimensional-vortices 1/1

Numerical studies of relaxing two-dimensional vortices with non-axisymmetric states

L. A. BarbaDepartment of Aerospace Engineering

University of BristolBristol BS8 1TR UK

A. LeonardGraduate Aeronautical Labs

California Institute of TechnologyPasadena CA 91125 USA

One of the relevant questions apropos of two-dimensional vortices with elliptical shapes is whether they decay toan axisymmetric state. The process of “axisymmetrization” is recognized as one of two fundamental processes inthe evolution of two-dimensional turbulent ows, the other being vortex merging [6]. These processes participatein the notorious evolution of 2D turbulence to form isolated, “coherent” vortices, that live for many eddy turn-overperiods [5]. The relaxation of linearly perturbed, large- Re Lamb-Oseen vortices was studied numerically in [2], andit was seen that the non-axisymmetric perturbations decay much faster than the viscous timescale. A mechanismof shear-diffusion averaging [4] is active, whereby the shearing along streamlines causes the winding-up of the non-axisymmetric vorticity into spiral structures, which are then rapidly homogenized due to viscous diffusion. Theaxisymmetric state is approached on a Re 1 / 3 timescale, as shown in [4].

We compute the evolution of Gaussian vortex monopoles, with quadrupolar perturbations. Fully nonlinear sim-ulations of this ow were previously presented in [8], where it was observed that whereas for small-amplitude non-axisymmetric perturbations the ow relaxes to an axisymmetric state, for large enough amplitudes of the perturba-

tion, the ow relaxes instead to a quasi-steady, rotating tripole (only three perturbation amplitudes were computed,however). The shear-diffusion mechanism is still active, but only the positive portions of the quadrupolar pertur-bation are mixed while the negative parts form persistent inclusions. It was suggested that there exists a thresholdamplitude separating the domains of attraction of the monopole and tripole.

On the other hand, the study of [3] using asymptotic methods leads the author to conjecture that “the thresholdin perturbation amplitude for the existence of the non-axisymmetric state would decrease with increasing Reynoldsnumbers”, and to lament the lack of calculations with different Re in [8].

Presently, a parametric study has been performed by carry-

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410

−5

10−4

10−3

10−2

10−1

Re = 105

Re = 10 3

Re = 3x10 3

Re = 10 4

| ω m

i n /

ω m a x

| a

t t = T

δ , amplitude of the perturbation

Tripole

No tripole

Figure 1: |ωmin /ω max | at T vs. δ for different Re .

ing out calculations for various Re and perturbation amplitudeδ. The method used is a fully mesh-less vortex method, devel-oped in [1]. To quantify axisymmetrization, the magnitude of the minimum (negative) to maximum vorticity, |ωmin /ω max | , af-ter several turn-over times is measured. This is shown in Figure1, for nal time T = 800. Visualization of vorticity (plots and an-imations will be shown at the oral presentation), indicates thatwhen this ratio is smaller than about 10 − 2 there is no tripolestructure: the negative vorticity spirals around the core, result-ing in axisymmetrization. This is indicated by the horizontal linein Figure 1. As conjectured by [3], we conrm that the ampli-tude of the perturbation for the appearance of a tripole decreaseswith increasing Reynolds number. The difference in relaxationbehaviour around this threshold amplitude is dramatic, like a‘jump’ in the min-max ratio of ω, but it is not clear whether this

is a bifurcation or a continuous (but fast!) transition from one state to the other. Results of more than 50 simulationswill be summarized and visualized during the oral presentation for the discussion of these interesting issues.

References[1] L. A. Barba. Vortex method for computing high-Reynolds number ows: Increased accuracy with a ful ly mesh-less formu-

lation . PhD thesis, California Institute of Technology, 2004.

[2] A. J. Bernoff and J. F. Lingevitch. Rapid relaxation of an axisymmetric vortex. Phys. Fluids , 6(11):3717–3723, 1994.

[3] S. Le Dizes. Non-axisymmetric vortices in two-dimensional ows. J. Fluid Mech. , 406:175–198, 2000.

[4] T. S. Lundgren. Strained spiral vortex model for turbulent ne structures. Phys. Fluids , 25(12):2193–2203, 1982.

[5] J. C. McWilliams. The emergence of isolated coherent vortices in turbulent ow. J. Fluid Mech. , 146:21–43, 1984.

[6] M. V. Melander, J. C. McWilliams, and N. J. Zabusky. Axisymmetrization and vorticity-gradient intensication of anisolated two-dimensional vortex through lamentation. J. Fluid Mech. , 178:137–159, 1987.

[7] L. F. Rossi. Resurrecting core spreading vortex methods: A new scheme that is both deterministic and convergent. SIAM J. Sci. Comput. , 17:370–397, 1996.

[8] L. F. Rossi, J. F. Lingevitch, and A. J. Bernoff. Quasi-steady monopole and tripole attractors for relaxing vortices. Phys.Fluids , 9(8):2329–2338, 1997.

1