l.a. barba- emergence of tripoles in nonlinearly perturbed planar vortices: a numerical study

8/3/2019 L.A. Barba- Emergence of tripoles in nonlinearly perturbed planar vortices: a numerical study

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Emergence of tripoles in nonlinearly perturbed

planar vortices: a numerical study

L. A. BARBA

Department of Mathematics, University of Bristol, Bristol U.K. BS84TW

Summary

Since the observation of spontaneous generation of vorticity concentrations from random 2Dturbulence, a huge interest has been stimulated on so-called coherent structures. The tripoleis a much less common one than the monopole and dipole in numerical and experimentalobservations. It was first observed in both domains in the mid-late 1980s, and only in 1991

was a tripolar structure detected in the oceans. Laboratory tripoles occur as a result ofthe growth of a perturbation of azimuthal wavenumber 2 in unstable axisymmetric shieldedvortices (a vortex surrounded by a ring of opposite sign vorticity). For this reason, shieldedmonopoles have also been the subject of several numerical studies.

In the present study, in contrast, the tripole is seen to emerge from a non-shielded,Gaussian monopole (which is stable), with a large nonlinear quadrupolar perturbation (whichdestabilizes it). In this flow, the tripole was first observed by Rossi et al. [7]. We extend theresult by performing a parameter study, spanning values of the amplitude of perturbationand Reynolds number, with numerous simulations using a meshless vortex method that wasintroduced by Barba et al.[1]. One of the goals is to determine whether there is a threshold

amplitude that separates two asymptotic states (axisymmetric and tripolar). The possiblerelationship between this threshold amplitude and Reynolds number is sought, and severalobservations are made regarding the nonlinear, long-time evolution of the structure.

Initial condition. The flow under study is initiated by a Gaussian vortex with a super-posed m = 2 (quadrupolar) perturbation. The initial condition is given by

o(x) =1

4exp

|x|2

4

, (x) =

4|x|2 exp

|x|2

4

cos2, (1)

where o stands for the base vorticity, for the perturbation, and = arg x. The main

case discussed in Rossi et al.[7] corresponds to = 0.25 and Re = 104, with Re = / (totalcirculation divided by the viscosity). Figure 1 shows the initial vorticity for this case, (c),as well as plots of the base vortex, (a), the perturbation, (b), and the tripole obtained afterthe flow self-organizes for approximately 5 turn over times, (d).

Threshold for emergence of the tripole. In Figure 2, the logarithm of the vorticity isplotted in a grey colormap, thus accentuating the region of the domain where the vorticitychanges sign. It can be seen that the self-organization of the flow consists in the zero-contourpinching, leaving the satellites of negative vorticity in place. For the case of a smaller initial


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suggests that there might be an inviscid limit, where the flow always tends to the tripolesolution above a threshold perturbation amplitude.

Le Dizes[3] performed an asymptotic study

103

104

101

Re2/3

slope

Re1/3

slope

Figure 4: Threshold amplitude of the initialperturbation, , versus Reynolds number: forpinching of zero-contour (solid line), or forhalf-turn or full-turn satellite survival (dash-dot lines above).

of non-axisymmetric vortices, and he predicteda Re2/3 behavior for the amplitude of the non-axisymmetric correction. This amplitude, how-ever, refers to the streamfunction of the finalstate. Thus, it is not linearly related to the pa-rameter that we have used, corresponding tothe initial non-axisymmetric component (whichis very much controlled by our type of initialcondition).

Stability of the tripole. Once formed, thetripole is very robust and can live for a very longtime. For example, with = 0.25, the satellitessurvive for 6 turn-over times for Re = 3 103,and more than 25 turn-over times for Re = 104

(after the pinching of the level of zero vorticity).To test the robustness of the structure, we takethe output of one simulation at t = 800 and in-

troduce a random perturbation to the location of the vortex particles (the computationalelements in the method), consisting in a random walk with step /2. Performing a continu-ation run from this perturbed state, one finds that the flow relaxes back to the unperturbed

tripole solution. Figure 5 shows the contours of the perturbation, , for = 0 and = 0.1which is of the order of the initial inter-particle spacing (a resolution parameter of the nu-merical method). After about 5 turn-over times, there is practically no trace of the randomdisturbance.

t = 800

6 4 2 0 2 4 66

4

2

0

2

4

6

(a) = 0

t = 800

6 4 2 0 2 4 66

4

2

0

2

4

6

(b) = 0.1

t = 1600

6 4 2 0 2 4 66

4

2

0

2

4

6

(c) = 0

t = 1600

6 4 2 0 2 4 66

4

2

0

2

4

6

(d) = 0.1

Figure 5: Perturbation vorticity, , for tripole continuation runs with perturbed output; Re =104, = 0.25; (a) and (b) initial conditions; (c) and (d) after 5 turn-over times. Contour levels[0.0 4 : 0.02 : 0.14] (higher contours not present due to decay of perturbation vorticity).


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Time scales of the self-organization and relaxation. In the previous work of Rossiet al. [7], the authors were preoccupied that they were unable to detect shear-diffusion timescales. Shear-enhanced diffusion homogenizes perturbations on a time scale ofO(Re1/3); this

was proved for the case of passive scalars in [6] and for weak vorticity perturbations in [4, 2].The results for vorticity, however, rely on the assumption that perturbations become rapidlyvarying in the radial direction (due to differential rotation), thus cancelling the couplingof streamfunction and vorticity at leading order. This in fact occurs only for sufficientlysmall perturbations, and thus when the perturbation is large there is no reason to expect anO(Re1/3) time scale. By plotting adequate diagnostics of the decay of the tripolar structure,we find that indeed an O(Re1/3) time scale can be extracted for small, but for larger valuesthe scaling does not work. One can, however, educe quite well the mixing time given fromthe theory of Meunier and Villermaux[5], when looking at the decay of the positive part ofthe perturbation. This is the time when the perturbation just starts to decay, and in the

theory it follows an error function behavior. It is suggested that mixing processes are activein the early stages of the relaxation, after which nonlinear effects become dominant. Theseresults cannot be shown here due to lack of space, but will be shown in the talk.

In conclusion, this is a parametric study of the flow produced by superposing a quadrupoleand a Gaussian monopole, spanning a range of Reynolds number and amplitude of thequadrupole. It was partly motivated by previous work suggesting a threshold amplitudeseparating the domains of attraction of the monopole and the tripole. This threshold hasbeen determined quite precisely for several Reynolds numbers, and it is suggested that aninviscid limit might exist. Furthermore, it was observed that the tripole is very robust,quickly recovering from a random disturbance, and it decays slowly due to diffusion.

References

[1] L. A. Barba, A. Leonard, and C. B. Allen. Advances in viscous vortex methods meshless spatialadaption based on radial basis function interpolation. Int. J. Num. Meth. Fluids, 47(5):387421,2005.

[2] A. J. Bernoff and J. F. Lingevitch. Rapid relaxation of an axisymmetric vortex. Phys. Fluids,6(11):37173723, 1994.

[3] S. Le Dizes. Non-axisymmetric vortices in two-dimensional flows. J. Fluid Mech., 406:175198,2000.

[4] T. S. Lundgren. Strained spiral vortex model for turbulent fine structures. Phys. Fluids,25(12):21932203, 1982.

[5] P. Meunier and E. Villermaux. How vortices mix. J. Fluid Mech., 476:213222, 2003.

[6] P. B. Rhines and W. R. Young. How rapidly is a passive scalar mixed within closed streamlines?J. Fluid Mech., 133:133145, 1983.

[7] L. F. Rossi, J. F. Lingevitch, and A. J. Bernoff. Quasi-steady monopole and tripole attractorsfor relaxing vortices. Phys. Fluids, 9(8):23292338, 1997.

l.a. barba- emergence of tripoles in nonlinearly perturbed planar vortices: a numerical study

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