Point Vortex Dynamics in Two Dimensions - · PDF filePoint Vortex Dynamics in Two Dimensions...

11
Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 19 April to 2 May, 2009 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei Zhu Supervised by: Professor Keith Moffatt

Transcript of Point Vortex Dynamics in Two Dimensions - · PDF filePoint Vortex Dynamics in Two Dimensions...

Page 1: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

Spring School on Fluid Mechanics and Geophysics of

Environmental Hazards

19 April to 2 May, 2009

Point Vortex Dynamics in Two Dimensions

Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang,

Wei Zhu

Supervised by: Professor Keith Moffatt

Page 2: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

Introduction

Some theory on point vortices

The vorticity of a fluid is defined as its circulation, thus it is given mathematically by the curl of thevelocity vector field:

ω = ∇× u

In this project we consider only two dimensional flows, as such the vorticity vector is always directedperpendicularly to the plane of the flow. We denote the strength of the vortex as ‘κ’. Furthermore,we consider idealised point vortices such that the fluid is irrotational everywhere except at a pointconcentration. Stoke’s theorem may be written:∫∫

S

ω =

C

u.dl.

Thus, for a circular loop around a single point vortex, the above integral may be evaluated as

κ = 2πrvθ

where vθ is the tangential velocity arising from the point vortex along a circle of radius ‘r’.

The vorticity equation is derived by taking the curl of the Navier-Stokes equation:

Dt= ω.∇u + ν∇2ω

However, in two dimensional inviscid flows both terms on the right vanish, yielding the importantresult that in such flows vorticity is carried with the fluid. Consequently, the motion of a point vortexis seen to depend only upon the velocity of the fluid in which it is embedded. If the fluid is stationarythen the vortex will induce a circular velocity field, but will itself remain fixed in space. The interactionof point vortices, therefore, is governed by the fluid velocities that each induce at the position of theothers. Given the radial velocity distribution induced by a point vortex earlier derived, the velocityof a vortex situated at (xα, yα), may be found as the vector sum of the velocities induced by each ofthe vorticies at that point:

dxα

dt= −

∑β

κβ(yα − yβ)

2πr2

αβ

,dyα

dt=

∑β

κβ(xα − xβ)

2πr2

αβ

where the summation is performed for α 6= β. A side-effect of the point vortex idealisation is thatthe velocity field at the location of the point vortex is singular. This has the unhappy consequence ofcausing the kinetic energy of the system to become infinite. Furthermore integrals of momentum andangular momentum become infinite in the limit as r → ∞. As such, new invariants must be devised inorder to provide some reasonable physical constraints to the system. The first constraint is providedby the Hamiltonian for the system, which is given by:

H = −1

N∑α,β=1

κακβ ln rαβ

1

Page 3: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

This is a measure of the ‘energy of interaction’ of the vortices. Momentum and angular momentumlike quantities are given by:

P = 1

2ρ(Σκαyα,−Σκαxα, 0) = cst

M = 1

3ρ(0, 0,−Σκα(x2

α + y2

α)) = cst

In the case that Σκα 6= 0, a fixed ‘centre of vorticity’ and dispersion, D, of the vortex distributionmay be defined:

x =Σκαxα

Σκα, y =

Σκαyα

Σκα

D = Σκα[(xα − x)2 + (yα − y)2] = cst

In general it is not possible to integrate a system of N point vortices where N > 3, however, thesystem is deterministic and solutions may be calculated numerically. In this project, code was writtento numerically integrate the positions of an arbitrary number of vortices forward in time. The aboveinvariants were used to confirm that the numerical procedure was working correctly.

Results of numerical simulations

Testing the code: Two point vortices

In the case that each vortex is of similar strength, the motion of the vortex pair is determined bythe relative signs of the vortices. Vortices of opposing sign are observed to propagate together alonga line perpendicular to their bisector. Note that in this case Σκα = 0, implying that the system isnot constrained to have a fixed centre of vorticity or dispersion. However, similar signed vortices areconstrained in this manner and they are observed to move in a circle around their fixed centre ofvorticity. Figure 1 shows the results of the numerical solutions for these cases.

0

0

-10

-20

-30

-40

-50

-60

-70

-80

-1.5 -1 -0.5 0.5 1 1.5

(a) Two vortices, different signs

0

0

-1

-1

-0.5

-0.5

0.5

0.5

1

1

(b) Two vortices, same signs

Figure 1: Some numerical results

2

Page 4: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

Three point vortices on three verticies of a square

κ = 1

κ = 2

κ = 3

r12 r23

r31

-0.20

0

0.2

0.4

0.5-0.5

0.6

0.8

1

1

-1

1.2

Figure 2: Initial configuration of vortices

Three vortices were initially arranged as in Figure 2 such that they formed a right angled trianglewith two sides having equal length. The chosen values of vortex strength mean that the system hasa fixed centre of vorticity, and this is clear from the resulting vortex tracks (Figure 3) which showthe vortices moving in a biperiodic manner around the point (1

3, 1

3). The periodic motion of the three

vortices through phase space is illustrated in Figure 3b) where the values of the separations of thevortices have been plotted at each timestep.

-1

-1

-0.5

-0.5

0

0

0.5

0.5

1

1

1.5

1.5

(a) Vortex tracks

r12

r23r31

1

1.5 1.5

2.2

2

21.9

1.8

1.8

1.81.7

1.7

1.6

1.61.6

1.4

1.4 1.41.3

1.2

1.2

(b) Phase space

Figure 3: Three point vortices around a square

Instabilities in symmetrical three vortex configurations

Three vortices arranged in an equilateral triangle will rotate around a fixed centre of vorticity providedthat the sum of the vorticities is non-zero (i.e. the dispersion invariant is defined). However, thestability of their motion is determined by the value of the invariant J , a function of P

2 and M given

3

Page 5: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

by:

J = 1

2(3

ρ(κ1 + κ2 + κ3)|M| − 4

ρ2 |P|2)

= 1

2R2(κ2κ3 + κ3κ1 + κ1κ2)

Theory states that the configuration should be stable to perturbations when J > 0, unstable whenJ < 0 and neutrally stable when J = 0. In the following simulations, the strengths of the pointvortices were set such that J satisfied each of the above cases. In the unperturbed run, the vorticeswere arranged exactly at the vertices of an equilateral triangle. In the perturbed run, one of thevortices was shifted by 0.0001 units in the x direction.

-1 -0.5

-0.5

0

0

0.5

0.5

1

1

1.5

1.5

(a) Vortex tracks: note that two of the vortices follow

identical trajectories

r12

r23

r13

2.001

2.0012.001

2.002

2.0022.002

2.003

2.0032.003

2.004

2.004

2.004

2.000

2.000

(b) Separation space of vortices

Figure 4: J > 0 case, perturbed and unperturbed cases are identical

In the case where J > 0, the system is theoretically stable. After one million iterations with atimestep of 0.001, the perturbed and unperturbed cases were indistinguishable by eye. In both casesthe vortices exhibit periodic motion tracing circles with radii determined by their relative strengths.Note the gradual drift in the separations of the vortices arising from numerical error.

-3 -2.5 -2 -1.5 -1

-1

-0.5

-0.5

0

0

0.5

0.5

1

1

1.5

2

2.5

3

3.5

(a) Vortex tracks: unperturbed

r12

r23

r13

2.000

2.000

2.002

2.002 2.002

2.004

2.0042.004

2.006

2.0062.006

2.008

2.008

2.008

(b) Separation space: unperturbed

Figure 5: Neutrally stable, unperturbed configuration: J = 0

4

Page 6: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

-3.5 -3 -2.5 -2 -1.5 -1

-1

-0.5

-0.5

0

0

0.5

0.5

1

1

1.5

2

2.5

3

3.5

(a) Vortex tracks: perturbed

r12

r23

r31

2.00

2.00

2.02

2.02 2.02

2.04

2.04 2.04

2.06

2.06 2.06

2.08

2.08 2.08

2.10

2.10 2.10

2.12

2.122.12

(b) Separation space: perturbed

Figure 6: Neutrally stable, perturbed configuration: J = 0

J = 0 is neutrally stable. The unperturbed vortex tracks are shown in Figure 5. Perturbationsto this state caused the vortices to slowly spiral further apart, as shown in Figure 6, however theoverall track pattern is largely unchanged. The configuration where (J < 0) is unstable and a smallperturbation to the initial position causes a dramatic change in the behaviour of the vortices overtime. Note that in the case illustrated in Figure 8, κ1 = 1, κ2 = −2 and κ3 = 1, therefore there is nofixed centre of vorticity and the dispersion of the system is not invariant.

-50 -40 -30 -20 -10-1

0

0

1

2

3

4

5

(a) Vortex tracks, unperturbed

1.3 1.41.6 1.7 1.8 1.9

2.1 2.2

r12

r23

r13

1

1.5 1.5

2

2

22.5

3

33.5

4

44.5

5

5

6

(b) Separation space, unperturbed

-35 -25 -15 -5-40 -30 -20 -10 0

0

1

2

3

4

5

(c) Vortex tracks, perturbed

r12

r23

r13

1.5

2

2 2

2.5

2.5 2.5

3

33

3.5

3.53.5

4

44

4.5

4.54.5

5

5

5.56

(d) Separation space, perturbed

Figure 7: Unstable, J < 0. Note dramatic change in vortex tracks and separations arising from small pertur-bation to original position of one vortex.

5

Page 7: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

Instabilities in 6, 7 and 8 vortex configurations

A further interesting case arises in the stability of N equal vortices which are regularly distributedaround the circumference of a circle. If N equal vortices are placed at the vertices of a regular polygon,then a rotating steady state is obviously possible. The question is: what would happen if one of thevortices is perturbed? Detailed investigation shows that the configuration is stable if N ≤ 6 andunstable if N ≥ 8. The case N = 7 is neutrally stable. The following figures illustrate the effects ofperturbation on the stability of 6, 7 and 8 equal vortices located on the vertices of a regular polygon.It can be seen that a small perturbation has no significant effect on the trajectories of 6 and 7 vortices,but it completely changes the trajectories of 8 vortices.

(a) 6 vortices, unperturbed (b) 6 vortices, perturbed

(c) 7 vortices, unperturbed (d) 7 vortices, perturbed (e) 8 vortices, unperturbed (f) 8 vortices, perturbed

Figure 8: Vortex tracks for vortices arranged around circle. Note stability of 6 and 7 vortex cases to smallperturbations whilst the 8 vortex case demonstrates significant instability.

6

Page 8: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

The three vortex collapsing configuration

Figure 9: Initial configuration of vortices

A special, neutrally stable (J = 0) three vortex configuration exists wherein the vortices collapseto a point within a finite time. Let κ1 = 1, κ2 = 1

2, κ3 = −1

3be the strength of the vortices, and a0,

b0 and c0 be their initial separations r23, r31 and r12. It can be shown that the area of the trianglecreated by these vortices will go to zero in a finite time if the following constraints are satisfied:

1

κ1

+1

κ2

+1

κ3

= 0

a20

κ1

+b20

κ2

+c20

κ3

= 0

These constraints correspond to setting J = 0 and ensuring that the shape of the triangle is constantin time (though its area changes). In this configuration, each vortex follows a spiral path towards thecentre of vorticity. Figure 10 shows the vortex trajectories and the vortex separations as a functionof time, and illustrates that the separations go to zero in finite time.

-2 -1.5 -1 -0.5

-0.5

0

0

0.5

1

(a) Vortex tracks

00

0.5

1

1.5

2

500 1000 1500 2000

separ

atio

n

t

(b) Vortex separations

Figure 10: Collapsing vortices, J = 0. Note that simulation stops prior to collapse due to numerical instability

Coherent structures in a 4 vortex system

Let’s investigate a system of four point vortices. The total vorticity of the system is not zero (Σκα 6= 0).As you know, it means that we can define centre of vorticity by using earlier stated formulas. Wecan also define a momentum of the system and as you can see on Figure 11 it does not coincide

7

Page 9: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

-1.5-1.5

-1

-1

-0.5

-0.5

0

0

0.5

0.5

1

1

1.5

1.5

(0, 1), κ = 0.5

(0,−1), κ = 0.5

(−1, 0), κ = −1 (1, 0), κ = 1

(a) Vortex configuration

-1 0

0

1 2 3 4 5 6 7 8

-2

-4

-6

-8

-10

(b) Vortex tracks

Figure 11: Initial vortex configuration and vortex tracks

with direction of propagation of the vortices. This is the first but not last interesting feature of thesystem. Velocity of the first point vortex with κ = 1 reduced to nearly zero, while another vorticescontinue moving with the same speed. It is well known that velocity of the point vortex depends onlyon interaction with the other ones within the system, in other words, it depends on structure of thesystem and vorticity. But we cannot say the same about acceleration of vortex. According to M.Rast and J-F Pinton work at smallest temporal increment acceleration can be modified by vorticityreconfiguration within the system.

In modern theory of the systems of point vortices it has often been proposed that coherent structuresplay an important role in decreasing or increasing of velocities of the point vortices. On Figure 11there is exactly such structure created by 3 vortices with total vorticity equal to zero. And we supposeto think that first vortices’ velocity reduction can be caused by the coherent structure created here.Much deeper analysis which includes solution of the system of four nonlinear equations is required inthis problem.

4-vortex chaos and the Liapunov exponent

The three vortex problem may be integrated using the system invariants to constrain the solution.However, there are insufficient invariants available to integrate the four vortex problem in general, andas such solutions must be computed numerically. Of particular interest is the emergence of chaoticbehaviour in four vortex configurations. Though not all configurations are chaotic, many are, and inthis section we identify a chaotic system and find the Liapunov exponent for that system.

A chaotic system is one where tiny perturbations to the initial conditions of the system result indramatically different solutions, where the difference between solutions grows quasi-exponentially withtime. For point vortex systems as are considered here, we may define the distance between solutions asd(t) = r12perturbed

(t) − r12unperturbed(t), then, for a chaotic system this distance will grow exponentially

at first until the physical constraints on the system (such as fixed dispersion) force the distance to

8

Page 10: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

eventually decrease as 1/t. Initially at least, a chaotic system behaves as::

d(t) = d0eµt.

d0 is the distance between the solutions at t = 0, thus it is the initial perturbation. If µ ≤ 0 then thedistance between the solutions tends to zero (or a constant), and the system is not chaotic. However,for µ > 0 the distance between the solutions grows exponentially and the system exhibits deterministicchaos. Rearranging the above, and taking the limit as t → ∞, we obtain the following expression forµ:

µ = limt→∞

1

t(ln d(t) − ln(d0)).

Four vortices were arranged as in Figure 12 and the solution was iterated for one million timesteps.The first vortex (having strength κ = 1) was then perturbed in space by 0.0001 and the solution wasrerun. The resulting vortex tracks are shown in Figure 13 and it is clear that the small perturbationhas caused a significant change to the motion of the vortices.

Figure 12: Initial configuration of vortices

-6

-6

-4

-4

-2

-2

0

0

2

2

4

4

6

6

(a) Unperturbed vortex tracks

-8

-6

-6

-4

-4

-2

-2

0

0

2

2

4

4

6

6

(b) Perturbed vortex tracks

Figure 13: Vortex tracks for perturbed and unperturbed 4 vortex chaotic system

The Liapunov exponent was evaluated at each timestep and is shown in Figure 14. Initially theexponent exhibits some high frequency variation, but over time it smooths out and reaches a positivenon-zero limit, demonstrating that the system is chaotic. It should be noted that in this case thedispersion of the system is constrained, thus there is a maximum separation, d(t), that the vorticesmay attain. This means that in the limit that t → ∞, µ will be seen to decrease gradually towardszero in this case.

9

Page 11: Point Vortex Dynamics in Two Dimensions -  · PDF filePoint Vortex Dynamics in Two Dimensions Ruth Musgrave, ... Note that in this case Σκ ... -20-30-40-50-60-70-80

200000 400000 600000 800000 1e+06

-6

-4

-2

0

0

2

4

(a) Separation between vortex tracks

200000 400000 600000 800000 1e+060

0

-1e-5

-0.5e-5

0.5e-5

1e-5

(b) Liapunov exponent as a function of time

Figure 14: Track separations exhibiting quasi-exponential behaviour in four vortex case

References

Professor Keith Moffatt: Notes and personal communication.

Phys. Rev. E 79, 046314 (2009): Point-vortex model for Lagrangian intermittency in turbulence(Mark Peter Rast, Jean-Francois Pinton)

10