Point Vortex Dynamics in Two Dimensions

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1 Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 9 April to May, 9 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei Zhu Supervised by: Professor Keith Moffatt

2 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 the velocity vector field: ω = u In this project we consider only two dimensional flows, as such the vorticity vector is always directed perpendicularly 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 point concentration. Stoke s theorem may be written: ω = u.dl. S Thus, for a circular loop around a single point vortex, the above integral may be evaluated as C κ = πrv θ where v θ 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: Dω Dt = ω. u + ν ω However, in two dimensional inviscid flows both terms on the right vanish, yielding the important result that in such flows vorticity is carried with the fluid. Consequently, the motion of a point vortex is seen to depend only upon the velocity of the fluid in which it is embedded. If the fluid is stationary then the vortex will induce a circular velocity field, but will itself remain fixed in space. The interaction of point vortices, therefore, is governed by the fluid velocities that each induce at the position of the others. Given the radial velocity distribution induced by a point vortex earlier derived, the velocity of a vortex situated at (x α,y α ), may be found as the vector sum of the velocities induced by each of the vorticies at that point: dx α dt = β κ β (y α y β ) πr αβ, dy α dt = β κ β (x α x β ) πr αβ where the summation is performed for α β. A side-effect of the point vortex idealisation is that the velocity field at the location of the point vortex is singular. This has the unhappy consequence of causing the kinetic energy of the system to become infinite. Furthermore integrals of momentum and angular momentum become infinite in the limit as r. As such, new invariants must be devised in order to provide some reasonable physical constraints to the system. The first constraint is provided by the Hamiltonian for the system, which is given by: H = π N α,β= κ α κ β ln r αβ

3 This is a measure of the energy of interaction of the vortices. Momentum and angular momentum like quantities are given by: P = ρ(σκ αy α, Σκ α x α,) = cst M = ρ(,, Σκ α(x α + y α)) = cst In the case that Σκ α, a fixed centre of vorticity and dispersion, D, of the vortex distribution may be defined: x = Σκ αx α Σκ α, ȳ = Σκ αy α Σκ α D = Σκ α [(x α x) + (y α ȳ) ] = cst In general it is not possible to integrate a system of N point vortices where N >, however, the system is deterministic and solutions may be calculated numerically. In this project, code was written to numerically integrate the positions of an arbitrary number of vortices forward in time. The above invariants 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 by the relative signs of the vortices. Vortices of opposing sign are observed to propagate together along a line perpendicular to their bisector. Note that in this case Σκ α =, implying that the system is not constrained to have a fixed centre of vorticity or dispersion. However, similar signed vortices are constrained in this manner and they are observed to move in a circle around their fixed centre of vorticity. Figure shows the results of the numerical solutions for these cases (a) Two vortices, different signs (b) Two vortices, same signs Figure : Some numerical results

4 Three point vortices on three verticies of a square. κ =.8.. r r. κ = r κ = Figure : Initial configuration of vortices Three vortices were initially arranged as in Figure such that they formed a right angled triangle with two sides having equal length. The chosen values of vortex strength mean that the system has a fixed centre of vorticity, and this is clear from the resulting vortex tracks (Figure ) which show the vortices moving in a biperiodic manner around the point (, ). The periodic motion of the three vortices through phase space is illustrated in Figure b) where the values of the separations of the vortices have been plotted at each timestep r r r (a) Vortex tracks (b) Phase space Figure : 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 provided that the sum of the vorticities is non-zero (i.e. the dispersion invariant is defined). However, the stability of their motion is determined by the value of the invariant J, a function of P and M given

5 by: J = ( ρ (κ + κ + κ ) M ρ P ) = R (κ κ + κ κ + κ κ ) Theory states that the configuration should be stable to perturbations when J >, unstable when J < and neutrally stable when J =. In the following simulations, the strengths of the point vortices were set such that J satisfied each of the above cases. In the unperturbed run, the vortices were arranged exactly at the vertices of an equilateral triangle. In the perturbed run, one of the vortices was shifted by. units in the x direction (a) Vortex tracks: note that two of the vortices follow identical trajectories.. r r..... r. (b) Separation space of vortices Figure : J > case, perturbed and unperturbed cases are identical In the case where J >, the system is theoretically stable. After one million iterations with a timestep of., the perturbed and unperturbed cases were indistinguishable by eye. In both cases the 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.... r r..... r.8 (a) Vortex tracks: unperturbed (b) Separation space: unperturbed Figure : Neutrally stable, unperturbed configuration: J =

6 (a) Vortex tracks: perturbed...8 r r r. (b) Separation space: perturbed Figure : Neutrally stable, perturbed configuration: J = J = is neutrally stable. The unperturbed vortex tracks are shown in Figure. Perturbations to this state caused the vortices to slowly spiral further apart, as shown in Figure, however the overall track pattern is largely unchanged. The configuration where (J < ) is unstable and a small perturbation to the initial position causes a dramatic change in the behaviour of the vortices over time. Note that in the case illustrated in Figure 8, κ =, κ = and κ =, therefore there is no fixed centre of vorticity and the dispersion of the system is not invariant r.. r r..... (a) Vortex tracks, unperturbed (b) Separation space, unperturbed r r.... r (c) Vortex tracks, perturbed (d) Separation space, perturbed Figure 7: Unstable, J <. Note dramatic change in vortex tracks and separations arising from small perturbation to original position of one vortex.

7 Instabilities in, 7 and 8 vortex configurations A further interesting case arises in the stability of N equal vortices which are regularly distributed around 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 the vortices is perturbed? Detailed investigation shows that the configuration is stable if N and unstable if N 8. The case N = 7 is neutrally stable. The following figures illustrate the effects of perturbation on the stability of, 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 and 7 vortices, but it completely changes the trajectories of 8 vortices. (a) vortices, unperturbed (b) 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 and 7 vortex cases to small perturbations whilst the 8 vortex case demonstrates significant instability.

8 The three vortex collapsing configuration Figure 9: Initial configuration of vortices A special, neutrally stable (J = ) three vortex configuration exists wherein the vortices collapse to a point within a finite time. Let κ =, κ =, κ = be the strength of the vortices, and a, b and c be their initial separations r, r and r. It can be shown that the area of the triangle created by these vortices will go to zero in a finite time if the following constraints are satisfied: κ + κ + κ = a κ + b κ + c κ = These constraints correspond to setting J = and ensuring that the shape of the triangle is constant in time (though its area changes). In this configuration, each vortex follows a spiral path towards the centre of vorticity. Figure shows the vortex trajectories and the vortex separations as a function of time, and illustrates that the separations go to zero in finite time... separation (a) Vortex tracks t (b) Vortex separations Figure : Collapsing vortices, J =. Note that simulation stops prior to collapse due to numerical instability Coherent structures in a vortex system Let s investigate a system of four point vortices. The total vorticity of the system is not zero (Σκ α ). As you know, it means that we can define centre of vorticity by using earlier stated formulas. We can also define a momentum of the system and as you can see on Figure it does not coincide 7

9 . (,), κ =.. - (,), κ = (,), κ = (, ), κ = (a) Vortex configuration (b) Vortex tracks Figure : Initial vortex configuration and vortex tracks with direction of propagation of the vortices. This is the first but not last interesting feature of the system. Velocity of the first point vortex with κ = reduced to nearly zero, while another vortices continue moving with the same speed. It is well known that velocity of the point vortex depends only on interaction with the other ones within the system, in other words, it depends on structure of the system 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 vorticity reconfiguration within the system. In modern theory of the systems of point vortices it has often been proposed that coherent structures play an important role in decreasing or increasing of velocities of the point vortices. On Figure there is exactly such structure created by vortices with total vorticity equal to zero. And we suppose to 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 in this problem. -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, and as such solutions must be computed numerically. Of particular interest is the emergence of chaotic behaviour in four vortex configurations. Though not all configurations are chaotic, many are, and in this 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 in dramatically different solutions, where the difference between solutions grows quasi-exponentially with time. For point vortex systems as are considered here, we may define the distance between solutions as d(t) = r perturbed (t) r unperturbed (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

10 eventually decrease as /t. Initially at least, a chaotic system behaves as:: d(t) = d e µt. d is the distance between the solutions at t =, thus it is the initial perturbation. If µ then the distance between the solutions tends to zero (or a constant), and the system is not chaotic. However, for µ > the distance between the solutions grows exponentially and the system exhibits deterministic chaos. Rearranging the above, and taking the limit as t, we obtain the following expression for µ: µ = lim t t (ln d(t) ln(d )). Four vortices were arranged as in Figure and the solution was iterated for one million timesteps. The first vortex (having strength κ = ) was then perturbed in space by. and the solution was rerun. The resulting vortex tracks are shown in Figure and it is clear that the small perturbation has caused a significant change to the motion of the vortices. Figure : Initial configuration of vortices (a) Unperturbed vortex tracks (b) Perturbed vortex tracks Figure : Vortex tracks for perturbed and unperturbed vortex chaotic system The Liapunov exponent was evaluated at each timestep and is shown in Figure. Initially the exponent exhibits some high frequency variation, but over time it smooths out and reaches a positive non-zero limit, demonstrating that the system is chaotic. It should be noted that in this case the dispersion of the system is constrained, thus there is a maximum separation, d(t), that the vortices may attain. This means that in the limit that t, µ will be seen to decrease gradually towards zero in this case. 9

11 e-.e e- - 8 e+ (a) Separation between vortex tracks -e- 8 e+ (b) Liapunov exponent as a function of time Figure : Track separations exhibiting quasi-exponential behaviour in four vortex case References Professor Keith Moffatt: Notes and personal communication. Phys. Rev. E 79, (9): Point-vortex model for Lagrangian intermittency in turbulence (Mark Peter Rast, Jean-Francois Pinton)

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