F. Kaplanski and Y. Rudi Tallinn University of Technology, Faculty of Science,

1. Transport Processes in an Oscillating Vortex Ring and Pair F. Kaplanski and Y. Rudi Tallinn University of Technology, Faculty of Science, Laboratory of Multiphase Media Physics

2. Basic idea Trajectories of fluid particles in steady flows can never cross - particles have no opportunity to become mixed. The simplest way to improve this is to force the flow in a periodic manner (see for details J. Ottino, Sci. Am. 1989) . plus periodic perturbations.

3. Important remarks • Particle motion is generally more complex than the underlying fluid dynamics: while the motion of three point vortices in an unbounded domain is integrable, particle motion in this flow can be chaotic (Aref, 1983). • In the case of two-dimensional time – periodic motions the equations describing the trajectories of fluid particles are formally identical with those describing a Hamiltonian system.

4. Earlier studiesof mixing with introducing of periodic perturbations • The flow due to two blinking vortices (Aref, 1984) • The two-dimensional flow fields generated by time – periodic motion of either eccentric cylinders (Aref&Balachandar 1986) • The sidewalls of a cavity flow (Leong&Ottino 1989) • The inviscid vortex pair flow (Rom –Kedar et.al, 1990 )

5. Relatively recent studies • Sotiropoulus et. al.(2002) - The vortex –breakdown bubbles inside cylinder with rotated lid. The study of the role of swirl intensity using the Hill vortex ring • Tsega and Michaelides (2001) - The study of the motion of small spherical particles inside the Kelvin cat eyes flow

6. Aim • to find how a time-periodic strain field affected the particle motion in the velocity field induced by the viscous vortex pair and ring flows (to expandresults by Rom –Kedar et. al. for inviscid vortex pair on the viscous case)

7. Contents • An analytical model for the vortex ring and pair • Analysis of mixing processes in the perturbed vortex pair

8. A model for the viscous vortex ring and pair

9. Giant vortex ring generated in laboratory (Australia).

10. Schematic representation of the vortex ring and pair

11. Stokes solution for the viscous vortex ring (Berezovski & Kaplanski, 1988) where - is the ring translational velocity. • Can be considered as the second –order solution • (Rott & Cantwell , 1993). • In the short-time limit tends to the Gaussian distribution. This is typical for vortex rings generated by piston/cylinder arrangements.

12. Properties of the viscous vortex ring (Kaplanski & Rudi,1999) Energy Translation velocity Circulation Impulse

13. Streamfunction Fourier- Hankel integral transform Inverse Fourier- Hankel integral transform The integration with respect to gives

14. Validation of the model for high values of t (formation stage). Gharib (1998)

15. Slug model Following (Mohseni and Gharib (1998) and Linden and Turner (2001)),, we can write for the slug of fluid with the diameter D, length L, and with uniform velocity at the pinch off moment the following expressions for the circulation, impulse and kinetic energy per unit density This allows us to get the dimensionless stroke length in the following form

16. Comparison with experiment The dimensionless energy ( Gharib et al, 1998).

17. The analogue of the viscous vortex ring-vortex pair (Berezovski & Kaplanski, 1988) Here and are the longitudinal and transversal coordinates, respectively, exponental inegral function. The approximate value of the asymptotic drift velocity for the viscous vortex pair (Rott &Cantwell 1988)

18. Streamfunction Oscillating viscous vortex pair Note that vortex pair properties depend on a single parameter , which represents the concentration of vorticity and allows us to describe thick and thin vortex pairs. This means that the presented model can be considered as theviscous analogyto the Norbury vortices (1973). Further we will consider fixed.

19. Isolines of the streamfunction for the vortex pair in the movingcoordinate systemfor =4.

20. Equations of particle motion The velocity components are determined by the derivatives and are not presented because of theirs inconvenience. DIMENSIONLESS VARIABLES PARAMETERS e-perturbation constant, q-driving frequency, t-concentration of vorticity

21. Numerical algorithm • Perturbed vortex pair flow is studied via the Poincare map • Poincare calculations involved 24 initial positions of fluid particles located on the axis s near the centre of pair. Points were sampled every period (based on perturbation frequency) for 4000 iterations.

22. Effect of perturbation constant Poincare sections for two systems with the same perturbation frequency q=0.5, but different perturbation constants e=0.05(blue)and e=0.01(red) for t =2. Points were sampled every 10 periods.

23. Concept of chaos. I- enclosed area, T(I) –period or time needed for particle to make complete circuit along streamline. J=T(I)/(2pg) – integer or irrational rotation number. Firstly, the invariant curves associated with integer J, are the first to be destroyed when a perturbation is introduced to the system. Half of the periodic orbits will be stable and half will be unstable. The stable and unstable manifolds may intersect transversely, yielding chaotic particle motions.

24. Effect of perturbation frequency Poincare sections for perturbation frequency q=0.1 ( perturbation constant e=0.05 ,t=2). Points were sampled every 10 periods.

25. Effect of perturbation frequency Poincare sections for perturbation frequency q=0.5 ( perturbation constant e=0.05 ,t=2). Points were sampled every 10 periods.

26. Effect of perturbation frequency Poincare sections for perturbation frequency q=1. ( perturbation constant e=0.05 ,t=2). Points were sampled every period.

27. Effect of perturbation frequency Poincare sections for perturbation frequency q=1.5 ( perturbation constant e=0.05 ,t=2). Points were sampled every 10 periods.

28. Effect of perturbation frequency Poincare sections for perturbation frequency q=2 (perturbation constant e=0.05, t=2). Points were sampled every 10 periods.

29. KAM theory As the perturbation increased, an increase in the number of manifold intersections is exhibited and the motion become more complicated. When the perturbation frequency is increased further, small islands chains start appearing on the pattern. These chains show better organization as frequency is increased to 0.1 and transform into invariant curves. Such invariant curve is referred to as a KAM torus after Kolmogorov, Arnol’d and Moser theorem. KAM tori represent total barries to fluid motion and correspond to the irrational rotation numbers. For special case of the irrationality so called cantorus start appear, which are similar to the dynamics on the KAM torus. However, the cantorus contains gaps which permit the (possible very slow) passage of fluid. As the perturbation frequency is increased further, all the invariant tori are reformed and the Poincare section looks similar to the slightly perturbed system.

30. Effect of vorticity concentration Poincare sections for perturbation q=1 and perturbation constant e=0.05 with t=2 ( a) and t= 4 (b). Points were sampled every 10 periods. (a) (b)

31. Conclusions • It is shown that the flow inside the viscous vortex pair for t>1 can display Lagrangian chaos when the pair is under the influence of a periodic perturbation. It is generally believed that this phenomena is associated with better mixing and transport. • However, an increase of the perturbation frequency causes the appearing of the regions where a bounded quasi-periodic motion occurs (known as the KAM tori).These regions represent total barries to fluid motion and hence strongly influence transport. • An attempt to study mixing process in the viscous vortex ring flow is the focus of our future study.