F. Kaplanski and Y. Rudi

Tallinn University of Technology,

Faculty of Science,

Laboratory of Multiphase Media Physics

Transport Processes in an Oscillating Vortex Ring and

Pair

Basic ideaTrajectories 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.

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.

Earlier studies of 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 )

• 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

Relatively recent studies

),2(Sinrux 3

1

,vr

,

),(Sinux1

),(Sinvr2

, t

•Tsega and Michaelides (2001) - The study of the motion of small spherical particles inside the Kelvin cat eyes flow

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 expand results by Rom –Kedar et. al. for inviscid vortex pair on the viscous case)

Contents

• An analytical model for the vortex ring and pair

• Analysis of mixing processes in the perturbed vortex pair

A model for the viscous vortex ring and pair

Giant vortex ring generated in laboratory (Australia).

Schematic representation of the vortex ring

and pair

Stokes solution for the viscous vortex ring

(Berezovski & Kaplanski, 1988)

 

where - is the ring translational velocity.

, I21

exp 1222

, 0 ,

R= ,

)t(x-x= ,

r oo

td)t(xd

)t(U 0

• 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.

Energy

3

0

2

2

m R2

ME

2

22

22/1

,3,2

5,

2

3,

2

3F

12

1

Translation velocity

)

2(I

2exp3

R4

MU

2

1

2

2/1

3

0

2m

2

22

22/1

,3,2

5,

2

3,

2

3F

12

1

2

222

2/1

,27

,2,25

,23

F5

3

Circulation

220

m exp1RM

Impulse

MIm

Properties of the viscous vortex ring (Kaplanski & Rudi,1999)

StreamfunctionFourier- Hankel integral transform

I2

1exp

1

222

1

22

J2

exp

/f

rrr

1

xr 2

2

2

2

122

22

J2exp

f

Inverse Fourier- Hankel integral transform

)/f( f

0 0 1122

22

2/1

3

0 dd)cos(JJ2exp

2

L2

The integration with respect to gives

0

0

11

0

0

2

0

dR

rJJ)

R

)xx(,(F

R4

rM

Validation of the model for high values of (formation stage).

Gharib (1998)

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

,UL21

p 2p

2 D21

ULD41

I

,I2

UULD

81

E p2p

2

This allows us to get the dimensionless stroke length in the following form

,E

I2D

L 2/12/3

2/1m

2/3m

m

I

EE~

The dimensionless energy ( Gharib et al, 1998).

Comparison with experiment

The analogue of the viscous vortex ring-vortex pair

(Berezovski & Kaplanski, 1988)

  , sh

21

exp 222

))(sLn)(sE(-))sLn()(sE(( 221111

222

221 τ)(σ(η

21

s,)τ)(σ(η21

s

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)  

)R16/(U 02

tr

1E

Streamfunction

,))1(σ(η21

s,))1(σ(η21

s

),tsin(xrrU

))(sLn)(sE(-))sLn()(sE((

2222

2221

1tr

221111

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 the viscous analogy to the Norbury vortices (1973). Further we will consider fixed.

Oscillating viscous vortex pair

000 R/)xx(,R/r

-10 -5 0 5 10

-10

-5

0

5

10

Isolines of the streamfunction for the vortex pair in the moving

coordinate system for =4.rU~tr

)T(sin)(vdTd

s

)T(sin)(uudTd

s

;/1;/

),R/(,/Ruu

,/,R/tT200tr

120

Equations of particle motion

The velocity components are determined by the derivatives and are not presented because of theirs inconvenience.DIMENSIONLESS VARIABLES

PARAMETERSperturbation constant, -driving frequency, concentration of vorticity

ss v,u

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 near the centre of pair. Points were sampled every period (based on perturbation frequency) for 4000 iterations.

))2(r),2(x())(r),(x(

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

Concept of chaos.

I- enclosed area, T(I) –period or time needed for particle to make complete circuit along streamline.

J=T(I)/(2– 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.

Effect of perturbation frequencyPoincare sections for perturbation frequency =0.1 ( perturbation constant =0.05 ,Points were sampled every 10 periods.

Effect of perturbation frequencyPoincare sections for perturbation frequency =0.5 ( perturbation constant =0.05 ,Points were sampled every 10 periods.

Effect of perturbation frequencyPoincare sections for perturbation frequency =1. ( perturbation constant =0.05 ,Points were sampled every period.

Effect of perturbation frequencyPoincare sections for perturbation frequency =1.5 ( perturbation constant =0.05 ,Points were sampled every 10 periods.

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

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.

KAM theory

Effect of vorticity concentration Poincare sections for perturbation =1 and perturbation constant =0.05 with ( a) andbPoints were sampled every 10 periods.

(a)

(b)

Conclusions

• It is shown that the flow inside the viscous vortex pair for >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.