1. Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady world Greg King University of Warwick (UK)
Collaborators:
Murray Rudman (CSIRO)
George Rowlands (Warwick)
Thanasis Yannacopoulos (Aegean)
Katie Coughlin (LLNL)
Igor Mezic (UCSB)

2. To understand this lecture you need to know
Some fluid dynamics
Some Hamiltonian dynamics
Something about phase space
Poincare sections
Need > 2D phase space to get chaos
Symmetry can reduce the dimensionality of phase space
Some knowledge of diffusion
A “friendly” applied mathematician !!

3. Phase Space

4. Dynamical Systems and Phase Space

5. Classical Mechanics and Phase Space
Hamiltonian
Dissipative

6. Fluid Dynamics and Phase Space
2D incompressible fluid
3D incompressible fluid
Phase Space
No chaos here
Symmetries -- can reduce phase space

7. Poincare Sections (Experimental – i.e., light sheet)

8. Eccentric Couette Flow Chaiken, Chevray, Tabor and Tan, Proc Roy Soc 1984 ?? Illustrates “Significance” of KAM theory

9. Fountain et al, JFM 417, 265-301 (2000) Stirring creates deformed
vortex

11. Fountain et al, Experiment (light sheet) Numerical Particle Tracking
JFM 417, (2000)
Experiment
(light sheet)
Numerical
Particle Tracking
(“light sheet”)

12. Taylor-Couette Radius Ratio:   = a/b Reynolds Number: Re = a(b-a)/

13. Engineering Applications
Chemical reactors
Bioreactors
Blood – Plasma separation
etc

14. Taylor-Couette regime diagram
(Andereck et al)
Rein
Reout

15. Some Possible Flows Taylor Wavy vortices vortices Twisted vortices
Spiral
vortices

16. Taylor Vortex Flow TVF --
Centrifugal instability of circular Couette flow.
Periodic cellular structure.
Three-dimensional, rotationally symmetric:
u = u(r,z)
Flat inflow and outflow boundaries are barriers to inter-vortex transport.

17. Rotational Symmetry 3D  2D Phase Space
“Light Sheet”
Radius Z
/2
inner
cylinder
outer
nested
streamtubes

18. Wavy Vortex Flow
wavy vortex flow
Taylor vortex flow
Rec

19. The Leaky Transport Barrier
Wavy vortex flow is a deformation of rotationally symmetric Taylor vortex flow.
Flow is steady
in co-moving frame
Dividing stream surface breaks up => particles can migrate from vortex to vortex
Dividing stream surface
Increase Re
Poincare Sections

20. Methods
Solve Navier-Stokes equations numerically to obtain wavy vortex flow.
Finite differences (MAC method);
Pseudo-spectral (P.S. Marcus)
Integrate particle path equations (20,000 particles) in a frame rotating with the wave (4th order Runge-Kutta).

21. Poincare Section near onset of waves
Wavy Vortex Flow
Poincare Section near onset of waves
6 vortices 1 2 3 4 5 6
inner
cylinder
outer
cylinder

22. At larger Reynolds numbers (Rudman, Metcalfe, Graham: 1998)

23. Effective Diffusion Coefficient
Characterize the migration of particles from vortex to vortex
Rudman, AIChE J 44 (1998)
Initialization:
Uniformly distribute
20,000 particles
(dimensionless)
Taylor vortices
Wavy vortices

24. Dz
Size of mixing region
(dimensionless)

25. Dz

26. An Eulerian Approach Symmetry Measures
Theoretical Fact
A three dimensional phase space is necessary for chaotic trajectories.
The Idea (Mezic):
Deviation from certain continuous symmetries can be used to measure the local deviation from 2D
For Wavy Vortex Flow
rotational symmetry
and dynamical symmetry :
If either is zero, then flow is locally integrable, so as a diagnostic we consider the product

27. Dynamical Symmetry
Steady incompressible Navier-Stokes equations in the form
Equation of motion for B
B is a constant of the motion if

28. Measure for Rotational Symmetry
155
162
324
486
648
Reynolds Number

29. Measure for Dynamical Symmetry
155
162
324
486
648
Reynolds Number

30. X Rotational Dynamical f = 155 162 324 486 648 Reynolds Number
Looks interesting, but correlation does not look strong !

31. Averaged Symmetry Measures
and partial averages

32. Dz
Size of chaotic region fq fn fD

33. King, Rudman, Rowlands and Yannacopoulos Physics of Fluids 2000
Serendipity !
King, Rudman, Rowlands and Yannacopoulos Physics of Fluids 2000

34. Effect of Radius Ratio (Mind the Gap)
Dz/
Re/Rec 

35. Effect of Flow State : Axial wavelength m: Number of waves

Re/Rec 

36. Effect of Flow State Dz
Re/Rec 

37. Summary Dz is highly correlated with <><n>
The correlation is not perfect.
The symmetry arguments are general
Yannacopoulos et al (Phys Fluids ) show that Melnikov function,
M ~ < >< n >.
Is it good for anything else?

38. 2D Rotating Annulus u(r,z,t) Richard Keane’s results (see poster)
Symmetry measure:
FSLE
Log(FSLE)
Log(<|d/dt|>)

39. Prandtl-Batchelor Flows (Batchelor, JFM 1, 177 (1956)
Steady Navier-Stokes equations in the form
Integrating N-S equation around a closed streamline s yields

40. Break-up of Closed Streamlines Yannacopoulos et al, Phys Fluids 14 2002 (see also Mezic JFM 2001)
This is the Melnikov function