Wavy Vortex Flow: A Tale of Chaos, Symmetry, and Serendipity in a Steady World

1. Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady worldGreg KingUniversity 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. 2D incompressible fluid 3D incompressible fluid Phase Space No chaos here Fluid Dynamics and Phase Space Symmetries -- can reduce phase space

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

8. Eccentric Couette FlowChaiken, 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, JFM 417, 265-301 (2000) Experiment (light sheet) Numerical Particle Tracking (“light sheet”)

12. b a 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 Wavy vortices Taylor 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. nested streamtubes /2 Z 0 Radius outer cylinder inner cylinder Rotational Symmetry3D  2D Phase Space “Light Sheet”

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

19. Dividing stream surface Dividing stream surface breaks up => particles can migrate from vortex to vortex The Leaky Transport Barrier Wavy vortex flow is adeformation of rotationally symmetric Taylor vortex flow. Flow is steady in co-moving frame 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 (20,000particles) in a frame rotating with the wave (4th order Runge-Kutta).

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

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

23. Taylor vortices Wavy vortices Effective Diffusion Coefficient Characterize the migration of particles from vortex to vortex Rudman, AIChE J44 (1998) 1015-26. Initialization: Uniformly distribute 20,000 particles (dimensionless)

24. Size of mixing region Dz (dimensionless)

25. Dz

26. An Eulerian ApproachSymmetry 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 rotationalsymmetry and dynamical symmetry : • If either is zero, then flow is locally integrable, so as a diagnostic we consider the product

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

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

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

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

31. Averaged Symmetry Measures andpartialaverages

32. Size of chaotic region fq Dz fD fn

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

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

35. Effect of Flow State: Axial wavelengthm: 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 14 2002) 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 StreamlinesYannacopoulos et al, Phys Fluids 14 2002(see alsoMezic JFM 2001) This is the Melnikov function