1. Fluid Mixing Greg Voth Wesleyan University Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)

2. Why study fluid mixing? Nigel listed three fundamental processes that engineers need to optimize that depend on turbulence:  Turbulent Combustion  Environmental Transport  Drag on transportation vehicles I would argue that each of these is primarily a problem of transport and mixing:  Turbulent Combustion is a transport and mixing of fuel, oxidizer, and thermal energy  Environmental Transport is obviously a mixing problem.  Drag on transportation vehicles is even the turbulent transport of momentum.

3. Equations for Passive Scalar Transport Advection Diffusion: Navier-Stokes : Incompressibility:

4. Equations for Passive Scalar Transport Advection Diffusion: Navier-Stokes : Incompressibility: New Dimensionless Parameter: Peclet Number

5. Equations for Passive Scalar Transport Advection Diffusion: Navier-Stokes : Incompressibility: For small diffusivity, the advection diffusion equation reduces to conservation of the scalar along Lagrangian trajectories.

6. Scalar Dissipative Anomaly Doniz, Sreenivasan and Yeung JFM 532:199 (2005) In turbulence, the energy dissipation rate is independent of the viscosity (when the viscosity is reasonably small) even though the viscosity enters the definition of the energy dissipation rate: Similarly, the scalar dissipation rate is independent of the diffusivity (when the diffusivity is reasonably small) even though the viscosity enters its definition:

7. Kolmogorov-Obukhov-Corrsin scaling for passive scalar statistics Scalar Spectrum in the inertial range: Scalar Structure Functions in the inertial range: Actually: Warhaft Annu. Rev. Fluid Mech. 32:203 (2000) Intermittency of the passive scalar field is stronger than that of the velocity field. (For high Re and Pe)

8. Scalar Anisotropy Measurements in a wind tunnel with a mean scalar gradient up to R = 460 show the odd moments of the scalar derivative do not go to zero at small scales, indicating persistent anisotropy. Warhaft. Annu. Rev. Fluid Mech. 32:203–240 (2000) Need still higher Re? Intermittency effects? Active Grid Turbulence? In any case, scalar fields generally require higher Reynolds numbers to see isotropy or Kolmogorov scaling.

9. Lagrangian Descriptions Fluid mixing is fundamentally a Lagrangian phenomenon…but traditional analysis of turbulent mixing has analyzed the instantaneous spatial structure of the scalar field. Why? -Primarily, Lagrangian data has simply been unavailable This has changed in the last 25 years…with the availability of numerical simulations and experimental tools for particle tracking. -But the theory was developed before any reliable data was available…why was the Lagrangian description of mixing ignored? Kolmogorov’s second mistake…see readings for Thursday

10. Outline of my talks this week Rest of this talk: Lagrangian desciptions of chaotic mixing Patterns in fluid mixing Stretching fields and the Cauchy strain tensors What controls mixing rates Thursday morning and afternoon: Lagrangian descriptions of turbulent flows Lagrangian Kolmogorov Theory Tools for measuring particle trajectories Motion of non-tracer particles in turbulence

11. Brandeis University, 2002 Lagrangian descriptions of chaotic mixing Magnet Array Dense, conducting lower layer (glycerol, water, and salt, 3 mm thick) Electrodes Less dense, non-conducting upper layer (glycerol and water, 1 mm thick) Top View:Periodic forcing:

12. Evolution of dye concentration fieldSame data updated once per period. Persistent Patterns

13. Dye pattern develops filaments which are stretched and folded until they are small enough that diffusion removes them. A persistent pattern develops in which transport and stretching balances diffusion. The overall contrast decays, while the spatial pattern remains unchanged. Image can be decomposed into a function of space times a function of time. Questions: What determines the geometry of the persistent pattern? What controls the decay rate? Observations

14. Raw Particle Tracking Data l ~ 800 fluorescent particles tracked simultaneously. Positions are found with 40  m accuracy. l ~15,000 images: 40-80 images per period of forcing, and 240 periods. l Phase Averaging: 800*240 = 10 5 particles tracked at each phase. l The flow is time periodic and so exactly the same flow can be used in both dye imaging and particle tracking measurements.

15. Velocity Fields: Phase averaging allows us to obtain highly accurate time-resolved velocity fields 0.9 cm/sec 0 cm/sec (p=5, Re=56)

16. Lines connect position of each measured particle with its position one period later: Poincaré Map. Color codes for distance traveled in a period: Blue Small Distance Red Large Distance Particle Displacement Map

17. Structures in the Poincaré Map Hyperbolic Fixed Points Elliptic Fixed Points

18. Manifolds of Hyperbolic Fixed Points Unstable Manifold Stable Manifold

19. Hamiltonian Chaos Henri Poincaré first identified the hyperbolic fixed points and their manifolds as central to understanding chaos in Hamiltonian systems in a memoir published in 1890. His interest was in planetary motion and the three body problem, but structures like these are seen in many other problems: Charged particles in magnetic fields Quantum systems But why do these different systems exhibit the same organizing structures? Henri Poincaré (1854-1912) ( from Barrow-Green, Poincaré and the three body problem, AMS 1997)

20. Why do these systems show similar structures? Fluid MixingHamiltonian System y x Real Space Phase Space Generalized Momentum, p Generalized Position, q Stream Function Equations: Hamilton’s Equations: (Aref, J. Fluid Mech, 1984)

21. Manifold Structure and Chaos Regular (Non-chaotic)Chaotic

22. Can we extract manifolds in experiments? These manifolds have been hard to extract from experiments. They are fundamentally Lagrangian structures. We could simply search for fixed points and construct the manifolds of each fixed point, but there is a more elegant way: The manifolds consist of fluid elements that experience large stretching (Haller, Chaos 2000)... So, we want to measure the stretching fields experienced by fluid elements

23. Calculating Stretching L0L0 L Stretching = lim (L/L 0 ) L 0 0

24. Practice with the Cauchy Strain Tensor What is the Right Cauchy Green Strain Tensor for a uniform strain field:

25. Practice with the Cauchy Strain Tensor What is the Right Cauchy Green Strain Tensor for a uniform strain field:

26. Finite Time Lyapunov Exponent What is the Right Cauchy Green Strain Tensor for a uniform strain field:

27. Stretching Field Re=45, p=1,  t=3  Stretching is organized in sharp lines.  Stretching Field labels the unstable manifold.  Structure in the stretching field are sometimes called Lagrangian Coherent Structures

28. Unstable manifold and the dye concentration field

29. Brandeis University, 2002 Unstable manifold and the dye concentration field

30. Brandeis University, 2002 Animation of manifold and dye field  Lines of large past stretching (unstable manifold) are aligned with the contours of the concentration field.  This is true at every time (phase).

31. Brandeis University, 2002 Fixed points and stretching Fixed points dominate the stretching field because particles remain near them for a long time and so are stretched in a single direction. So points near the unstable manifold have large past stretching, and points near the stable manifold have large future stretching.

32. Definition of Stretching Stretching = lim (L/L 0 ) L0L0 L L 0 0 Past Stretching Field: Stretching that a fluid element has experienced during the last  t. Future Stretching Field: Stretching that a fluid element will experience in the next  t.

33. Future and Past Stretching Fields  Future Stretching Field (Blue) marks the stable manifold  Past Stretching Field (Red) marks the unstable manifold  This pattern is appropriately named a “heteroclinic tangle”.

34. Finding Hyperbolic Fixed Points

35. Following a lobe

36. At Larger Reynolds Number Re=100, p=5  Stretching fields continue to form sharp lines that mark the manifolds of the flow.  Contours of dye concentration field continue to be aligned by the stretching field.

37. Application to 2D Turbulent Flows Quasi-2D turbulence in a rotating tank Mathur et al, PRL 98:144502 (2007)

38. Monterey Bay Lekien Couliette and Shadden NY Times September 28, 2009

39. Gulf of Mexico (Deep Water Horizon Spill)

40. Summary so far: What determines the geometry of the scalar patterns observed in fluid mixing? The orientation of the striations in the patterns aligns with lines of large Lagrangian stretching. In 2D time periodic flows the lines of large stretching match the manifolds that have been the focus of a large amount of work in dynamical systems and chaos. The Lagrangian stretching can be extracted experimentally with careful optical particle tracking. But what controls the decay rate?

41. Contrast Decay Animation (p=2, Re=65, 110 periods)

42. Decay of the Dye Concentration Field The functional form can be adequately parameterized by an exponential plus constant. (p=5)

43. Measured Mixing Rates vs Re

44. Predicting Mixing Rates There is a theory that has been successful in predicting mixing rates in simulated flows: Antonsen et al. (Phys. Fluids 8, 3094, 1996) Takes as input the distribution of Finite Time Lyapunov Exponents of the flow, P(h,t). Calculates the rate at which scalar variance is transferred to smaller scales by stretching: Since we have measured the Lyapunov exponents in our flow, we can directly calculate the predicted mixing rate … But it is larger than the observed mixing rate by a factor of 10. Why? The problem is that transport down scale by stretching is not the rate limiting step in our flow.

45. Evolution of the Horizontal Concentration Profile Dye pattern approaches a sinusoidal horizontal profile… which is the solution of the diffusion equation in a closed domain. A simple effective diffusion process might be a better model for the mixing rate. t=0, dotted line t = 6 periods, solid line t=36 periods, bold line

46. Measuring the Effective Diffusivity p=5, Re=100 p=2, Re=100 Then use to find the decay rate of the slowest decaying mode:

47. Comparison of experiment with predictions from effective diffusivity So the mixing rate is determined by effective diffusion, which is a measure of system scale transport, not by stretching which controls the small scale structure of the scalar field. There is an important lesson here: Physicists like the small scales of turbulence. They sometimes shows elegant universality. But often, the quantities that matter are controlled by the large scales.

48. Source of the Persistent Patterns The persistent patterns in this system were observed to be But two very different processes are both contributing to : Small Scale: Stretching leads to alignment of the contours of concentration with the unstable manifold. Large Scale: Effective diffusion leads to a sinusoidal pattern with one half wavelength across the system. Both processes individually create persistent patterns. The large scale pattern decays with time. Rothstein et al, Nature, 401:770 (1999)

49. Surprises in the Mixing Rates (p=5, Re=115) No dramatic change in mixing rate when flow bifurcates to period 2.

50. Surprises in the Mixing Rates (p=5, Re=170) Or when it becomes turbulent (loses time periodicity).

51. Brandeis University, 2002 Summary Traditional analysis of the spatial structure of passive scalar fields has produced a detailed phenomenology of turbulent mixing, but a Lagrangian analysis allows new and more direct insights. Lagrangian analysis of chaotic mixing The dynamics of the spatial patterns in fluid mixing can be understood as a reflection of the invariant manifolds of the flow Invariant manifolds can be extracted experimentally from the stretching fields in the flow.

52. Brandeis University, 2002 End

53. At higher Reynolds Number Re=100, p=5  Stretching fields continue to form sharp lines that mark the manifolds of the flow.

54. Brandeis University, 2002 Control Parameters  Reynolds Number: Ratio of Inertia of the fluid to viscous drag  Path Length: Typical distance traveled by the fluid during one period, divided by the magnet spacing

55. Brandeis University, 2002 Poincaré Map at different phases of the periodic flow

56. Brandeis University, 2002 Probability Distribution of Stretching / Probability Density Stretching over one periodLog(stretching) (Finite Time Lyapunov Exponents) (Re=100,p=5) Solid Line: Re=45, p=1, =1.9 periods -1 Dotted Line: Re=100, p=5, =6.4 periods -1

57. Brandeis University, 2002 Mixing Rate vs. Path Length (Re=80)