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Fluid Mixing. Greg Voth Wesleyan University. Voth et al. Phys Rev Lett 88:254501 (2002). Chen & Kraichnan Phys. Fluids 10:2867 (1998). Why study fluid mixing?. Nigel listed three fundamental processes that engineers need to optimize that depend on turbulence: Turbulent Combustion

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### Persistent Patterns statistics

### Particle Displacement Map accurate time-resolved velocity fields

### Future and Past Stretching Fields accurate time-resolved velocity fields

### Finding Hyperbolic Fixed Points accurate time-resolved velocity fields

Fluid Mixing

Greg Voth Wesleyan University

Voth et al. Phys Rev Lett 88:254501 (2002)

Chen & Kraichnan Phys. Fluids 10:2867 (1998)

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.

Equations for Passive Scalar Transport

Advection Diffusion:

Navier-Stokes :

Incompressibility:

New Dimensionless Parameter:

Peclet Number

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.

Scalar Dissipative Anomaly

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:

Doniz, Sreenivasan and Yeung JFM 532:199 (2005)

Kolmogorov-Obukhov-Corrsin scaling for passive scalar statistics

Scalar Spectrum in the inertial range:

(For high Re and Pe)

Scalar Structure Functions in the inertial range:

Actually:

Intermittency of thepassive scalar field is stronger than that of thevelocity field.

Warhaft Annu. Rev. Fluid Mech. 32:203 (2000)

Scalar Anisotropy statistics

Measurements in a wind tunnel with a mean scalar gradient up to Rl = 460 show the odd moments of the scalar derivative do not go to zero at small scales, indicating persistent anisotropy.

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.

Warhaft. Annu. Rev. Fluid Mech. 32:203–240 (2000)

Lagrangian Descriptions statistics

- 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

Outline of my talks this week statistics

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

Lagrangian descriptions of chaotic mixing statistics

Dense, conducting lower layer

(glycerol, water, and salt, 3 mm thick)

Less dense, non-conducting upper layer

(glycerol and water, 1 mm thick)

Electrodes

Magnet Array

Top View:

Periodic forcing:

Brandeis University, 2002

Evolution of dye concentration field

Same data updated once per period.

Observations statistics

- 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?

Raw Particle Tracking Data statistics

- ~ 800 fluorescent particles tracked simultaneously.
- Positions are found with 40mm accuracy.
- ~15,000 images: 40-80 images per period of forcing, and 240 periods.
- Phase Averaging: 800*240 = 105 particles tracked at each phase.
- The flow is time periodic and so exactly the same flow can be used in both dye imaging and particle tracking measurements.

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)

- 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

Structures in the Poincaré Map accurate time-resolved velocity fields

Hyperbolic Fixed

Points

Elliptic Fixed

Points

Manifolds of Hyperbolic Fixed Points accurate time-resolved velocity fields

Unstable

Manifold

Stable

Manifold

Hamiltonian Chaos accurate time-resolved velocity fields

- 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)

Why do these systems show similar structures? accurate time-resolved velocity fields

Fluid Mixing

Hamiltonian System

Real Space

Phase Space

GeneralizedMomentum,

p

y

x

Generalized Position, q

Stream Function Equations:

Hamilton’s Equations:

(Aref, J. Fluid Mech, 1984)

Can we extract manifolds in experiments? accurate time-resolved velocity fields

- 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

Practice with the Cauchy Strain Tensor accurate time-resolved velocity fields

- What is the Right Cauchy Green Strain Tensor for a uniform strain field:

Practice with the Cauchy Strain Tensor accurate time-resolved velocity fields

- What is the Right Cauchy Green Strain Tensor for a uniform strain field:

Finite Time Lyapunov Exponent accurate time-resolved velocity fields

- What is the Right Cauchy Green Strain Tensor for a uniform strain field:

Stretching Field accurate time-resolved velocity fields

- Stretching is organized in sharp lines.
- Stretching Field labels the unstable manifold.
- Structure in the stretching field are sometimes called Lagrangian Coherent Structures

Re=45, p=1, Dt=3

Unstable manifold and the accurate time-resolved velocity fieldsdye concentration field

Unstable manifold and the accurate time-resolved velocity fieldsdye concentration field

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Animation of manifold and dye field accurate time-resolved velocity fields

- Lines of large past stretching (unstable manifold) are aligned with the contours of the concentration field.
- This is true at every time (phase).

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Fixed points and stretching accurate time-resolved velocity fields

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.

Brandeis University, 2002

Definition of Stretching accurate time-resolved velocity fields

Stretching = lim (L/L0)

L

L0 0

Past Stretching Field: Stretching that a fluid element has experienced during the last Dt.

Future Stretching Field: Stretching that a fluid element will experience in the next Dt.

L0

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

Following a lobe accurate time-resolved velocity fields

At Larger Reynolds Number accurate time-resolved velocity fields

- 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.

Re=100, p=5

Application to 2D Turbulent Flows accurate time-resolved velocity fields

Quasi-2D turbulence in a rotating tank

Mathur et al, PRL 98:144502 (2007)

Monterey Bay accurate time-resolved velocity fields

Lekien Couliette and Shadden NY Times September 28, 2009

Gulf of Mexico (Deep Water Horizon Spill) accurate time-resolved velocity fields

Summary so far: accurate time-resolved velocity fields

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?

Contrast Decay Animation accurate time-resolved velocity fields(p=2, Re=65 , 110 periods)

Decay of the Dye Concentration Field accurate time-resolved velocity fields

(p=5)

The functional form can be adequately parameterized

by an exponential plus constant.

Measured Mixing Rates vs Re accurate time-resolved velocity fields

Predicting Mixing Rates accurate time-resolved velocity fields

- There is a theory that has been successful in predicting mixing rates in simulated flows:
Antonsen et al. (Phys. Fluids8, 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.

Evolution of the Horizontal accurate time-resolved velocity fields 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

Measuring the Effective Diffusivity accurate time-resolved velocity fields

Then use to find the decay rate of the slowest decaying mode:

p=5, Re=100

p=2, Re=100

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.

Source of the Persistent Patterns diffusivity

- 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)

Surprises in the Mixing Rates diffusivity

No dramatic change in mixing rate when flow bifurcates to period 2.

(p=5, Re=115)

Surprises in the Mixing Rates diffusivity

Or when it becomes turbulent (loses time periodicity).

(p=5, Re=170)

Summary diffusivity

- 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.

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End diffusivity

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At higher Reynolds Number diffusivity

- Stretching fields continue to form sharp lines that mark the manifolds of the flow.

Re=100, p=5

Control Parameters diffusivity

- 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

Brandeis University, 2002

Poincaré Map at diffusivitydifferent phases of the periodic flow

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Probability Distribution of Stretching diffusivity

Stretching over one period

Log(stretching)

(Finite Time Lyapunov Exponents)

Probability Density

l / < l >

Solid Line: Re=45, p=1, <l>=1.9 periods-1

Dotted Line: Re=100, p=5, <l>=6.4 periods-1

(Re=100,p=5)

Brandeis University, 2002

Mixing Rate vs. Path Length (Re=80) diffusivity

Brandeis University, 2002

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