fluid mixing greg voth wesleyan university chen & kraichnan phys. fluids 10:2867 (1998)voth et...

Fluid Mixing

Greg Voth Wesleyan University

Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)

Why study fluid mixing?

Nigel listed three fundamental processes that engineers need to optimize that depend on turbulence:Turbulent CombustionEnvironmental TransportDrag 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 energyEnvironmental Transport is obviously a mixing problem. Drag on transportation vehicles is even the turbulent transport of momentum.

Equations for Passive Scalar Transport

2Du

Dt t

21Du uu u P u

Dt t

Advection Diffusion:

Navier-Stokes :

0u

Incompressibility:

Equations for Passive Scalar Transport

2Du

Dt t

21Du uu u P u

Dt t

Advection Diffusion:

Navier-Stokes :

0u

Incompressibility:

New Dimensionless Parameter:

Peclet NumberuL

Pe

Equations for Passive Scalar Transport

2Du

Dt t

21Du uu u P u

Dt t

Advection Diffusion:

Navier-Stokes :

0u

Incompressibility:

For small diffusivity, the advection diffusion equation reduces to conservation of the scalar along Lagrangian trajectories.

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:

2 ij ijs s

1

2ji

ijj i

dudus

dx dx

32u u

C C uL L

Similarly, the scalar dissipation rate is independent of the diffusivity (when the diffusivity is reasonably small) even though the viscosity enters its definition:

2i ix x

2 uC

L

Kolmogorov-Obukhov-Corrsin scaling for passive scalar statistics

1/3 5/3( )F k C k Scalar Spectrum in the inertial range:

Scalar Structure Functions in the inertial range:

/3n nr r

Actually:

nn

r r

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

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

(For high Re and Pe)

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.

3

3/22( )

y

y

S y

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

Outline of my talks this week

Rest of this talk:Lagrangian desciptions of chaotic mixing

Patterns in fluid mixingStretching fields and the Cauchy strain tensorsWhat controls mixing rates

Thursday morning and afternoon:Lagrangian descriptions of turbulent flows

Lagrangian Kolmogorov Theory Tools for measuring particle trajectoriesMotion of non-tracer particles in turbulence

Brandeis University, 2002

Lagrangian descriptions of chaotic mixing

Magnet Array

Dense, conducting lower layer(glycerol, water, and salt, 3 mm thick)

Electrodes

ft)sin(2 I(t) 0 I

Less dense, non-conducting upper layer(glycerol and water, 1 mm thick)

Top View: Periodic forcing:

Evolution of dye concentration field Same data updated once per period.

Persistent Patterns

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

Raw Particle Tracking Data

~ 800 fluorescent particles tracked simultaneously.

Positions are found with 40 m 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

0.9cm/sec

0cm/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

Particle Displacement Map

Structures in the Poincaré Map

Hyperbolic Fixed Points

Elliptic Fixed Points

Manifolds of Hyperbolic Fixed Points

UnstableManifold

StableManifold

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)

Why do these systems show similar structures?

Fluid Mixing Hamiltonian System

y

x

Real Space Phase Space

Gen

eral

ized

Mom

entu

m,

p

Generalized Position, q

dxdt y

,dydt x

Stream Function Equations: Hamilton’s Equations:

dq Hdt p

,dp Hdt q

(Aref, J. Fluid Mech, 1984)

Manifold Structure and Chaos

Regular (Non-chaotic) Chaotic

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

Calculating Stretching

L0 L

Stretching = lim (L/L0)L0 0

Right Cauchy Green Strain Tensor k kij

i j

Cx x

max eigenvalue( )Stretching = ijC

Practice with the Cauchy Strain Tensor

Right Cauchy Green Strain Tensor k kij

i j

Cx x

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

0 0ˆ ˆkt ktx y ky x e x y e yu kx u

Practice with the Cauchy Strain Tensor

k kij

i j

Cx x

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

0 0ˆ ˆ( ) ( )k t k tx y ky x t e x y t e yu kx u

0

0

k ti

k t

j

e

ex

2

2

0

0

k tk k

k t

i j

e

ex x

max eigenvalue( )=Stretching = k tij eC

Finite Time Lyapunov Exponent

k kij

i j

Cx x

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

= Stretching k te

1= log(stretching)

=

Finite Time Lyapunov Exponent

Finite Time Lyapunov Exponent t

k

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

Unstable manifold and the dye concentration field

Brandeis University, 2002

Unstable manifold and the dye concentration field

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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).

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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.

Definition of Stretching

Stretching = lim (L/L0)

L0

LL0 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.

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”.

Finding Hyperbolic Fixed Points

Following a lobe

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.

Application to 2D Turbulent FlowsQuasi-2D turbulence in a rotating tank

Mathur et al, PRL 98:144502 (2007)

Monterey Bay

Lekien Couliette and Shadden NY Times September 28, 2009

Gulf of Mexico (Deep Water Horizon Spill)

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?

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

Decay of the Dye Concentration Field

-1.5

-1.0

-0.5

0.0

Lo

g o

f S

tan

dar

d D

evia

tio

n o

f D

ye In

ten

sity

50403020100Time (periods)

Re=25 Re=55 Re=85 Re=100 Re=115 Re=145 Re=170

The functional form can be adequately parameterizedby an exponential plus constant.

(p=5)

Measured Mixing Rates vs Re

0.20

0.15

0.10

0.05

0.00

Mix

ing

Ra

te (

pe

rio

ds-1

)

200150100500

Reynolds Number

p=2 p=5 p=8

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.

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 linet = 6 periods, solid linet=36 periods, bold line

Measuring the Effective Diffusivity

p=5, Re=100p=2, Re=100

2 2 effx t

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

eff

2

2Mixing Rate eff L

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

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.

( , ) ( ) ( )I r t f r g t

Rothstein et al, Nature, 401:770 (1999)

( )f r

Surprises in the Mixing Rates

(p=5, Re=115)

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

0.20

0.15

0.10

0.05

0.00

Mix

ing

Ra

te (

pe

rio

ds-1

)

200150100500

Reynolds Number

p=2 p=5 p=8

Surprises in the Mixing Rates

(p=5, Re=170)

Or when it becomes turbulent (loses time periodicity).

0.20

0.15

0.10

0.05

0.00

Mix

ing

Ra

te (

pe

rio

ds-1

)

200150100500

Reynolds Number

p=2 p=5 p=8

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.

Brandeis University, 2002

End

At higher Reynolds Number

Re=100, p=5

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

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

LVRe

Magnet spacing

Velocity scaleKinematic Viscosity

LV

forcing freq.f VfL

p

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Poincaré Map at different phases of the periodic flow

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Probability Distribution of Stretching

/ < >Pr

obab

ilit

y D

ensi

ty

Stretching over one period Log(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

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Mixing Rate vs. Path Length (Re=80)