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 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
2
D
u
Dt t
2
1
Du u
u u P u
Dt t
Advection Diffusion:
Navier

Stokes :
0
u
Incompressibility:
Equations for Passive Scalar Transport
2
D
u
Dt t
2
1
Du u
u u P u
Dt t
Advection Diffusion:
Navier

Stokes :
0
u
Incompressibility:
New Dimensionless Parameter:
Peclet Number
uL
Pe
Equations for Passive Scalar Transport
2
D
u
Dt t
2
1
Du u
u u P u
Dt t
Advection Diffusion:
Navier

Stokes :
0
u
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 ij
s s
1
2
j
i
ij
j i
du
du
s
dx dx
3
2
u u
C C u
L L
Similarly, the scalar dissipation rate is independent of the
diffusivity (when the diffusivity is reasonably small) even though
the viscosity enters its definition:
2
i i
x x
2
u
C
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:
/3
n
n
r
r
Actually:
n
n
r
r
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)
Scalar Anisotropy
Measurements in a wind tunnel with a mean scalar gradient up
to R
l
= 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
3/2
2
( )
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 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
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
l
~ 800 fluorescent particles
tracked simultaneously.
l
Positions are found with
40
m
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.
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
Particle Displacement Map
Structures in the Poincaré Map
Hyperbolic Fixed
Points
Elliptic Fixed
Points
Manifolds of Hyperbolic Fixed Points
Unstable
Manifold
Stable
Manifold
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
Generalized
Momentum,
p
Generalized Position, q
dx
dt y
,
dy
dt x
Stream Function Equations:
Hamilton’s Equations:
dq H
dt p
,
dp H
dt 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
L
0
L
Stretching = lim (L/L
0
)
L
0
0
Right Cauchy Green Strain Tensor
k k
ij
i j
C
x x
max eigenvalue( )
Stretching =
ij
C
Practice with the Cauchy Strain Tensor
Right Cauchy Green Strain Tensor
k k
ij
i j
C
x x
What is the Right Cauchy Green Strain Tensor for a
uniform strain field:
0 0
ˆ ˆ
kt kt
x y
ky x e x y e y
u kx u
Practice with the Cauchy Strain Tensor
k k
ij
i j
C
x x
What is the Right Cauchy Green Strain Tensor
for a uniform strain field:
0 0
ˆ ˆ
( ) ( )
k t k t
x y
ky x t e x y t e y
u kx u
0
0
k t
i
k t
j
e
e
x
2
2
0
0
k t
k k
k t
i j
e
e
x x
max eigenvalue( )=
Stretching =
k t
ij
e
C
Finite Time Lyapunov Exponent
k k
ij
i j
C
x x
What is the Right Cauchy Green Strain Tensor
for a uniform strain field:
=
Stretching
k t
e
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
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).
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
.
Definition of Stretching
Stretching = lim (L/L
0
)
L
0
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.
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 Flows
Quasi

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
Log of Standard Deviation of Dye Intensity
50
40
30
20
10
0
Time (periods)
Re=25
Re=55
Re=85
Re=100
Re=115
Re=145
Re=170
The functional form can be adequately parameterized
by an exponential plus constant.
(p=5)
Measured Mixing Rates vs Re
0.20
0.15
0.10
0.05
0.00
Mixing Rate (periods
1
)
200
150
100
50
0
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 line
t = 6 periods, solid line
t=36 periods, bold line
Measuring the Effective Diffusivity
p=5, Re=100
p=2, Re=100
2
2
eff
x t
Then use to find the
decay rate of the slowest
decaying mode:
eff
2
2
Mixing 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
Mixing Rate (periods
1
)
200
150
100
50
0
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
Mixing Rate (periods
1
)
200
150
100
50
0
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
LV
Re
Magnet spacing
Velocity scale
Kinematic Viscosity
L
V
forcing freq.
f
V
fL
p
Brandeis University, 2002
Poincaré Map at
different phases of the periodic flow
Brandeis University, 2002
Probability Distribution of Stretching
l
/ <
l
>
Probability Density
Stretching over one period
Log(stretching)
(Finite Time Lyapunov Exponents)
(Re=100,p=5)
Solid Line: Re=45, p=1, <
l
>=1.9 periods

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

1
Brandeis University, 2002
Mixing Rate vs. Path Length (Re=80)
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