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Transcript of Fluid Mixing Greg Voth Wesleyan University Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al....
![Page 1: Fluid Mixing Greg Voth Wesleyan University Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)](https://reader031.fdocuments.net/reader031/viewer/2022032205/56649e895503460f94b8e08c/html5/thumbnails/1.jpg)
Fluid Mixing
Greg Voth Wesleyan University
Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)
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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.
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Equations for Passive Scalar Transport
2Du
Dt t
21Du uu u P u
Dt t
Advection Diffusion:
Navier-Stokes :
0u
Incompressibility:
![Page 4: Fluid Mixing Greg Voth Wesleyan University Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)](https://reader031.fdocuments.net/reader031/viewer/2022032205/56649e895503460f94b8e08c/html5/thumbnails/4.jpg)
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
![Page 5: Fluid Mixing Greg Voth Wesleyan University Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)](https://reader031.fdocuments.net/reader031/viewer/2022032205/56649e895503460f94b8e08c/html5/thumbnails/5.jpg)
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.
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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
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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)
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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
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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
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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
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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:
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Evolution of dye concentration field Same data updated once per period.
Persistent Patterns
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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
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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.
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Velocity Fields: Phase averaging allows us to obtain highly accurate time-resolved velocity fields
0.9cm/sec
0cm/sec
(p=5, Re=56)
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•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
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Structures in the Poincaré Map
Hyperbolic Fixed Points
Elliptic Fixed Points
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Manifolds of Hyperbolic Fixed Points
UnstableManifold
StableManifold
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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)
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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)
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Manifold Structure and Chaos
Regular (Non-chaotic) Chaotic
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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
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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
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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
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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
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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
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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
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Unstable manifold and the dye concentration field
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Brandeis University, 2002
Unstable manifold and the dye concentration field
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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).
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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.
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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.
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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”.
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Finding Hyperbolic Fixed Points
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Following a lobe
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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.
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Application to 2D Turbulent FlowsQuasi-2D turbulence in a rotating tank
Mathur et al, PRL 98:144502 (2007)
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Monterey Bay
Lekien Couliette and Shadden NY Times September 28, 2009
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Gulf of Mexico (Deep Water Horizon Spill)
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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?
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Contrast Decay Animation (p=2, Re=65 , 110 periods)
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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)
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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
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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.
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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
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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
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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.
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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
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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
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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
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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.
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Brandeis University, 2002
End
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At higher Reynolds Number
Re=100, p=5
Stretching fields continue to form sharp lines that mark the manifolds of the flow.
![Page 54: Fluid Mixing Greg Voth Wesleyan University Chen & Kraichnan Phys. Fluids 10:2867 (1998)Voth et al. Phys Rev Lett 88:254501 (2002)](https://reader031.fdocuments.net/reader031/viewer/2022032205/56649e895503460f94b8e08c/html5/thumbnails/54.jpg)
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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Brandeis University, 2002
Poincaré Map at different phases of the periodic flow
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Brandeis University, 2002
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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Brandeis University, 2002
Mixing Rate vs. Path Length (Re=80)