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Nonlinear Dynamics of Fluid Motion
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
“Nonlinear Dynamics” has deepened our understanding of fluid dynamics, and taught us many lessons about nonlinear phenomena.
No movies in this version; some may be found at www.haverford.edu/physics-astro/Gollub/lab.html
Thanks to Collaborators
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Postdocs and graduate students: G. Voth, JC Tsai, W. Losert, A. Kudrolli, D. Vallette, B. Gluckman, J. Liu, O. Mesquita, T. Solomon, F. Simonelli, A. Dougherty, M. Heutmaker, M. Lowe, S. Ciliberto.Undergraduate collaborators: E. Henry, B. Pier, Antony Bak, David G.W. Cooper, David Rothstein, David Schalk, Mark Buckley, and Ben Bigger and many previous students.
NSF - DMR-0072203 (Haverford) and DMR-0072203 (Penn)
Central Ideas from Nonlinear Dynamics
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Using phase space to describe fluid systems: mode amplitudes and their evolution.Geometrical thinking: attracting and repelling fixed points; multiple attractors; limit cycle attractors; chaotic attractors representing nonperiodic flows.Understanding instabilities as bifurcations or qualitative changes in the phase space of a system, e.g. birth or death of fixed points.
Central ideas - II
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Common features shared by disparate systems:Supercritical vs. subcritical transitionsBifurcation scenarios The central role of symmetryPatterns and spatiotemporal chaos described by generic amplitude equations in large systems
A Few Examples
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
The role of stretching in fluid mixingClustering of particles suspended in a liquid.Nonlinear waves on fluid interfacesMultiple stability in granular flow
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
1. Stretching and Fluid Mixing
Measuring fluid stretching can illuminate mixing and illustrate nonlinear dynamics.
Greg Voth, George Haller, Greg Dobler, Tim Saint.
Types of Fluid Mixing
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Turbulent mixing: Random structures produced by fluid instability at high Reynolds number Re stretch and fold fluid elements. Chaotic mixing: Some laminar flows at modest Re can produce complex distributions of material. Fundamental process: Non-reversible stretching and folding of fluid elements, with diffusion at small scales. Chaos in real space.
Background on Stretching in Chaotic Mixing
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Chaotic mixing has been well studied, and the importance of stretching was emphasized and investigated by J. Ottino and colleagues Khakhar, Swanson, and Muzzio.Others (Rom-Kedar, Leonard, Wiggins) have illuminated the connection between nonlinear dynamics and mixing theoretically.What is new in our work is the experimental measurement of space and time resolved stretching fields.
Apparatus: 2D Magnetically Driven Fluid Layer
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Magnet Array
Electrodes
ft)sin(2 I(t) 0 πI=
Glycerol and water a few mm thick, containing fluorescent dye.
Periodic forcing:Top View:
Precise Particle Tracking
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
~ 800 fluorescent particles tracked simultaneously.
Velocity Fields
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
0.9 cm/sec
0
(p=5, Re=56)
Why Do Periodic Flows Mix? Breaking Time Reversal Symmetry
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
To mix, the flow can be time periodic, but must not be time reversible:
relative to any starting time.Finite Reynolds numberRe=VL/ν can break time reversal symmetry:
( ) ( )t t≠ − −V V
Velocity at equal intervals before and after the moment of minimum velocity.
Breaking of Time Reversal Symmetry
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
10
8
6
4
2
0
< ((
v(t)
+ v(
-t))2
>1/2
(mm
/s)
200150100500
Reynolds Number
Vx
Vy
Structures in the Poincaré Map of the Flow
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Displacements in one period, color coded.Hyperbolic Fixed Points
A small part (6%) of the flow.
Elliptic Fixed Points
0 cm 1.7 cm
Manifolds of Hyperbolic Fixed Points
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
UnstableManifold
StableManifold
Typical of Hamiltonian Chaos
Definition of Stretching
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Stretching = lim (L/L0)
L0
LL0 0
Past Stretching Field: Stretching that a fluid element has experienced during the last ∆t. (Large near unstable man.)
Future Stretching Field: Stretching that a fluid element will experience in the next ∆t. (Large near stable man.)
Past Stretching Field
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Stretching is organized in sharp lines.
Re=45, p=1, ∆t=3
Future and Past Stretching Fields
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Future Stretching Field (Blue) marks the stable manifold.
Past Stretching Field (Red) marks the unstable manifold.
Circles mark hyperbolic points. A “heteroclinic tangle”.
Understanding the Dye Concentration Field
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Unstable manifold (past stretching field) and the dye concentration field
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Lines of large past stretching (unstable manifold) are aligned with the contours of the concentration field.
This is true at every time (phase).
Stretching is Inhomogeneous: PDF
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
λ / < λ >Pr
obab
ility
Stretching over one periodLog(stretching)(Finite Time Lyap. Exp.)
Solid: Re=45, p=1, <λ>=1.9 per-1
Dotted: Re=100, p=5, <λ>=6.4 per-1(Re=100,p=5)
Decay of the Dye Concentration Field
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
-1.5
-1.0
-0.5
0.0
Log
of S
tand
ard
Dev
iatio
n of
Dye
Inte
nsity
50403020100Time (periods)
Re=25 Re=55 Re=85 Re=100 Re=115 Re=145 Re=170
Log
Contrast
Can Mixing Rates Be Predicted?
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Antonsen et al. predict the long time decay of concentration variance based on the prob. dist. P(h,t) of finite time Lyapunov exponents (stretching)Using our measured P(h,t) and their theory, we predict mixing (variance decay) rates a factor of 10 larger than is observed.
Predicting Mixing Rates - (cont.)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Why is mixing so slow? Large system Transport over long distances is important. Enhanced or “eddy” diffusion: By watching particles diffuse, we can succesfully predict the rate of decay of a scalar field.
Future
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
The onset of non-periodic flow (2D turbulence) does not dramatically change mixing rates, to our surprise. Are the main features of non-periodic mixing captured by the periodic case discussed here? What will be the topological properties of stretching fields for non-periodic flows?Papers: G.A. Voth et al, Phys. Rev. Lett. (2002) and Phys. Fluids (2003)
2. Structure and Dynamics of HydrodynamicallyMediated Particle Clusters
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Fluid mediated interactions between particles lead to forces between them.
A variety of tunable patterns are produced, including chaotic states.
Greg Voth, Charles Thomas, Ben Bigger, Mark Buckley, Michael Brenner, Howard Stone, and JPG.
Observations of Clustering
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
f=50 HzΓ=4.5
Related Phenomena
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Sedimentation – when particles fall in a fluid, they influence each other.Fluidized beds, widely used for catalysis.
Here: An example of physical self-assembly of ordered particulate structuresChaos in a system of interacting particles
Rayleigh Streaming
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Streaming flow for a particle in a fluid - no boundary From Van Dyke, An Album of Fluid Motion
Origin of Long Range Attraction
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Rayleigh: steady streaming flow away from poles and toward the equator generated outside the viscous boundary layer (0.2 mm) surrounding the particle. Predicted inflow velocity
(via collaborators M. Brenner and H. Stone).Particles follow this flow as they approach each other.However, see Voth and Otto, KC004, Tuesday 08:39
2 3( ) 0.53 /V r Aa rων= −
Approach Curves for 2 Particles
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
F=20 Hz; Γ=2.9; A=0.4 mm (periodic)
F=50 Hz; Γ=4.6; A=0.15 mm (chaotic)
•Dashed: theory; Solid:experiments;
t = 0 at end of approach.
•At higher excitation amplitude, the particles do not touch. Repulsive force also.
-2 s 0.0 stime
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
y po
sitio
n (m
m)
x position (mm)
Particle TwoParticle One
Particle Three
Particle Four
N=4, Γ=2.97.(a) trajectories; (b) spacings vs.
time. Note transitions.
0 200 400 600 800 1000 12000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
pair
sepa
ratio
ns (m
m)
time (s)
Particles 2 & 1 Particles 3 & 1 Particles 4 & 2 Particles 4 & 3
Trapezoidal Cluster - Chaotic Motion
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
(a) Trajectories (Γ=2.96) (b) separations vs. time. Nonperiodicoscillations.
0 200 400 600 800 1000 12000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Particles 2 & 1 Particles 3 & 1 Particles 3 & 2 Particles 4 & 1 Particles 4 & 3 Particles 5 & 2 Particles 5 & 3 Particles 5 & 4
pair
sepa
ratio
n (m
m)
time (s)
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Particle Five
Particle Four
Particle Three
Particle Two
y po
sitio
n (m
m)
x position (mm)
Particle One
Fluctuating Five Particle Cluster
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
Γ=3.44
x position (mm)
y po
sitio
n (m
m)
x position (mm)
y po
sitio
n (m
m)
Γ=5.53
Large cluster trajectories, for several accelerations.
Large Cluster Trajectories
Main Conclusions - Interacting Particles
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Many different structures, ordered, disordered, chaoticObservations suggest an effective interaction potential for two particles. But pairwiseinteractions do not suffice for many particles.Large and small clusters behave differently. More: Voth et al. PRL 88, 234301 (2002); C.C. Thomas & JPG, in preparation.
Complex wave patterns
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003Kudrolli, Pier, Edwards, JPG
Quasicrystalline wave pattern
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Sheared Granular Flow
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Shearing induces order; order modifies
Conclusions
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Ideas from nonlinear dynamics contribute to understanding fluid phenomena.Equally, fluids illuminate nonlinear dynamics and can be used to teach it.Fluids ought to play a larger role in physics teaching. I make a case for this in Physics Today, December 2003: “One of the oddities of contemporary physics education is the nearly complete absence of continuum mechanics…”Movies: www.haverford.edu/physics-astro/Gollub/lab.html
END
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003