Transcript The Buzz in Convection

QUIZ
Which of these convection patterns
is
non-Boussinesq?
Homological Characterization
Of
Convection Patterns
Kapilanjan Krishan
Marcio Gameiro
Michael Schatz
Konstantin Mischaikow
School of Physics
School of Mathematics
Georgia Institute of Technology
Supported by:
DOE, DARPA, NSF
Patterns and Drug Delivery
Caffeine in Polyurethane Matrix
D. Saylor et al., (U.S. Food and Drug Administration)
Patterns and Strength of Materials
Maximal Principal Stresses in Alumina
E. Fuller et al., (NIST)
Patterns and Convection
Light
Source
Camera
Reduced Rayleigh number
e=(T-Tc)/ Tc
Convection cell
e=0.125
Spiral Defect Chaos
Homology
Using algebra to determine topology
Representations
Simplicial
Cubical
Elementary Cubes and Chains
0-cube
1-cube
2-cube
 v̂
 ê
 fˆ
0-chain
1-chain
2-chain
Boundary Operator
e5
e1
e4
f

e2
e6
e8
e3
e7
fˆ  eˆ1  eˆ2  eˆ3  eˆ4
(eˆ1  eˆ2  eˆ3  eˆ4 )  0
(eˆ5  eˆ6  eˆ7  eˆ8 )  0
dimension
# of Loops enclosing holes =
of
homology group H1
Homology Summary
• Patterns are described by H i
• Dimension of H i = i , the ith Betti number
• Homology: Computable topology
Reduction to Binary Representation
Experiment image
Hot flow
Cold flow
Number of Components
Zeroth Betti number = 34
Hot flows vs. Cold flows
Hot flow
Cold flow
Spiral Defect Chaos
e~1
e~2
Number of distinct components
Hot flow vs. Cold flow
Time ~ 103 tn
e~1
Time ~ 103 tn
e~2
Number of holes
First Betti number = 13
Number of distinct holes
Hot flow vs. Cold flow
Time ~ 103 tn
e~1
Time ~ 103 tn
e~2
Betti numbers vs Epsilon
Hot flow and Cold flow
0
Betti
numbers
1
0
1
e
Asymmetry between hot and cold regions
Non-Boussinesq effects ?
Which of these convection patterns
is
non-Boussinesq?
Simulations (SF6 )
(Madruga and Riecke)
Boussinesq
Non-Boussinesq
e1.4
e1.4, Q=4.5
Boussinesq Simulations (SF6 )
Time Series
Components
0 0
Holes
1 1
Non-Boussinesq Simulations (SF6 )
Time Series
Components
0
Holes
1
0
1
Simulations (CO2 ) at Experimental Conditions
e2, Q=0.7
Components
Holes
0
1
0
1
Boundary Influence
Number of connected components
Hot flow vs. Cold flow
Time ~ 103 tn
e~1
Time ~ 103 tn
e~2
Percentage of connected components
Hot flow vs. Cold flow
Time ~ 103 tn
e~1
Time ~ 103 tn
e~2
Convergence to Attractor
Frequency
of
occurrence
cold0 : Number of cold
flow components ( e~1 )
Entropy
Joint Probability
P(hot0 ,cold0 , hot1 , cold1)
Entropy
(-S Pi log Pi)
Bifurcations?
e
Entropy vs epsilon
e1.00
e1.13
Entropy=8.3
Entropy=7.9
e1.25
Entropy=8.9
Space-Time Topology
Space
1-D Gray-Scott model
Time
Time Series—First Betti number
Exhbits Chaos
Summary
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Homology characterizes complex patterns
Underlying symmetries detected in data
Alternative measure of boundary effects
Detects transitions between complex states
Space-time topology may reveal new insights
Homology source codes available at:
http://www.math.gatech.edu/~chomp