FRACTAL PATTERNS IN CHAOTIC FLUID MIXINGa_a_a/Public/Old_Projects/ddays2006... · 2010. 12. 13. ·...
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FRACTAL PATTERNS IN CHAOTIC FLUID MIXING Amir Ali Ahmadi, Jennifer Rieser, Yue-Kin Tsang Edward Ott, and Thomas M. Antonsen INTRODUCTION We study passive scalar advection-diffusion in chaotic fluid flows. An every day example: mixing cream in coffee. BACKGROUND φ κ φ φ 2 / ∇ = ∇ ⋅ + ∂ ∂ v t Other examples: the ozone in the atmosphere, pollutants in the ocean, plankton in the upper ocean, and the mixing of paints. The density of the passive scalar , , obeys: ( ,) xt φ Where : molecular diffusion coefficient κ (,) vxt : fluid velocity field We simulate a particular incompressible chaotic fluid flow given by the velocity field: Integrating the velocity field over one time period T, we obtain an iterative map expressing the position of the fluid elements (x n , y n ) at time n = t/T : Where n and n : random numbers between [0, 2 ] L f : spatial period of the flow T : time period of the flow for 1 2 0 otherwise 0 ( / ) cos 2 2 [( / ) ] cos[( / ) ] (,) f n f n nT t n T U U x yL y xL vxt π α π β ≤ < + = + + L d = M L f : spatial period of the passive scalar M : positive integer 1 { ( / 2)cos[(2 / ) ]} mod n n n f n d x x UT y L L π α + = + + 1 1 { ( / 2)cos[(2 / ) ]} mod n n n f n d y y UT x L L π β + + = + + NUMERICAL SIMULATIONS Passive scalar initial conditions We numerically determine the values of on a 2-D grid with periodic boundary conditions (periodicity= M Lf ) as a function of time n. Passive scalar at n=1 with M=1 Passive scalar at n=5 with M=1 Passive scalar at n=125 with M=1 Passive scalar at n=1 with M=4 Passive scalar at n=5 with M=4 Passive scalar at n=125 with M=4 n ln variance( ) The decay rate of the variance of As time increases, chaotic mixing causes to develop variations on a finer and finer scale. Therefore, we consider the possibility that the density concentrates on a fractal set. As M increases, it takes longer for to develop fine scale structure. It has previously been shown in several papers that the slower decay rate at larger M is controlled by processes at the largest possible length scales allowed by the system size (M L f ). Basically, variance slowly leaks from the largest length scales to shorter scales. We are interested in the case of a small decay rate. While the decay rate is determined by processes at long wavelength, fractal properties are short wavelength phenomena. Using large M is not feasible for the numerical simulations that we perform. Thus we desire to use M=1 while simulating a situation with large M (i.e. slow damping) by incorporating an exponentially decaying source at the largest length scale: 2 2 sin / ( / ) f t t v e x L γ φ φ κ φ π − ∂ ∂+ ⋅∇ = ∇ + Where is the decay rate of the source γ By incorporating a source with exponential decay rate =0.044, we achieve a slow damping rate that mimics a simulation with M=4. γ n ln variance( ) The variance decay rate with =0.044 γ MAIN CONTRIBUTIONS OF OUR WORK We obtain high resolution numerical results for the fractal properties of the decaying scalar in the long time limit. We obtain theoretical predictions for the fractal properties. We compare the numerical and theoretical results. Our study of the fractal structure of the passive scalar is in terms of its “Spectrum of Fractal Dimension, D q ”. D q is a means of quantitatively characterizing the observed fine scale pattern behavior. To calculate D q , we cover the space with a grid of boxes of size and determine the “measure”, i , of each box: NUMERICAL APPROACH TO CALCULATING D q | | | | d d Box i i L L dxdy dxdy λ λ μ φ φ × = 1 2 where or λ = D q is then defined by: Diffusive cutoff << << L f scale ( 1) q q D q i i μ ε − ∝ ) / ( log 2 d L ε Slope=D q 044 . 0 1, = = γ λ Since D q varies with q, by definition the decaying scalar has multifractal structure. For a nonfractal smooth density D q would be 2 independent of q. 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.85 1.875 1.9 1.925 1.95 1.975 2 0 0.25 0.5 0.75 1 1.25 1.5 1.7 1.75 1.8 1.85 1.9 1.95 2 q D q D q q 044 . 0 1 , = = γ λ 044 . 0 2, = = γ λ Finite resolution limits the accuracy of D q : With a constant source ( = 0) D q approaches 2.0 as the resolution is increased. γ 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.85 1.88 1.91 1.94 1.97 2 1024 2 2048 2 4096 2 8192 2 16384 2 D q q 0 , 1 = = γ λ 0 0.25 0.5 0.75 1 1.25 1.5 1.85 1.875 1.9 1.925 1.95 1.975 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2 At the 16384x16384 experiment, we correct for the finite resolution by adding the gap between 2.0 and D q for = 0 to the dimension for = 0.044 γ γ Numerical results after correction for the finite resolution q D q D q q 044 . 0 1 , = = γ λ 044 . 0 2, = = γ λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 n=10 n=20 n=40 n=80 h P(h|n) We compute the Lyapunov exponent (h) of the trajectories of 16384 initial conditions uniformly distributed over the region of the flow. At finite time, the dependence of h on initial conditions results in a conditional density function P(h| n): THEORETICAL APPROACH TO CALCULATING D q A mean value of = 0.30 ensures that the flow is chaotic as the trajectories of nearby fluid elements diverge exponentially in time. h The time-independent function G(h) is concave up and has the minimum value zero, occurring at h= . h For large enough n: ) ( ) | ( h nG e n h P − ≈ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 n=20 n=40 n=80 h G(h) () (1) 2 1 q Fq qF D q − = − − where 0 () () max h q Gh Fq h γ ≥ − = A theoretical analysis (not given here) yields a result for D q (with = 1) in terms of G(h): λ A similar result is obtained for D q with = 2 λ 0 0.25 0.5 0.75 1 1.25 1.5 1.8 1.85 1.9 1.95 2 Theory with = 0 Theory with = 0.044 q D q 2 = λ 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.9 1.92 1.94 1.96 1.98 2 Theory with = 0 Theory with = 0.044 D q q 1 = λ γ γ γ γ COMPARISON OF NUMERICAL AND THEORETICAL RESULTS 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1.9 1.915 1.93 1.945 1.96 1.975 1.99 2 Numerical Theoretical 0 0.25 0.5 0.75 1 1.25 1.5 1.8 1.825 1.85 1.875 1.9 1.925 1.95 1.975 2 Numerical Theoretical q D q D q q 044 . 0 1 , = = γ λ 044 . 0 2, = = γ λ CONCLUSIONS The decaying passive scalar has fractal properties and is characterized by D q . We have developed a theory which relates D q to the distribution of finite-time Lyapunov exponents of the chaotic flow. We provide high resolution numerical evidence for our theoretical predictions.
Transcript of FRACTAL PATTERNS IN CHAOTIC FLUID MIXINGa_a_a/Public/Old_Projects/ddays2006... · 2010. 12. 13. ·...
FRACTAL PATTERNS IN CHAOTIC FLUID MIXINGAmir Ali Ahmadi, Jennifer Rieser, Yue-Kin Tsang
Edward Ott, and Thomas M. Antonsen
INTRODUCTIONWe study passive scalar advection-diffusion in chaotic fluid flows.
An every day example: mixing cream in coffee.
BACKGROUND
φκφφ 2/ ∇=∇⋅+∂∂ vt
Other examples: the ozone in the atmosphere, pollutants in the ocean, plankton in the upper ocean, and the mixing of paints.
The density of the passive scalar , , obeys:( , )x tφ
Where : molecular diffusion coefficientκ( , )v x t : fluid velocity field
We simulate a particular incompressible chaotic fluid flow given by the velocity field:
Integrating the velocity field over one time period T, we obtain an iterative map expressing the position of the fluid elements (xn, yn) at time n = t/T :
Where n and n : random numbers between [0, 2 ]Lf : spatial period of the flow T : time period of the flow
for 1 20
otherwise0
( / )cos 22
[( / ) ]cos[( / ) ]
( , ) f n
f n
nT t n TU
U
x y Ly x L
v x t π απ β
≤ < +=
++
Ld = M Lf : spatial period of the passive scalarM : positive integer
1 { ( / 2)cos[(2 / ) ]} modn n n f n dx x UT y L Lπ α+ = + +
1 1{ ( / 2)cos[(2 / ) ]} modn n n f n dy y UT x L Lπ β+ += + +
NUMERICAL SIMULATIONS
Passive scalar initial conditions
We numerically determine the values of on a 2-D grid with periodic boundary conditions (periodicity= M Lf ) as a function of time n.
Passive scalar at n=1 with M=1
Passive scalar at n=5 with M=1
Passive scalar at n=125 with M=1
Passive scalar at n=1 with M=4
Passive scalar at n=5 with M=4
Passive scalar at n=125 with M=4
n
ln v
aria
nce(
)
The decay rate of the variance of
As time increases, chaotic mixing causes to develop variations on a finer and finer scale.
Therefore, we consider the possibility that the density concentrates on a fractal set.
As M increases, it takes longer for to develop fine scale structure.
It has previously been shown in several papers that the slower decay rate at larger M is controlled by processes at the largest possible length scales allowed by the system size (M Lf ).
Basically, variance slowly leaks from the largest length scales to shorter scales.
We are interested in the case of a small decay rate.
While the decay rate is determined by processes at long wavelength, fractal properties are short wavelength phenomena.
Using large M is not feasible for the numerical simulations that we perform.
Thus we desire to use M=1 while simulating a situation with large M (i.e. slow damping) by incorporating an exponentially decaying source at the largest length scale:
2 2sin/ ( / )ftt v e x Lγφ φ κ φ π−∂ ∂ + ⋅ ∇ = ∇ +
Where
is the decay rateof the source
γ
By incorporating a source with exponential decay rate=0.044, we achieve a slow damping rate that mimics a
simulation with M=4.γ
n
ln v
aria
nce(
)
The variance decay rate with =0.044γ
MAIN CONTRIBUTIONS OF OUR WORK
We obtain high resolution numerical results for the fractal properties of the decaying scalar in the long time limit.
We obtain theoretical predictions for the fractal properties.
We compare the numerical and theoretical results.
Our study of the fractal structure of the passive scalar is in terms of its “Spectrum of Fractal Dimension, Dq”.
Dq is a means of quantitatively characterizing the observed fine scale pattern behavior.
To calculate Dq , we cover the space with a grid of boxes of size and determine the “measure”, i , of each box:
NUMERICAL APPROACH TO CALCULATING Dq
| |
| |d d
Box ii
L L
dxdy
dxdy
λ
λμ
φ
φ×
=
1 2 where orλ =
Dq is then defined by:
Diffusivecutoff << << Lfscale
( 1) qq Dqi
i
μ ε −∝
)/(log2 dLε
Slope=Dq
044.01, == γλ
Since Dq varies with q, by definition the decaying scalar has multifractal structure.
For a nonfractal smooth density Dq would be 2 independent of q.
0 0.25 0.5 0.75 1 1.25 1.5 1.75 21.85
1.875
1.9
1.925
1.95
1.975
2
0 0.25 0.5 0.75 1 1.25 1.51.7
1.75
1.8
1.85
1.9
1.95
2
q
DqDq
q
044.01, == γλ 044.02, == γλ
Finite resolution limits the accuracy of Dq :
With a constant source ( = 0) Dq approaches 2.0 as the resolution is increased.
γ
0 0.25 0.5 0.75 1 1.25 1.5 1.75 21.85
1.88
1.91
1.94
1.97
2
10242
20482
40962
81922
163842
Dq
q
0,1 == γλ
0 0.25 0.5 0.75 1 1.25 1.51.85
1.875
1.9
1.925
1.95
1.975
2
0 0.25 0.5 0.75 1 1.25 1.5 1.75 21.93
1.94
1.95
1.96
1.97
1.98
1.99
2
At the 16384x16384 experiment, we correct for the finite resolution by adding the gap between 2.0 and Dqfor = 0 to the dimension for = 0.044γ γ
Numerical results after correction for the finite resolution
q
DqDq
q
044.01, == γλ 044.02, == γλ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8n=10n=20n=40n=80
h
P(h|n)
We compute the Lyapunov exponent (h) of the trajectories of 16384 initial conditions uniformly distributed over the region of the flow.
At finite time, the dependence of h on initial conditions results in a conditional density function P(h|n):
THEORETICAL APPROACH TO CALCULATING Dq
A mean value of = 0.30 ensures that the flow is chaotic as the trajectories of nearby fluid elements diverge exponentially in time.
h
The time-independent function G(h) is concave up and has the minimum value zero, occurring at h= .h
For large enough n:)()|( hnGenhP −≈
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.05
0.1
0.15
0.2
0.25
0.3
0.35n=20n=40n=80
h
G(h
)
( ) (1)21q
F q qFDq−= −−
where0
( )( ) maxh
q G hF qh
γ≥
−=
A theoretical analysis (not given here) yields a result for Dq(with = 1) in terms of G(h):λ
A similar result is obtained for Dq with = 2λ
0 0.25 0.5 0.75 1 1.25 1.51.8
1.85
1.9
1.95
2
Theory with = 0Theory with = 0.044
q
Dq 2=λ
0 0.25 0.5 0.75 1 1.25 1.5 1.75 21.9
1.92
1.94
1.96
1.98
2
Theory with = 0Theory with = 0.044
Dq
q
1=λ
γγ γ
γ
COMPARISON OF NUMERICAL AND THEORETICAL RESULTS
0 0.25 0.5 0.75 1 1.25 1.5 1.75 21.9
1.915
1.93
1.945
1.96
1.975
1.99
2
NumericalTheoretical
0 0.25 0.5 0.75 1 1.25 1.51.8
1.825
1.85
1.875
1.9
1.925
1.95
1.975
2
NumericalTheoretical
q
DqDq
q
044.01, == γλ 044.02, == γλ
CONCLUSIONS
The decaying passive scalar has fractal properties and is characterized by Dq.
We have developed a theory which relates Dq to the distribution of finite-time Lyapunov exponents of the chaotic flow.
We provide high resolution numerical evidence for our theoretical predictions.
