Turbulent flow drives a fluid through a
repertoire of unstable patterns.
As we watch a turbulent system evolve,
every so often we catch a glimpse of a familiar pattern.
For any finite spatial resolution, for a finite time
the system follows approximately
a pattern belonging to a finite
alphabet of admissible patterns, and the long term dynamics can be thought
of as a walk through the space of such patterns,
just as chaotic dynamics with a low dimensional
attractor can be thought of as a succession of nearly periodic (but
unstable) motions.
Here we apply this vision to
the flame flutter of gas burning
on your kitchen stove.
We are happy if in a few days of experimentation we succeed in simulating
the system numerically, and develop some intuition about turbulence.
Please read the chapter "Turbulence?" [1]
A flame front is
described by the Kuramoto-Sivashinsky [KS] equation
u_{t} = -
1
2
(u^{2})_{x}-u_{xx}-u_{xxxx} .
(1)
Here t ³ 0 is the time and
x Î [0,L] is the periodic space coordinate.
In what follows we use interchangeably the "dimensionless
system size" [L\tilde], or the periodic domain size L = 2p[L\tilde],
as the system parameter.
The subscripts x and t denote the partial derivatives with respect to
x and t;
u_{t} = du/dt, u_{xxxx} stands for 4th spatial
derivative of the "velocity of the flame front"
u=u(x,t) at position x and time t.
As the "flame front velocity"
u(x,t)=u(x+2p,t) is periodic on the x Î [0,2p] interval,
expand it in a spatial Fourier basis:
u(x,t)=
+¥ å
k=-¥
a_{k} (t) e^{ i k x /[L\tilde]} .
(2)
Since u(x,t) is real,
a_{k}=a_{-k}^{*} .
(3)
Substituting (2) into (1) yields
the infinite ladder of evolution equations for
the complex Fourier coefficients a_{k}(t):
×
a
k
= v_{k}(a) = ( (k/
~
L
)^{2} - ( k/
~
L
)^{4} ) a_{k} - i
k
2
~
L
+¥ å
m=-¥
a_{m} a_{k-m} .
(4)
As [a\dot]_{0}=0, the solution integrated over space is
constant in time. We set this average velocity to
zero, a_{0} = ò dx u(x,t) = 0.
The coefficients a_{k} are in general complex functions of time.
The constant solution u(x,t)=0 is an
equilibrium point of (1). For this "laminar" equilibrium the stability matrix
is diagonal,
A_{kj}(a) =
æ è
k^{2}/
~
L
2
- k^{4}/
~
L
4
ö ø
d_{kj} ,
(5)
and
so is the Jacobian matrix
J^{t}_{kj} = d_{kj} e^{(k/[L\tilde])2(1- (k/[L\tilde])2)t} .
From
(5) it follows that the |k| < [L\tilde]
long wavelength modes of this
equilibrium are linearly unstable, and the
|k| > [L\tilde] short wavelength modes are stable.
For [L\tilde] < 1, u(x,t)=0 is the globally attractive stable equilibrium,
i.e., the dissipation is so strong that any flame front burns out.
A theory of turbulence should
predict measurable properties of turbulent flows, such as
their mean energies and their energy dissipation rates.
The time-dependent average velocity-squared
E =
1
L
ó õ
L
0
dx
u^{2}
2
(6)
has a physical interpretation
as the average "kinetic energy" density of the flame front.
Its time evolution is given by the power/dissipation energy rate equation
×
E
= P - D , P =
á (u_{x})^{2}
ñ , D =
á (u_{xx})^{2}
ñ.
(7)
KS is a far-from equilibrium system:
the power P
pumped in by the anti-diffusion u_{xx}
is
balanced by the hypervicosity u_{xxxx}
dissipation rate D. In principle,
these are experimentally observable quantities,
used
in what follows as flow diagnostics.
From here on we turn to numerical experimentation.
Take L sufficiently large so that the
dynamics can be spatiotemporally chaotic, but not so large that we would be
overwhelmed by many short wavelength modes needed in order to accurately
represent the dynamics.
A bit of advice: start on terra firma,
small system size [L\tilde]
= 1,
and increase [L\tilde] a little bit, integrate until the
trajectory has settled down; then increase [L\tilde] a little bit again,
restart from the trajectory just computed,
integrate until has settled down.
Repeat. Sometimes stop incrementing
the trajectory, increment N instead and check how sensitive is your
attractor to truncation number N. This "adiabatic"
approach has advantage of (almost) always starting you close to
the attractor, thus avoiding long transients typical of random
starting conditions.
The problem with high-dimensional truncations of (4)
is that the dynamics is difficult to visualize.
We visualize a trajectory by its projections onto any three
2 or 3 basis vectors (states of the system):
see Davidchack code for examples.
Figure 1:
The energy (6) of all equilibria that exist up
to [L\tilde]
= 4.5
plotted as a function of the system size
[L\tilde] (from ref. [3]).
Solid curves denote n-cell solutions,
dotted curves GLMRT, the dash-dotted curve the
`giant states', and dashed curves the relative equilibria.
Open circles indicate Hopf bifurcations.
The color of a branch indicates the number of unstable
eigenvalues: (red) 2 unstable eigenvalues, (blue) 1
unstable eigenvalue, (green) stable.
3.1 Bifurcations with the increase of system size L
Explore dynamics for different system sizes L
by plotting long trajectories both in the state space and
in color-coded space-time plots of u(x,t). If the long
time trajectory is settling on an
attractive equilibrium or periodic orbit, eliminate the initial
transient by starting the plots only after the
transients have died out.
Classical papers on the equilibria of
Kuramoto-Sivashinsky equation, bifurcations and symmetry analysis are
Kevrekidis, Nicolaenko and Scovel [4],
and Greene and Kim [3]. You might be able to
identify your states in the
fig. 3 bifurcation diagram.
To get the right E, multiply
by 4; the horizontal axis is [L\tilde]
= Ö{a/4}.
Here is what the class fished out, listed as
series of numerical experiments ordered by [L\tilde] parameter.
Please send you .png files to
Matt Marshall, raequin[snail]gmail.com
1.5915494: L = 10 (Holt)
stable E_{1} 1.7603981: [L\tilde]
= 1+p/4-0.025 (Marshall)
What's this bifurcation? Narrow the gap ±0.025
to see whether it really happens at
[L\tilde]
= 1+p/4 ±e. Why would that happen?
1.8103981: [L\tilde]
= 1+p/4+0.025 (Marshall)
Iacobucco and Freeman have interesting ways of scanning
energy as function of L.
With this they have explored bifurcation sequences
by plotting (inter alia) the time-averaged values
([`E],[`P],[`D]) of (E,P,D)
defined in (7), for small increments of
1 < L < 25.
Plot several long trajectories for L = 22,
different initial conditions,
using the same vector basis as Davidchack. Is your dynamics
qualitatively the same as in his plots?
[Holt has nice long time runs]
Do you get any stable periodic orbits for [L\tilde]
=22?
If you do, we would love to see them - have not found any.
[Nobody saw anything different, and no stable orbits
were observed. Only Sharon tried a set of different initial conditions,
but al close-by]
Kirill Davydychev: I left my laptop overnight to compute this
flame front movie:
FlameFront [avi format].
From it
one can guess the stable/unstable cycles, the
period for them seems to be around 7L, with 3.5 L separation.
you mean period in time, or that by increasing [L\tilde] by
1 you see another bifurcation?
Also, here is a geeky approach to the onset of chaos:
consider the file
sizes of each frame. In a lossless format such as .png, the
compression ratio is an indicator of the complexity of the
structure, see graph of L vs. the file size.
It correlates with the chaotic behavior of this system
(not to mention that it's also completely useless).
Now, have a Carlsberg, perhaps the best beer
in some parts of Copenhagen, and a good summer.
P. Cvitanovi\'c, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay.
Chaos: Classical and Quantum.
Niels Bohr Institute, Copenhagen, 2005.
ChaosBook.org.
I. G. Kevrekidis, B. Nicolaenko, and J. C. Scovel.
Back in the saddle again: a computer assisted study of the
Kuramoto-Sivashinsky equation.
SIAM J. Appl. Math., 50(3):760-790, 1990.
File translated from
T_{E}X
by
T_{T}H,
version 3.77. On 06 May 2007, 03:17.