Movies of plane Couette flow
An example: Statespace on left, velocity field on right. See Dual views section for details.
See also:
My research, in plain words and
Note on video formats.
What these movies illustrate
There are two kinds of animations here:
 3D movies of velocity fields, which show turbulent flows
visiting unstable coherent structures, and
 Statespace trajectories, which show
 that the coherent structures in the 3D movies result from close
passes to unstable equilibrium solutions of NavierStokes, and
 that these solutions and their unstable manifolds impart a rigid
structure to state space, which organizes the turbulent dynamics.
The Dual views section shows some of these animations
sidebyside. In these, you can see coherent structures appear in the
turbulent velocity field as the statespace trajectory makes close passes to
equilibrium solutions of NavierStokes.
In the 3D velocity fields, the top wall moves toward viewer, and the bottom wall moves
away with equal and opposite velocity. The velocity field is coded by color
and 2D vectors. Red: fluid moving toward viewer. Blue: fluid moving away
from the viewer. Arrows: fluid velocity transverse to the direction of wall
motion. The top half of the fluid is cut away to show the midplane velocity
field, halfway between the upper and lower walls.
Plane Couette, from experimental to idealized flows
These animations show a sequence of simplifications, starting with a
flow with close to experimental conditions and ending with a qualitatively similar
flow whose dynamics are simpler to analyze.
 Plane Couette turbulence in a periodic cell with large aspect ratio (20070911)
explanation of setup and color coding
In both cases the initial velocity field is small, smooth, and
incompressible. meets the boundary conditions, and satisfies none of
the plane Couette symmetries. The periodic orbit is 3 x 4 copies of
the P35 orbit, slightly stretched and scaled to
fit [L_{x}, L_{y}, L_{z}] = [15, 2, 15].
The Reynolds number is 400, the spatial grid is 96 x 33 x 128, and the
time step is dt = 0.03125.
What is striking about these simulations is that all nonlaminar
initial conditions appear to settle into the similar patterns of
behavior, consisting of episodes of highly ordered motion among
streamwise counterrotating rolls, interspersed by periods of
less ordered, more turbulent motions.
This observation motivates the empirical search for small aspect ratios
large enough to accommodate a pair of unstable counterrotating
rolls, following Hamilton, Kim and Waleffe.
 Plane Couette turbulence in a small periodic cell.
The flow domain here is the "HKW" cell, L_{x} = 7π/4 = 5.50,
L_{z} = 6π/5 = 3.76, found empirically by
Hamilton, Kim and Waleffe to be the smallest
that appears to sustain turbulence. The initial condition is a random
perturbation from laminar flow (20070904)
 "Symmetric turbulence," within an invariant symmetric subspace of the HKW cell. (20070903)
The initial condition is a field from the previous simulation,
translated streamwise, spanwise to optimize
the s_{1},s_{2} symmetries (to within 2% in the energy norm),
and then projected to lie exactly
in the symmetric subspace. The s_{1},s_{2} symmetries are

s_{1} : [u,v,w](x,y,z) > [u,v,w](x+L_{x}/2, y, z)

s_{2} : [u,v,w](x,y,z) > [u,v,w](x+L_{x}/2, y, z+L_{z}/2)
The symmetric subspace is space of velocity fields
left invariant under both s_{1} and s_{2}.
All the exact invariant solutions (equilibria and their
stable/unstable manifolds, periodic orbits) shown in what follows
belong to this symmetric subspace. What is surprising about the above
simulations is that even though a typical initial state of a fluid
has a zero probability of being within (or entering) the symmetric
subspace the steady state turbulent dynamics seems to stays close to
it for all times. This observation motivates what follows, a closer
investigation of the geometry of the statespace flow within the
symmetric subspace.
Dual views of velocity field and statespace evolution
The following movies show a plane Couette velocity field evolving
under NavierStokes on the right and the corresponding statespace
trajectory on the left. The evolution of the field is computed with a
CFD algorithm. The flow and simulation parameters are
[L_{x}, L_{y}, L_{z}] = [5.51, 2, 2.51],
Re = 400, 32 x 49 x 32, and dt = 0.03125. The fields in this section are
within the s_{1},s_{2} invariant subspace defined above.
The statespace projection is from the 10^5 dimensional space of free variables
in the CFD algorithm onto a 2d plane e1,e2 formed from linear combinations of
the upperbranch equilibrium and its halfcell translations, with the laminar
equilibrium as the origin. For example,
e_{1} = (1 + τ_{x} + τ_{z} + τ_{xz}) u_{UB},
normalized to e_{1} = 1, where τ_{x} is a translation operator that
shifts a velocity field by half the cell length in x. The norm and projection operator are
defined by the L2 inner product. For example
a_{1}(t) = 1/V ∫_{V} u(x,t) ∙ e_{1}(x) dx, where V is
the cell volume.
Why does this make sense? See the Gibson, Halcrow and Cvitanović paper.
The labeled points are
LM
LB
UB
NB
u(t):

Laminar equilibrium
Lowerbranch equilibrium(Nagata, Waleffe)
Upperbranch equilibrium (Nagata, Waleffe)
"Newbie" equilibrium (Gibson, Halcrow, Cvitanović)
the timevarying velocity field, evolving under NavierStokes

Transient turbulence
 Transient turbulence from the unstable manifold of the NB equilibrium.
Other encodings:
Legend: The statespace trajectory is plotted against
three exact equilibria of plane Couette flow:
the lowerbranch (LB, blue), the upperbranch (UB, green),
and the "newbie" (NB, red) equilibria, plus the laminar solution (LM, black)
(see Equilibria).
The halfcell translations in x and z of the equilibria
are points indicated with prefix "τx", "τz", and "τxz".
The blue and green solid lines show the 1 and 2dimensional unstable manifolds
of the LB and UB equilibria. The moving red dot is the state of the fluid velocity
field evolving under the NavierStokes equations.
Partial narrative:
 t=0150 : The initial condition is a small perturbation in the unstable
manifold of the NB equilibrium. The perturbation is small enough that no visible
changes occur until about t=150.
 t=150280 : An oscillation around the NB equilibrium, governed by the NB's
unstable complex eigenvalue.
 t=280300 : A close pass to the halfcell ztranslated LB equilibrium,
escape back toward turbulence along LB's 1d unstable manifold.
 t=3001200 : Close passes to UB unstable manifold,
equilibria still waiting to be discovered, zshift LB, etc.
 t=12001350 : Decay to laminar flow.

Transient turbulence from the unstable manifold of the
UB equilibrium.
(12 MB)
This movie shows several close passes the lowerbranch equilibrium.
Equilibria teach us a lot about dynamics, but as they are stationary, no turbulence takes
place there. As the following dualview movies show, the time dependence of typical
unstable structures seen in turbulence is better captured by unstable periodic orbits.
Periodic orbits in symmetric subspace of narrow box
 T=35.86 periodic orbit (1.3 MB)
This is a relative periodic orbit: the field returns to a
streamwise halfcell
translation in x of its original state after one period (T=35.86),
and then to its original state after two periods (T=71.72).
 T=97.08 periodic orbit (1.3 MB)
The P97 orbit is a true periodic orbit close to a twoperiod sequence of P35
orbits, with an extra wiggle.
Periodic orbits in symmetric subspace of HKW cell
 My current collection of periodic orbits in the "HKW" box, Lx=1.75 pi, Lz=1.2pi. The P41p36
was calculated by Divakar Viswanath, and it appears to be the same as Kawahara and Kida's
"gentle" orbit. P85p27 orbits was calculated by Viswanath and independently by J. Halcrow,
P. Cvitanovic, and myself. Data for these orbits can be found in the
channeflow database. All these orbits were computed with
Viswanath's NewtonhookstepGMRES search algorithm
from close recurrences in a single u(x,t) trajectory. Each is a
true periodic orbit in the s_{1},s_{2} symmetric subspace (no
translations).
P19p02,
P41p36,
P46p23,
P75p35,
P76p82,
P76p85,
P85p27,
P87p89,
P88p90,
P121p4
 A turbulent trajectory making a close pass to the P68p07 orbit.
Equilibria and their stability
These plots show equilibria (steady states) of plane Couette flow for
[L_{x}, L_{y}, L_{z}] = [5.51, 2, 2.51] and Re=400 computed with
Viswanath's NewtonhookstepGMRES search algorithm. Discretization is on a
32 x 35 x 32 dealiased grid. Truncated spectral coefficients are on the order
1e6. Each field is a fixed point of the CFD algorithm to about the same accuracy.
 uLM is the classic laminar equilibrium a linear profile (u,v,w) = (y,0,0).
 uLB,UB are the Nagata lower and upper branch equilibria
(computed also by Waleffe, and refined here with Viswanath's
algorithm).
 uNB, uNB2, and uEQ5 are new equilibria found by
Gibson, Halcrow, and Cvitanović.
Here are the eigenvalues of the above equilibria, computed with Arnoldi
iteration. The left plot shows the complete set of eigenvalues; on the right are
the eienvalues in the s_{1},s_{2}symmetric subspace. Accuracy is about 1e7.
Streak instabilities and the selfsustaining process
The following movies illustrate pieces of the selfsustaining process theory presented in
Waleffe Phys Fluids 1997. Please refer to that paper for details.
The movies show the growth of streamwisevarying instabilities of streamwiseconstant
streaks. Under the linearized dynamics with the streaks held constant, the
instabilities evolve as
u(t) = u0 + ε e^{μ t} (v_{r} cosωt  v_{i} sinωt)
where λ = μ + i ω is the complex eigenvalue and v_{r} and v_{i}
are realvalued superpositions of the complex eigenfunction.
 An unstable eigenfunction of frozen streaks
(2.2 MB) does not show dynamics, rather shows realvalued superpositions
u0 + ε (v_{r} cos&theta  v_{i} sin&theta) for varying θ, which
goes from 0 to 2 π over the course of the movie.
 Linear dynamics around frozen rollstreak
(1.8 MB) shows the initial condition u0 + ε v_{r} evolving as
u(t) = u0 + ε e^{μ t} (v_{r} cosωt  v_{i} sinωt)
under the linearized dynamics. &epsilon v_{r} = 0.01 u0.
 DNS of streak instability (6.9 MB)
shows the same initial condition u0 + ε v_{r} evolving under
the NavierStokes equations, computed with DNS.
 DNS of streak instability, slightly different
initial condition (5.3 MB) shows a similar movie to the one above, but starting with slightly
weaker streaks and a slightly larger eigenfunction perturbation. If you have a little traiing in nonlinear
dynamics you can probably can see where I'm going with this...
These are preliminary research results intended to whip up a frenzy of enthusiasm among my colleagues.
Details will follow in publications. (20080207)
Channelflow.org
All computations were performed with CFD software from
www.channelflow.org. Channelflow
provides welltested and easytouse implementations of a spectral CFD
algorithm, Viswanath's algorithms for computing equilibria, traveling
waves, periodic orbits, and linear stability modes, and a number of
utilities for manipulating velocity fields and producing
visualizations.
The animations were produced as uncompressed AVI files in with custom
Matlab visualization scripts and then compressed and packaged as mp4s
with the "mencoder" and "mp4creator" video utilities in Linux. The
simulation software and scripts for producing movies can be downloaded
from
www.channelflow.org/download.
References

M. Nagata, "Threedimensional finiteamplitude solutions in plane Couette
flow: bifurcation from infinity," J. Fluid Mech. 217 (1990).
Calculation of a lower/upper branch pair of equilibrium solutions to
plane Couette flow.

J. M. Hamilton and J. Kim and F. Waleffe, "Regeneration mechanisms of nearwall turbulence
structures," J. Fluid Mech. 287 (1995). Identification and numerical
study of organized dynamics in lowReynolds plane Couette flow in small aspect
ratios.

F. Waleffe, "On a selfsustaining process in shear flows"
Phys. Fluids 9 (4) (1997). Analysis of how in plane Couette flow,
(a) rolls create streaks, (b) streamwisevarying instabilities grow on streaks,
(c) instabilities interact with rollstreaks to sustain the rolls.

F. Waleffe, "Homotopy of exact coherent structures in plane shear flows,"
Phys. Fluids 15 (2003). In this work, Waleffe constructs
numerically exact solutions to plane Couette flow with a variety of boundary
conditions. Waleffe provided us with the lower branch and upper branch
solutions; these served as the starting points in our investigations.

D. Viswanath, "Recurrent motions within plane Couette turbulence,"
J. Fluid Mech. 580 (2007)
[arXiv:/physics/0604062].
Viswanath presents a novel
combination of the Newtonhookstep trustregion minimization algorithm
with the GMRES algorithm for iterative solution of highdimensional linear
algebra problems, and uses it to compute periodic and relative periodic
orbits of plane Couette flow, using a CFD algorithm to evaluate NavierStokes
dynamics and close recurrences for initial guesses.

J. F. Gibson, J. Halcrow, P. Cvitanović, "Visualizing
the geometry of state space in plane Couette flow," accepted by
J. Fluid Mech. pending revisions
[arXiv:0705.3957].
John F. Gibson,
in collaboration with:
Jonathan Halcrow,
Predrag Cvitanović,
Fabian Waleffe, and
Divakar Viswanath.
emacs tag: Last modified: Mon Dec 3 00:12:13 IST 2007
subversion tag: $Author: gibson $  $Date: 20080408 09:39:47 0400 (Tue, 08 Apr 2008) $