|John F. Gibson||Center for Nonlinear Science||Georgia Institute of Technology|
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.
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.
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 [Lx, Ly, Lz] = [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 non-laminar initial conditions appear to settle into the similar patterns of behavior, consisting of episodes of highly ordered motion among streamwise counter-rotating 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 counter-rotating rolls, following Hamilton, Kim and Waleffe.
The symmetric subspace is space of velocity fields left invariant under both s1 and s2.
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 state-space flow within the symmetric subspace.
The state-space 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 upper-branch equilibrium and its half-cell translations, with the laminar equilibrium as the origin. For example, e1 = (1 + τx + τz + τxz) uUB, normalized to ||e1|| = 1, where τx 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 a1(t) = 1/||V|| ∫V u(x,t) ∙ e1(x) dx, where V is the cell volume.
Why does this make sense? See the Gibson, Halcrow and Cvitanović paper.
The labeled points are
Legend: The state-space trajectory is plotted against three exact equilibria of plane Couette flow: the lower-branch (LB, blue), the upper-branch (UB, green), and the "newbie" (NB, red) equilibria, plus the laminar solution (LM, black) (see Equilibria). The half-cell 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 2-dimensional unstable manifolds of the LB and UB equilibria. The moving red dot is the state of the fluid velocity field evolving under the Navier-Stokes equations.
Equilibria teach us a lot about dynamics, but as they are stationary, no turbulence takes place there. As the following dual-view 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
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.
emacs tag: Last modified: Mon Dec 3 00:12:13 IST 2007
subversion tag: $Author: gibson $ - $Date: 2008-04-08 09:39:47 -0400 (Tue, 08 Apr 2008) $