Movies of plane Couette flow

An example: State-space on left, velocity field on right. See Dual views section for details.

• What these movies illustrate
• Plane Couette, from experimental to idealized flows
• Dual views of velocity field and state-space evolution
  • Transient turbulence
  • Periodic orbits in a narrow box
  • Periodic orbits in HKW box
• Equilibria and eigenvalues
• Streak instabilities and the self-sustaining process
• Channelflow.org
• References

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.

1. Plane Couette turbulence in a periodic cell with large aspect ratio (2007-09-11)
   explanation of setup and color coding
   
   
   
   • large perturbation from laminar flow (2007-09-11, 20 MB)
   • small perturbation from periodic orbit (2007-09-11, 25 MB)
   
   

   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 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.

2. Plane Couette turbulence in a small periodic cell.
   • random initial condition (15 MB)
     
   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 (2007-09-04)

3. "Symmetric turbulence," within an invariant symmetric subspace of the HKW cell. (2007-09-03)
   • t=0-200 (2007-09-03, 3 MB)
   The initial condition is a field from the previous simulation, translated streamwise, spanwise to optimize the s₁,s₂ symmetries (to within 2% in the energy norm), and then projected to lie exactly in the symmetric subspace. The s₁,s₂ symmetries are
   s₁ : [u,v,w](x,y,z) -> [u,v,-w](x+L_x/2, y, z)
   s₂ : [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 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 + τ_x + τ_z + τ_xz) u_UB, normalized to ||e₁|| = 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 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

LM LB UB NB u(t):

Laminar equilibrium Lower-branch equilibrium(Nagata, Waleffe) Upper-branch equilibrium (Nagata, Waleffe) "Newbie" equilibrium (Gibson, Halcrow, Cvitanović) the time-varying velocity field, evolving under Navier-Stokes

Transient turbulence

1. Transient turbulence from the unstable manifold of the NB equilibrium.
   Other encodings:
   • (MPEG-4, h264 encoding, 15 MB)
   • (Quicktime .mov, h264 encoding, 18 MB)
   • (Microsoft .avi, wmv2 encoding, 22 MB)
   • (Xiph .ogg, theora encoding, 15 MB)

   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.

   Partial narrative:

   • t=0-150 : 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=150-280 : An oscillation around the NB equilibrium, governed by the NB's unstable complex eigenvalue.
   • t=280-300 : A close pass to the half-cell z-translated LB equilibrium, escape back toward turbulence along LB's 1d unstable manifold.
   • t=300-1200 : Close passes to UB unstable manifold, equilibria still waiting to be discovered, z-shift LB, etc.
   • t=1200-1350 : Decay to laminar flow.

2. Transient turbulence from the unstable manifold of the UB equilibrium. (12 MB)
   This movie shows several close passes the lower-branch equilibrium.

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.

1. T=35.86 periodic orbit (1.3 MB)
   This is a relative periodic orbit: the field returns to a streamwise half-cell 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).

2. T=97.08 periodic orbit (1.3 MB)
   The P97 orbit is a true periodic orbit close to a two-period sequence of P35 orbits, with an extra wiggle.

1. 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 Newton-hookstep-GMRES search algorithm from close recurrences in a single u(x,t) trajectory. Each is a true periodic orbit in the s₁,s₂ symmetric subspace (no translations).
   
   P19p02, P41p36, P46p23, P75p35, P76p82, P76p85, P85p27, P87p89, P88p90, P121p4
   
   
   
2. A turbulent trajectory making a close pass to the P68p07 orbit.
   

• 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.

1. An unstable eigenfunction of frozen streaks (2.2 MB) does not show dynamics, rather shows real-valued 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.
   
2. Linear dynamics around frozen roll-streak (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|.
   
3. DNS of streak instability (6.9 MB) shows the same initial condition u0 + ε v_r evolving under the Navier-Stokes equations, computed with DNS.
4. 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...

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.

• M. Nagata, "Three-dimensional finite-amplitude 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 near-wall turbulence structures," J. Fluid Mech. 287 (1995). Identification and numerical study of organized dynamics in low-Reynolds plane Couette flow in small aspect ratios.

• F. Waleffe, "On a self-sustaining process in shear flows" Phys. Fluids 9 (4) (1997). Analysis of how in plane Couette flow, (a) rolls create streaks, (b) streamwise-varying instabilities grow on streaks, (c) instabilities interact with roll-streaks 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 Newton-hookstep trust-region minimization algorithm with the GMRES algorithm for iterative solution of high-dimensional linear algebra problems, and uses it to compute periodic and relative periodic orbits of plane Couette flow, using a CFD algorithm to evaluate Navier-Stokes 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].

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) $