movies

# Movies of plane Couette flow

These movies are designed to convey the main ideas of my research in plane Couette flow. For more details, please see my papers.

## Visualization scheme

The movies show plane Couette flow in a rectangular box of size [Lx, Ly, Lz] with solid walls on the top and bottom (y = -Ly/2 and Ly/2). The top wall and the upper half of the fluid are cut away to show what happens at the midplane y=0. The walls slide at constant speeds in opposite directions along x, the top wall slides towards the back of the box, the bottom towards the front.

Arrows indicate in-plane velocity. The color indicates the streamwise velocity, that is, the speed of the fluid in the direction of the wall motion:

• Red indicates fluid moving in the positive streamwise direction (away from the viewer)
• Blue indicates fluid moving in the negative streamwise direction (towards the viewer)

The x,y,z directions are streamwise, wall-normal, and spanwise. The rectangular cell is periodic in x and z, so that the front and back slices match, and the left and the right.

# Coherent structures in shear flows

This movie show the formation of `coherent structures' in plane Couette flow, with fairly large aspect-ratio cell: [Lx,Ly,Lz] = [16, 2, 16]. The initial condition is a random perturbation of laminar flow that meets boundary and divergence-free conditions, has roughly the spectral characteristics of turbulent fields, and is about 10% in magnitude of the laminar flow (or 1% in energy).

Observe

• 0 < t < 10 : the random perturbations grow and no apparent order
• 10 < t < 100 : there is little discernable order
• 100 < t < 200 : the flow organizes into alternating +/- streamwise-moving streaks (red/blue) associated with `rolls' visible in the front y,z plane, which draw the + streamwise (red) fluid down from the top wall and - streamwise (blue) up from the bottom
• 200 < t < 250 : an instability grows and destroys the system of streaks and rolls
• 250 < t < 300 : less organized flow ensues, with roll-streak patterns emerging here and there, now and then

# Turbulent dynamics in a 'minimal flow unit'

The dynamics of the system above are complex, so for the time being we focus on a cell with smaller aspect ratios, just big enough to contain one pair of alternating roll-streak structures. The cell size of [1.75 π, 2, 1.2 π] and Reynolds number of 400 is from Hamilton, Kim, and Waleffe (1995), an important paper that identified the dynamics seen above as a 'self-sustaining process' in plane Couette flow. Observe this repetitive but nonperiodic cycle of behavior

1. streaks and rolls that are nearly uniform in x, the streamwise direction
2. growth of a roughly sinusoidal-in-x instability in the roll-streak structures
3. destruction of the structures, finer scale fluctuations, and higher dissipation
4. reformation of the roll-streak structures

# Periodic orbits

We have computed a number of exact periodic orbits in the system seen above. Two are shown above; there are more in the channelflow database of exact solutions. The periodic orbits repeat themselves exactly after a finite time. This opens up a number of interesting possibilities for dynamical analysis of turbulence. For example, we can compute the eigenvalues and eigenfunctions of the orbits and so determine the linear stability of turbulent trajectories. The orbits also do quite well in capturing first and second-order statistics of the turbulent flow, i.e. the mean flow and Reynolds stresses.