Date: Tue, 10 Dec 1996 23:20:01 GMT Server: NCSA/1.4.2 Content-type: text/html Last-modified: Fri, 22 Mar 1996 03:32:37 GMT Content-length: 3765 Rayleigh-Benard Convection

A stable state




Abstract. This experiment models the spatial patterns formed by fluid convection close to onset. You are looking down at a large thin square box, filled with (an infinitely viscous) slow-moving fluid. The dimensions of the box are twenty by twenty by one unit deep.

The areas shaded blue represent fluid moving downwards, and those shaded red represent fluid moving up. Green is motionless fluid; at the boundary the fluid is constrained to be motionless, so we will eventually see all of the patterns fading out at the boundary.

The convection rolls that ultimately form have diameter comparable to the depth of the box.


Frame 0. We start with some initial conditions, velocities small and distributed randomly.

Frame 1. As time progresses the fluid begins to develop local coherence.


Frame 2. Distinct local convection rolls form, of measurable diameter.
Frame 3. The rolls become locally parallel, but defects form in the center of the box.
Frame 4. The center defects are resolved, leaving defects only at the boundary.
Frame 5. The fluid eventually reaches a steady state.
These evolutions also make spiffy movies. Unfortunately, to make them small enough to be reasonable over the network, the geometries are so small that the fluid motion is a little contrived. But the pictures are pretty cute anyway. Here's one with rigid boundary conditions (as above, only smaller).

Rigid/rigid boundaries (picture) (Quicktime format, 2.1Mb.)

The final frames from some other movies still under construction:

Rigid/periodic boundaries(picture) (Picture only.)


The second picture plainly shows the specific effect we're trying to pin down, the bending of the convection rolls as they leave the boundary. This effect was first described by S. Zaleski et als., "Optimal merging of rolls near a plane boundary", 29 Phys. Rev. A. 366 (1984).
The numerical problems are challenging principally because the steady state takes an enormously long time to achieve. There is hope for improved techniques, because the successive differences from one state to another have considerable coherence. Here are final frames (the movies are still under construction) of the successive differences between frames of the above two simulations.

Rigid/rigid boundaries (picture) (Picture only.)

Rigid/periodic boundaries (picture) (Picture only.)


eric@cs.washington.edu
26 Mar 1996