Fluid dynamics · from rolls to chaos

Rayleigh–Bénard Convection

When does a fluid layer heated from below start to convect? When the Rayleigh number, which compares buoyancy with viscous and thermal diffusion, passes a critical value. Above it, any small disturbance grows into steady convection rolls, and stronger heating can make the rolls reverse again and again. With no disturbance at all, the layer keeps conducting heat without moving.

Start from a small perturbation and integrate three convection modes. Rotate the view to watch hot and cold fluid trade places and tracers follow the flow, then compare the temperature profile, heat transport and phase-space trajectory as steady rolls form and lose stability.

Model time

\tau

$0$

Rayleigh

\mathrm{Ra}

$3287.6$

Circulation

X

$0.1$

Plate heat flux

\mathrm{Nu}_p

$1$

Bulk transport

\mathrm{Nu}_v

$1$

Onset growth

\lambda_+

$2.882$

Nearby-state distance

\delta

$10^{-6}$

Linear stabilitySteady branches stable

Thermal workbench

Let a perturbation evolve

Tracers & pathsIntegrated through the same evolving velocity field; colors show local temperature.Temperature slicesThree identical slices of a 2D field; no spanwise dynamics.Linked diagnosticsHeat-flux history, mean temperature and modal phase portrait.Relative buoyancy drive

r=\mathrm{Ra}/\mathrm{Ra}_c

5.0

Momentum / heat diffusion

\mathrm{Pr}=\nu/\kappa

10.0

Initial modal seed

X_0=Y_0

0.10

Model time per playback second

d\tau/dt_{\mathrm{play}}

1.00

The playback rate changes only how fast the clock runs, not the equations. Changing a physical control restarts the run from its initial seed and leaves a paused run paused.

Model source: E. N. Lorenz, Deterministic Nonperiodic Flow (1963), equations 23–27.

The Lorenz truncation uses stress-free boundaries at fixed temperature, with $a^2=1/2,\quad \mathrm{Ra}_c=27\pi^4/4\approx657.5$ . The 3D view stretches a 2D field across the depth; the Lab does not solve for rigid walls or 3D turbulence.

Physics tutorial

Convection: a perturbation becomes steady rolls—or reversals

BackgroundHeating from below makes the lower fluid lighter, but viscosity and thermal diffusion resist motion. This Lab does not draw ready-made rolls. It starts from a perturbation, integrates three coupled modes and builds the velocity and temperature fields from them.

Why it mattersConvection ties together heat transport, linear instability and deterministic chaos. Every trajectory you see can be checked against the same equations, the steady solutions and the heat-flux readouts.

Start with the essentials

Focus question: Why do some perturbations disappear, some settle into circulation, and others reverse repeatedly?
One-sentence intuition: The drive ratio $r$ decides whether pure conduction is stable, but instability cannot create a disturbance from nothing. With no seed, the fluid stays in the exact conducting state; only a nonzero disturbance can grow.

Core mathematical model

Buoyancy, damping and geometry

\mathrm{Ra}=\frac{g\alpha\Delta T H^3}{\nu\kappa},\quad r=\frac{\mathrm{Ra}}{\mathrm{Ra}_c},\quad \mathrm{Pr}=\frac{\nu}{\kappa},\quad \mathrm{Ra}_c=\frac{27\pi^4}{4}

For stress-free boundaries held at fixed temperature, the chosen horizontal wave number gives the lowest onset threshold. Rigid walls have a different threshold.

Three modes that evolve together

\frac{dX}{d\tau}=\mathrm{Pr}(Y-X),\quad \frac{dY}{d\tau}=rX-Y-XZ,\quad \frac{dZ}{d\tau}=XY-bZ,\quad b=\frac{8}{3}

The circulation, the hot–cold contrast and the distortion of the mean temperature profile evolve together. RK4 advances them with a fixed step in model time. A companion run starts with a tiny extra amount of circulation.

Velocity field from a streamfunction

\psi=\frac{\sqrt{2}\kappa(1+a^2)}{a}X\sin\!\left(\frac{\pi ax}{H}\right)\sin\!\left(\frac{\pi z}{H}\right),\quad u=-\partial_z\psi,\quad w=\partial_x\psi,\quad a^2=\frac12

Writing the flow with a streamfunction guarantees zero divergence, and no fluid passes through either plate. The flow is periodic in the horizontal direction. The spatial view stretches the same field across the depth and adds no motion in the third dimension.

Temperature field, with no clipping

\frac{T-T_c}{\Delta T}=1-\frac{z}{H}+\frac{\sqrt{2}Y\cos(\pi ax/H)\sin(\pi z/H)-Z\sin(2\pi z/H)}{\pi r}

Both plate temperatures are fixed. Outside the range between them only the color scale saturates; the temperature values are not clipped. The reduced model can overshoot that range, especially under strong drive.

Two measures of heat transport

\mathrm{Nu}_p=1+\frac{2Z}{r},\quad \mathrm{Nu}_v=1+\frac{2XY}{br},\quad \mathrm{Nu}_v-\mathrm{Nu}_p=\frac{2}{br}\frac{dZ}{d\tau}

While the heat stored in the layer is still changing, the temperature gradient at the plate and the volume-averaged total heat flux differ. On a steady branch they agree. Both are computed from the current modes.

Steady solutions and time scale

X_*=Y_*=\pm\sqrt{b(r-1)},\quad Z_*=r-1,\quad \mathrm{Nu}_*=3-\frac{2}{r},\quad \tau=\pi^2(1+a^2)\frac{\kappa t}{H^2}

The nonzero steady branches exist only above onset. The clock shows dimensionless model time, not seconds. Under strong drive, the Lorenz dynamics do not predict laboratory turbulence quantitatively.

Common difficulties

Boundary conditions set the threshold

Typical misconceptionEvery heated layer shares one universal onset value.

Better mental modelThis model has stress-free boundaries at fixed temperature and a fixed horizontal wave number. Its onset value does not apply to rigid no-slip walls.

A phase portrait is not a particle path

Typical misconceptionThe two-wing modal trajectory depicts two physical vortices.

Better mental modelEach point in the portrait gives mode amplitudes that describe the whole flow. The tracers in the apparatus mark real positions. Both are driven by the same state, but they are plotted in different spaces.

Instability does not mean real turbulence

Typical misconceptionOnce the steady branches become unstable, the picture shows fully resolved three-dimensional turbulence.

Better mental modelThree modes leave out the small-scale structure of strongly driven convection. The chaotic reversals belong to this truncated system and do not give a quantitative law for turbulent heat transfer.

Run the experiment

01
Let a perturbation decay
Choose Perturbation decays and advance several model-time units.
What to observe: The circulation fades, the mean temperature returns to the straight-line conduction profile, and both heat-flux values approach one.
02
Build steady rolls
Choose Steady rolls and compare early and late heat-flux history.
What to observe: The two heat-flux values differ at first. As the orbit in mode space settles, both approach the dashed steady-state value.
03
Watch deterministic reversals
Choose Lorenz reversals and keep playing, or increase playback rate.
What to observe: The circulation keeps reversing direction, and the trajectory loops around both branches. A tiny difference in the starting state can grow. Playback rate changes only the clock, not the equations.
04
Remove the seed
Choose No seed and advance, then set a nonzero initial perturbation.
What to observe: The exact zero state never starts circulating by itself. Setting a seed restarts the experiment from time zero.