Fluid dynamics · from rolls to chaos

Rayleigh–Bénard Convection

Integrate three convection modes from a small perturbation. Orbit the hot–cold exchange and tracer paths; connect temperature profiles, heat transport and phase-space trajectories 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

Playback rate changes only the clock, not the equations. Physical controls restart the seeded experiment and preserve pause state.

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

The Lorenz truncation uses stress-free isothermal boundaries with $a^2=1/2,\quad \mathrm{Ra}_c=27\pi^4/4\approx657.5$ . The spatial display extrudes a 2D field; it is not a rigid-wall or 3D turbulence solver.

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 $r$ determines the stability of conduction, but instability does not create a seed from nothing. No seed retains the exact conducting solution; nonzero disturbances 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}

The selected horizontal wave number minimizes onset for stress-free isothermal boundaries. Rigid walls have a different threshold.

Three evolving modes

\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}

Circulation, hot–cold contrast and distortion of the mean temperature profile evolve together. RK4 uses a fixed model-time step; the companion starts with a minute circulation offset.

Recover the incompressible field

\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

A streamfunction enforces zero divergence and no penetration at both plates. Horizontal boundaries are periodic. Spanwise extrusion adds no third-dimensional dynamics.

Recover temperature without 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. Only the display palette saturates outside the plate range; the reduced model can overshoot, especially under strong drive.

Two heat-transport diagnostics

\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}

The plate gradient and volume-averaged total heat flux differ during transient storage. On a steady branch they agree; neither is an empirical curve pasted onto the animation.

Steady-state benchmark 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 branches exist above onset. The displayed clock is dimensionless model time, not physical seconds. Strong-drive Lorenz dynamics are not quantitative predictions of laboratory turbulence.

Common difficulties

Boundary conditions set the threshold

Typical misconceptionEvery heated layer shares one universal onset value.

Better mental modelThis model has stress-free isothermal interfaces and a fixed periodic wave number. Its onset must not be confused with rigid no-slip walls.

A phase portrait is not a particle path

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

Better mental modelThe portrait records amplitudes describing the entire field. Tracers in the apparatus record spatial positions. They share a dynamic state, not a coordinate space.

Unstable branches do not certify turbulence

Typical misconceptionPast the branch instability, the image resolves three-dimensional turbulent flow.

Better mental modelThree modes omit the smaller spatial scales of strongly driven convection. Chaotic reversals belong to the truncated system, not a quantitative turbulent heat-transfer law.

Run the experiment

01
Let a perturbation decay
Choose Perturbation decays and advance several model-time units.
What to observe: Circulation fades, mean temperature approaches the linear profile, and both fluxes approach one.
02
Build steady rolls
Choose Steady rolls and compare early and late heat-flux history.
What to observe: The two fluxes differ during the transient; they converge toward the dashed steady benchmark as the modal orbit settles.
03
Watch deterministic reversals
Choose Lorenz reversals and keep playing, or increase playback rate.
What to observe: Circulation changes sign and the trajectory circles both branches. A minute initial offset can amplify; playback rate never modifies the equations.
04
Remove the seed
Choose No seed and advance, then set a nonzero initial perturbation.
What to observe: The exact zero state does not spontaneously circulate. Setting a seed restarts the experiment from time zero.