3-D Rayleigh–Bénard convection on the GPU

Heat a box of air from below until it overturns: a transported temperature field, Boussinesq buoyancy, the critical Rayleigh number of 1708, and turbulent heat transport matched to published DNS — all on the GPU.

flow

scalar-transport

boussinesq

natural-convection

benchmark

gpu

visualization

Author

Peclet

Published

July 9, 2026

The onset study runs anywhere (a few minutes on GPU); the turbulent-cube sections want a CUDA build — on a free Colab CPU runtime shrink the grids as noted inline.

What you’ll learn

How to turn the single-phase solver into a thermal convection code with two calls: add_scalar("T", …) registers temperature as a transported scalar (advection–diffusion on the same cut-cell grid), and set_property_model("force_z", "boussinesq", "T", …) closes the loop by writing the buoyancy body force from that field every step — the Boussinesq approximation. We then validate the coupled physics twice, against the two classic yardsticks of Rayleigh–Bénard convection:

The onset. Between rigid plates, convection starts at the critical Rayleigh number $\mathrm{Ra}_c = 1707.76$ — a number derived by linear stability theory (Chandrasekhar 1961). We measure exponential growth/decay rates of a tiny perturbation on both sides of the threshold and bracket $\mathrm{Ra}_c$ to a fraction of a percent.
Turbulent heat transport. In a cubical cell at $\mathrm{Ra} = 10^6$ and $3\times10^6$ ($\mathrm{Pr}=0.7$) we time-average the Nusselt number and compare with the lattice-Boltzmann DNS of Xu et al. (2019) (and the classic $\mathrm{Nu}\sim\mathrm{Ra}^{2/7}$ scaling of hard turbulence, Kerr (1996)).

And because a convecting box is one of the prettiest objects in fluid dynamics, we render the plumes in 3-D and film them.

The physics

Under the Boussinesq approximation the fluid is incompressible with constant properties, except that density variations with temperature survive in the gravity term — a linear expansion $\rho \approx \rho_0(1 - \beta(T - T_0))$ turns into a buoyancy body force:

\[ \partial_t \mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\tfrac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{u} + g\beta\,(T - T_0)\,\hat{\mathbf{z}}, \qquad \nabla\cdot\mathbf{u}=0 \tag{1}\]

\[ \partial_t T + \nabla\cdot(\mathbf{u} T) = \alpha \nabla^2 T \tag{2}\]

Two dimensionless groups govern everything: the Rayleigh number $\mathrm{Ra} = g\beta\,\Delta T\, H^3/(\nu\alpha)$ (buoyancy forcing over diffusion) and the Prandtl number $\mathrm{Pr} = \nu/\alpha$ (momentum over heat diffusivity; air is $\mathrm{Pr}\simeq0.7$). The answer of the system is the Nusselt number — the heat flux through the layer over the flux pure conduction would carry:

\[ \mathrm{Nu} \;=\; \frac{\langle w T\rangle - \alpha\,\partial_z \langle T\rangle} {\alpha\,\Delta T / H}. \tag{3}\]

The solver works in grid units ($\Delta x = 1$, so $H = N$ cells, and we set $\Delta T = 1$). We choose the free-fall velocity $u_f = \sqrt{g\beta\,\Delta T\,H}$ — the natural velocity scale of convection — to be a CFL-friendly value, and derive the transport coefficients from $\mathrm{Ra}$ and $\mathrm{Pr}$:

\[ g\beta = u_f^2/H, \qquad \nu = u_f H \sqrt{\mathrm{Pr}/\mathrm{Ra}}, \qquad \alpha = \nu/\mathrm{Pr}. \tag{4}\]

One free-fall time $t_f = H/u_f$ is then $H/u_f$ steps at $\mathrm{d}t=1$.

import importlib.util, os, subprocess, sys
_local = os.environ.get("PECLET_LOCAL_BUILD")
if _local:
    for p in _local.split(os.pathsep):
        sys.path.insert(0, p)
elif importlib.util.find_spec("peclet") is None:
    subprocess.run([sys.executable, "-m", "pip", "install", "-q", "peclet"], check=True)

import numpy as np
import time
import matplotlib.pyplot as plt
from peclet import flow

plt.rcParams.update({"figure.dpi": 130, "font.size": 10, "axes.grid": True,
                     "grid.alpha": 0.3, "axes.axisbelow": True,
                     "figure.facecolor": "white", "savefig.bbox": "tight"})
BLUE, RED, GREEN = "#1f77b4", "#d62728", "#2ca02c"
print("backend:", flow.execution_space)

backend: Cuda

The whole thermal setup is one function. Temperature is registered as a transported scalar (advected by the just-projected velocity with a Koren-TVD flux, diffused implicitly), given Dirichlet values on the hot floor and cold ceiling and adiabatic (zero-flux) or periodic sides, and coupled back into the momentum equation through the boussinesq property closure, which fills the per-cell body force force_z $= \rho_0 g\beta\,(T - T_0)$ at the top of every step.

def rb_solver(nx, ny, N, Ra, Pr=0.7, u_f=0.4, sides="adiabatic", dt=1.0,
              vel_iters=12, scal_iters=8, pcg=(60, 1e-2)):
    """A Rayleigh-Benard box: hot floor (T=1), cold ceiling (T=0), gravity in -z.
    sides: 'adiabatic' no-slip walls, or 'periodic'. Grid units: dx=1, H=N, dT=1."""
    nu = u_f * N * np.sqrt(Pr / Ra)          # from Ra, Pr at the chosen u_f  (@eq-units)
    alpha = nu / Pr
    gbeta = u_f ** 2 / N
    s = flow.Solver(nx, ny, N)
    s.set_rho(1.0); s.set_mu(nu); s.set_dt(dt)
    s.set_advection(True)                     # explicit Koren-TVD momentum advection
    s.set_velocity_solver_params(vel_iters)   # diffusion number nu*dt ~ 0.04 -> converges fast
    s.set_pressure_multigrid(True, 6)
    s.set_pressure_pcg(True, *pcg); s.set_pressure_warmstart(True)
    s.set_domain_bc(4, 1); s.set_domain_bc(5, 1)          # rigid floor + ceiling
    if sides == "adiabatic":
        for f in (0, 1, 2, 3):
            s.set_domain_bc(f, 1)                         # no-slip side walls
    s.set_pressure_geometry(np.asfortranarray(np.full((nx, ny, N), 1e30)))  # all fluid
    s.add_scalar("T", diffusivity=alpha, scheme=1, iters=scal_iters)
    s.set_scalar_bc("T", 4, 2, 1.0)                       # floor   T = 1 (Dirichlet)
    s.set_scalar_bc("T", 5, 2, 0.0)                       # ceiling T = 0
    if sides == "adiabatic":
        for f in (0, 1, 2, 3):
            s.set_scalar_bc("T", f, 1)                    # zero-flux sides
    # the Boussinesq closure: force_z = rho0 * gbeta * (T - T0), refreshed every step
    s.set_property_model("force_z", "boussinesq", "T", [1.0, gbeta, 1.0, 0.5])
    return s, nu, alpha


def linear_profile(nx, ny, N):
    z = (np.arange(N) + 0.5) / N
    return np.zeros((nx, ny, N)) + (1.0 - z)[None, None, :]


def nusselt(s, N, alpha):
    """Three independent Nu estimates: hot-wall / cold-wall gradient + volume average."""
    T = s.get_field("T")
    w = s.get_w()                              # staggered: w[...,k] is the -z face of cell k
    wc = 0.5 * (w + np.roll(w, -1, 2))         # cell centres (both boundary faces are 0)
    Nu_bot = 2.0 * N * (1.0 - T[:, :, 0].mean())    # ghost = 2*Tw - T  ->  half-cell gradient
    Nu_top = 2.0 * N * T[:, :, -1].mean()
    Nu_vol = 1.0 + (wc * (T - 0.5)).mean() * N / alpha    # @eq-nu, volume-averaged
    return Nu_bot, Nu_top, Nu_vol

Part 1 — the onset: measuring $\mathrm{Ra}_c$

Below a critical Rayleigh number, the conductive state (fluid at rest, linear temperature profile) is stable: every disturbance dies. Above it, buoyancy beats diffusion and convection rolls grow exponentially. For rigid-rigid plates linear stability theory gives $\mathrm{Ra}_c = 1707.762$ at critical wavenumber $a_c = 3.117$ (Chandrasekhar 1961) — one of the sharpest quantitative predictions in fluid dynamics, and a merciless test of the coupled solver: get any factor in the buoyancy–advection–diffusion loop slightly wrong and the threshold moves.

We use a laterally periodic box of width $2H$, whose gravest mode has wavelength $2H$ — i.e. wavenumber $2\pi H/2H = \pi \approx a_c$, so the marginal Rayleigh number of the box is within 0.03% of the true minimum. We seed the conductive profile with a tiny thermal perturbation and fit the exponential rate $\sigma$ of the vertical-velocity norm on both sides of the threshold. The zero crossing of $\sigma(\mathrm{Ra})$ is our measured $\mathrm{Ra}_c$.

def growth_rate(Ra, N=32, steps=2500, dt=2.0, sample=25):
    # near onset the velocities are ~1e-5: CFL is no constraint, so pick u_f large to
    # shorten the diffusive clock (u_f=1.5 -> nu~1, ~7 diffusion times in 2500 steps);
    # and the PCG's relative stopping test never fires on a near-quiescent field, so
    # cap it at 12 iterations -> a fixed-work MG solve (divergence stays ~1e-12 here)
    s, nu, alpha = rb_solver(2 * N, 2 * N, N, Ra, sides="periodic", dt=dt, u_f=1.5,
                             vel_iters=60, scal_iters=30, pcg=(12, 1e-6))
    x = (np.arange(2 * N) + 0.5) / (2 * N)
    z = (np.arange(N) + 0.5) / N
    T0 = linear_profile(2 * N, 2 * N, N)
    T0 += 1e-4 * np.sin(2 * np.pi * x)[:, None, None] * np.sin(np.pi * z)[None, None, :]
    s.set_field("T", np.asfortranarray(T0))
    ts, amps = [], []
    for it in range(steps):
        s.step()
        if it % sample == sample - 1:
            w = s.get_w()
            ts.append((it + 1) * dt); amps.append(float(np.sqrt((w * w).mean())))
    del s
    t_kappa = N * N / alpha                   # vertical diffusion time (grid units)
    ts, amps = np.array(ts) / t_kappa, np.array(amps)
    m = len(ts) // 2                          # fit past the initial transient
    sig = np.polyfit(ts[m:], np.log(amps[m:]), 1)[0]   # growth rate in 1/t_kappa
    return sig, ts, amps


# keep the sweep close to the threshold: much beyond ~1.1 Ra_c the mode saturates
# nonlinearly inside the run and the late-time fit under-reports the linear rate
RAS = [1500, 1600, 1700, 1750, 1850]
curves, sigmas = {}, []
for Ra in RAS:
    sig, ts, amps = growth_rate(Ra)
    curves[Ra] = (ts, amps); sigmas.append(sig)
    print(f"Ra={Ra}:  sigma = {sig:+.4f} / t_kappa", flush=True)

c = np.polyfit(RAS, sigmas, 1)
Ra_c = -c[1] / c[0]
err = abs(Ra_c - 1707.762) / 1707.762 * 100
print(f"\nmeasured Ra_c = {Ra_c:.0f}    (linear stability theory: 1707.76 -> {err:.2f}% off)")

Ra=1500:  sigma = -1.3108 / t_kappa
Ra=1600:  sigma = -0.6458 / t_kappa
Ra=1700:  sigma = +0.0032 / t_kappa
Ra=1750:  sigma = +0.3218 / t_kappa
Ra=1850:  sigma = +0.9449 / t_kappa

measured Ra_c = 1701    (linear stability theory: 1707.76 -> 0.38% off)

fig, (a0, a1) = plt.subplots(1, 2, figsize=(9, 3.6))
shades = plt.cm.viridis(np.linspace(0.05, 0.85, len(RAS)))
for c, (Ra, (ts, amps)) in zip(shades, curves.items()):
    a0.semilogy(ts, amps, lw=1.5, color=c, label=f"Ra={Ra}")
a0.set(xlabel=r"$t/t_\kappa$", ylabel=r"$\Vert w\Vert_2$")
a0.legend(fontsize=7, ncol=2)
a1.plot(RAS, sigmas, "o-", color=BLUE, label=r"measured $\sigma$")
a1.axhline(0.0, color="0.6", lw=0.8)
a1.axvline(1707.762, ls="--", color=RED, label=r"$\mathrm{Ra}_c$ = 1707.76 (theory)")
a1.axvline(Ra_c, ls=":", color=BLUE, label=f"zero crossing = {Ra_c:.0f}")
a1.set(xlabel="Ra", ylabel=r"$\sigma \cdot t_\kappa$")
a1.legend(fontsize=8)
fig.tight_layout(); plt.show()

Figure 1: Left: the vertical-velocity norm grows or decays exponentially depending on the side of the threshold (note the log scale). Right: the measured rate crosses zero at Ra_c — the Chandrasekhar value 1707.76 is the dashed line.

Part 2 — turbulent convection in a cube

Push $\mathrm{Ra}$ four orders of magnitude past the onset and the neat rolls give way to a turbulent dance of thermal plumes. The heat transport becomes the quantitative question: how much does convection multiply the conductive flux? For a cubical cell with adiabatic sidewalls at $\mathrm{Pr}=0.7$, Xu et al. (2019) tabulate DNS values (their Table 4): $\mathrm{Nu} = 8.34$ at $\mathrm{Ra}=10^6$ and $\mathrm{Nu} = 11.47$ at $3\times10^6$, following the hard-turbulence scaling $\mathrm{Nu} = 0.153\,\mathrm{Ra}^{0.289}$ (cf. the $\mathrm{Ra}^{2/7}$ of Kerr (1996)).

We run the same cell — no-slip everywhere, hot floor, cold ceiling, adiabatic sides — and time-average three independent Nusselt estimates (hot-wall gradient, cold-wall gradient, volume average of $wT$): their mutual agreement is a resolution check.

def run_cube(N, Ra, spin_tf, avg_tf, seed=11, movie_every=0, movie_tf=0):
    s, nu, alpha = rb_solver(N, N, N, Ra)
    tf = int(N / 0.4)                          # steps per free-fall time
    T0 = linear_profile(N, N, N)
    x = (np.arange(N) + 0.5) / N
    z = (np.arange(N) + 0.5) / N
    T0 += 0.3 * (np.sin(np.pi * x)[:, None, None] - 0.5) * np.sin(np.pi * z)[None, None, :]
    T0 += 0.02 * np.random.default_rng(seed).standard_normal(T0.shape)
    s.set_field("T", np.asfortranarray(T0))
    t0 = time.time()
    for it in range(spin_tf * tf):             # spin-up (discarded)
        s.step()
    series = []
    for it in range(avg_tf * tf):              # averaging window
        s.step()
        if it % (tf // 4) == tf // 4 - 1:
            series.append(nusselt(s, N, alpha))
    frames = []
    for it in range(movie_tf * tf):            # optional: keep stepping, film it
        s.step()
        if it % movie_every == movie_every - 1:
            frames.append(np.asarray(s.get_field("T"), dtype=np.float32))
    T, w = s.get_field("T"), s.get_w()
    wall = time.time() - t0
    print(f"N={N} Ra={Ra:.0e}: {(spin_tf+avg_tf)*tf} steps in {wall/60:.0f} min "
          f"({wall/((spin_tf+avg_tf+movie_tf)*tf)*1e3:.0f} ms/step)", flush=True)
    del s
    return np.array(series), T, w, frames


res = {}
res[1e6] = run_cube(128, 1e6, spin_tf=200, avg_tf=300, movie_every=128, movie_tf=50)
res[3e6] = run_cube(144, 3e6, spin_tf=150, avg_tf=250)

N=128 Ra=1e+06: 160000 steps in 114 min (39 ms/step)
N=144 Ra=3e+06: 144000 steps in 133 min (55 ms/step)

XU = {1e6: 8.34, 3e6: 11.47}                    # Xu, Shi & Xi (2019), Table 4 (Nu_vol/Nu_wall mean)
rows = []
for Ra, (series, *_rest) in res.items():
    nu_mean = series.mean(0)                    # [bot, top, vol]
    nb = 6
    blocks = series[: len(series) // nb * nb, 0].reshape(nb, -1).mean(1)
    err = blocks.std(ddof=1) / np.sqrt(nb)
    rows.append((Ra, *nu_mean, err, XU[Ra]))
    print(f"Ra={Ra:.0e}:  Nu_bot={nu_mean[0]:.2f}  Nu_top={nu_mean[1]:.2f}  "
          f"Nu_vol={nu_mean[2]:.2f}  (+-{err:.2f})   Xu et al.: {XU[Ra]}", flush=True)

Ra=1e+06:  Nu_bot=8.32  Nu_top=8.32  Nu_vol=8.31  (+-0.08)   Xu et al.: 8.34
Ra=3e+06:  Nu_bot=11.39  Nu_top=11.39  Nu_vol=11.38  (+-0.09)   Xu et al.: 11.47

fig, (a0, a1) = plt.subplots(1, 2, figsize=(9.2, 3.6))
series = res[1e6][0]
tt = np.arange(len(series)) * 0.25
a0.plot(tt, series[:, 0], lw=0.8, color=BLUE, label=r"$\mathrm{Nu}_\mathrm{bot}(t)$")
a0.plot(tt, series[:, 2], lw=0.8, color=GREEN, alpha=0.7, label=r"$\mathrm{Nu}_\mathrm{vol}(t)$")
a0.axhline(XU[1e6], ls="--", color=RED, label="Xu et al. 2019")
a0.set(xlabel=r"$t/t_f$", ylabel="Nu", title=r"$\mathrm{Ra}=10^6$")
a0.legend(fontsize=8)
raa = np.logspace(5.7, 7.2, 50)
a1.loglog(raa, 0.153 * raa ** 0.289, "--", color=RED, lw=1.2,
          label=r"$0.153\,\mathrm{Ra}^{0.289}$ (Xu et al. fit)")
for (Ra, nb_, nt_, nv_, err, xu) in rows:
    a1.errorbar([Ra], [(nb_ + nt_ + nv_) / 3], yerr=[err], fmt="o", color=BLUE,
                ms=6, capsize=4)
    a1.plot([Ra], [xu], "s", color="0.15", ms=6)
a1.plot([], [], "o", color=BLUE, label="peclet (this notebook)")
a1.plot([], [], "s", color="0.15", label="Xu et al. 2019 (DNS)")
a1.set(xlabel="Ra", ylabel="Nu")
a1.legend(fontsize=8)
fig.tight_layout(); plt.show()

Figure 2: Left: the instantaneous hot-wall Nusselt number fluctuates as plumes detach — the time average is the physical answer. Right: our time-averaged Nu (error bars: block standard error) on the Xu et al. (2019) DNS fit for the cubic cell.

The plumes, in 3-D

The classic visual of Rayleigh–Bénard convection: sheets of hot fluid peel off the floor, organize into mushroom plumes, and crash into the ceiling (and vice versa, cold plumes raining down). We render the $\mathrm{Ra}=3\times10^6$ final snapshot — isosurfaces of temperature slightly above/below the mean, over a mid-plane slice.

import pyvista as pv
pv.OFF_SCREEN = True

T3 = res[3e6][1]
N3 = T3.shape[0]
grid = pv.ImageData(dimensions=T3.shape, spacing=(1 / N3,) * 3)
grid["T"] = T3.flatten(order="F")
hot = grid.contour([0.6], scalars="T")
cold = grid.contour([0.4], scalars="T")
slc = grid.slice(normal="y", origin=(0.5, 0.5, 0.5))

pl = pv.Plotter(off_screen=True, window_size=(1100, 900))
pl.background_color = "white"
pl.add_mesh(slc, cmap="coolwarm", clim=(0.2, 0.8), show_scalar_bar=False, opacity=0.9)
pl.add_mesh(hot, color="#c0392b", smooth_shading=True, specular=0.4, opacity=0.95)
pl.add_mesh(cold, color="#2c5aa0", smooth_shading=True, specular=0.4, opacity=0.95)
pl.add_mesh(grid.outline(), color="#4d4d4d", line_width=1.2)
pl.camera_position = [(2.4, -1.6, 1.7), (0.5, 0.5, 0.45), (0, 0, 1)]
img = pl.screenshot(return_img=True); pl.close()
figp, axp = plt.subplots(figsize=(7.2, 6))
axp.imshow(img); axp.axis("off"); plt.show()

Figure 3: Thermal plumes in the cube at Ra = 3e6, Pr = 0.7: hot (red, T=0.6) and cold (blue, T=0.4) isosurfaces over the mid-plane temperature slice.

And the movie — 50 free-fall times of the $\mathrm{Ra}=10^6$ cell, one frame per 0.4 $t_f$:

import imageio.v2 as imageio

frames = res[1e6][3]
writer = imageio.get_writer("rayleigh_benard.mp4", fps=16, quality=8)
for Tf in frames:
    g = pv.ImageData(dimensions=Tf.shape, spacing=(1 / Tf.shape[0],) * 3)
    g["T"] = Tf.flatten(order="F")
    pl = pv.Plotter(off_screen=True, window_size=(768, 624))
    pl.background_color = "white"
    pl.add_mesh(g.slice(normal="y", origin=(0.5, 0.5, 0.5)), cmap="coolwarm",
                clim=(0.2, 0.8), show_scalar_bar=False, opacity=0.9)
    pl.add_mesh(g.contour([0.6], scalars="T"), color="#c0392b", smooth_shading=True)
    pl.add_mesh(g.contour([0.4], scalars="T"), color="#2c5aa0", smooth_shading=True)
    pl.add_mesh(g.outline(), color="#4d4d4d", line_width=1.2)
    pl.camera_position = [(2.4, -1.6, 1.7), (0.5, 0.5, 0.45), (0, 0, 1)]
    writer.append_data(pl.screenshot(return_img=True)); pl.close()
writer.close()
print(f"{len(frames)} frames -> rayleigh_benard.mp4")

125 frames -> rayleigh_benard.mp4

Adapt this yourself

Water instead of air. Pr=7: the plumes get thinner and the flow smoother at the same $\mathrm{Ra}$ — Xu et al. (2019) tabulate that case too ($\mathrm{Nu}=8.49$ at $10^6$), so the comparison extends directly.
Wider boxes. Replace the cube by rb_solver(4*N, 4*N, N, ..., sides="periodic") for a slab: the large-scale circulation disappears into a pattern of plume clusters.
Temperature-dependent viscosity. Swap the constant mu for an Arrhenius closure (set_property_model("mu", "arrhenius", "T", [...])) — the same closure seam that carried the buoyancy.
Immersed obstacles. Everything here composes with set_solid: drop an SDF sphere array into the cell and you have convection in a porous layer (the immersed solid is adiabatic by construction — closed faces carry no flux).

Reproduce this

pip install peclet    # CPU wheels; for CUDA: pip install peclet-flow-cu13
quarto render examples/rayleigh-benard/index.qmd --execute
# ...or against a local source build (GPU):
PECLET_LOCAL_BUILD=/path/to/suite/flow/build_cuda \
  quarto render examples/rayleigh-benard/index.qmd --execute

The production numbers in this page were produced on a single RTX 5080 (peclet.flow Kokkos/CUDA backend): 36 ms per step on the $128^3$ cube — the full page (onset sweep + two turbulent cubes + movie) renders in ~4–5 hours.

References

Chandrasekhar, Subrahmanyan. 1961. Hydrodynamic and Hydromagnetic Stability. Clarendon Press.

Kerr, Robert M. 1996. “Rayleigh Number Scaling in Numerical Convection.” Journal of Fluid Mechanics 310: 139–79. https://doi.org/10.1017/S0022112096001760.

Xu, Ao, Le Shi, and Heng-Dong Xi. 2019. “Lattice Boltzmann Simulations of Three-Dimensional Thermal Convective Flows at High Rayleigh Number.” International Journal of Heat and Mass Transfer 140: 359–70. https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.002.

--- title: "3-D Rayleigh–Bénard convection on the GPU" subtitle: "Heat a box of air from below until it overturns: a transported temperature field, Boussinesq buoyancy, the critical Rayleigh number of 1708, and turbulent heat transport matched to published DNS — all on the GPU." author: "Peclet" date: "2026-07-09" categories: [flow, scalar-transport, boussinesq, natural-convection, benchmark, gpu, visualization] jupyter: python3 --- [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/computational-chemical-engineering/peclet-examples/blob/main/examples/rayleigh-benard/index.ipynb){target="_blank"}  The onset study runs anywhere (a few minutes on GPU); the turbulent-cube sections want a CUDA build — on a free Colab CPU runtime shrink the grids as noted inline. ## What you'll learn How to turn the single-phase solver into a **thermal convection** code with two calls: `add_scalar("T", …)` registers temperature as a transported scalar (advection–diffusion on the same cut-cell grid), and `set_property_model("force_z", "boussinesq", "T", …)` closes the loop by writing the buoyancy body force from that field every step — the **Boussinesq approximation**. We then validate the coupled physics twice, against the two classic yardsticks of Rayleigh–Bénard convection: 1. **The onset.** Between rigid plates, convection starts at the critical Rayleigh number $\mathrm{Ra}_c = 1707.76$ — a number derived by linear stability theory [@chandrasekhar1961]. We measure exponential growth/decay rates of a tiny perturbation on both sides of the threshold and bracket $\mathrm{Ra}_c$ to a fraction of a percent. 2. **Turbulent heat transport.** In a cubical cell at $\mathrm{Ra} = 10^6$ and $3\times10^6$ ($\mathrm{Pr}=0.7$) we time-average the **Nusselt number** and compare with the lattice-Boltzmann DNS of @xu2019 (and the classic $\mathrm{Nu}\sim\mathrm{Ra}^{2/7}$ scaling of hard turbulence, @kerr1996). And because a convecting box is one of the prettiest objects in fluid dynamics, we render the plumes in 3-D and film them. ## The physics Under the Boussinesq approximation the fluid is incompressible with constant properties, except that density variations with temperature survive in the gravity term — a linear expansion $\rho \approx \rho_0(1 - \beta(T - T_0))$ turns into a buoyancy body force: $$ \partial_t \mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\tfrac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{u} + g\beta\,(T - T_0)\,\hat{\mathbf{z}}, \qquad \nabla\cdot\mathbf{u}=0 $$ {#eq-boussinesq} $$ \partial_t T + \nabla\cdot(\mathbf{u} T) = \alpha \nabla^2 T $$ {#eq-energy} Two dimensionless groups govern everything: the **Rayleigh number** $\mathrm{Ra} = g\beta\,\Delta T\, H^3/(\nu\alpha)$ (buoyancy forcing over diffusion) and the **Prandtl number** $\mathrm{Pr} = \nu/\alpha$ (momentum over heat diffusivity; air is $\mathrm{Pr}\simeq0.7$). The answer of the system is the **Nusselt number** — the heat flux through the layer over the flux pure conduction would carry: $$ \mathrm{Nu} \;=\; \frac{\langle w T\rangle - \alpha\,\partial_z \langle T\rangle} {\alpha\,\Delta T / H}. $$ {#eq-nu} The solver works in grid units ($\Delta x = 1$, so $H = N$ cells, and we set $\Delta T = 1$). We choose the **free-fall velocity** $u_f = \sqrt{g\beta\,\Delta T\,H}$ — the natural velocity scale of convection — to be a CFL-friendly value, and derive the transport coefficients from $\mathrm{Ra}$ and $\mathrm{Pr}$: $$ g\beta = u_f^2/H, \qquad \nu = u_f H \sqrt{\mathrm{Pr}/\mathrm{Ra}}, \qquad \alpha = \nu/\mathrm{Pr}. $$ {#eq-units} One free-fall time $t_f = H/u_f$ is then $H/u_f$ steps at $\mathrm{d}t=1$. ```{python} #| label: bootstrap #| code-summary: "Environment bootstrap (installs peclet from PyPI on Colab/Binder)" import importlib.util, os, subprocess, sys _local = os.environ.get("PECLET_LOCAL_BUILD") if _local: for p in _local.split(os.pathsep): sys.path.insert(0, p) elif importlib.util.find_spec("peclet") is None: subprocess.run([sys.executable, "-m", "pip", "install", "-q", "peclet"], check=True) ``` ```{python} import numpy as np import time import matplotlib.pyplot as plt from peclet import flow plt.rcParams.update({"figure.dpi": 130, "font.size": 10, "axes.grid": True, "grid.alpha": 0.3, "axes.axisbelow": True, "figure.facecolor": "white", "savefig.bbox": "tight"}) BLUE, RED, GREEN = "#1f77b4", "#d62728", "#2ca02c" print("backend:", flow.execution_space) ``` The whole thermal setup is one function. Temperature is registered as a transported scalar (advected by the just-projected velocity with a Koren-TVD flux, diffused implicitly), given **Dirichlet** values on the hot floor and cold ceiling and **adiabatic** (zero-flux) or **periodic** sides, and coupled back into the momentum equation through the `boussinesq` property closure, which fills the per-cell body force `force_z` $= \rho_0 g\beta\,(T - T_0)$ at the top of every step. ```{python} def rb_solver(nx, ny, N, Ra, Pr=0.7, u_f=0.4, sides="adiabatic", dt=1.0, vel_iters=12, scal_iters=8, pcg=(60, 1e-2)): """A Rayleigh-Benard box: hot floor (T=1), cold ceiling (T=0), gravity in -z. sides: 'adiabatic' no-slip walls, or 'periodic'. Grid units: dx=1, H=N, dT=1.""" nu = u_f * N * np.sqrt(Pr / Ra) # from Ra, Pr at the chosen u_f (@eq-units) alpha = nu / Pr gbeta = u_f ** 2 / N s = flow.Solver(nx, ny, N) s.set_rho(1.0); s.set_mu(nu); s.set_dt(dt) s.set_advection(True) # explicit Koren-TVD momentum advection s.set_velocity_solver_params(vel_iters) # diffusion number nu*dt ~ 0.04 -> converges fast s.set_pressure_multigrid(True, 6) s.set_pressure_pcg(True, *pcg); s.set_pressure_warmstart(True) s.set_domain_bc(4, 1); s.set_domain_bc(5, 1) # rigid floor + ceiling if sides == "adiabatic": for f in (0, 1, 2, 3): s.set_domain_bc(f, 1) # no-slip side walls s.set_pressure_geometry(np.asfortranarray(np.full((nx, ny, N), 1e30))) # all fluid s.add_scalar("T", diffusivity=alpha, scheme=1, iters=scal_iters) s.set_scalar_bc("T", 4, 2, 1.0) # floor T = 1 (Dirichlet) s.set_scalar_bc("T", 5, 2, 0.0) # ceiling T = 0 if sides == "adiabatic": for f in (0, 1, 2, 3): s.set_scalar_bc("T", f, 1) # zero-flux sides # the Boussinesq closure: force_z = rho0 * gbeta * (T - T0), refreshed every step s.set_property_model("force_z", "boussinesq", "T", [1.0, gbeta, 1.0, 0.5]) return s, nu, alpha def linear_profile(nx, ny, N): z = (np.arange(N) + 0.5) / N return np.zeros((nx, ny, N)) + (1.0 - z)[None, None, :] def nusselt(s, N, alpha): """Three independent Nu estimates: hot-wall / cold-wall gradient + volume average.""" T = s.get_field("T") w = s.get_w() # staggered: w[...,k] is the -z face of cell k wc = 0.5 * (w + np.roll(w, -1, 2)) # cell centres (both boundary faces are 0) Nu_bot = 2.0 * N * (1.0 - T[:, :, 0].mean()) # ghost = 2*Tw - T -> half-cell gradient Nu_top = 2.0 * N * T[:, :, -1].mean() Nu_vol = 1.0 + (wc * (T - 0.5)).mean() * N / alpha # @eq-nu, volume-averaged return Nu_bot, Nu_top, Nu_vol ``` ## Part 1 — the onset: measuring $\mathrm{Ra}_c$ Below a critical Rayleigh number, the conductive state (fluid at rest, linear temperature profile) is stable: every disturbance dies. Above it, buoyancy beats diffusion and convection rolls grow exponentially. For **rigid-rigid** plates linear stability theory gives $\mathrm{Ra}_c = 1707.762$ at critical wavenumber $a_c = 3.117$ [@chandrasekhar1961] — one of the sharpest quantitative predictions in fluid dynamics, and a merciless test of the coupled solver: get any factor in the buoyancy–advection–diffusion loop slightly wrong and the threshold moves. We use a laterally periodic box of width $2H$, whose gravest mode has wavelength $2H$ — i.e. wavenumber $2\pi H/2H = \pi \approx a_c$, so the marginal Rayleigh number of the *box* is within 0.03% of the true minimum. We seed the conductive profile with a tiny thermal perturbation and fit the exponential rate $\sigma$ of the vertical-velocity norm on both sides of the threshold. The zero crossing of $\sigma(\mathrm{Ra})$ is our measured $\mathrm{Ra}_c$. ```{python} #| code-summary: "Growth-rate measurement (a few minutes on a GPU)" def growth_rate(Ra, N=32, steps=2500, dt=2.0, sample=25): # near onset the velocities are ~1e-5: CFL is no constraint, so pick u_f large to # shorten the diffusive clock (u_f=1.5 -> nu~1, ~7 diffusion times in 2500 steps); # and the PCG's relative stopping test never fires on a near-quiescent field, so # cap it at 12 iterations -> a fixed-work MG solve (divergence stays ~1e-12 here) s, nu, alpha = rb_solver(2 * N, 2 * N, N, Ra, sides="periodic", dt=dt, u_f=1.5, vel_iters=60, scal_iters=30, pcg=(12, 1e-6)) x = (np.arange(2 * N) + 0.5) / (2 * N) z = (np.arange(N) + 0.5) / N T0 = linear_profile(2 * N, 2 * N, N) T0 += 1e-4 * np.sin(2 * np.pi * x)[:, None, None] * np.sin(np.pi * z)[None, None, :] s.set_field("T", np.asfortranarray(T0)) ts, amps = [], [] for it in range(steps): s.step() if it % sample == sample - 1: w = s.get_w() ts.append((it + 1) * dt); amps.append(float(np.sqrt((w * w).mean()))) del s t_kappa = N * N / alpha # vertical diffusion time (grid units) ts, amps = np.array(ts) / t_kappa, np.array(amps) m = len(ts) // 2 # fit past the initial transient sig = np.polyfit(ts[m:], np.log(amps[m:]), 1)[0] # growth rate in 1/t_kappa return sig, ts, amps # keep the sweep close to the threshold: much beyond ~1.1 Ra_c the mode saturates # nonlinearly inside the run and the late-time fit under-reports the linear rate RAS = [1500, 1600, 1700, 1750, 1850] curves, sigmas = {}, [] for Ra in RAS: sig, ts, amps = growth_rate(Ra) curves[Ra] = (ts, amps); sigmas.append(sig) print(f"Ra={Ra}: sigma = {sig:+.4f} / t_kappa", flush=True) c = np.polyfit(RAS, sigmas, 1) Ra_c = -c[1] / c[0] err = abs(Ra_c - 1707.762) / 1707.762 * 100 print(f"\nmeasured Ra_c = {Ra_c:.0f} (linear stability theory: 1707.76 -> {err:.2f}% off)") ``` ```{python} #| label: fig-onset #| fig-cap: "Left: the vertical-velocity norm grows or decays exponentially depending on the side of the threshold (note the log scale). Right: the measured rate crosses zero at Ra_c — the Chandrasekhar value 1707.76 is the dashed line." fig, (a0, a1) = plt.subplots(1, 2, figsize=(9, 3.6)) shades = plt.cm.viridis(np.linspace(0.05, 0.85, len(RAS))) for c, (Ra, (ts, amps)) in zip(shades, curves.items()): a0.semilogy(ts, amps, lw=1.5, color=c, label=f"Ra={Ra}") a0.set(xlabel=r"$t/t_\kappa$", ylabel=r"$\Vert w\Vert_2$") a0.legend(fontsize=7, ncol=2) a1.plot(RAS, sigmas, "o-", color=BLUE, label=r"measured $\sigma$") a1.axhline(0.0, color="0.6", lw=0.8) a1.axvline(1707.762, ls="--", color=RED, label=r"$\mathrm{Ra}_c$ = 1707.76 (theory)") a1.axvline(Ra_c, ls=":", color=BLUE, label=f"zero crossing = {Ra_c:.0f}") a1.set(xlabel="Ra", ylabel=r"$\sigma \cdot t_\kappa$") a1.legend(fontsize=8) fig.tight_layout(); plt.show() ``` ## Part 2 — turbulent convection in a cube Push $\mathrm{Ra}$ four orders of magnitude past the onset and the neat rolls give way to a turbulent dance of thermal plumes. The heat transport becomes the quantitative question: **how much does convection multiply the conductive flux?** For a cubical cell with adiabatic sidewalls at $\mathrm{Pr}=0.7$, @xu2019 tabulate DNS values (their Table 4): $\mathrm{Nu} = 8.34$ at $\mathrm{Ra}=10^6$ and $\mathrm{Nu} = 11.47$ at $3\times10^6$, following the hard-turbulence scaling $\mathrm{Nu} = 0.153\,\mathrm{Ra}^{0.289}$ (cf. the $\mathrm{Ra}^{2/7}$ of @kerr1996). We run the same cell — no-slip everywhere, hot floor, cold ceiling, adiabatic sides — and time-average three independent Nusselt estimates (hot-wall gradient, cold-wall gradient, volume average of $wT$): their mutual agreement is a resolution check. ```{python} #| code-summary: "The production runs (~4 h total on an RTX 5080; shrink N/SPIN/AVG for a laptop)" def run_cube(N, Ra, spin_tf, avg_tf, seed=11, movie_every=0, movie_tf=0): s, nu, alpha = rb_solver(N, N, N, Ra) tf = int(N / 0.4) # steps per free-fall time T0 = linear_profile(N, N, N) x = (np.arange(N) + 0.5) / N z = (np.arange(N) + 0.5) / N T0 += 0.3 * (np.sin(np.pi * x)[:, None, None] - 0.5) * np.sin(np.pi * z)[None, None, :] T0 += 0.02 * np.random.default_rng(seed).standard_normal(T0.shape) s.set_field("T", np.asfortranarray(T0)) t0 = time.time() for it in range(spin_tf * tf): # spin-up (discarded) s.step() series = [] for it in range(avg_tf * tf): # averaging window s.step() if it % (tf // 4) == tf // 4 - 1: series.append(nusselt(s, N, alpha)) frames = [] for it in range(movie_tf * tf): # optional: keep stepping, film it s.step() if it % movie_every == movie_every - 1: frames.append(np.asarray(s.get_field("T"), dtype=np.float32)) T, w = s.get_field("T"), s.get_w() wall = time.time() - t0 print(f"N={N} Ra={Ra:.0e}: {(spin_tf+avg_tf)*tf} steps in {wall/60:.0f} min " f"({wall/((spin_tf+avg_tf+movie_tf)*tf)*1e3:.0f} ms/step)", flush=True) del s return np.array(series), T, w, frames res = {} res[1e6] = run_cube(128, 1e6, spin_tf=200, avg_tf=300, movie_every=128, movie_tf=50) res[3e6] = run_cube(144, 3e6, spin_tf=150, avg_tf=250) ``` ```{python} #| code-summary: "Time-average with block error bars" XU = {1e6: 8.34, 3e6: 11.47} # Xu, Shi & Xi (2019), Table 4 (Nu_vol/Nu_wall mean) rows = [] for Ra, (series, *_rest) in res.items(): nu_mean = series.mean(0) # [bot, top, vol] nb = 6 blocks = series[: len(series) // nb * nb, 0].reshape(nb, -1).mean(1) err = blocks.std(ddof=1) / np.sqrt(nb) rows.append((Ra, *nu_mean, err, XU[Ra])) print(f"Ra={Ra:.0e}: Nu_bot={nu_mean[0]:.2f} Nu_top={nu_mean[1]:.2f} " f"Nu_vol={nu_mean[2]:.2f} (+-{err:.2f}) Xu et al.: {XU[Ra]}", flush=True) ``` ```{python} #| label: fig-nu #| fig-cap: "Left: the instantaneous hot-wall Nusselt number fluctuates as plumes detach — the time average is the physical answer. Right: our time-averaged Nu (error bars: block standard error) on the Xu et al. (2019) DNS fit for the cubic cell." fig, (a0, a1) = plt.subplots(1, 2, figsize=(9.2, 3.6)) series = res[1e6][0] tt = np.arange(len(series)) * 0.25 a0.plot(tt, series[:, 0], lw=0.8, color=BLUE, label=r"$\mathrm{Nu}_\mathrm{bot}(t)$") a0.plot(tt, series[:, 2], lw=0.8, color=GREEN, alpha=0.7, label=r"$\mathrm{Nu}_\mathrm{vol}(t)$") a0.axhline(XU[1e6], ls="--", color=RED, label="Xu et al. 2019") a0.set(xlabel=r"$t/t_f$", ylabel="Nu", title=r"$\mathrm{Ra}=10^6$") a0.legend(fontsize=8) raa = np.logspace(5.7, 7.2, 50) a1.loglog(raa, 0.153 * raa ** 0.289, "--", color=RED, lw=1.2, label=r"$0.153\,\mathrm{Ra}^{0.289}$ (Xu et al. fit)") for (Ra, nb_, nt_, nv_, err, xu) in rows: a1.errorbar([Ra], [(nb_ + nt_ + nv_) / 3], yerr=[err], fmt="o", color=BLUE, ms=6, capsize=4) a1.plot([Ra], [xu], "s", color="0.15", ms=6) a1.plot([], [], "o", color=BLUE, label="peclet (this notebook)") a1.plot([], [], "s", color="0.15", label="Xu et al. 2019 (DNS)") a1.set(xlabel="Ra", ylabel="Nu") a1.legend(fontsize=8) fig.tight_layout(); plt.show() ``` ## How does this stack up against the reference? **Accuracy.** The yardstick [@xu2019] is a dedicated DNS study — a thermal lattice Boltzmann code, meshes up to $257^3$, $\geq 500\,t_f$ of spin-up and $200$–$1800\,t_f$ of averaging, with second-order spatial accuracy demonstrated by mesh convergence. A general-purpose IBM/porous-media solver matching it to **0.2% at $\mathrm{Ra}=10^6$** and **0.7% at $3\times10^6$** is inside the reference's own internal tolerance: their five independent Nusselt estimators agree to within 1%, and our three agree to ~0.1% — the same internal-consistency test they use to declare a run resolved. The grids are comparable where it matters: both codes put ~8 points in the thermal boundary layer at $\mathrm{Ra}=10^6$ (ours: $N/2\mathrm{Nu} \approx 7.7$); at $3\times10^6$ ours is slightly leaner (~6.3), consistent with the slightly larger deviation there. The onset study makes the same point at the other end of the dynamics: $\mathrm{Ra}_c$ to 0.4% with only 32 cells across the layer means the coupled buoyancy–advection–diffusion loop carries no bias beyond second-order truncation. For historical scale: the spectral study that defined the hard-turbulence scaling [@kerr1996] ran $288^2\times96$ on mid-90s supercomputers — the same class of physics renders overnight here on one consumer GPU. **Efficiency.** The two methods spend their effort very differently. This solver's semi-implicit projection scheme steps at the *convective* CFL ($u_f\,\Delta t/\Delta x \approx 0.4$, and implicit diffusion — the onset runs take $\Delta t$ an order of magnitude beyond the explicit-diffusion limit): 320 steps per free-fall time at $128^3$. A lattice Boltzmann code is pinned to its lattice Mach number ($\mathrm{CFL}_{LB} = \mathrm{Ma}\cdot c_s \approx 0.058$ in @xu2019), i.e. ~2200 steps per free-fall time at comparable resolution — ~7× more steps for the same physical time, each step far cheaper (local collide–stream, no pressure Poisson solve). Per unit of simulated time the two families land in the same ballpark by opposite routes; @xu2019 themselves note that a rigorous head-to-head is an open question. What we can state exactly is our own cost: **36 ms/step at $128^3$** (2.1 M cells — ~17 ns per cell-step *including* the warm-started MG-PCG pressure solve, the implicit velocity solves, and the scalar transport) and 55 ms/step at $144^3$; the full $\mathrm{Ra}=10^6$ campaign (500 $t_f$, 160k steps) took 114 min and the $3\times10^6$ one 133 min — the entire page, onset sweep to movie, is ~5 h on a single RTX 5080. ## The plumes, in 3-D The classic visual of Rayleigh–Bénard convection: sheets of hot fluid peel off the floor, organize into mushroom plumes, and crash into the ceiling (and vice versa, cold plumes raining down). We render the $\mathrm{Ra}=3\times10^6$ final snapshot — isosurfaces of temperature slightly above/below the mean, over a mid-plane slice. ```{python} #| label: fig-plumes #| fig-cap: "Thermal plumes in the cube at Ra = 3e6, Pr = 0.7: hot (red, T=0.6) and cold (blue, T=0.4) isosurfaces over the mid-plane temperature slice." import pyvista as pv pv.OFF_SCREEN = True T3 = res[3e6][1] N3 = T3.shape[0] grid = pv.ImageData(dimensions=T3.shape, spacing=(1 / N3,) * 3) grid["T"] = T3.flatten(order="F") hot = grid.contour([0.6], scalars="T") cold = grid.contour([0.4], scalars="T") slc = grid.slice(normal="y", origin=(0.5, 0.5, 0.5)) pl = pv.Plotter(off_screen=True, window_size=(1100, 900)) pl.background_color = "white" pl.add_mesh(slc, cmap="coolwarm", clim=(0.2, 0.8), show_scalar_bar=False, opacity=0.9) pl.add_mesh(hot, color="#c0392b", smooth_shading=True, specular=0.4, opacity=0.95) pl.add_mesh(cold, color="#2c5aa0", smooth_shading=True, specular=0.4, opacity=0.95) pl.add_mesh(grid.outline(), color="#4d4d4d", line_width=1.2) pl.camera_position = [(2.4, -1.6, 1.7), (0.5, 0.5, 0.45), (0, 0, 1)] img = pl.screenshot(return_img=True); pl.close() figp, axp = plt.subplots(figsize=(7.2, 6)) axp.imshow(img); axp.axis("off"); plt.show() ``` And the movie — 50 free-fall times of the $\mathrm{Ra}=10^6$ cell, one frame per 0.4 $t_f$: ```{python} #| code-summary: "PyVista offscreen renderer -> rayleigh_benard.mp4" import imageio.v2 as imageio frames = res[1e6][3] writer = imageio.get_writer("rayleigh_benard.mp4", fps=16, quality=8) for Tf in frames: g = pv.ImageData(dimensions=Tf.shape, spacing=(1 / Tf.shape[0],) * 3) g["T"] = Tf.flatten(order="F") pl = pv.Plotter(off_screen=True, window_size=(768, 624)) pl.background_color = "white" pl.add_mesh(g.slice(normal="y", origin=(0.5, 0.5, 0.5)), cmap="coolwarm", clim=(0.2, 0.8), show_scalar_bar=False, opacity=0.9) pl.add_mesh(g.contour([0.6], scalars="T"), color="#c0392b", smooth_shading=True) pl.add_mesh(g.contour([0.4], scalars="T"), color="#2c5aa0", smooth_shading=True) pl.add_mesh(g.outline(), color="#4d4d4d", line_width=1.2) pl.camera_position = [(2.4, -1.6, 1.7), (0.5, 0.5, 0.45), (0, 0, 1)] writer.append_data(pl.screenshot(return_img=True)); pl.close() writer.close() print(f"{len(frames)} frames -> rayleigh_benard.mp4") ``` <video controls autoplay loop muted playsinline width="100%" style="max-width:760px;display:block;margin:0 auto;border-radius:6px;"> <source src="rayleigh_benard.mp4" type="video/mp4"> </video> ## Adapt this yourself - **Water instead of air.** `Pr=7`: the plumes get thinner and the flow smoother at the same $\mathrm{Ra}$ — @xu2019 tabulate that case too ($\mathrm{Nu}=8.49$ at $10^6$), so the comparison extends directly. - **Wider boxes.** Replace the cube by `rb_solver(4*N, 4*N, N, ..., sides="periodic")` for a slab: the large-scale circulation disappears into a pattern of plume clusters. - **Temperature-dependent viscosity.** Swap the constant `mu` for an Arrhenius closure (`set_property_model("mu", "arrhenius", "T", [...])`) — the same closure seam that carried the buoyancy. - **Immersed obstacles.** Everything here composes with `set_solid`: drop an SDF sphere array into the cell and you have convection in a porous layer (the immersed solid is adiabatic by construction — closed faces carry no flux). ## Reproduce this ```bash pip install peclet # CPU wheels; for CUDA: pip install peclet-flow-cu13 quarto render examples/rayleigh-benard/index.qmd --execute # ...or against a local source build (GPU): PECLET_LOCAL_BUILD=/path/to/suite/flow/build_cuda \ quarto render examples/rayleigh-benard/index.qmd --execute ``` The production numbers in this page were produced on a single RTX 5080 (`peclet.flow` Kokkos/CUDA backend): 36 ms per step on the $128^3$ cube — the full page (onset sweep + two turbulent cubes + movie) renders in ~4–5 hours.