Fluid That Warms Up While Standing Still — Natural Convection and Rayleigh 1708

Light a flame underneath and the fluid refuses to budge for a while. The floor is hot, the ceiling cold, yet the liquid stands still and only passes heat upward. Then, at some instant, a whole field of striped vortices switches on at once. This post follows why that "threshold" is the Rayleigh number 1708, and goes all the way to the Nusselt correlations that set how well a warmed wall cools. By the end you'll push past the critical value yourself and compute the heat loss from a vertical plate in Python.

Where buoyancy and viscosity collide#

Natural convection — flow driven by density differences with no external pump — is a fight among three forces. Heat the floor and the fluid above it expands and lightens. A lighter parcel wants to rise. That is buoyancy.

If buoyancy were alone, flow would start instantly. But two things hold it back. Viscosity drags the rising parcel down through friction with its neighbors. Thermal diffusion quickly blurs the hot parcel's temperature, erasing the very temperature difference that drives the buoyancy.

So the verdict is a simple contest. If the two brakes — viscosity plus thermal diffusion — beat buoyancy, the fluid stays put. If buoyancy wins, convection switches on. Answering which side wins with a single number is the goal of this post.

The Boussinesq approximation — density change in a single term#

When density varies with temperature the equations get messy. The Boussinesq approximation — treating density as variable only inside the gravity term — lifts that weight. Everywhere else density is a constant; only the buoyancy term reacts to temperature.

Define the volumetric thermal expansion coefficient (the fractional drop in density per degree of warming).

Here $\beta$ is the expansion coefficient, $\rho$ the density, $T$ the temperature. For an ideal gas it drops cleanly to $\beta = 1/T$.

This definition trades the density difference for a temperature difference.

$\rho_\infty$ and $T_\infty$ are the density and temperature of the still fluid far away. That one line plants a buoyancy source in the momentum equation of a vertical-plate boundary layer.

$u$, $v$ are velocity components, $g$ gravity, $\nu$ kinematic viscosity. The first term on the right is buoyancy, the second viscous resistance. The energy equation carries temperature along by diffusion.

$\alpha$ is the thermal diffusivity (how fast heat spreads). This pair — momentum and energy tied together through buoyancy — is the skeleton of natural convection.

Grashof and Rayleigh — a division of labor#

Nondimensionalize and the buoyancy term collapses into a single number. That is the Grashof number.

$T_s$ is the wall temperature, $L$ the characteristic length. Grashof takes on the job that the Reynolds number does in forced convection. It measures how badly buoyancy overwhelms viscosity.

But viscosity is not buoyancy's only enemy. Thermal diffusion also erases the temperature gap. To count that second brake, multiply by the Prandtl number (momentum diffusion over heat diffusion, $\mathrm{Pr} = \nu/\alpha$). The result is the Rayleigh number.

Both viscosity $\nu$ and thermal diffusion $\alpha$ now sit in the denominator. That is the key. Whether convection ignites is set not by Gr alone but by Ra. Buoyancy must beat the product of the two brakes for flow to live.

The threshold called 1708#

Feed a layer heated from below into a linear stability analysis and a clean critical value appears. With solid walls on top and bottom the critical Rayleigh number is $\mathrm{Ra}_c = 1708$. Below it, every disturbance is eaten by viscosity and thermal diffusion and dies. The fluid stays still and heat moves by pure conduction.

The moment Ra crosses 1708 a pattern switches on: regularly spaced counter-rotating rolls — Bénard cells. The cell width is about twice the layer depth.

Play with the simulation below.

Drop Ra below 1708 and the particles nearly freeze while the temperature field splits cleanly top to bottom. Push past 1708 and counter-rotating cells switch on, sending the particles circulating. Raise Ra further and the circulation grows fierce while the boundary layers thin.

Python — tracking the Nusselt number of a vertical plate#

How well does a warmed wall cool? The answer is the Nusselt number (convective over conductive heat transfer, $\mathrm{Nu} = hL/k$). For a vertical isothermal plate there is a Churchill–Chu correlation that holds across the entire range. Let's compute a 60 °C wall standing in 20 °C air.

import numpy as np
 
def rayleigh_number(Ts, Tinf, L, beta, nu, alpha, g=9.81):
    # ratio of buoyancy to (viscosity x thermal diffusion)
    return g * beta * (Ts - Tinf) * L**3 / (nu * alpha)
 
def churchill_chu_vertical(Ra, Pr):
    # vertical isothermal plate -- valid over the whole range
    denom = (1.0 + (0.492 / Pr)**(9 / 16))**(8 / 27)
    return (0.825 + 0.387 * Ra**(1 / 6) / denom)**2
 
# air properties at the film temperature
nu, alpha, Pr, k = 1.85e-5, 2.60e-5, 0.71, 0.027
Ts, Tinf, L = 60.0, 20.0, 0.30          # 60C wall, 0.3 m tall
beta = 1.0 / (0.5 * (Ts + Tinf) + 273.15)  # ideal gas, 1/Tf
 
Ra = rayleigh_number(Ts, Tinf, L, beta, nu, alpha)
Nu = churchill_chu_vertical(Ra, Pr)
h = Nu * k / L
q = h * (Ts - Tinf)                     # heat loss per unit area
 
print(f"Ra = {Ra:.2e}")
print(f"Nu = {Nu:.1f},  h = {h:.2f} W/m^2K")
print(f"q  = {q:.1f} W/m^2")
 
# Ra = 7.03e+07
# Nu = 54.9,  h = 4.94 W/m^2K
# q  = 197.6 W/m^2

With $\mathrm{Ra} \approx 7\times10^7$ we're still in the laminar regime. One square meter of plate sheds about 198 watts. Change the temperature gap or the height and Ra moves immediately.

The chart below spreads the same correlation out along the Ra axis.

Lower Pr (toward liquid metals) and the curve drops; raise it (toward oils) and it climbs. Drag the Ra marker past $10^9$ and the slope switches from the laminar $\mathrm{Ra}^{1/4}$ to the turbulent $\mathrm{Ra}^{1/3}$.

A bundle of Nusselt correlations#

Each geometry has its own correlation. Here are the common ones.

• Vertical plate: $\mathrm{Nu} = 0.59\,\mathrm{Ra}^{1/4}$ (laminar, $10^4$–$10^9$), $\mathrm{Nu} = 0.10\,\mathrm{Ra}^{1/3}$ (turbulent, $10^9$–$10^{13}$). Whole range: Churchill–Chu.
• Horizontal heated plate, facing up: $\mathrm{Nu} = 0.54\,\mathrm{Ra}^{1/4}$ ($10^4$–$10^7$), $0.15\,\mathrm{Ra}^{1/3}$ ($10^7$–$10^{11}$).
• Horizontal heated plate, facing down: $\mathrm{Nu} = 0.27\,\mathrm{Ra}^{1/4}$.
• Enclosure heated from below: above $\mathrm{Ra}_c = 1708$, Bénard cells.

For horizontal plates the characteristic length is area over perimeter, $L = A_s/P$. The exponent $1/3$ in the turbulent regime hides a meaning. Since $\mathrm{Nu} \propto \mathrm{Ra}^{1/3}$ and $\mathrm{Ra} \propto L^3$, the $L$ cancels in $h = \mathrm{Nu}\,k/L$. The heat-transfer coefficient of turbulent natural convection is independent of size.

What to remember#

• Buoyancy's enemies are two: viscosity and thermal diffusion. So the onset of convection is set not by Gr alone but by Ra, which multiplies the two in.
• For $\mathrm{Ra} < 1708$ the fluid stands still (pure conduction); above it, Bénard cells switch on.
• $\mathrm{Nu}$ scales as $\mathrm{Ra}^{1/4}$ in the laminar regime and $\mathrm{Ra}^{1/3}$ in the turbulent one. In the latter the heat-transfer coefficient no longer depends on plate size.