Grashof Number Simulator — Natural Convection Characteristic Number
Evaluate the Grashof number Gr = g B dT L^3 / nu^2, the ratio of buoyancy to viscous forces in free convection, in real time. From the temperature difference, characteristic length, thermal-expansion coefficient and kinematic viscosity the tool reports the Rayleigh number Ra = Gr Pr, the Churchill-Chu Nusselt number, and the laminar / turbulent regime, and visualizes the vertical-plate boundary layer and the Ra-Nu log-log chart to teach natural-convection physics.
Parameters
Temperature difference dT
K
Characteristic length L
m
Thermal-expansion coefficient B
x10^-3 /K
Kinematic viscosity nu
x10^-5 m^2/s
With the defaults (dT = 75 K, L = 0.5 m, B = 3.0e-3 / K, nu = 1.6e-5 m^2 / s, Pr = 0.7 for air) Gr is about 1.08e9, Ra about 7.55e8, Nu about 113.2 and the regime is Laminar. Pr is fixed at 0.7 (air) and the laminar / turbulent transition occurs at Ra = 1e9.
Results
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Grashof number
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Rayleigh number
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Nusselt number
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Flow regime
Natural convection on a vertical plate
A heated vertical plate (red) drives a rising plume of warm air (red arrows) while cooler ambient air (blue arrows) flows in from the side. The yellow envelope is the thermal boundary layer of thickness about L Ra^(-1/4), thinner at larger Ra and thus driving stronger heat transfer. The color gradient on the plate represents the temperature gradient.
Churchill-Chu Ra-Nu chart (log-log)
Horizontal axis: Rayleigh number Ra (log10, 10^3 to 10^12). Vertical axis: Nusselt number Nu (log10). Blue solid curve: Churchill-Chu correlation. Red vertical line: Ra = 1e9 laminar / turbulent boundary. Yellow marker: current (Ra, Nu) operating point. The slope is Nu proportional to Ra^(1/4) in laminar flow and Nu proportional to Ra^(1/3) in turbulent flow.
Theory & Key Formulas
Grashof number: the dimensionless ratio of buoyancy (caused by density differences from temperature variation) to viscous force in free convection.
The Rayleigh number is the Grashof number multiplied by the Prandtl number $\mathrm{Pr}=\nu/\alpha$, and it is the true governing parameter of natural convection:
$$\mathrm{Ra} = \mathrm{Gr}\cdot\mathrm{Pr}$$
For natural convection on a vertical plate the Churchill-Chu correlation expresses Nu as a single formula over the full Ra range:
$g$ is gravity (9.81 m/s²), $\beta$ is the thermal-expansion coefficient (1/T for an ideal gas), $\Delta T$ is the surface-to-ambient temperature difference, $L$ is the characteristic length (plate height for a vertical plate), $\nu$ is the kinematic viscosity and $\alpha$ is the thermal diffusivity. The flow is laminar for Ra below 10⁹ and turbulent for Ra at or above 10⁹.
What is the Grashof Number Simulator?
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My heat transfer course keeps talking about the Grashof number for natural convection, but I am not sure how it differs from the Reynolds number. They are both dimensionless, so when do I use which?
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Good question. The Reynolds number Re = rho U L / mu is the ratio of imposed inertial force to viscous force, and it characterizes forced convection where a fan or pump drives the flow. The Grashof number Gr = g beta dT L^3 / nu^2 is the ratio of buoyancy (caused by a temperature-induced density difference) to viscous force, and it characterizes free convection where the flow is driven by the fluid itself. With the tool defaults (air, dT = 75 K, L = 0.5 m, beta = 3e-3 / K, nu = 1.6e-5 m^2/s) you should see Gr about 1.08e9 — that is a typical value for "standing in front of a home heater" with a "vertical plate of about 50 cm". Fanless natural air cooling of IT equipment and the thermal performance of building envelopes are dominated by Gr.
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Then why do we multiply by Pr to get Ra = Gr Pr? Is Gr alone not enough?
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A fundamental question. Gr only captures how much buoyancy overcomes viscosity. But the onset of natural convection also depends on whether heat diffuses faster than momentum: if heat conducts away too quickly, buoyancy never builds up. That is captured by Pr = nu / alpha, and Ra = Gr Pr is the ratio of buoyancy drive to combined viscous and thermal damping. So Ra is the real governing parameter. In air (Pr = 0.7) Ra is about 0.7 Gr, in water (Pr = 7) Ra is about 7 Gr — water gets natural convection much more easily. With the tool defaults Ra is about 7.55e8, just below the laminar / turbulent transition Ra = 1e9. Increase dT to 100 K and the regime switches to turbulent right in front of your eyes.
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The red vertical line on the Ra-Nu chart is Ra = 1e9 and the blue curve is the Churchill-Chu Nu. Why does the curve change slope there?
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Nice observation. In laminar natural convection the boundary layer gives Nu proportional to Ra^(1/4), but once Ra crosses 1e9 the flow becomes turbulent and Nu scales as Ra^(1/3), so the slope steepens. The Churchill-Chu correlation Nu = {0.825 + 0.387 Ra^(1/6) / [1+(0.492/Pr)^(9/16)]^(8/27)}^2 smoothly spans both regimes in a single expression. With the tool defaults (Ra about 7.55e8, Pr = 0.7) you get Nu about 113.2, and the corresponding heat-transfer coefficient h = Nu k / L is about 5.9 W per square meter and kelvin for air — right in the middle of the "5 to 25" range typical of natural convection in air. Press the sweep button and watch the yellow marker cross the red line into the turbulent regime.
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Raising the thermal-expansion coefficient B from 3 to 10 more than triples Gr. What is B physically and how different is it for air and water?
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Important point. The thermal-expansion coefficient beta = -(1/rho)(d rho / d T) at constant pressure quantifies how much the fluid expands per kelvin of temperature rise, and a larger beta yields larger buoyancy for the same dT. For an ideal gas beta = 1/T (absolute), so air at 300 K has beta about 3.3e-3 / K. Water at 20 C has beta about 0.21e-3 / K, which is about one-sixteenth of air, but its density is roughly 800 times higher so the absolute buoyancy force in water is much bigger and you can feel strong convection from small temperature differences. Glycol antifreeze has beta about 0.5e-3 / K, engine oil about 0.7e-3 / K. The B slider in this tool shows how the same dT and L give very different Gr for different fluids — always check beta of your actual fluid when designing.
Frequently Asked Questions
The Grashof number Gr = g beta dT L^3 / nu^2 is the dimensionless ratio of buoyancy force (caused by density differences from temperature variation) to viscous force in free convection. Here g is gravity, beta is the fluid thermal-expansion coefficient (1/T for ideal gases), dT is the surface-to-ambient temperature difference, L is the characteristic length (plate height for a vertical plate), and nu is the kinematic viscosity. With the default values of this tool (dT = 75 K, L = 0.5 m, beta = 0.003 / K, nu = 1.6e-5 m^2 / s, air) the result is Gr about 1.08e9. If the Reynolds number is the characteristic number of forced convection, the Grashof number is its counterpart for free convection.
The Rayleigh number is defined as Ra = Gr Pr, that is the Grashof number multiplied by the Prandtl number Pr = nu / alpha (momentum diffusivity over thermal diffusivity). Ra is the combined ratio of buoyancy drive to damping by both viscosity and heat diffusion, and it is the genuine governing parameter for the onset and transition of natural convection. On a vertical plate the flow is laminar for Ra below 1e9 and turbulent for Ra at or above 1e9. With the tool defaults (air with Pr = 0.7) Ra is about 7.55e8, just inside the laminar regime, and a small increase in dT or L pushes the flow into the turbulent regime where the slope of Nu changes.
The Churchill-Chu correlation is an empirical equation that expresses the Nusselt number for natural convection on a vertical plate as a single formula over the entire range of Ra, from laminar to turbulent flow. It reads Nu = { 0.825 + 0.387 Ra^(1/6) / [ 1 + (0.492/Pr)^(9/16) ]^(8/27) }^2. With the tool defaults (Ra about 7.55e8, Pr = 0.7) Nu comes out around 113.2. The corresponding heat-transfer coefficient h = Nu k / L is about 5.9 W per square meter and kelvin for air (k about 0.026 W / (m K)), which is a typical natural-convection value.
The Grashof number is used to evaluate free-convection heat transfer where there is no fan or pump and the flow is driven entirely by buoyancy of the fluid itself. Typical examples are: (1) fanless heat sinks in electronic equipment, (2) heat loss by air around building walls and windows, (3) insulation and heat rejection of pipes, (4) buoyancy-driven mixing in chemical reactors, and (5) passive natural-circulation cooling in nuclear reactors during loss-of-power events. In mixed convection where forced (Re) and free (Gr) effects coexist, the governing mechanism is judged by Gr / Re^2 (above 10 buoyancy dominates, below 0.1 forced convection dominates).
Real-World Applications
Fanless heat sinks for electronics: In silent servers and embedded systems a CPU heat sink (fin height L = 50 mm, dT = 40 K) gives Gr about 4e5 and Ra about 3e5, and the tool returns Nu about 12 and h about 6 W per square meter and kelvin. To reject 30 W you need fin area A = Q / (h dT) = 30 / (6 * 40) = 0.125 m^2, so Grashof-number reasoning ties directly to fin geometry optimization.
Building envelope and insulation: A house exterior wall (height L = 3 m, dT = 20 K with indoor 22 C and outdoor 2 C) gives Gr about 1.8e10 and Ra about 1.3e10, well inside turbulent natural convection. The tool reports Nu about 230 and h about 2 W per square meter and kelvin, so a 100 m^2 wall loses about 4 kW to ambient air. Double glazing and insulation reshape Gr and Ra to cut this loss — the core of low-energy building design.
Passive natural-circulation cooling in reactors: Since Fukushima, "natural-circulation cooling during station blackout" has been a focus area. Decay heat from the reactor pressure vessel creates a density difference and drives a coolant loop without any pump. For a core of height L = 4 m, dT = 50 K, with pressurized water (beta about 1e-3 / K, nu about 1.3e-7 m^2/s) Gr reaches about 7e13, giving strong turbulent natural convection that can remove decay heat. The tool's nu slider stops at 0.1e-5, so reproduce this case with a slightly higher setting.
Temperature uniformity in chemical reactors: A large batch reactor (diameter L = 2 m, dT = 10 K) with an aqueous solution (beta about 0.3e-3 / K) gives Gr about 1.4e11, strong enough that natural convection mixes the fluid without an impeller. The flipside is thermal-runaway risk in exothermic reactions (Frank-Kamenetskii analysis), so Grashof-number evaluation directly informs the choice of agitator and cooling jacket. Try L = 2 m and dT = 10 K in this tool to see the case.
Common Pitfalls and Notes
The most common misconception is "the Grashof number alone is enough to characterize natural convection". In reality Gr only captures buoyancy versus viscosity, and the true governing parameter is the Rayleigh number Ra = Gr Pr, which includes the Prandtl number for heat diffusion. Air (Pr = 0.7) and water (Pr = 7) differ by an order of magnitude in Ra for the same Gr and behave very differently. This tool fixes Pr = 0.7 for air; for water or oil a Pr-specific Churchill-Chu evaluation or a Pr slider is needed. Textbooks routinely report Gr and Ra together.
The next pitfall is "above Ra = 1e9 the flow is always turbulent". In practice the transition depends on geometry (vertical plate, horizontal cylinder, horizontal disk, sphere) and boundary condition (isothermal wall, uniform heat flux). For an isothermal vertical plate the transition is near Ra = 1e9, for a horizontal cylinder near Ra = 1e7, for a horizontal heated surface facing up near Ra = 1e7, and a downward-facing heated surface is so strongly suppressed that Nu is nearly 1 (almost pure conduction). This tool covers Churchill-Chu for a vertical plate, but a correlation matched to the actual geometry is needed in design.
The last pitfall is "forced and free convection can be evaluated separately". Real equipment often sees mixed convection where both coexist, and the dominant mechanism is judged by Gr / Re^2. Above 10 buoyancy dominates, below 0.1 forced convection dominates, and in between a combined correlation such as Nu_mix^3 = Nu_forced^3 plus Nu_natural^3 is required. Indoor HVAC at low fan speed, outdoor piping in light wind, and heated rooms are all classic mixed-convection cases where evaluating Gr alone with this tool would underestimate the heat transfer.