Library / Dimensional analysis and similarity / DIM-09
Dimensional analysis and similarity
Grashof number
Ratio of buoyancy to viscous forces in natural convection.
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- DIM-09
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- v1
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- Dimensional analysis and similarity
- Result
- Gr
Gr = gβΔT L³/ν² measures the buoyancy force produced by a temperature difference against the viscous force that resists the motion. It plays the role in natural convection that the Reynolds number plays in forced flow: the flow is laminar when Gr is small and turbulent when it is large. The thermal form assumes a small fractional change in density and uses ΔT as a magnitude, so it gives the size of the buoyancy effect, not its direction. For gases, β is about 1/T with T in kelvin (about 3.4 × 10⁻³ per K at room temperature).
Ratio of buoyancy to viscous forces in natural convection. It plays the role in free convection that the Reynolds number plays in forced flow.
Calculator
- Buoyancy velocity scale, √(gβΔT L)0.1716 m/s
- Rayleigh number for air, Pr = 0.71, Ra = 0.71 Gr928,700
- Gr with the length doubled1.046×107
- Gr with the temperature difference halved654,000
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For a 0.1 m surface 10 K warmer than the fluid, Gr = 1,308,000. Together with the Prandtl number this gives the Rayleigh number, which decides the natural-convection regime and the Nusselt number.
- Gr rises with the cube of the length, so a body twice as tall has eight times the Grashof number: large surfaces are far more likely to be turbulent in natural convection.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Characteristic length (for a vertical plate, its height) | 0.001 to 10 m | 1.308 | 1.308e12 | 1.308e12 | +10% | +33.1 % |
| ν Kinematic viscosity | 0.1 to 1,000 mm²/s | 294.3 | 2.943e10 | 2.943e10 | +10% | −17.4 % |
| ΔT Temperature difference (magnitude) | 0.1 to 500 K | 13,080 | 6.54e7 | 6.539e7 | +10% | +10 % |
| β Volumetric thermal expansion coefficient | 0.01 to 20 10⁻³/K | 4,360 | 8,720,000 | 8,716,000 | +10% | +10 % |
| g Gravitational acceleration | 1 to 30 m/s² | 133,300 | 4,000,000 | 3,867,000 | +10% | +10 % |
Sweep of Characteristic length (for a vertical plate, its height)
| Characteristic length (for a vertical plate, its height) (m) | Grashof number |
|---|---|
| 0.001 | 1.308 |
| 0.003162 | 41.36 |
| 0.01 | 1,308 |
| 0.03162 | 41,360 |
| 0.1 | 1,308,000 |
| 0.3162 | 4.136e7 |
| 1 | 1.308e9 |
| 3.162 | 4.136e10 |
| 10 | 1.308e12 |
Sweep one variable
Variables and how each one contributes
Grashof number
What the equation gives you. Everything else feeds it.
Gravitational acceleration
The gravitational acceleration, in m/s² (9.81 on Earth). Raises the Grashof number in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 1,308,000 to 1,439,000.
Volumetric thermal expansion coefficient
The volumetric thermal expansion coefficient, in units of 10⁻³ per K: about 3.4 for air at room temperature and about 0.2 for water. Raises the Grashof number in direct proportion: a fluid that expands more is more buoyant. Raising it by 10 % (to 3.3) moves the result by +10 %, from 1,308,000 to 1,439,000.
Temperature difference (magnitude)
The magnitude of the temperature difference between the surface and the fluid, in K. Raises the Grashof number in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 1,308,000 to 1,439,000.
Characteristic length (for a vertical plate, its height)
The characteristic length, in m: for a vertical plate or wall, its height. Raises the Grashof number with the cube of the length, so size is the strongest lever. Raising it by 10 % (to 0.11) moves the result by +33.1 %, from 1,308,000 to 1,741,000.
Kinematic viscosity
The kinematic viscosity, in mm²/s (air about 15, water about 1). Lowers the Grashof number with the square of ν: a more viscous fluid damps the motion strongly. Raising it by 10 % (to 16.5) moves the result by −17.4 %, from 1,308,000 to 1,081,000.
Worked example with the values above
- Convert: β = 3 × 10⁻³ /K, ν = 15 mm²/s = 1.5×10−5 m²/s.
- Numerator: gβΔT L³ = 9.81 × 0.003 × 10 × 0.1³ = 2.943×10−4.
- Gr = 2.943×10−4 / (1.5×10−5)² = 1,308,000.
Assumptions and limits
- Natural convection with a small fractional density variation (the Boussinesq approximation).
- Properties evaluated at a representative film temperature, with ΔT as a magnitude.
Common mistakes
- Entering β in 1/K instead of the 10⁻³/K used here: 0.0034 /K is 3.4.
- Entering ν in m²/s instead of mm²/s: 1.5 × 10⁻⁵ m²/s is 15 mm²/s.
- Using a different length from the one in the correlation, for instance the width instead of the height.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (g × 1, β × 1, ΔT × 1, L × 3, ν × −2). The length (cubed) and the viscosity (squared) dominate. Properties vary with temperature, so evaluate β and ν at the film temperature, the mean of the surface and fluid temperatures.
Where it is used
- Judging whether natural convection on a plate, wall or cylinder is laminar or turbulent.
- Calculating the Rayleigh number and then the natural-convection heat-transfer coefficient.
Maths you need
- Thermal expansion: density falls as temperature rises
- Kinematic viscosity: dynamic viscosity divided by density