Library / Dimensional analysis and similarity / DIM-10
Dimensional analysis and similarity
Rayleigh number
Buoyancy driving relative to the combined damping of momentum and heat diffusion.
- Ref
- DIM-10
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Ra
Ra = gβΔT L³/(να) combines buoyancy with both diffusive effects that resist it, momentum diffusion (ν) and heat diffusion (α). It equals the Grashof number times the Prandtl number, Pr = ν/α. Convection starts only when Ra exceeds a threshold that depends on the geometry, orientation and boundary conditions; for a thin horizontal layer between rigid plates heated from below it is about 1708, based on the layer depth. On a vertical plate, the boundary layer typically turns turbulent near Ra ≈ 10⁹.
Buoyancy driving relative to the combined damping of momentum and heat diffusion. It is the Grashof number times the Prandtl number, and decides whether natural convection starts and what regime it is in.
Calculator
- Prandtl number, ν/α0.75
- Grashof number, Ra/Pr1,308,000
- Ra with the length doubled7,848,000
- Ra divided by the layer-onset value 1708574.4
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 these conditions Ra = 981,000, well below the 10⁹ at which a vertical-plate boundary layer typically turns turbulent, so the flow is laminar.
- Nusselt-number correlations for natural convection are written in Ra (and sometimes Pr), so this is usually the step before finding the heat-transfer coefficient h = Nu k/L.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Characteristic length | 0.001 to 10 m | 0.981 | 9.81e11 | 9.81e11 | +10% | +33.1 % |
| α Thermal diffusivity | 0.01 to 1,000 mm²/s | 19,620 | 1.962e9 | 1.962e9 | +10% | −9.09 % |
| ν Kinematic viscosity | 0.1 to 1,000 mm²/s | 14,720 | 1.472e8 | 1.471e8 | +10% | −9.09 % |
| ΔT Temperature difference (magnitude) | 0.1 to 500 K | 9,810 | 4.905e7 | 4.904e7 | +10% | +10 % |
| β Volumetric thermal expansion coefficient | 0.01 to 20 10⁻³/K | 3,270 | 6,540,000 | 6,537,000 | +10% | +10 % |
| g Gravitational acceleration | 1 to 30 m/s² | 100,000 | 3,000,000 | 2,900,000 | +10% | +10 % |
Sweep of Temperature difference (magnitude)
| Temperature difference (magnitude) (K) | Rayleigh number |
|---|---|
| 0.1 | 9,810 |
| 0.29 | 28,450 |
| 0.8409 | 82,490 |
| 2.438 | 239,200 |
| 7.071 | 693,700 |
| 20.5 | 2,012,000 |
| 59.46 | 5,833,000 |
| 172.4 | 1.691e7 |
| 500 | 4.905e7 |
Sweep one variable
Variables and how each one contributes
Rayleigh number
What the equation gives you. Everything else feeds it.
Gravitational acceleration
The gravitational acceleration, in m/s² (9.81 on Earth). Raises the Rayleigh number in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 981,000 to 1,079,000.
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 Rayleigh number in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 981,000 to 1,079,000.
Temperature difference (magnitude)
The magnitude of the temperature difference across the fluid, in K. Raises the Rayleigh number in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 981,000 to 1,079,000.
Characteristic length
The characteristic length, in m: the layer depth for a heated layer, or the height of a vertical surface. Raises the Rayleigh 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 981,000 to 1,306,000.
Kinematic viscosity
The kinematic viscosity, in mm²/s (air about 15, water about 1). Lowers the Rayleigh number in inverse proportion. Raising it by 10 % (to 16.5) moves the result by −9.09 %, from 981,000 to 891,800.
Thermal diffusivity
The thermal diffusivity, in mm²/s (air about 20, water about 0.14). Lowers the Rayleigh number in inverse proportion. Raising it by 10 % (to 22) moves the result by −9.09 %, from 981,000 to 891,800.
Worked example with the values above
- Numerator: gβΔT L³ = 9.81 × 0.003 × 10 × 0.1³ = 2.943×10−4.
- Denominator: να = 1.5×10−5 × 2×10−5 = 3×10−10 m⁴/s².
- Ra = 981,000, which is Gr × Pr = 1,308,000 × 0.75.
Assumptions and limits
- Natural convection with a small fractional density variation (the Boussinesq approximation).
- The threshold for convection depends on the geometry and boundary conditions, and L is defined as in the correlation used.
Common mistakes
- Using the Rayleigh number without the Prandtl number correction where a correlation needs Pr as well.
- Mixing α and ν in different units.
- Applying the 1708 onset value to a vertical surface or to different boundary conditions.
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 × −1, α × −1). Length (cubed) dominates, and the fluid properties change with temperature, so evaluate them at the film temperature.
Where it is used
- Predicting whether a heated fluid layer will start to convect, and in which regime.
- Finding the natural-convection heat-transfer coefficient from a Nusselt correlation in Ra.
Maths you need
- Prandtl number: ratio of momentum to thermal diffusivity
- Thermal diffusivity: thermal conductivity over density times specific heat