Library / Dimensional analysis and similarity / DIM-11
Dimensional analysis and similarity
Bulk Richardson number for mixed convection
Ratio of buoyancy to imposed-flow inertia.
- Ref
- DIM-11
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Ri
Ri = gβΔT L/U² compares buoyancy with the inertia of an imposed flow of speed U. With Re = UL/ν it equals Gr/Re², so it shows which kind of convection controls the heat transfer. A value near 1 means buoyancy and the imposed flow are comparable (mixed convection); much less than 1 means forced convection dominates, and much more than 1 means natural convection does. A common rule of thumb is forced convection for Ri < 0.1 and natural convection for Ri > 10. The thermal form uses ΔT as a magnitude, so record separately whether buoyancy aids or opposes the flow.
Ratio of buoyancy to imposed-flow inertia. A value near 1 means natural and forced convection are comparable; much smaller means forced convection dominates, and much larger means buoyancy dominates.
Calculator
- Buoyancy velocity scale, √(gβΔT L)0.1716 m/s
- Speed at which Ri = 10.1716 m/s
- Ri with the flow speed doubled0.1839
- Flow speed divided by the buoyancy velocity, U/√(gβΔT L) = 1/√Ri1.166
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.
- With Ri = 0.736 the buoyancy and the imposed flow are of comparable importance, so this is mixed convection: neither a forced-convection nor a natural-convection correlation alone applies.
- The speed at which Ri = 1 is 0.172 m/s here. Below it buoyancy matters; well above it, the flow can be treated as forced convection.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Imposed flow speed | 0.01 to 20 m/s | 7.357e-5 | 294.3 | 294.3 | +10% | −17.4 % |
| L Characteristic length | 0.001 to 10 m | 0.007357 | 73.57 | 73.57 | +10% | +10 % |
| ΔT Temperature difference (magnitude) | 0.1 to 500 K | 0.007357 | 36.79 | 36.78 | +10% | +10 % |
| β Volumetric thermal expansion coefficient | 0.01 to 20 10⁻³/K | 0.002452 | 4.905 | 4.903 | +10% | +10 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.075 | 2.25 | 2.175 | +10% | +10 % |
Sweep of Imposed flow speed
| Imposed flow speed (m/s) | Richardson number |
|---|---|
| 0.01 | 294.3 |
| 0.02586 | 44.01 |
| 0.06687 | 6.581 |
| 0.1729 | 0.9841 |
| 0.4472 | 0.1471 |
| 1.156 | 0.022 |
| 2.991 | 0.00329 |
| 7.734 | 4.92e-4 |
| 20 | 7.357e-5 |
Sweep one variable
Variables and how each one contributes
Richardson number
What the equation gives you. Everything else feeds it.
Gravitational acceleration
The gravitational acceleration, in m/s² (9.81 on Earth). Raises the Richardson number in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 0.7357 to 0.8093.
Volumetric thermal expansion coefficient
The volumetric thermal expansion coefficient, in units of 10⁻³ per K: about 3.4 for air at room temperature. Raises the Richardson number in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 0.7357 to 0.8093.
Temperature difference (magnitude)
The magnitude of the temperature difference, in K. Raises the Richardson number in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 0.7357 to 0.8093.
Characteristic length
The characteristic length, in m. Raises the Richardson number in direct proportion. Raising it by 10 % (to 0.11) moves the result by +10 %, from 0.7357 to 0.8093.
Imposed flow speed
The speed of the imposed flow, in m/s. Lowers the Richardson number with the square of the speed: a faster flow makes buoyancy less important. Raising it by 10 % (to 0.22) moves the result by −17.4 %, from 0.7357 to 0.6081.
Worked example with the values above
- Buoyancy term: gβΔT L = 9.81 × 0.003 × 10 × 0.1 = 0.02943 m²/s².
- Flow inertia term: U² = 0.2² = 0.04 m²/s².
- Ri = 0.02943 / 0.04 = 0.73575.
Assumptions and limits
- A bulk thermal form with a single length, speed and temperature difference.
- A small fractional density variation, and a clear imposed flow with a known speed.
Common mistakes
- Confusing this thermal form with the gradient Richardson number used for stratified shear flows.
- Ignoring whether buoyancy aids or opposes the flow: the magnitude form cannot show this.
- Using Ri without checking that the length and speed are consistent with the Reynolds number used.
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 × 1, U × −2). Speed dominates, and in low-speed flows it is hard to measure well, so the Richardson number can be quite uncertain near the transition between regimes.
Where it is used
- Deciding whether a heated or cooled flow in a duct, room, channel or electronics enclosure is forced, natural or mixed convection.
- Choosing which heat-transfer correlation to apply.
Maths you need
- Grashof number: buoyancy relative to viscous forces
- Reynolds number: inertia relative to viscous forces