Library / Dimensional analysis and similarity / DIM-03
Dimensional analysis and similarity
Model test speed for Reynolds-number matching
Flow speed at which a geometrically similar scale model has the same Reynolds number as the full-size prototype, so that its force coefficient is the same under the stated assumptions.
- Ref
- DIM-03
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Vm [m/s]
For geometrically similar bodies in a steady incompressible flow, the force coefficient depends only on the Reynolds number Re = ρVℓ/μ. A scale model therefore reproduces the prototype’s force coefficient when its Reynolds number matches: ρmVmℓm/μm = ρpVpℓp/μp. Solving for the model speed gives Vm = Vp(ρp/ρm)(ℓp/ℓm)(μm/μp). A smaller model (ℓm < ℓp) needs a faster flow in the same fluid, which is often impractical, so a fluid with a lower kinematic viscosity μ/ρ, such as water instead of air, is used.
Flow speed at which a geometrically similar scale model has the same Reynolds number as the full-size prototype, so that its force coefficient is the same under the stated assumptions. A small model needs a faster flow, or a fluid with a lower kinematic viscosity, which is why small models are often tested in water.
Calculator
- Reynolds number matched, ρpVpℓp/μp6,163,000
- Speed ratio, Vm/Vp0.3437
- Length ratio, ℓm/ℓp0.2
- Model speed if the model were tested in the prototype fluid300 m/s
- Kinematic viscosity ratio, νm/νp0.06875
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.
- A model 5 times smaller than the prototype needs a test speed of 20.62 m/s in the model fluid to match the Reynolds number. In the prototype fluid it would need 300 m/s, which is why a different test fluid is used.
- Matching the Reynolds number can clash with other requirements: in air a very fast model test becomes compressible (the Mach number changes), and with a free surface the Froude number matters. Often only the most important group is matched and the force coefficient is checked for its dependence on Reynolds number.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρm Model fluid density | 0.1 to 2,000 kg/m³ | 10.29 | 205,800 | 205,800 | +10% | −9.09 % |
| ρp Prototype fluid density | 0.1 to 2,000 kg/m³ | 1.684 | 33,670 | 33,670 | +10% | +10 % |
| ℓp Prototype length | 0.1 to 100 m | 1.375 | 1,375 | 1,374 | +10% | +10 % |
| μp Prototype fluid dynamic viscosity | 1 to 2,000 µPa·s | 0.1845 | 369 | 368.8 | +10% | −9.09 % |
| ℓm Model length | 0.02 to 50 m | 0.1237 | 309.4 | 309.2 | +10% | −9.09 % |
| Vp Prototype speed | 1 to 400 m/s | 0.3437 | 137.5 | 137.2 | +10% | +10 % |
| μm Model fluid dynamic viscosity | 1 to 2,000 µPa·s | 0.02058 | 41.17 | 41.15 | +10% | +10 % |
Sweep of Model length
| Model length (m) | Model test speed (m/s) |
|---|---|
| 0.02 | 309.4 |
| 0.05318 | 116.3 |
| 0.1414 | 43.75 |
| 0.3761 | 16.45 |
| 1 | 6.187 |
| 2.659 | 2.327 |
| 7.071 | 0.875 |
| 18.8 | 0.3291 |
| 50 | 0.1237 |
Sweep one variable
Variables and how each one contributes
Model test speed
What the equation gives you. Everything else feeds it.
Prototype speed
The speed of the full-size prototype, in m/s. Raises the model test speed in direct proportion. Raising it by 10 % (to 66) moves the result by +10 %, from 20.62 to 22.69 m/s.
Prototype length
The characteristic length of the prototype, in m, such as a wing chord or hull length. Raises the model test speed in direct proportion: a larger prototype relative to the model needs a faster model flow. Raising it by 10 % (to 1.65) moves the result by +10 %, from 20.62 to 22.69 m/s.
Model length
The corresponding length of the model, in m. Use the same feature as for the prototype. Lowers the model test speed in inverse proportion: a smaller model needs a faster flow. Raising it by 10 % (to 0.33) moves the result by −9.09 %, from 20.62 to 18.75 m/s.
Prototype fluid density
The density of the fluid the prototype operates in, in kg/m³. Raises the model test speed in direct proportion. Raising it by 10 % (to 1.348) moves the result by +10 %, from 20.62 to 22.69 m/s.
Prototype fluid dynamic viscosity
The dynamic viscosity of the prototype fluid, in µPa·s (air about 17.9 at 15 °C). Lowers the model test speed in inverse proportion. Raising it by 10 % (to 19.68) moves the result by −9.09 %, from 20.62 to 18.75 m/s.
Model fluid density
The density of the fluid in the model test, in kg/m³ (water about 998). Lowers the model test speed in inverse proportion: a denser test fluid needs a lower speed. Raising it by 10 % (to 1,098) moves the result by −9.09 %, from 20.62 to 18.75 m/s.
Model fluid dynamic viscosity
The dynamic viscosity of the model test fluid, in µPa·s (water about 1002 at 20 °C). Raises the model test speed in direct proportion. Raising it by 10 % (to 1,102) moves the result by +10 %, from 20.62 to 22.69 m/s.
Worked example with the values above
- Ratios: ρp/ρm = 1.225 / 998 = 0.001227; ℓp/ℓm = 1.5 / 0.3 = 5; μm/μp = 1,002 / 17.89 = 56.01.
- Multiply them with the prototype speed: 60 × 0.001227 × 5 × 56.01.
- Vm = 20.62 m/s.
Assumptions and limits
- Geometric similarity: the model has the same shape and orientation, with every length scaled by one factor, and ideally scaled roughness.
- Steady incompressible flow in which the force coefficient depends only on the Reynolds number, with no free-surface or compressibility effects.
Common mistakes
- Entering the two viscosities in different units: both must be in µPa·s here.
- Mixing dynamic and kinematic viscosity: the formula uses dynamic viscosity with density.
- Using different features for the prototype and model lengths.
Statistics and uncertainty
The speed depends on a ratio of five property and length values, so its relative error is about the root sum of squares of their relative errors. Viscosity is the weak point: water viscosity changes by about 2 % per kelvin near room temperature, so the tunnel temperature must be measured and the viscosity taken at that temperature.
Where it is used
- Setting the speed of a water-tunnel or wind-tunnel test of a model aircraft, vehicle, hull or building.
- Checking whether Reynolds-number matching is practical or whether a different test approach is needed.
Maths you need
- Reynolds number: ratio of inertial to viscous forces, ρVℓ/μ
- Geometric similarity: same shape with every length scaled by one factor
Related formulas
Practise this formulaOpen the dimension solverOpen the model test planner