Library / Dimensional analysis and similarity / DIM-04
Dimensional analysis and similarity
Full-scale force from a model measurement
Force on the full-size prototype predicted from a force measured on a scale model, when the two are dynamically similar: same shape and orientation, the same Reynolds number, and so the same force coefficient.
- Ref
- DIM-04
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Fp [N]
When a model and a prototype are dynamically similar (same shape and orientation, same Reynolds number), they have the same force coefficient π1 = F/(ρV²ℓ²). Equating π1 for the two gives Fp = Fm(ρpVp²ℓp²)/(ρmVm²ℓm²). The ratio is the ratio of the force scales ρV²ℓ², so a model measured at a different speed, fluid and size can be scaled directly to the full-size prediction. The scaling is only valid if the Reynolds numbers match; the Model test speed entry gives the speed that does so.
Force on the full-size prototype predicted from a force measured on a scale model, when the two are dynamically similar: same shape and orientation, the same Reynolds number, and so the same force coefficient. It is the force coefficient π₁ applied to the prototype conditions.
Calculator
- Force coefficient π1, the same for both0.003148
- Scaling factor, Fp/Fm0.2603
- Drag coefficient if the reference area is ℓ², 2π10.006297
- Prototype force scale ρpVp²ℓp²9,923 N
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 force of 120 N measured at 20.6 m/s on a 0.3 m model corresponds to 31.24 N on the 1.5 m prototype at 60 m/s. The coefficient π1 = 0.003148 is the same for both.
- The prediction is only as good as the similarity. If the model Reynolds number is lower than the prototype’s, the force coefficient may not carry over, particularly where the flow separates or changes from laminar to turbulent, so check the Reynolds numbers first.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρm Model fluid density | 0.1 to 2,000 kg/m³ | 15.59 | 311,800 | 311,700 | +10% | −9.09 % |
| ℓp Prototype length | 0.1 to 100 m | 0.1388 | 138,800 | 138,800 | +10% | +21 % |
| Vm Model test speed | 0.5 to 400 m/s | 0.08285 | 53,030 | 53,030 | +10% | −17.4 % |
| ρp Prototype fluid density | 0.1 to 2,000 kg/m³ | 2.55 | 51,000 | 51,000 | +10% | +10 % |
| Fm Force measured on the model | 0.1 to 100,000 N | 0.02603 | 26,030 | 26,030 | +10% | +10 % |
| ℓm Model length | 0.02 to 50 m | 0.001125 | 7,029 | 7,029 | +10% | −17.4 % |
| Vp Prototype speed | 1 to 400 m/s | 0.008677 | 1,388 | 1,388 | +10% | +21 % |
Sweep of Prototype speed
| Prototype speed (m/s) | Predicted full-scale force (N) |
|---|---|
| 1 | 0.008677 |
| 50.88 | 22.46 |
| 100.8 | 88.08 |
| 150.6 | 196.9 |
| 200.5 | 348.8 |
| 250.4 | 544 |
| 300.3 | 782.3 |
| 350.1 | 1,064 |
| 400 | 1,388 |
Sweep one variable
Variables and how each one contributes
Predicted full-scale force
What the equation gives you. Everything else feeds it.
Force measured on the model
The force measured on the model, in N. Raises the predicted full-scale force in direct proportion. Raising it by 10 % (to 132) moves the result by +10 %, from 31.24 to 34.36 N.
Prototype fluid density
The density of the fluid the prototype operates in, in kg/m³. Raises the predicted force in direct proportion. Raising it by 10 % (to 1.348) moves the result by +10 %, from 31.24 to 34.36 N.
Prototype speed
The prototype speed, in m/s. Raises the predicted force with the square of the speed. Raising it by 10 % (to 66) moves the result by +21 %, from 31.24 to 37.8 N.
Prototype length
The characteristic length of the prototype, in m. Raises the predicted force with the square of the length. Raising it by 10 % (to 1.65) moves the result by +21 %, from 31.24 to 37.8 N.
Model fluid density
The density of the model test fluid, in kg/m³. Lowers the predicted force in inverse proportion: a denser test fluid gives a bigger model force for the same coefficient. Raising it by 10 % (to 1,098) moves the result by −9.09 %, from 31.24 to 28.4 N.
Model test speed
The speed in the model test, in m/s. Lowers the predicted force with the square of the model speed. Raising it by 10 % (to 22.66) moves the result by −17.4 %, from 31.24 to 25.82 N.
Model length
The characteristic length of the model, in m. Lowers the predicted force with the square of the model length. Raising it by 10 % (to 0.33) moves the result by −17.4 %, from 31.24 to 25.82 N.
Worked example with the values above
- Force scales: prototype ρV²ℓ² = 1.225 × 60² × 1.5² = 9,922.5; model = 998 × 20.6² × 0.3² = 38,116.
- Their ratio is 0.2603.
- Fp = 120 × 0.2603 = 31.24 N.
Assumptions and limits
- Dynamic similarity: same shape and orientation, same Reynolds number, and the force coefficient depends only on the Reynolds number.
- Steady, incompressible flow with no free-surface effects, and the model force already corrected for tunnel blockage and support interference.
Common mistakes
- Scaling forces without checking that the Reynolds numbers match.
- Using different features for the model and prototype lengths.
- Using the model speed where the prototype speed is needed (or the reverse): the speeds usually differ a lot.
Statistics and uncertainty
Speeds and lengths enter squared, so their relative errors count double: the relative error of the prediction is roughly the root sum of squares of the force and density errors plus twice the speed and length errors. A 2 % error in each of the two speeds and two lengths alone gives about 8 %. Corrections for tunnel walls and supports, which are not included here, can be larger.
Where it is used
- Predicting full-scale lift, drag and loads from wind-tunnel and water-tunnel model tests.
- Sizing a model test so that the measured force is large enough to resolve.
Maths you need
- Force coefficient: force divided by ρV²ℓ², the same at matched Reynolds number
- Dynamic similarity: matching the dimensionless groups so the flows scale
Related formulas
Practise this formulaOpen the dimension solverOpen the model test planner