Library / Dimensional analysis and similarity / DIM-08
Dimensional analysis and similarity
Euler number
Ratio of a pressure difference to the inertial stress ρU².
- Ref
- DIM-08
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Eu
Eu = Δp/(ρU²) compares a chosen pressure difference with the inertial stress ρU². For a pressure loss use the upstream pressure minus the downstream pressure, so a positive Eu is a loss. Some authors normalise by the dynamic pressure ½ρU² instead, which gives a loss coefficient K = 2Eu; always check which one a source uses. At matched Reynolds numbers a component in a model and the prototype have the same Euler number, so a measured pressure drop can be scaled.
Ratio of a pressure difference to the inertial stress ρU². It is the pressure-loss coefficient of a component or the pressure coefficient in a flow, when normalised by ρU² rather than ½ρU².
Calculator
- Dynamic pressure, ½ρU²2,000 Pa
- Loss coefficient K = Δp/(½ρU²) = 2Eu0.25
- Head loss, Δp/(ρg)0.05097 m
- Pressure drop at twice the speed and the same Eu2,000 Pa
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 pressure drop of 500 Pa at 2 m/s in a fluid of 1,000 kg/m³ gives Eu = 0.125, a loss coefficient K = 0.25 when based on the dynamic pressure.
- Because Eu is size-independent, a pressure drop measured on a small model of a valve or bend predicts the full-size one: multiply Eu by the prototype ρU².
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Characteristic speed | 0.01 to 100 m/s | 5e-5 | 5,000 | 5,000 | +10% | −17.4 % |
| Δp Pressure difference (upstream minus downstream for a loss) | 1 to 1,000,000 Pa | 2.5e-4 | 250 | 250 | +10% | +10 % |
| ρ Fluid density | 1 to 2,000 kg/m³ | 0.0625 | 125 | 124.9 | +10% | −9.09 % |
Sweep of Characteristic speed
| Characteristic speed (m/s) | Euler number |
|---|---|
| 0.01 | 5,000 |
| 0.03162 | 500 |
| 0.1 | 50 |
| 0.3162 | 5 |
| 1 | 0.5 |
| 3.162 | 0.05 |
| 10 | 0.005 |
| 31.62 | 5e-4 |
| 100 | 5e-5 |
Sweep one variable
Variables and how each one contributes
Euler number
What the equation gives you. Everything else feeds it.
Pressure difference (upstream minus downstream for a loss)
The pressure difference, in Pa: upstream minus downstream for a loss across a component. Raises the Euler number in direct proportion. Raising it by 10 % (to 550) moves the result by +10 %, from 0.125 to 0.1375.
Fluid density
The fluid density, in kg/m³. Lowers the Euler number in inverse proportion. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 0.125 to 0.1136.
Characteristic speed
The characteristic speed, in m/s, usually the mean speed in the pipe or approach flow. Lowers the Euler number with the square of the speed, so speed is the strongest lever. Raising it by 10 % (to 2.2) moves the result by −17.4 %, from 0.125 to 0.1033.
Worked example with the values above
- Inertial stress: ρU² = 1,000 × 2² = 4,000 Pa.
- Divide the pressure difference by it: 500 / 4,000.
- Eu = 0.125.
Assumptions and limits
- A single well-defined pressure difference and a representative speed, with the same convention for both model and prototype.
- For scaling, geometric similarity and matched Reynolds number.
Common mistakes
- Dividing by ½ρU² and calling it the Euler number: that doubles the value.
- Using downstream minus upstream, which gives a negative loss.
- Using a speed from one section for a pressure difference defined at another.
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 (Δp × 1, ρ × −1, U × −2). Speed dominates, and a pressure difference from two separate readings carries the error of each: measure the difference directly where you can.
Where it is used
- Expressing the pressure loss of a valve, bend, orifice or heat-exchanger bank as a size-independent number.
- Scaling a measured pressure drop from a model to full size.
Maths you need
- Dynamic pressure: ½ρU², the inertial stress in a flow
- Pressure drop: upstream pressure minus downstream pressure