Library / Dimensional analysis and similarity / DIM-02
Dimensional analysis and similarity
Dimensionless force coefficient
Dimensionless force coefficient that comes out of the π analysis of a body in a steady incompressible flow: the force divided by density times speed squared times length squared.
- Ref
- DIM-02
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- π1
For a body of fixed shape and orientation in a steady incompressible flow, the force F depends on the density ρ, the speed V, a length ℓ and the viscosity μ. These five variables with three base dimensions (n = 5, r = 3) give two dimensionless groups. Taking ρ, V and ℓ as the repeating variables, the force group F ρaVbℓc is dimensionless when mass (1 + a = 0), time (−2 − b = 0) and length (1 − 3a + b + c = 0) balance, which gives a = −1, b = −2 and c = −2. So π1 = F/(ρV²ℓ²). The viscosity group is the inverse of the Reynolds number, so the full result is π1 = f(Re). For geometrically similar bodies A/ℓ² is constant, and the drag coefficient CD = 2F/(ρV²A) differs from π1 only by a constant factor.
Dimensionless force coefficient that comes out of the π analysis of a body in a steady incompressible flow: the force divided by density times speed squared times length squared. It is the first of the two groups for drag (the other is the Reynolds number). For a reference area equal to ℓ² it is half the drag coefficient.
Calculator
- Drag coefficient if the reference area is ℓ², 2π10.2016
- Dynamic pressure, ½ρV²2,205 Pa
- Force scale ρV²ℓ²9,923 N
- Force at the same π1 with twice the speed4,000 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 force of 1,000 N on a body of length 1.5 m at 60 m/s in a fluid of density 1.225 kg/m³ gives π1 = 0.1008. If the reference area were ℓ², that would be a drag coefficient of 0.2016.
- Because π1 = f(Re), the same value applies at any size, speed or fluid that has the same Reynolds number: the force then scales as ρV²ℓ². At the same π1, twice the speed means four times the force.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| V Flow speed | 0.5 to 400 m/s | 0.002268 | 1,451 | 1,451 | +10% | −17.4 % |
| ℓ Characteristic length | 0.05 to 100 m | 2.268e-5 | 90.7 | 90.7 | +10% | −17.4 % |
| F Force on the body | 1 to 100,000 N | 1.008e-4 | 10.08 | 10.08 | +10% | +10 % |
| ρ Fluid density | 0.1 to 2,000 kg/m³ | 6.173e-5 | 1.235 | 1.235 | +10% | −9.09 % |
Sweep of Flow speed
| Flow speed (m/s) | Force coefficient |
|---|---|
| 0.5 | 1,451 |
| 50.44 | 0.1426 |
| 100.4 | 0.03601 |
| 150.3 | 0.01606 |
| 200.3 | 0.009048 |
| 250.2 | 0.005796 |
| 300.1 | 0.004028 |
| 350.1 | 0.002961 |
| 400 | 0.002268 |
Sweep one variable
Variables and how each one contributes
Force coefficient
What the equation gives you. Everything else feeds it.
Force on the body
The force on the body in the flow direction (the drag), in N. Raises the force coefficient in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 0.1008 to 0.1109.
Fluid density
The density of the fluid, in kg/m³ (air about 1.2, water about 1000). Lowers the coefficient in inverse proportion: a denser fluid gives a larger force for the same coefficient. Raising it by 10 % (to 1.348) moves the result by −9.09 %, from 0.1008 to 0.09162.
Flow speed
The flow speed relative to the body, in m/s. Lowers the coefficient with the square of the speed: the force normally rises roughly as V², so π1 changes only slowly. Raising it by 10 % (to 66) moves the result by −17.4 %, from 0.1008 to 0.08329.
Characteristic length
The characteristic length of the body, in m. Use the same length as in the Reynolds number. Lowers the coefficient with the square of the length. Raising it by 10 % (to 1.65) moves the result by −17.4 %, from 0.1008 to 0.08329.
Worked example with the values above
- Force scale: ρV²ℓ² = 1.225 × 60² × 1.5² = 9,922.5 N.
- Divide the force by it: 1,000 / 9,922.5.
- π1 = 0.1008.
Assumptions and limits
- Steady, incompressible flow past a body of fixed shape and orientation, with negligible surface roughness and no free surface.
- The force depends on the fluid only through density and viscosity, so π1 varies only with the Reynolds number.
Common mistakes
- Using a different length here and in the Reynolds number.
- Comparing π1 values from tests at different Reynolds numbers as if they were the same.
- Confusing π1 with the drag coefficient: CD = 2F/(ρV²A) has a factor ½ and uses an area.
Statistics and uncertainty
The speed and the length both enter squared, so their relative errors count double: a 2 % error in speed and 2 % in length together give about 5.7 % in π1 when the force and density are exact. Measure speed and length carefully, and use the density at the actual temperature and pressure.
Where it is used
- Reducing wind-tunnel and water-tunnel force measurements to a size-independent coefficient.
- Predicting the full-scale force from a model: F = π1ρV²ℓ² at the prototype conditions.
Maths you need
- Dimensional analysis: combining variables so every unit cancels
- Reynolds number: ratio of inertial to viscous forces, ρVℓ/μ