Library / Dimensional analysis and similarity / DIM-07
Dimensional analysis and similarity
Strouhal number
Dimensionless frequency: the oscillation frequency multiplied by the time a fluid particle takes to travel one length.
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- DIM-07
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- Dimensional analysis and similarity
- Result
- Sr
Sr = fL/U relates an oscillation frequency f to the flow time L/U, the time a fluid particle takes to travel one length. For vortex shedding from a circular cylinder, use the cylinder diameter for L and the upstream speed for U; Sr is then about 0.2 over a wide range of Reynolds numbers, so the shedding frequency is about 0.2U/D. If the frequency is given as an angular frequency in rad/s, divide by 2π to get hertz first.
Dimensionless frequency: the oscillation frequency multiplied by the time a fluid particle takes to travel one length. For vortex shedding behind a circular cylinder it is about 0.2 over a wide range of Reynolds numbers.
Calculator
- Oscillation period, 1/f25 ms
- Flow time, L/U5 ms
- Angular frequency, 2πf251.3 rad/s
- Shedding frequency at Sr = 0.240 Hz
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 body of 50 mm shedding at 40 Hz in a 10 m/s stream has Sr = 0.2, close to the 0.2 typical of a circular cylinder.
- Compare the shedding frequency with the natural frequency of the structure: if the two come close, the vortex shedding can drive large vibrations (lock-in), as in chimneys, cables and risers.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Upstream flow speed | 0.01 to 400 m/s | 0.005 | 200 | 200 | +10% | −9.09 % |
| f Oscillation or shedding frequency | 0.1 to 5,000 Hz | 5e-4 | 25 | 25 | +10% | +10 % |
| L Characteristic length (for a cylinder, its diameter) | 0.1 to 5,000 mm | 4e-4 | 20 | 20 | +10% | +10 % |
Sweep of Upstream flow speed
| Upstream flow speed (m/s) | Strouhal number |
|---|---|
| 0.01 | 200 |
| 0.03761 | 53.18 |
| 0.1414 | 14.14 |
| 0.5318 | 3.761 |
| 2 | 1 |
| 7.521 | 0.2659 |
| 28.28 | 0.07071 |
| 106.4 | 0.0188 |
| 400 | 0.005 |
Sweep one variable
Variables and how each one contributes
Strouhal number
What the equation gives you. Everything else feeds it.
Oscillation or shedding frequency
The oscillation or shedding frequency, in Hz (cycles per second, not rad/s). Raises the Strouhal number in direct proportion. Raising it by 10 % (to 44) moves the result by +10 %, from 0.2 to 0.22.
Characteristic length (for a cylinder, its diameter)
The characteristic length, in mm: the cylinder diameter for vortex shedding. Raises the Strouhal number in direct proportion. Raising it by 10 % (to 55) moves the result by +10 %, from 0.2 to 0.22.
Upstream flow speed
The upstream flow speed, in m/s. Lowers the Strouhal number in inverse proportion: the same frequency at a higher speed means a smaller Sr. Raising it by 10 % (to 11) moves the result by −9.09 %, from 0.2 to 0.1818.
Worked example with the values above
- Convert the length to metres: L = 50 mm = 0.05 m.
- Multiply by the frequency: fL = 40 × 0.05 = 2 m/s.
- Sr = fL/U = 2 / 10 = 0.2.
Assumptions and limits
- Periodic flow with one dominant frequency, defined with a consistent L and U.
- For the typical value 0.2, a circular cylinder in a steady cross-flow at moderate to high Reynolds number.
Common mistakes
- Putting an angular frequency in rad/s into the frequency box: divide by 2π first.
- Using the radius for L when the correlation uses the diameter.
- Taking the local speed near the body instead of the upstream speed.
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 (f × 1, L × 1, U × −1). Frequency is the easiest to measure precisely, so the uncertainty is usually in the flow speed, and in whether L is the same dimension used in the correlation you compare against.
Where it is used
- Estimating the shedding frequency of a cylinder, mast, cable, chimney or heat-exchanger tube.
- Checking a structure for flow-induced resonance, and scaling an unsteady flow from a model.
Maths you need
- Frequency: cycles per second, in hertz
- Vortex shedding: alternate vortices leaving a bluff body