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# Strouhal number

In dimensional analysis, the **Strouhal number** (**St**) is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind.^{[1]}^{[2]} The Strouhal number is an integral part of the fundamentals of fluid mechanics.

The Strouhal number is often given as;

- <math> \mathrm{St}= {f L\over U}, </math>

where *f* is the frequency of vortex shedding, *L* is the characteristic length (for example hydraulic diameter, or chord length) and *U* is the flow velocity. In certain cases like heaving (plunging) flight, this characteristic length is the amplitude of oscillation. This selection of characteristic length can be used to present a distinction between Strouhal number and Reduced Frequency.

- <math> \mathrm{St}= {k a\over \pi c}, </math>

where *k* is the reduced frequency and *a* is amplitude of the heaving oscillation.

For large Strouhal numbers (order of 1), viscosity dominates fluid flow, resulting in a collective oscillating movement of the fluid "plug". For low Strouhal numbers (order of 10^{−4} and below), the high-speed, quasi steady state portion of the movement dominates the oscillation. Oscillation at intermediate Strouhal numbers is characterized by the buildup and rapidly subsequent shedding of vortices.^{[3]}

For spheres in uniform flow in the Reynolds number range of 800 < Re < 200,000 there co-exist two values of the Strouhal number. The lower frequency is attributed to the large-scale instability of the wake and is independent of the Reynolds number Re and is approximately equal to 0.2. The higher frequency Strouhal number is caused by small-scale instabilities from the separation of the shear layer.^{[4]}^{[5]}

## Applications

In metrology, specifically axial-flow turbine meters, the **Strouhal number** is used in combination with the Roshko number to give a correlation between flow rate and frequency. The advantage of this method over the freq/viscosity versus K-factor method is that it takes into account temperature effects on the meter.

- <math> \mathrm{St}= {f\over U}{C^3} </math>

f = meter frequency, U = flow rate, C = linear coefficient of expansion for the meter housing material

This relationship leaves Strouhal dimensionless, although a dimensionless approximation is often used for C^{3}, resulting in units of pulses/volume (same as K-factor).

In animal flight or swimming, propulsive efficiency is high over a narrow range of Strouhal constants, generally peaking in the 0.2 < St < 0.4 range.^{[6]} This range is used in the swimming of dolphins, sharks, and bony fish, and in the cruising flight of birds, bats and insects.^{[6]} However, in other forms of flight other values are found.^{[6]} Intuitively the ratio measures the steepness of the strokes, viewed from the side (e.g., assuming movement through a stationary fluid) – *f* is the stroke frequency, *L* is the amplitude, so the numerator *fL* is half the vertical speed of the wing tip, while the denominator *V* is the horizontal speed. Thus the graph of the wing tip forms an approximate sinusoid with aspect (maximum slope) twice the Strouhal constant.^{[7]}

## See also

## References

**^**Strouhal, V. (1878) "Ueber eine besondere Art der Tonerregung" (On an unusual sort of sound excitation),*Annalen der Physik und Chemie*, 3rd series,**5**(10) : 216–251.**^**White, Frank M. (1999).*Fluid Mechanics*(4th ed.). McGraw Hill. ISBN 0-07-116848-6.**^**Sobey, Ian J. (1982). "Oscillatory flows at intermediate Strouhal number in asymmetry channels".*Journal of Fluid Mechanics***125**: 359–373. Bibcode:1982JFM...125..359S. doi:10.1017/S0022112082003371.**^**Kim, K. J.; Durbin, P. A. (1988). "Observations of the frequencies in a sphere wake and drag increase by acoustic excitation".*Physics of Fluids***31**(11): 3260–3265. Bibcode:1988PhFl...31.3260K. doi:10.1063/1.866937.**^**Sakamoto, H.; Haniu, H. (1990). "A study on vortex shedding from spheres in uniform flow".*Journal of Fluids Engineering***112**(December): 386–392. Bibcode:1990ATJFE.112..386S. doi:10.1115/1.2909415.- ^
^{a}^{b}^{c}(Taylor, Nudds & Thomas 2003) **^**See illustrations at (Corum 2003)

- Taylor, Graham K.; Nudds, Robert L.; Thomas, Adrian L. R. (2003). "Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency".
*Nature***425**(6959): 707–711. PMID 14562101. doi:10.1038/nature02000. - Corum, Jonathan (2003). "The Strouhal Number in Cruising Flight".
*<span />*13pt,*<span />*. Retrieved 2012-11-13– depiction of Strouhal number for flying and swimming animals