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# Strouhal number

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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] The Strouhal number is an integral part of the fundamentals of fluid mechanics.

The Strouhal number is often given as;

\mathrm{St}= {f L\over V},

where f is the frequency of vortex shedding, L is the characteristic length (for example hydraulic diameter, or chord length) and V is the velocity of the fluid. 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.

\mathrm{St}= {k a\over \pi c},

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

Strouhal number as a function of the Reynolds number for a long cylinder

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.[2]

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.[3][4]

## 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.

\mathrm{St}= {f\over U}{C^3}

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 C3, 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.[5] This range is used in the swimming of dolphins, sharks, and bony fish, and in the cruising flight of birds, bats and insects.[5] However, in other forms of flight other values are found.[5] 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.[6]

## References

1. ^ White, Frank M. (1999). Fluid Mechanics (4th ed.). McGraw Hill.
2. ^ Sobey, Ian J. (1982). "Oscillatory flows at intermediate Strouhal number in asymmetry channels".
3. ^ Kim, K. J.; Durbin, P. A. (1988). "Observations of the frequencies in a sphere wake and drag increase by acoustic excitation".
4. ^ Sakamoto, H.; Haniu, H. (1990). "A study on vortex shedding from spheres in uniform flow". Journal of Fluids Engineering 112 (December): 386–392.
5. ^ a b c
6. ^ See illustrations at (Corum 2003)
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