Strouhal number
In dimensional analysis, the Strouhal number (St, or sometimes Sr to avoid the conflict with the Stanton number) 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
where f is the frequency of vortex shedding, L is the characteristic length (for example, hydraulic diameter or the airfoil thickness) 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:
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 8×102 < Re < 2×105 there co-exist two values of the Strouhal number. The lower frequency is attributed to the large-scale instability of the wake, 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
Metrology
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 frequency/viscosity versus K-factor method is that it takes into account temperature effects on the meter.
where
- 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).
Animal locomotion
In swimming or flying animals, Strouhal number is defined as
where,
- f = oscillation frequency (tail-beat, wing-flapping, etc.),
- U = flow rate,
- A = peak-to-peak oscillation amplitude.
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, A is the amplitude, so the numerator fA 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 (maximal slope) twice the Strouhal constant.[7]
See also
- Aeroelastic flutter – Interactions among inertial, elastic, and aerodynamic forces
- Froude number – Dimensionless number; ratio of a fluid's flow inertia to the external field
- Kármán vortex street – Repeating pattern of swirling vortices
- Mach number – Ratio of speed of an object moving through fluid and local speed of sound
- Reynolds number – Ratio of inertial to viscous forces acting on a liquid
- Rossby number – Ratio of inertial force to Coriolis force
- Weber number – Dimensionless number in fluid mechanics
- Womersley number – Dimensionless expression of the pulsatile flow frequency in relation to viscous effects
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 978-0-07-116848-9.
- ^ 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, 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. Bibcode:2003Natur.425..707T. doi:10.1038/nature02000. PMID 14562101.
- ^ Corum, Jonathan (2003). "The Strouhal Number in Cruising Flight". Retrieved 2012-11-13– depiction of Strouhal number for flying and swimming animals
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