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
\( {\displaystyle {\text{St}}={\frac {fL}{U}},} \)
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:
\( {\displaystyle {\text{St}}={\frac {ka}{\pi c}},} \)
where k is the reduced frequency, and a is amplitude of the heaving oscillation.
Strouhal number (Sr) as a function of the Reynolds number (R) for a long circular 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.[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.
\( {\displaystyle {\text{St}}={\frac {f}{U}}C^{3},} \)
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
S \({\displaystyle {\text{St}}={\frac {f}{U}}A,} \)
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
Froude number – A dimensionless number defined as the ratio of the flow inertia to the external field
Kármán vortex street – Repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies
Mach number – Ratio of speed of object moving through fluid and local speed of sound
Reynolds number – Dimensionless quantity used to help predict fluid flow patterns
Rossby number – The ratio of inertial force to Coriolis force
Weber number – A dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids
Womersley number – A 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.
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
vte
Dimensionless numbers in fluid mechanics
Archimedes Atwood Bagnold Bejan Biot Bond Brinkman Capillary Cauchy Chandrasekhar Damköhler Darcy Dean Deborah Dukhin Eckert Ekman Eötvös Euler Froude Galilei Graetz Grashof Görtler Hagen Iribarren Kapitza Keulegan–Carpenter Knudsen Laplace Lewis Mach Marangoni Morton Nusselt Ohnesorge Péclet Prandtl
magnetic turbulent Rayleigh Reynolds
magnetic Richardson Roshko Rossby Rouse Schmidt Scruton Sherwood Shields Stanton Stokes Strouhal Stuart Suratman Taylor Ursell Weber Weissenberg Womersley
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
