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

${\displaystyle {\text{St}}={\frac {fL}{U}},}$

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

${\displaystyle {\text{St}}={\frac {f}{U}}A,}$

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]

• 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

1. ^ 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.
2. ^ White, Frank M. (1999). Fluid Mechanics (4th ed.). McGraw Hill. ISBN 978-0-07-116848-9.
3. ^ 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.
4. ^ 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.
5. ^ 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.
6. ^ 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.
7. ^ Corum, Jonathan (2003). "The Strouhal Number in Cruising Flight". Retrieved 2012-11-13– depiction of Strouhal number for flying and swimming animalsCS1 maint: postscript (link)