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 in Hertz,^[3] 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.

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.^[5]

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.^[6][7]