Strouhal number
The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind.
The Strouhal number is often given as 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.
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.
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] Knowing Newton’s Second Law stating force is equivalent to mass times acceleration, or
If the net external forces are predominantly elastic, we can use Hooke’s Law to see where, Assuming
[8] In the medical field, microrobots that use swimming motions to move may make micromanipulations in unreachable environments.
Considering pulsating methane-air coflow jet diffusion flames, we get where, For a small Strouhal number (St=0.1) the modulation forms a deviation in the flow that travels very far downstream.
In swimming or flying animals, Strouhal number is defined as where, 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.
[11] This range is used in the swimming of dolphins, sharks, and bony fish, and in the cruising flight of birds, bats and insects.
[11] 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.
[12] The Strouhal number is most commonly used for assessing oscillating flow as a result of an object's motion through a fluid.
The Strouhal number reflects the difficulty for animals to travel efficiently through a fluid with their cyclic propelling motions.
For instance, the value represents the constraints to achieve greater propulsive efficiency, which affects motion when cruising and aerodynamic forces when hovering.
[13] Greater reactive forces and properties that act against the object, such as viscosity and density, reduce the ability of an animal's motion to fall within the ideal Strouhal number range when swimming.
Its significance is demonstrated through the motion of alcids as it passes through different mediums (air to water).
The assessment of alcids determined the peculiarity of being able to fly under the efficient Strouhal number range in air and water despite a high mass relative to their wing area.
The dual-medium motion demonstrates how alcids had two different flight patterns based on the stroke velocities as it moved through each fluid.
[14] However, as the bird travels through a different medium, it has to face the influence of the fluid’s density and viscosity.
When considered in the context of microrobotics and nanorobotics, size is the factor of interest when performing scale analysis.
is density, we can see mass is directly related to size as volume scales with length (L).
, so considering all three relationships, we get Length (L) already denotes size and remains L. Taking all of this together, we get With the Strouhal number relating the mass to inertial forces, this can be expected as these two factors will scale proportionately with size and neither will increase nor decrease in significance with respect to their contribution to the body’s behavior in the cyclic motion of the fluid.
