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. The Strouhal number is an integral part of the fundamentals of fluid mechanics.

The Strouhal number is often given as

<math display=block> \text{St} = \frac{f L}{U}, </math>

where f is the frequency of vortex shedding in Hertz, L is the characteristic length (for example, hydraulic diameter or airfoil thickness), and U is the average flow speed in meters per second. 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:

<math display=block> \text{St} = \frac{k A}{\pi c}, </math>

where k is the reduced frequency, and A is amplitude of the heaving oscillation.

upright=3|thumb|alt=Plot showing the variation of Strouhal number with Reynolds number for a circular cylinder in crossflow for Reynolds numbers from 50 to 10 million based on aggregated experimental data|Strouhal number variation with Reynolds number for a cylinder in cross-flow for Reynolds numbers based on aggregated experimental data

In the case of uniform flow past a fixed cylinder, the cylinder's diameter is the characteristic length. In that case, the Strouhal number is a function of the Reynolds number based on diameter, <math>\mathrm{Re}_D = \rho V D/\mu</math>, where <math>\rho</math> is the fluid's density (kg/m<sup>3</sup>) and <math>\mu</math> [kg-m/s] is the fluid's dynamic viscosity. Over four orders of magnitude in Reynolds number, from 10<sup>2</sup> to 10<sup>5</sup>, the value of the Strouhal number remains close to 0.2 (see figure).

For spheres in uniform flow in the Reynolds number range of 8×10<sup>2</sup> < Re < 2×10<sup>5</sup> 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.

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<sup>−4</sup> 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.

Derivation

Knowing Newton's second law stating force is equivalent to mass times acceleration, or <math>F=ma</math>, and that acceleration is the derivative of velocity, or <math>\tfrac{U}{t}</math> (characteristic speed/time) in the case of fluid mechanics, we see

:<math> F=\dfrac{mU}{t}</math>,

Since characteristic speed can be represented as length per unit time, <math>\tfrac{L}{t}</math>, we get

:<math> F=\dfrac{mU^2}{L}</math>,

where,

: m = mass,

: U = characteristic speed,

: L = characteristic length.

Dividing both sides by <math>\tfrac{mU^2}{L}</math>, we get

:<math> \tfrac{FL}{mU^2}=1=\text{constant}</math> ⇒ <math>\tfrac{mU^2}{FL}=1=\text{constant}</math>,

where,

: m = mass,

: U = characteristic speed,

: F = net external forces,

: L = characteristic length.

This provides a dimensionless basis for a relationship between mass, characteristic speed, net external forces, and length (size) which can be used to analyze the effects of fluid mechanics on a body with mass.

If the net external forces are predominantly elastic, we can use Hooke's law to see

:<math> F=k\Delta L</math>,

where,

: k = spring constant (stiffness of elastic element),

: ΔL = deformation (change in length).

Assuming <math>\Delta L\propto L</math>, then <math>F\approx kL</math>. With the natural resonant frequency of the elastic system, <math>\omega_0^2</math>, being equal to <math>\tfrac{k}{m}</math>, we get

:<math> \dfrac{mU^2}{FL}=\dfrac{mU^2}{kL^2}=\dfrac{U^2}{\omega_0^2L^2}</math>,

where,

: m = mass,

: U = characteristic speed,

: <math>\omega_0</math> = natural resonant frequency,

: ΔL = deformation (change in length).

Given that cyclic motion frequency can be represented by <math>f=\tfrac{\omega_0^2L}{U}</math> we get,

:<math>\dfrac{U^2}{\omega_0^2L^2}=\dfrac{U}{fL}=\text{constant}=\dfrac{fL}{U}=\text{St (Strouhal Number)}</math>,

where,

: f = frequency,

: L = characteristic length,

: U = characteristic speed.

Applications

Micro/Nanorobotics

In the field of micro and nanorobotics, the Strouhal number is used alongside the Reynolds number in analyzing the impact of an external oscillatory fluidic flow on the body of a microrobot. When considering a microrobot with cyclic motion, the Strouhal number can be evaluated as

:<math> \text{St} = \dfrac{fL}{U}</math>,

where,

: f = cyclic motion frequency,

: L = characteristic length of robot,

: U = characteristic speed.

The analysis of a microrobot using the Strouhal number allows one to assess the impact that the motion of the fluid it is in has on its motion in relation to the inertial forces acting on the robot–regardless of the dominant forces being elastic or not.

Medical

In the medical field, microrobots that use swimming motions to move may make micromanipulations in unreachable environments.

The equation used for a blood vessel:

  • Vincenc Strouhal, Ueber eine besondere Art der Tonerregung