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

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

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.^[6]

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:^[7]

${\displaystyle {\text{St}}={\dfrac {fD}{V}}}$,

where,

f = oscillation frequency of the microbot swimming motion

D = blood vessel diameter

V = unsteady viscoelastic flow

The Strouhal number is used as a ratio of the Deborah number (De) and Weissenberg number (Wi):^[7]

${\displaystyle {\text{St}}={\dfrac {\text{De}}{\text{Wi}}}}$.

The Strouhal number may also be used to obtain the Womersley number (Wo). The case for blood flow can be categorized as an unsteady viscoelastic flow, therefore the Womersley number is^[7]

${\displaystyle {\text{Wo}}={\sqrt {{\dfrac {\pi }{2}}*{\text{Re}}*{\text{St}}}}}$,

Or considering both equations,

${\displaystyle {\text{Wo}}={\sqrt {{\dfrac {\pi }{2}}*{\text{Re}}*{\dfrac {\text{De}}{\text{Wi}}}}}}$.

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).

This relationship between flow and frequency can also be found in the aeronautical field. Considering pulsating methane-air coflow jet diffusion flames, we get

${\displaystyle {\text{St}}={\dfrac {aw_{j}}{U_{j}}}}$,

where,

a = fuel jet radius

w = the modulation frequency

U = exit velocity of the fuel jet

For a small Strouhal number (St=0.1) the modulation forms a deviation in the flow that travels very far downstream. As the Strouhal number grows, the non-dimensional frequency approaches the natural frequency of a flickering flame, and eventually will have greater pulsation than the flame.^[8]

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.^[9] This range is used in the swimming of dolphins, sharks, and bony fish, and in the cruising flight of birds, bats and insects.^[9] However, in other forms of flight other values are found.^[9] 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.^[10]

Scale Analysis

In order to determine significance of the Strouhal number at varying scales, one may perform scale analysis–a simplification method to analyze the impact of factors as they change with respect to some scale. When considered in the context of microrobotics and nanorobotics, size is the factor of interest when performing scale analysis.

Scale analysis of the Strouhal number allows for analysis of the relationship between mass and inertial forces as both change with respect to size. Taking its original underived form, ${\displaystyle {\tfrac {mU^{2}}{FL}}}$, we can then relate each term to size and see how the ratio changes as size changes.

Given ${\displaystyle m=V\rho }$ where m is mass, V is volume, and ${\displaystyle \rho }$ is density, we can see mass is directly related to size as volume scales with length (L). Taking the volume to be ${\displaystyle L^{3}}$, we can directly relate mass and size as

${\displaystyle m\approx L^{3}}$.

Characteristic speed (U) is in terms of ${\displaystyle {\tfrac {\text{distance}}{\text{time}}}}$, and relative distance scales with size, therefore

${\displaystyle U^{2}\approx L^{2}}$.

The net external forces (F) scales in relation to mass and acceleration, given by ${\displaystyle F=m\cdot a}$. Acceleration is in terms of ${\displaystyle {\tfrac {\text{distance}}{{\text{time}}^{2}}}}$, therefore ${\displaystyle a\approx L}$. The mass-size relationship was established to be ${\displaystyle m\approx L^{3}}$, so considering all three relationships, we get

${\displaystyle F\approx L^{4}}$.

Length (L) already denotes size and remains L.

Taking all of this together, we get

${\displaystyle {\dfrac {mU^{2}}{FL}}\approx {\dfrac {L^{3}L^{2}}{L^{4}L}}\approx {\dfrac {L^{5}}{L^{5}}}\approx L^{0}=1}$.

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.

Relationship with the Richardson number

The scaling relationship between the Richardson number and the Strouhal number is represented by the equation:^[13]

${\displaystyle {\text{St}}_{l}=b{\text{Ri}}_{l}^{a}}$,

where a and b are constants depending on the condition.

For round helium buoyant jets and plumes:^[13]

${\displaystyle {\text{St}}_{D}\sim {\text{Ri}}_{D}^{0.38}}$.

When ${\displaystyle {\text{Ri}}<100}$,

${\displaystyle {\text{St}}_{D}=0.8{\text{Ri}}_{D}^{0.38}}$.

When ${\displaystyle 100<{\text{Ri}}<500}$,

${\displaystyle {\text{St}}_{D}=2.1{\text{Ri}}_{D}^{0.28}}$.

For planar buoyant jets and plumes:^[13]

${\displaystyle {\text{St}}_{W}=0.55{\text{Ri}}_{W}^{0.45}}$.

For shape-independent scaling:^[13]

${\displaystyle {\text{St}}_{Rh}=e^{-1}{\text{Ri}}_{Rh}^{\tfrac {2}{5}}}$

Relationship with Reynolds number

The Strouhal number and Reynolds number must be considered when addressing the ideal method to develop a body made to move through a fluid. Furthermore, the relationship for these values is expressed through Lighthill's elongated-body theory, which relates the reactive forces experienced by a body moving through a fluid with its inertial forces.^[14] The Strouhal number was determined to depend upon the dimensionless Lighthill number, which in turn relates to the Reynolds number. The value of the Strouhal number can then be seen to decrease with an increasing Reynolds number, and to increase with an increasing Lighthill number.^[14]