Stokes number
The Stokes number (Stk), named after George Gabriel Stokes, is a dimensionless number characterising the behavior of particles suspended in a fluid flow.
The Stokes number is defined as the ratio of the characteristic time of a particle (or droplet) to a characteristic time of the flow or of an obstacle, or
[1] A particle with a low Stokes number follows fluid streamlines (perfect advection), while a particle with a large Stokes number is dominated by its inertia and continues along its initial trajectory.
In the case of Stokes flow, which is when the particle (or droplet) Reynolds number is less than about one, the particle drag coefficient is inversely proportional to the Reynolds number itself.
In that case, the characteristic time of the particle can be written as
[2] In experimental fluid dynamics, the Stokes number is a measure of flow tracer fidelity in particle image velocimetry (PIV) experiments where very small particles are entrained in turbulent flows and optically observed to determine the speed and direction of fluid movement (also known as the velocity field of the fluid).
Smaller Stokes numbers represent better tracing accuracy; for
[3] The Stokes number provides a means of estimating the quality of PIV data sets, as previously discussed.
However, a definition of a characteristic velocity or length scale may not be evident in all applications.
Thus, a deeper insight of how a tracking delay arises could be drawn by simply defining the differential equations of a particle in the Stokes regime.
will encounter a variable fluid velocity field as it advects.
Let's assume the velocity of the fluid, in the Lagrangian frame of reference of the particle, is
It is the difference between these velocities that will generate the drag force necessary to correct the particle path:
The first-order differential equation above can be solved through the Laplace transform method:
The solution above, in the frequency domain, characterizes a first-order system with a characteristic time of
The cut-off frequency and the particle transfer function, plotted on the side panel, allows for the assessment of PIV error in unsteady flow applications and its effect on turbulence spectral quantities and kinetic energy.
The bias error in particle tracking discussed in the previous section is evident in the frequency domain, but it can be difficult to appreciate in cases where the particle motion is being tracked to perform flow field measurements (like in particle image velocimetry).
A simple but insightful solution to the above-mentioned differential equation is possible when the forcing function
is a Heaviside step function; representing particles going through a shockwave.
is positioned where the shock wave is, and then integrate the previous equation to get:
In practice, this means a shock wave would look, to a PIV system, blurred by approximately this
Although a shock wave is the worst-case scenario of abrupt deceleration of a flow, it illustrates the effect of particle tracking error in PIV, which results in a blurring of the velocity fields acquired at the length scales of order
The preceding analysis will not be accurate in the ultra-Stokesian regime.
was defined by;[4] this describes the non-Stokesian drag correction factor,
Considering the limiting particle free-stream Reynolds numbers, as
Thus as expected there correction factor is unity in the Stokesian drag regime.
from the empirical correlation for drag on a sphere from Schiller & Naumann.
Thus overestimating the tendency for particles to depart from the fluid flow direction.
This will lead to errors in subsequent calculations or experimental comparisons.
For example, the selective capture of particles by an aligned, thin-walled circular nozzle is given by Belyaev and Levin[7] as:
