A particle is a low-pass filter

The tracer that is not one establishes the Stokes number and asks whether a particle follows the flow. This essay asks the question the way an instrument engineer would: not whether, but how much, and at what frequency — and the answer is a transfer function.

The response #

A small particle in a fluctuating flow obeys a first-order equation: its velocity relaxes towards the fluid’s over a time set by its own inertia and the drag on it. Forced at a frequency, that gives a response with a magnitude and a phase, both functions of one number — the frequency times the relaxation time, which is the Stokes number.

The magnitude is one over the root of one plus the Stokes number squared. At a Stokes number of one it is 0.707, and the phase lag is 45 degrees.

Those are the numbers of a first-order low-pass filter, and they are the same numbers a resistor and a capacitor give. A particle is a filter, its corner frequency is the reciprocal of its relaxation time, and everything that is true of one is true of the other.

What fidelity costs #

Working backwards from a stated fidelity gives the requirement:

• half the motion needs a Stokes number below 1.73;
• ninety per cent needs 0.48;
• ninety-nine per cent needs 0.14;
• and 99.9 per cent needs 0.045.

The error goes as the square of the Stokes number at small values, so halving a particle’s diameter — which quarters its relaxation time — reduces the error by a factor of sixteen. Read the other way: one per cent fidelity needs a Stokes number of 0.1425, a tenth of a per cent needs 0.04475, and a tenth of one per cent of error therefore costs a 3.2-fold reduction in the Stokes number and so a 1.8-fold reduction in diameter. That steepness is why seeding is chosen by diameter and why a small improvement in the seeding is worth so much.

Which frequency #

The awkward part of the definition is that the Stokes number contains a frequency, and a turbulent flow has all of them.

A one-micron water droplet in air has a relaxation time of three microseconds, so at a kilohertz its Stokes number is 0.019 and it follows everything. At a hundred kilohertz it would be 1.9 and it would follow half.

A fifty-micron droplet has a relaxation time of eight milliseconds. At a kilohertz its Stokes number is 48 and it is following two per cent of the motion — it is measuring its own inertia with a small correction for the flow.

Five cases, one seeding material:

The same ten-micron droplet reads 0.458 at a kilohertz and 0.0515 at ten — a factor of nine in fidelity across one decade of frequency, on one particle in one flow. So the same particle is an excellent tracer and a useless one depending on which part of the spectrum is being asked about, and a measurement of a turbulent flow with a single seeding is a measurement whose fidelity varies across its own bandwidth.

What the relaxation time is made of #

The corner frequency is the reciprocal of the particle’s relaxation time, so it is worth writing that time out, because every design decision follows from its ingredients.

For a sphere in Stokes drag it is the particle’s density times its diameter squared, divided by eighteen times the fluid’s viscosity. Three things follow immediately.

The diameter squared. Halving the diameter quarters the time and quarters the Stokes number, so size is by far the most powerful lever — and it is the only one a user of a seeding usually controls.

The density ratio. A particle of the fluid’s own density has a relaxation time that is not zero but is much smaller, and the added-mass and history terms then dominate its dynamics rather than its own inertia. That is why neutrally buoyant seeding behaves differently in kind rather than merely in degree.

And the viscosity, in the denominator. Seeding in a liquid has a much shorter relaxation time than the same particle in a gas, which is why liquid experiments can use much larger particles for the same fidelity — a practical convenience that has nothing to do with the flow being measured.

That combination is why gas-phase measurements use sub-micron droplets and liquid ones use tens of microns, and why the two literatures quote wildly different “adequate” particle sizes without contradicting each other.

The lag, which is the half nobody quotes #

Seeding requirements are almost always quoted as a magnitude — “the particle follows the flow to within one per cent” — and the phase is left out. It should not be.

A particle at a Stokes number of one lags by 45 degrees, which is an eighth of a period. For a measurement of a mean velocity that does not matter at all. For a measurement of a correlation — a Reynolds stress, a spectrum, a two-point statistic — it matters enormously, because a correlation is a product of two quantities and a phase error between them is a direct error in the product.

That is exactly the structure of what a mean profile cannot tell anybody: the first-order quantity is insensitive to a defect that the second-order one is dominated by. A seeding adequate for a mean profile can be inadequate for a stress by a large factor, and the criterion usually quoted does not distinguish them.

What the solver computed, and how it was checked #

The response is the first-order transfer function, evaluated across four decades of Stokes number, with the fidelity requirement obtained by inverting the magnitude and the case studies from Stokes drag on a water droplet in air. The phase is the half of it nobody quotes: at 50 per cent fidelity the particle lags by 60 degrees, at 90 per cent by 25.8, at 99 per cent by 8.11 and at 99.9 per cent still by 2.56 — a particle correct in magnitude to a part in a thousand is still a fortieth of a period late.

Three checks. That the magnitude at a Stokes number of one is one over root two, and that the phase there is minus 45 degrees, which are the filter’s defining values. And that one per cent fidelity requires a Stokes number below 0.2, which is the number a practitioner would carry.

Why the response is first order at all #

The order of the filter is worth justifying rather than assuming, because it is what makes the whole description this simple.

The particle has one state variable — its velocity — and one restoring mechanism — the drag pulling it towards the fluid’s velocity. One state and one relaxation gives a first-order system, and a first-order system has exactly one time constant and exactly one response shape.

That is the same counting argument that makes a viscoelastic fluid with one relaxation mode a first-order system, and a turbulence summarised by one anisotropy a first-order system. The order of the filter is the number of state variables carried, and the reason all these responses look alike is that all these models carry one.

The moment a second state is added the response gains a second corner and can resonate. A particle with its added mass and history included has, in effect, infinitely many — the history integral is a continuum of relaxation times — which is why its true response is a fractional-order one rather than a first-order one, and why the correction described below is qualitative rather than numerical.

Why this is the same essay as three others #

The transfer function above is one this collection has now derived four times from four different mechanisms, and the repetition is the point.

A wake’s lift deficiency is one at low reduced frequency and falls; the mechanism is shed vorticity convecting away — the lag that makes flutter possible.

A turbulent stress responds to a strain with the same shape; the mechanism is an eddy turnover — a closure with no memory at all.

A viscoelastic stress the same; the mechanism is a molecular relaxation — the fluid that has not finished its last deformation.

And a particle’s velocity the same; the mechanism is its own inertia against drag.

Four unrelated pieces of physics, one curve. The reason is that all four are first-order relaxations, and a first-order relaxation has exactly one frequency response whatever produced it. The physics decides the time constant and the mathematics decides everything else.

Where the first-order description fails #

It fails in one direction and it is worth knowing which.

The equation used here has a quasi-steady drag and nothing else. The full unsteady force has an added mass and a history term, and the drag that integrates a whole history measures the second at up to 39 per cent of the total.

Including it changes the response: the magnitude falls off more slowly than one over the Stokes number at high frequency, because the history term supplies a force that the quasi-steady one does not, and the phase does not approach 90 degrees.

So the filter above is the first-order approximation to a filter with a fractional-order term in it, and its error is in the direction of understating what a heavy particle follows. For a particle much denser than the fluid — a droplet in air — the correction is modest; for a neutrally buoyant one it is not, and the response is qualitatively different.

The curve this filter is one instance of is the last essay in this field, on the same machinery.

What to do about it #

Three practical statements, in order of how much they buy.

Compute the number, do not assert it. The Stokes number is a relaxation time times a frequency and both are available: the first from the particle’s size and density and the second from the flow’s own scales. It takes a minute, and it is the same minute what of order one is worth asks for before any approximation is adopted.

Choose the diameter against the highest frequency of interest, not the lowest. The requirement is a Stokes number below about 0.14 at the top of the band, and since the Stokes number is proportional to the frequency, the top of the band sets the particle.

Report the transfer function, not a yes. A seeding described as “adequate” is a claim about one frequency. Publishing the magnitude and phase against frequency is the same amount of information honestly presented, and it lets a reader judge which of the reported statistics are affected.

And correct if the response is known. A measured spectrum can be divided by the known magnitude response, which recovers the flow’s spectrum where the particle followed anything at all. That is standard in instrumentation and rare in fluid mechanics, and it fails where the response has fallen far enough that the correction amplifies noise.

The filter’s other end #

A low-pass filter has a second use, and in this case it is not a benefit.

At frequencies well above the corner the particle is not measuring the flow at all — it is measuring a smoothed version of it, with the smoothing set by its own inertia. Its trajectory is therefore a filtered flow, and any statistic computed from it inherits the filter.

That has a specific consequence for turbulence measurements. The dissipation is dominated by the smallest scales, which are the highest frequencies, which are exactly where the filter is cutting. A dissipation estimate from particle tracking is therefore biased low, by an amount that depends on the particle and on the flow’s own spectrum together.

The correction is possible in principle — divide the measured spectrum by the known response — and it is unreliable in the region that matters most, because there the response is small and the correction is a large multiplication of a small number.

Which is the standing difficulty with any inverse filtering, and is the same objection a dissipation correlated across every scale raises about estimating high moments from finite records: the quantity wanted lives where the measurement is worst.

What the picture cannot show #

The response is drawn as a magnitude and a phase and they are one complex number. A figure of that number — the Nyquist circle a first-order system traces — is the compact statement and is unreadable to anybody who has not met one.

Nothing here shows a particle in a flow. The transfer function is a property of a linearised equation, and the trajectories it implies — a heavy particle overshooting a bend, a light one following it — are what the collection draws elsewhere.

Where the same filter is a design objective #

A particle’s inertia is a defect for a tracer and it is the entire point of several devices, which is worth noticing because the same transfer function is being used deliberately.

A cyclone separator works because particles above a size cannot follow the flow round the bend, so they are thrown to the wall. Its cut size is exactly a Stokes number threshold, and its sharpness is the steepness of the same curve.

An impactor does the same in a jet: the flow turns and the particles that cannot follow are collected. A cascade impactor is a set of them at different Stokes numbers, which is a filter bank.

And an inertial separator in an engine intake protects a compressor by making the sand fail to follow a bend that the air follows.

In all three the design question is the same one this essay poses for a tracer, asked with the sign reversed: a tracer wants a Stokes number well below the corner and a separator wants one well above it, and both are reading the same curve.

That symmetry is worth carrying. Every criterion in this essay is a specification for a device somewhere else, and the number that decides is the same.

What to do when the number is too large #

The response curve is also a design instruction, and it has three levers with very different costs.

Make the particle smaller. The relaxation time goes as the square of the diameter, so halving the diameter quarters it. This is the lever everybody reaches for, and it runs out against the optical one: a particle small enough to follow the flow may be too small to scatter enough light to be seen.

Match the density. The relaxation time carries the density difference between particle and fluid, so a neutrally buoyant particle has no inertia to speak of whatever its size. That is available in water and is nearly unavailable in air, which is why gas-phase seeding is the hard case and why the smallest usable droplets there are around a micron.

Or accept the response and correct for it. The transfer function is known, so a measured spectrum can be divided by the magnitude squared. That works and it amplifies the noise at exactly the frequencies where the signal was already weakest, so it is a last resort rather than a fix.

Who found it, and when #

Stokes’ drag law is from 1851 and the relaxation time follows immediately from it. The systematic use of the Stokes number as a seeding criterion belongs to the development of laser Doppler and particle image velocimetry, from the 1970s onwards, and the transfer-function framing came with it from the instrumentation side rather than from fluid mechanics.

The unsteady corrections are Basset’s and Boussinesq’s from the 1880s, assembled into the Maxey-Riley equation in 1983, and the practical consequence — that the first-order filter understates a heavy particle’s fidelity — is a result of the 1990s.

Limits recorded rather than smoothed over #

Stokes drag, linearised. Valid at small particle Reynolds number and for small fluctuations. A particle slipping fast relative to the fluid has a nonlinear drag, and then there is no transfer function at all.

No history term, no added mass. Both are described and neither is included, and their omission biases the response in the direction stated.

One frequency at a time. A turbulent flow forces the particle at all frequencies simultaneously. Because the equation is linear the responses superpose, so the transfer function still applies — which is a genuine convenience and would not survive the nonlinear drag above.

One particle, no interaction. The response is for a single particle in an undisturbed flow. At seeding densities high enough to matter the particles modify the flow and each other, which is a different and much harder problem.

And gravity is absent. A heavy particle in a real flow also settles, which biases where it is as well as how it moves — a separate effect that whether the droplet turns is about.