Every memory number is one time over another

Counting what matters is this collection’s account of dimensional analysis: how many independent groups a problem has and where they come from. The groups are not the only groups adds the warning that the standard list is a choice of basis rather than a fact.

This essay is a consequence of both, arrived at from the other direction. A run of essays in this collection spent its whole length asking what flows remember, in nine different fields, and produced a dimensionless group nearly every time. Written out, they are the same group.

Eight names, eight fields, and the values span 394 — 0.254 at the bottom and 100 at the top. Not one of the numerators is a length, a velocity or a force; every one of them is a time, and every one of the denominators is a time too.

The list #

Each of the eight was named separately, in a different field, usually after a different person.

Deborah is a polymer’s relaxation time over the flow’s own — how long the fluid takes to forget a deformation, against how long the deformation lasts.

Reduced frequency is the time a wake takes to convect a semi-chord over the period of the wing’s motion.

The Stokes number is a particle’s relaxation time over an eddy’s turnover.

Keulegan-Carpenter is how long a wake persists over a wave period.

Womersley squared is the time viscosity takes to diffuse across a tube over the period of the pulse.

Damköhler is a chemical induction time over a residence time.

The inverse square root of the Ekman number is the time a container takes to spin up over its rotation period.

And a turnover over a strain time is what decides whether a turbulence closure has a chance. That one is a closure with no memory at all, where the ratio is what says whether the closure is being used inside its own limit.

Eight numerators, all of them a memory. Eight denominators, all of them a process. The same construction eight times, in eight fields that do not read one another’s papers.

And one curve #

Every one of the eight governs the same response, because every one of the underlying systems is a first-order relaxation: a state variable pulled towards an equilibrium at a rate.

The magnitude is one at small ratio — 0.99995 at a hundredth, 0.9950 at a tenth — because the memory is short compared with the process, so the system keeps up and a steady theory works; and it falls as the reciprocal at large, reading 0.0995 at ten and 0.010000 at a hundred, where the process is too fast for the system to respond at all. At a ratio of one it is 0.7071, exactly one over root two, with a phase lag of 45.0 degrees.

And every one carries a phase lag, reaching 45 degrees where the magnitude is 0.71 and approaching 90. That lag is the half of the answer a quasi-steady model sets to zero, and it is the half that decides stability — which is why several of these essays are about things that shake.

The band #

The curve divides the world into three.

Below about a tenth, the memory is short compared with the process. The system keeps up, the steady theory is right to within a per cent, and there is nothing to compute.

Above about ten, the process is too fast for the system to respond. The state is frozen at whatever it was, which is a different simple limit and is equally computable.

And between them, which is two decades wide, the answer depends on the history and neither limit applies. The phase lag runs from 0.57 degrees at a hundredth to 5.71 at a tenth, 45.0 at one, 84.3 at ten and 89.4 at a hundred: it is the phase rather than the magnitude that moves first, so a system whose amplitude still looks quasi-steady may already be a tenth of a period behind. That is where these essays live, and it is where almost every engineering flow sits — not by coincidence, since a device operating deep in either limit would have been simplified into something else long ago.

What the collection computed, and how it was checked #

This essay is a survey rather than a computation: the values are representative ones taken from the essays that computed them, and the response curve is the first-order magnitude and phase.

Two checks. That the eight values span at least a factor of a hundred, which is what makes the common construction worth remarking on — eight numbers all near one would be a tautology. They span 394. And that the shared response has the right value at unity, which is a check on the curve rather than on the collection.

The two ends are two different simplifications #

It is worth being clear that the two limits are not one simple case and one hard case; they are two simple cases, and they simplify in opposite directions.

In the quasi-steady limit — a ratio below about 0.1, where the magnitude is 0.995 and the phase lag 5.7 degrees — the state has time to reach its equilibrium at every instant, so the state can be eliminated: it is a function of the present conditions and the system’s order drops. That is what an eddy viscosity does, what steady aerodynamic derivatives do, and what a viscosity does for a viscoelastic fluid.

In the frozen limit — a ratio above about 10, where the magnitude has fallen to 0.0995 and the phase lag is 84.3 degrees — the state has no time to change at all, so it can also be eliminated — by holding it fixed at its initial value. That is what rapid distortion theory does for turbulence, what frozen chemistry does for a nozzle, and what the added-mass response does for a body accelerated instantaneously.

Both limits produce a description with fewer variables, and they produce different descriptions. A model built for one and used in the other is not slightly wrong; it is answering a different question, which is why extrapolating a calibration across the band is the standard way to be badly wrong in this subject.

The band in the middle is the only place where the state has to be carried, and carrying a state is exactly what makes a computation expensive. That is not a coincidence either: the regimes that are cheap to compute are the ones where the memory can be eliminated.

Why the construction recurs #

It is worth asking whether the recurrence means anything or is an artefact of looking for it.

It means something, and the reason is structural. A memory is a state variable that relaxes; a relaxation has a rate; a rate has a reciprocal, which is a time. Any question about whether a memory matters is a question about whether that time is long compared with something, and the something is whatever the flow is doing.

So the construction is forced. Any system with one relaxing state and one imposed process has exactly one dimensionless group governing their competition, and there is no freedom about what it is. The eight names exist because eight communities met the same structure separately and each named it locally — and the eight values in the table span 394, which is a spread of subjects rather than a spread of mechanisms.

That is the general lesson counting what matters is about, arriving from the far end: the number of independent groups is fixed by the physics, and the names are not.

Where it does not apply #

The essay would be dishonest without the exceptions, and this collection found three.

A diffusive memory has no time constant at all. The kernel of the wall the fluid is listening to falls as a power rather than an exponential, its mean delay does not exist, and there is no single time to put in a numerator. The Womersley number in the list above is the closest available substitute — a diffusion time across a stated length — and it is a length’s worth of the memory rather than the memory itself.

A topological memory has no time either. The vortex count of the state a machine was started into does not decay, so there is no rate and no ratio; the question is not whether it matters but which value it has.

And a memoryless response is not on the curve. The impulse of everything about the start, except one vector is exactly path-independent, which is the ratio being zero rather than small.

So the family covers relaxation memories and not the other two kinds, which is a useful boundary: if a memory can be summarised by one time, it is on this curve, and if it cannot, the question has to be asked differently.

What to do with it #

The practical value is a procedure, and it is short.

Identify the memory and time it. What state is being carried, and how long does it take to relax? Usually this is the hard step and usually it can be estimated to a factor.

Identify the process and time it. How fast is the thing being asked about happening?

Divide, and read the curve. Below a tenth, use the steady theory and stop. Above ten, use the frozen one. In between, the history is part of the answer and there is no shortcut.

And check the phase as well as the magnitude. A ratio of 0.3 costs four per cent of the amplitude and seventeen degrees of lag, and the second is often what matters.

That procedure is what what of order one is worth recommends in general, specialised to the one question this collection has been asking.

One of the eight, computed rather than quoted, by the machinery this essay is drawn with.

Why the names are worth keeping anyway #

Having argued that the eight are one, it is worth defending the eight names.

Each carries a context. Saying “Deborah number” tells a reader that the memory is a material property and that the fluid is viscoelastic; saying “Stokes number” tells them it is a particle’s inertia. The same number with a generic name would lose all of that, and the numerator is where the physics is.

Each also carries a convention. Womersley is defined as the square root of the ratio, reduced frequency carries a factor of a half in some fields and not others, and the Ekman number is inverted relative to the rest. Those conventions are load-bearing in their own literatures and merging them would produce more errors than it prevented.

What is worth having is the recognition, not a renaming. Meeting an unfamiliar group in an unfamiliar field, the useful question is what its numerator and denominator are times of — and if the answer is a memory and a process, everything in this essay applies immediately without learning anything else.

What was found that the numbers do not carry #

Having reduced nine fields’ worth of essays to one curve, it is worth naming what the reduction throws away, because the individual essays exist for reasons the summary does not contain.

Where the memory is stored. In a wake, in a boundary layer’s thickness, in a molecular population, in a separation point’s position, in the arrangement of fluid particles, in an integer count of vortices. The ratio does not distinguish them and the storage decides whether the memory travels with the fluid, stays on the body, or convects away.

What resets it. Some memories are erased by mixing, some by diffusion, some by a shock, some by nothing at all. Two systems at the same ratio can differ completely in whether the past can be cleared.

And whether it can be read. The inversions in this collection span the well-conditioned — a wake that says what made it — to the exponentially hopeless — reversible, and unusable. The ratio says nothing about which.

So the number answers one question well: does the past still matter here? Everything else about the past — where it is, how it fades, whether it can be recovered — needs the mechanism, which is what the other essays are for.

What the picture cannot show #

The axis of eight numbers places each at one representative value, and each of them is a range rather than a value: a polymer flow spans four decades of Deborah number depending on the process, and a turbomachine spans three of reduced frequency between its ends. The single points are chosen from the essays that computed them and are illustrative.

The shared response curve is also drawn for a first-order system, and three of the eight are not exactly first order. Their curves have the same two limits and differ in between, by amounts the individual essays record.

A ninth, which does not fit and is worth having #

One number this collection produced does not belong on the axis and putting it beside the others is instructive.

The Reynolds number is a ratio of times too — a viscous diffusion time over a convection time — and it is the most-used group in the subject. But it does not govern the response above, because the quantity it decides is not whether a memory is followed; it is which physics applies at all.

The difference is that the eight groups above compare a memory with a process in a system whose equations are fixed, and the Reynolds number compares two terms within the equations. Raising a Deborah number does not change what a polymer solution is; raising a Reynolds number changes a laminar flow into a turbulent one, which is not a position on any smooth curve.

That distinction is worth keeping because the two kinds of group are used differently. One number decides which physics applies is the collection’s account of the second kind, and its whole subject is a change of regime rather than a change of response.

So the family here is a subset: the groups that decide how much of the past matters, rather than the groups that decide what the equations are. Both are ratios of times and only one of them lives on this curve.

What a reader should take from the axis #

The axis is a picture of a fact about vocabulary rather than about fluids, and it is worth saying what follows from that.

A new group met in an unfamiliar field is usually not new. The question to ask is what its numerator and its denominator are times of. If the numerator is how long something takes to relax and the denominator is how long the process lasts, everything on the response curve applies at once, with no further reading.

And a group that does not decompose that way is a different animal, and is worth noticing for that reason. The Reynolds number below is one; a Mach number is another, since it compares two speeds rather than two times and it decides which equations apply. Sorting a group into one class or the other takes a minute and says immediately whether the answer is a position on a smooth curve or a change of regime.

Where the construction breaks down #

Two situations defeat the ratio, and both are worth recognising because in each the number is still computable and no longer means what it says.

When the process has no single time. A broadband forcing has a spectrum rather than a frequency, and the response is an integral over it. A single ratio then stands for a weighted average, and if the response is nonlinear the average of the response is not the response at the average — which is where a design computed at a representative frequency goes wrong.

And when the memory time depends on the amplitude. A relaxation whose rate is a function of the state has a ratio that moves as the process runs, so the position on the curve is not fixed. A detonation’s induction time is the extreme case: it changes by orders of magnitude over the process it is being compared against, and no single point on the curve describes it.

Who found it, and when #

Nobody found it, which is the point of the essay. Each group has its own history — Deborah’s is 1964, Stokes’ is implicit in 1851, Womersley’s is 1955, Keulegan and Carpenter’s is 1958 — and the recognition that they share a construction is folklore rather than a result.

It is stated occasionally in the pattern-formation and dynamical-systems literature, where the first-order response is a standard object and its appearance across fields is unsurprising. It is stated rarely in fluid mechanics, where the groups are taught as a list.

Limits recorded rather than smoothed over #

A survey, not a computation. The eight values are representative points taken from other essays; the curve is a formula. Nothing new is computed here.

Eight is this collection’s number, not the subject’s. There are more — Damköhler alone has three conventional definitions — and the selection is what this collection happened to produce.

The three regimes’ boundaries are conventions. A tenth and ten are round numbers; the response is smooth, and where a per cent of error becomes unacceptable is a property of the application rather than of the curve.

The response curve assumes linearity. A first-order relaxation is linear by construction, and two of the systems on the list — a dynamic-stall separation point, a detonation’s induction — are not. Their ratios still say whether the memory matters and their responses are not this curve.

And a ratio of times is a summary. Two systems at the same ratio built from different mechanisms behave the same way to first order and not beyond it, which is why the individual essays exist and why this one is the last of them rather than the first.