A puff that does not know how old it is

The number that is not a number is this collection’s account of why a critical Reynolds number for pipe flow is so hard to state: laminar pipe flow is linearly stable at every Reynolds number, so nothing about it becomes unstable and there is no threshold to compute.

The modern answer is that there is a threshold anyway, it is sharp, and it is not a stability boundary at all. It is the point at which two random processes change places — and the argument that gets there is about memory in an unusually direct way.

What a puff is #

Below about 2000, turbulence in a pipe is not a state of the flow. It is a set of localised patches — puffs — a few pipe diameters long, travelling at nearly the mean speed, with laminar flow on both sides of them.

A puff can do two things. It can decay, collapsing back to laminar flow and disappearing. Or it can split, growing long enough that a second puff detaches from its upstream end.

Whether a pipe stays turbulent is therefore a question about a population rather than about a flow field: turbulence persists if puffs multiply faster than they die.

The decay has no memory #

The first surprising result is that a puff’s death is a Poisson process.

Its survival probability is an exponential in time, which means its hazard rate — the probability of dying in the next instant, given that it is still alive — is constant at 0.000388 per advective unit whatever the age. A puff that has travelled a hundred diameters is exactly as likely to collapse in the next diameter as one that has just formed: after 344 units 87.5 per cent are still alive, and of those, the same 87.5 per cent survive the next 344.

The conditional survival for a further mean lifetime is the same number for a new puff and for one three lifetimes old, to 5.6·10⁻¹⁷ — the last bit of double precision, at a Reynolds number of 1,900 where the mean lifetime is 2,580 advective units and the hazard rate is a flat 0.000388 per unit whatever the age. That is not a smooth curve or a good approximation; it is what “memoryless” means, and the exponential distribution is the only continuous distribution with the property.

So a puff carries no age at all. Whatever internal state it has — and it has a great deal of structure — none of it accumulates in a way that changes its chance of survival.

That is remarkable rather than obvious. A puff is a complicated object with vortices in it and a definite front and back, and one might expect it to become more robust as its internal structure matures. It does not. What it does instead is wait, for an exponentially distributed time, for the particular rare fluctuation that kills it.

And the lifetime is savagely steep #

The mean lifetime rises super-exponentially with Reynolds number — an exponential of an exponential — so it goes from a few tens of advective units at 1700 to 10¹⁰ at 2100. Around 2000 a further hundred in Reynolds number multiplies it by twenty-six thousand.

That is why the critical Reynolds number was so hard to measure. An experiment run for a fixed length of pipe sees puffs surviving at some Reynolds numbers and not at others, and the apparent threshold depends entirely on how long the pipe is — because the lifetime is crossing the observation window rather than crossing infinity.

The two lifetimes, in advective units:

The decay time rises by a factor of 2.7·10¹⁷ across six hundred in Reynolds number, and the two curves cross at 2,040. Every measurement made before the 2000s was therefore measuring its own apparatus. A quantity that changes by three hundred and forty-fold per hundred in Reynolds number at 1,900 — and by fifty million per hundred at 2,100 — looks like a threshold wherever somebody happens to be looking.

Where the memory arrives #

The second process is splitting, and its lifetime moves the other way: at higher Reynolds number a puff splits sooner.

Below the crossing, decay is faster than splitting and the population dies out however it starts. Above it, splitting is faster and the population grows until the pipe is full. The crossing is at a Reynolds number of 2040, and it is sharp: it is the point where two exponentials of exponentials cross, so the transition from one behaviour to the other happens over a few tens.

The memory arrives with the population. An individual puff has none; a pipe full of puffs has a state — how many there are — which changes slowly and depends on everything that has happened. That is the same structural point what a mean profile cannot tell anybody makes about second moments: a property of an ensemble need not be a property of any member of it.

What the solver computed, and how it was checked #

The two lifetimes are fitted super-exponentials — the decay coefficients are of the published order, and the splitting ones are chosen so that the crossing lands on the measured 2040. Nothing here derives any of the four, and it would be dishonest to present the crossing as a prediction: it is an experimental result, and what is computed is the shape of the competition around it.

Three checks. That the conditional survival is independent of age, to the last bit — which is a statement about the exponential distribution and is exact rather than fitted. That the crossing lies between 1900 and 2200, which is what the fits were arranged to give and is checked so that a change to them cannot pass silently. And that the lifetime multiplies by at least a hundred for a hundred in Reynolds number, which is the steepness the essay’s argument about measurement rests on.

Why memorylessness is a strong statement #

It is worth dwelling on what an exponential survival distribution rules out, because the negative content is larger than it looks.

No ageing. A puff does not wear out. If it did — if its internal structure degraded — the hazard rate would rise with age and the survival would be a Weibull rather than an exponential.

No maturing. Nor does it settle in. If it did, the hazard would fall with age.

And no internal clock of any kind. Any process inside the puff that accumulated — a slow drift in its length, a build-up of some quantity, a progression through states — would show up as an age-dependent hazard.

What is left is a picture in which the puff sits in a fixed condition, and dies when a fluctuation of a particular kind happens to occur. The rate of those fluctuations is the hazard, and it is a property of the Reynolds number and nothing else.

That picture has a name in dynamical systems: the puff is a chaotic saddle — a set of states the flow wanders around on, with an escape route it finds at random. The escape rate is the hazard, and its constancy is the signature that the wandering has already lost track of where it started.

The two clocks, and which of them the pipe runs on #

There is a clean way to put the whole result, and it is worth extracting because it is this collection’s theme stated in an unusual place.

A single puff has a rate and no clock. Its behaviour is described by one number, the hazard, and that number is a function of the Reynolds number alone. Nothing about the puff’s past enters, so the puff is a Markov process in the strictest sense: its future depends on its present state and not at all on how it got there.

The population has a clock and no rate. How many puffs there are now depends on how many there were, how long ago, and what the birth and death rates have been since. That is a memory — the population’s state is an integral over its history — and it is the memory that makes a pipe’s behaviour depend on how it was started.

The transition is where control passes from one to the other. Below 2040 the population’s memory is irrelevant because the population goes to zero whatever it was; above it, the memory is everything, because a single surviving puff eventually fills the pipe.

That is the same structure as a critical point in any birth-and-death process, and its fluid-mechanical form is what makes the threshold sharp: the crossing of two rates is a point, however smoothly each rate varies.

What this makes the transition #

Putting the pieces together gives a description of transition that does not look like a stability problem at all.

The laminar flow is stable, at every Reynolds number, to infinitesimal disturbances. A finite disturbance can put the flow onto the chaotic saddle, where it will stay for an exponentially distributed time. Whether the pipe ends up turbulent is decided by whether the patches multiply faster than they leave the saddle.

So the threshold is a statistical one, not a dynamical one, and the object it is a property of is the ensemble rather than the flow. That is why linear stability analysis — which every mode decays and it grows anyway shows is not useless — gets the answer entirely wrong here while being entirely correct about what it computes.

What is measured, and how #

The lifetimes above are not easy numbers to get and it is worth saying how, because the method is a good example of measuring a distribution rather than a value.

Run many realisations. A puff is created, the pipe is watched, and the distance at which it decays is recorded. Repeating that hundreds of times gives a survival curve directly, and fitting an exponential to it gives the mean lifetime.

Then extend the range by extrapolation. At Reynolds numbers where the lifetime exceeds any practical pipe, no realisation decays. The super-exponential form is fitted at lower Reynolds numbers, where decays can be observed, and extrapolated — which is why the fits are quoted with the range they were made over.

And the crossing is found by measuring the splitting instead. Above the crossing the decay time is unmeasurable and the splitting time is not, so the two branches come from different experiments and meet in the middle. That is a good arrangement and it is why the number is now believed to a few units.

Why this belongs among essays about memory #

The essay sits oddly in a collection about what flows remember, and the oddness is the point.

Every other memory in this collection is a mechanism for carrying the past forward: a wake, a displacement, a stress, a vorticity distribution, a layer’s accumulated thickness. Each of them is a quantity that can be measured now and that encodes something that happened earlier.

A puff has none. It is the clearest example in the collection of a flow structure whose future is independent of its past, and it is worth having because it shows that a complicated unsteady three-dimensional object need not carry any record at all. Complexity is not memory.

What produces the memory here is the counting, and the counting happens outside any single structure. That is a general and useful separation: in a system made of many objects, the memory may live in the population rather than in the objects, and looking for it in the wrong place is how a question gets asked for a century.

The same reading applies to a boundary that only exists over a window, where the structure being sought is a property of an interval rather than of an instant. Here it is a property of an ensemble rather than of a member.

Where the same shape appears #

Radioactive decay is the standard memoryless process and the analogy is exact: a nucleus does not age either, and the two are described by the same distribution for the same reason — a fixed probability per unit time of a rare event.

A metastable state in any system. The escape rate from a potential well over a barrier is constant, so the residence time is exponential; the pipe’s saddle is the same structure without a potential.

And the intermittency in turbulent some of the time is what a population of these patches looks like from a fixed probe: a signal that is turbulent for exponentially distributed intervals and laminar between them.

What a practitioner takes from this #

Three statements, and the first is the one that changes how a measurement is read.

A transition Reynolds number is a property of the apparatus unless it is quoted with a length. Below the crossing every puff dies eventually, so a pipe long enough is laminar at the end whatever happens in the middle. The commonly quoted 2300 is a statement about pipes of ordinary length and about the disturbance level at their inlets.

Disturbing the flow harder does not lower the crossing. It makes puffs more likely to form, which changes where turbulence is observed, and it does not change the rates at which puffs die or split. Those are properties of the Reynolds number, so the 2040 is not something an experimenter can move.

And the practical threshold is the one worth designing to. A pipe operated at 2200 is above the crossing and will be turbulent given enough length; one at 1900 will relaminarise however it is disturbed. Between those the answer depends on the length, which is a design parameter rather than a fluid one.

The contrast is worth drawing, because every other memory in this field is a mechanism for carrying the past forward.

What the picture cannot show #

The lifetimes are drawn on an axis spanning thirty orders of magnitude, which is the only way to fit both branches on one plot and which flattens the steepness that is the whole point. The factor of twenty-six thousand per hundred in Reynolds number is a barely perceptible slope there and is the essay’s most consequential number.

Nothing here draws a puff. It is a three-dimensional localised structure with a sharp upstream front and a diffuse downstream one, and this collection’s solver cannot compute it — which is the same limitation the street this site cannot draw records for a vortex street.

It is worth naming the one other place in this collection where a threshold is a crossing of two rates rather than a value of one. The bath that was only ever a wait computes a residual swirl decaying past a contribution that does not decay, and the answer is a time rather than a force ratio. The puff’s threshold has that structure too — a decay rate against a splitting rate — and in both cases the number everybody quotes is the crossing rather than either of the two curves that make it.

Who found it, and when #

Reynolds saw the patches in 1883 and called them flashes. The recognition that their decay is memoryless is Faisst and Eckhardt’s and Peixinho and Mullin’s, in the early 2000s, from long experimental and numerical runs.

The super-exponential lifetime and the crossing with the splitting rate are Avila and colleagues’, from 2011, in a paper that combined a very long pipe with very long simulations. That work is what turned a century-old question with a range of answers between 1800 and 2300 into a number.

Limits recorded rather than smoothed over #

The fits are fits. Two exponentials of exponentials with four coefficients, chosen to be of the published order and to cross at the measured value. Nothing here derives them and no figure should be read as a prediction of the critical Reynolds number.

Pipe flow only. Plane Couette flow has the same structure with different numbers; a boundary layer does not, because it has a mean flow that carries structures away and a growing thickness that changes the Reynolds number as they travel.

No puff is computed. This collection’s solver does not resolve a three-dimensional localised turbulent structure, and the essay is about the statistics of an object it cannot draw.

Splitting is idealised as a rate. A splitting event has its own structure and takes time, and treating it as a Poisson process with a single rate is the same simplification the decay gets — better justified for the decay, which has been measured that way, than for the splitting, which has not been measured as thoroughly.

No account of the disturbance that starts a puff. Everything above begins with a puff existing. How one forms from a finite disturbance is a separate and harder question, and it is where the solutions stop being chosen picks the story up.

And “memoryless” is a statement about the decay. A puff has plenty of internal state and a well-defined structure; what the exponential says is that none of that state changes its escape rate, not that the state does not exist.