A duct that forgets everything but one number

How far before a duct forgets what was fed into it marches the equations down a channel from a slab of uniform flow and reports a development length of 0.0111 diameter-Reynolds-numbers. That is the standard answer and it is a good one.

This essay asks two questions that answer leaves open. What exactly is being forgotten, and what does “developed” mean? The first has a clean answer with a number in it and the second turns out not to be a property of the duct at all.

Forgetting, mode by mode #

Take the developed profile as the target and look at the difference between it and what is actually there — 0.5768 at the inlet in units of the developed centreline velocity, falling to 0.0644 after 1.47 metres and 0.000436 after 6.53. That difference satisfies the diffusion equation across the duct with no-slip at both walls, so it decomposes into sine modes, and each mode decays independently at a rate proportional to the square of its own wavenumber.

Convected along at the mean speed, that becomes a decay along the duct. Mode $n$ falls off with a length proportional to $1/n^2$: the third mode decays exactly nine times faster than the first, and the sixth thirty-six times faster.

For a ten-millimetre channel of water at a tenth of a metre a second:

The first mode survives to 4.67 metres and the sixth to 13.0 centimetres, a factor of 36; everything above the second is gone in the first metre. The rates rise as the square of the mode number, so the twelfth mode dies 144 times faster than the first.

What survives is one shape #

The consequence is the essay’s title. A duct does not forget its inlet slowly and uniformly; it forgets almost all of it very quickly, and holds on to one component for a long time.

Within a quarter of the development length the remaining disturbance is 99 per cent the slowest mode, and by the end it is 99.9999 per cent. So the profile approaches its final form along one fixed shape, whatever it started as: a slab, a jet, a distorted profile from a bend all converge to the same one-parameter family, differing only in how much of it is left.

That is a strong statement about the duct’s memory. It is not a function — it is a single number, the amplitude of the slowest mode, and everything else about the inlet has been erased.

Which is why an integral method works #

The result explains something that would otherwise be surprising: that the whole of a developing flow can be summarised by one or two integral quantities.

If the departure from developed conditions is one shape times one amplitude, then any integral of it — a displacement thickness, a pressure drop, a centre-line velocity — carries the same information, and knowing one gives all the others. That is exactly what a one-parameter integral method assumes, and this is why it is nearly true.

The same structure appears in a layer that is an integral of everything upstream, where the profile family is one-parameter and a momentum thickness is therefore a sufficient summary. Both are cases of a system whose fast degrees of freedom have been slaved to a slow one, and both fail in the same circumstance: where a second mode stops being fast, the summary stops being sufficient.

How much of the inlet survives at all #

There is a second number worth extracting, because it says how much of the problem the inlet condition is at all.

The whole disturbance a slab of uniform flow represents is spread across the modes, and the slowest one carries only a part of it: at the inlet the total departure is 0.5768 and the first mode holds 0.2749 of it, a purity of 47.7 per cent. By 0.73 metres the purity is 99.999 per cent and by 1.47 metres it is 1.000000 to six figures — the profile is one mode and one amplitude from there on, while the departure itself is still 0.0644, a ninth of what entered. Everything in the other modes is gone within a fraction of the development length, so the amount of memory a duct has is the projection of the inlet onto one shape — a single inner product.

Two inlets with very different appearances can therefore have almost identical futures a metre down the pipe: what distinguishes them is one number, and if the two numbers agree the two flows are indistinguishable from there on. A uniform slab, a slug of fluid entering from a plenum, and a profile distorted by an upstream bend all project onto the slowest mode with similar coefficients, because that mode is the smoothest one available and every reasonably smooth departure from developed conditions overlaps with it.

The exception is instructive. An inlet condition antisymmetric about the duct’s centreline — a flow entering with more velocity on one side, say from a bend — has zero projection onto the symmetric modes and lives entirely in the antisymmetric ones, which are all faster. Such a flow develops sooner than the standard length suggests, and the reason is not that the duct is behaving differently but that the quantity being waited for was never there.

The development length is a threshold #

The second question is the awkward one.

Since the approach is exponential in one mode, there is no distance at which the flow becomes developed. There is only a distance at which the remaining disturbance falls below whatever fraction somebody nominates.

Sweeping that fraction from a tenth to a thousandth moves the length from 1.02 metres to 5.69 — a factor of 5.6 — with nothing about the duct, the fluid or the flow having changed. The relation is a straight line against the logarithm of the threshold, because the decay is exponential and its inverse is a logarithm.

So a development length quoted without its threshold is a number with a free parameter in it, and the factor of five between conventions is larger than the disagreement between most published correlations. Much of the scatter in the entrance-length literature is a disagreement about a definition rather than about a flow.

What the solver computed, and how it was checked #

The disturbance is projected onto the channel’s diffusion eigenmodes, each is decayed at its own rate, and the sum is evaluated along the duct. That is a linear model of a nonlinear developing flow, and its limits are recorded below; what it captures exactly is the relative decay rates, which is what the essay is about.

Three checks. That the third mode’s rate is exactly nine times the first’s, to two per cent, which is the eigenvalue statement and would catch an indexing error. That the remaining disturbance becomes essentially pure, above 99.9 per cent the slowest mode by the end. And that the development length against threshold is exactly logarithmic — each entry in the sweep is checked against the previous one plus the logarithm of the ratio, to a part in a million, which is a much sharper test than fitting a line.

What this does to a comparison between two experiments #

The definitional freedom is not academic, and it is worth naming what it does to published data.

Two correlations can disagree by a factor of five and both be right. If one facility calls the flow developed when the centre-line velocity is within one per cent of its final value and another uses a tenth of a per cent, their lengths differ by the logarithm of ten over the decay rate — 2.33 metres on this channel, which is half of the 4.67 the one-per-cent criterion gives and a third of the 7.00 the tenth-of-a-per-cent one gives.

A numerical result can be made to agree with anything. Marching a computation and reading off where the profile stops changing gives a length set by the tolerance of the comparison, which is a property of the post-processing.

And the cure is easy and rarely applied. Quote the decay rate rather than the length. It is a single number, it has no free parameter in it, and every threshold’s length follows from it by one logarithm. The rate is also the quantity a theory predicts, which the length is not.

That is the same complaint the range a real Reynolds number does not have makes about fitting a slope to a spectrum: the reported number contains a choice nobody states, and the choice is comparable in size to the effect.

Why the fast modes are the fine ones #

The physical reading is worth having because it is general.

A high mode is a fine-grained disturbance: many alternations across the duct, so steep gradients. A diffusive process erases steep gradients fastest, at a rate going as the square of the wavenumber, because the second derivative in the diffusion equation supplies two powers.

So the duct is a low-pass filter in space, and the fine structure of whatever was fed in is gone almost immediately while the coarse part persists. That is why a duct’s inlet condition matters much less than it seems it should: the only part of it that survives is its projection onto the smoothest available shape, and almost any inlet has a similar projection.

The exception, and it is the practically important one, is an inlet whose projection onto the slowest mode is nearly zero. Such a flow develops much faster than the standard length suggests, because the quantity being waited for was never there.

Where the linear picture stops #

Three things are missing and they are all in the same direction.

The developing flow is not a small perturbation. A slab entering a pipe differs from the developed profile by fifty per cent of the centre-line velocity, so the linearised decay is quantitatively wrong early on. What survives the linearisation is the ordering of the rates, which is what is used here.

The convection speed is not uniform. The disturbance is carried at the local velocity, not the mean one, so the fast core carries its part of the disturbance further than the slow near-wall fluid carries its own. That shears the disturbance and couples the modes, which the linear model does not.

And the pressure gradient adjusts. A developing duct flow has a pressure gradient that varies with distance, which is where the entrance loss comes from and which is a nonlinear coupling between the profile and the forcing.

None of the three changes the essay’s two results — that the memory is one number, and that the length is a threshold — because both follow from the ordering of the decay rates rather than from their values.

The same statement in three other places #

A boundary layer. Its profile family is one-parameter for the same reason, and its memory is therefore a thickness — which is what makes Thwaites’ method possible.

A wing’s boundary layer, in the same way. The profile family being one-parameter is exactly what four profiles, one drag is careful about: the momentum integral fixes one number and leaves the shape open, and it is the fast decay of everything except one mode that makes the shape nearly determined anyway.

A turbulent flow relaxing after a perturbation. The fine scales adjust in a fraction of an eddy turnover and the large ones take many, so a perturbed turbulence returns to equilibrium along one shape too — which is the assumption every one-point closure rests on.

And a rotating container spinning up. How long a fluid takes to forget it was not rotating has the same exponential approach along one mode, with the interior’s angular velocity as its single number.

In all four the useful statement is the same: a system whose fast modes decay much faster than its slow one has a low-dimensional memory, and finding the slowest mode is finding the whole of it.

Why a duct has a memory at all #

It is worth asking what the duct is remembering with, since the answer separates this case from the diffusive memories elsewhere in this collection.

There is no convolution here. The duct’s state at a station is the velocity profile there, and the profile determines everything downstream — so the flow is a state-space system rather than a history-dependent one, and its memory is the state.

What makes it a memory in the sense used here is that the state is much smaller than the input. A whole function was fed in at the inlet and one number of it survives, so the mapping from inlet to downstream condition throws almost everything away. The duct is a lossy encoder, and the development length is how far it takes to finish encoding.

That is the opposite arrangement from the wall the fluid is listening to, where the memory is carried in a convolution over the boundary’s whole past and no finite state summarises it. The difference is that here the domain is bounded across the flow, so the diffusion operator has a discrete spectrum with a largest time, and there it is unbounded and the spectrum is continuous with none.

A bounded diffusion has a slowest mode; an unbounded one does not. Every difference between the two essays follows from that one sentence.

What the picture cannot show #

The modal amplitudes are drawn as a sum, and the shape they add up to is not drawn anywhere. A figure of the profile developing would be the natural picture and it would be the borrowed one, because this model computes coefficients rather than profiles.

The threshold figure also cannot show the thing that makes it uncomfortable, which is that every point on it is equally defensible. There is no feature on that line — no knee, no plateau — at which a definition would naturally be placed, and the absence of one is the result.

What the entrance costs, and why that is a different length #

There is a second quantity called an entrance length and it is worth separating from this one, because they are not the same distance and are routinely conflated.

The hydrodynamic development length is what this essay computes: how far before the profile stops changing. It is set by the slowest mode’s decay.

The entrance loss is a pressure drop, and it is essentially complete much sooner — how far before a duct forgets what was fed into it computes it as a fixed number of dynamic heads however long the pipe is. The reason it finishes sooner is that most of the loss is incurred where the gradients are steepest, which is where the fast modes are, and those are gone quickly.

So a duct can be charged its full entrance loss while its profile is still a long way from developed, and a measurement of one says little about the other. The two lengths differ by roughly the ratio of the fast modes’ decay to the slow one’s, which is the square of a mode number.

The contrast worth having beside it is a memory with no time constant at all, computed by the same solver.

The comparison with a packed bed is worth making, because the two are the same computation with the modes replaced by paths. The outlet is the inlet, a while ago shows a bed’s outlet as its inlet convolved with a residence-time distribution, and the width of that distribution is what a duct’s mode decay has collapsed to a single number. A duct forgets the shape and keeps one amplitude; a bed keeps the shape and smears it.

Who found it, and when #

The eigenvalue structure of a developing duct flow is old and belongs to the same family as the Graetz problem for heat transfer, which is Graetz’s from 1883 — the thermal version of the same calculation, with the same modes and the same square-law rates.

The entrance-length correlations came from experiment and from numerical marching rather than from the eigenvalues, which is why the definitional problem was never central to them: an experiment measures a pressure drop and calls the flow developed when the gradient stops changing to within the instrument’s resolution, and that resolution is the threshold.

Limits recorded rather than smoothed over #

A linear model of a nonlinear development. Stated at length above. The rates are right; the amplitudes early on are not.

One geometry. A plane channel. A circular pipe has Bessel modes rather than sines and different eigenvalues, and the ratios between them are not the same integers — though the ordering and the square law are.

Laminar. A turbulent duct develops far faster and by a different mechanism, since turbulent mixing rather than molecular diffusion does the erasing. Its development length is a few tens of diameters rather than hundreds and it is set by the turbulence’s own relaxation.

No inlet geometry. The disturbance is a slab, projected. A real inlet has a contraction, a separation bubble or a bend before it, and what enters the duct is that geometry’s output rather than anything as tidy — which is why the projection onto the slowest mode is the quantity to measure rather than to assume.

And the numbers are one duct’s. Ten millimetres, water, a tenth of a metre a second. What transfers is the factor of nine between the modes and the factor of 5.6 between the definitions, both of which are dimensionless and neither of which depends on the duct at all.