How far before a duct forgets what was fed into it

A duct is almost always drawn with its answer already in place: a parabola across the section, a wall shear that does not change with distance, a pressure gradient that is the same everywhere. All of that is a description of the end state, and how long a duct takes to reach it is a question with a surprising answer and a practical one.

The question is really about how far a boundary condition reaches. The wall says no slip, that information spreads inwards by diffusion, and the flow is developed when it has arrived at the middle and there is nothing left to spread.

What is solved, and what is not assumed #

The parabolised equations, marched downstream:

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{dp}{dx} + \nu\frac{\partial^2 u}{\partial y^2},$$

with continuity supplying $v$ and the flow rate held. No profile family is assumed anywhere. The usual treatment of this problem picks a shape — a quartic, or a Blasius layer with a uniform core — and integrates the momentum equation across it, which gives an answer that partly reflects the choice. Here the shape at every station is whatever the march produced, and the developed parabola has to arrive rather than be imposed.

That gives the check the whole computation is built around. The centre-line speed comes out at 1.4979 times the mean, against the parabola’s exact $3/2$; the momentum flux at 1.19952, against the parabola’s exact $6/5$; and the pressure gradient within a quarter of a per cent of the developed value. Nothing in the march was told any of those numbers.

The pressure gradient is solved for, not marched #

There is a device in this calculation worth pulling out, because it is the same device as a body’s Neumann problem wearing different clothes.

The flow rate is prescribed and the pressure gradient is not. But the momentum equation is linear in $dp/dx$ once the convection is fixed, so each station is solved twice — once with no pressure gradient and once with a unit one — and the combination that carries the right flow rate is found by a single division. The constraint is on an integral of the answer rather than on a value of it, and the unknown that satisfies it is a constant rather than a field.

That is why a duct’s pressure gradient is an output while a boundary layer’s is an input. In external flow the pressure is imposed by the outer stream; in internal flow it is whatever it has to be for the mass to fit through, and the mass is what was actually specified.

The flow rate is held to $10^{-9}$ at every marched station as a result, which is a constraint rather than a result and is reported as such.

The length, and why more viscous is sooner #

The length is proportional to the Reynolds number. A more viscous fluid develops in fewer diameters, which sounds backwards until the two lengths involved are separated: diffusion has to cross the same distance whatever the fluid, and while it does, the flow carries it downstream. Halve the viscosity and the crossing takes twice as long, so the flow carries it twice as far.

The practical consequences are worth writing down because they are unusually blunt.

A domestic water pipe at $Re_D = 2000$, just below transition, needs about twenty-two diameters of straight run before its profile is developed. For a 15-millimetre pipe that is a third of a metre, which is fine. A hydraulic line at $Re_D = 1000$ in oil needs eleven diameters. But a laboratory duct carrying air at $Re_D = 20{,}000$ — if it stayed laminar, which it would not — would need two hundred and twenty, and this is why every careful measurement of a laminar profile is made a very long way from anything.

And the count starts again at every fitting. A bend, a valve, a sudden expansion or a flow meter resets the profile, and a pipe run made of short sections joined by elbows may never be developed anywhere along its length. Handbook friction factors are for developed flow; a system of fittings is not that flow, and the difference is on the order of tens of per cent in the wall shear.

One more consequence of the linear-in-Reynolds scaling is worth stating, because it is the reverse of the usual intuition about viscosity. Two pipes carrying the same fluid at the same speed, one twice the diameter of the other, do not develop in the same number of diameters: the larger pipe has twice the Reynolds number and needs twice as many. Scaling a pipe up makes its entrance region longer in its own units, which is why large ducts in wind tunnels and power stations have honeycomb and screens at their inlets — not to shorten the development, which nothing can, but to start it from a profile that is closer to the end state than a slab is.

What the entrance costs, and it is a fixed number #

The second quantity a designer wants is the extra pressure the entrance takes, above what the developed gradient alone would have cost over the same distance.

It comes to 0.679 dynamic pressures for a parallel-plate channel, against the 0.674 in the handbooks — a number known as the incremental pressure-drop number, or the Hagenbach factor in the older literature, measured and tabulated long before anybody could march the equations.

Two mechanisms are in it, in roughly equal parts. The momentum flux rises from 1 (a slab) to 6/5 (a parabola), and the pressure has to supply that change; and the wall shear in the entrance is larger than the developed value, because the layer there is thinner than the developed one and the gradient across it is steeper.

The first of those has a satisfying property: it depends on the shape of the profile and on nothing else. A slab carries momentum flux $\rho U^2 A$; a parabola in a channel carries $\tfrac{6}{5}$ of that, and in a round pipe $\tfrac{4}{3}$. Those are exact numbers, they are properties of two polynomials, and they appear in a pressure drop.

There is a third contribution that is not in the number, and leaving it out is deliberate. A real inlet has a loss of its own — a sharp-edged pipe entrance costs about half a dynamic pressure in the fittings tables, and a well-rounded bellmouth costs almost nothing — and that loss is a separation phenomenon at the lip rather than a development cost. The two are added in practice and they are different physics: one is the price of turning a corner badly, the other is the price of changing the shape of a profile, and only the second is computed here.

The check that holds it together #

Every station has to satisfy the global momentum balance:

$$\frac{d}{dx}\int \rho u^2\,dy = -a\frac{dp}{dx} - \tau_w .$$

The rate at which the profile’s momentum flux changes is exactly what the pressure force and the wall friction supply between them. That is checked at every station, and it holds to 2.2 per cent of the size of the forces in it.

The reason it is quoted that way rather than as a fraction of the difference is worth a sentence, because the alternative is a trap. Far downstream the pressure gradient and the wall shear are equal and opposite to four figures and the momentum flux has stopped changing, so their difference is a rounding error’s worth of a number; dividing by it reports a hundred per cent error on a flow in perfect balance. A residual has to be measured against the terms it came from.

A development length is a logarithm, not a distance #

The figure above defines the length as the station where the centre line reaches ninety-nine per cent of its final value, and calls that a convention. It is worth saying what kind of convention, because the answer decides how much a different choice would move the number.

Far downstream the flow is a small departure from the developed parabola, and a small departure obeys the linearised problem — the developed solution perturbed. That is a Sturm–Liouville problem across the duct, its solutions are a discrete set of modes each decaying as $e^{-\lambda_n x/(a\,Re)}$, and after a short distance only the slowest survives. The approach is exponential, not algebraic, and every subsequent statement follows from that one word.

An exponential approach means that tightening the criterion costs a fixed distance rather than a factor. Going from ninety-nine per cent to ninety-nine point nine adds $\ln 10 / \lambda_1$ — the same increment again, whatever the Reynolds number, because the Reynolds number was scaled out of the exponent. So the handbook disagreement over entrance lengths is not a disagreement about the physics: 0.011, 0.013 and 0.016 diameter-Reynolds-numbers are three tolerances on one curve, and they are spaced the way three tolerances on an exponential are.

It also says which quantity to ask about. The centre-line speed, the wall shear and the pressure gradient all approach their developed values through the same slowest mode, so they all decay at the same rate — but they start with different coefficients, so they cross any given tolerance at different stations. The wall shear arrives last, because the entrance’s excess friction is largest where the layer is thinnest and the quantity has furthest to fall. A designer sizing a straight run for a flow meter and a designer sizing one for a shear measurement want different numbers off the same solve.

The momentum half of the loss is a property of two polynomials #

The entrance’s cost was split above into a momentum-flux change and an excess wall shear, in roughly equal parts. The first half generalises further than the calculation that produced it, and it is worth extracting because it needs no march at all.

The momentum flux of a profile is $\int \rho u^2\,dA$, and dividing it by $\rho \bar{u}^2 A$ gives a pure number — the momentum correction factor — that depends on the shape of the profile and on nothing else. A slab gives exactly 1. A channel parabola gives $6/5$. A pipe parabola gives $4/3$. Those are integrals of polynomials, they contain no fluid property and no Reynolds number, and the pressure has to supply the difference between whichever two apply.

Which explains a fact about turbulent pipes that otherwise looks like a coincidence. A turbulent profile is much fuller than a parabola — a seventh-power profile has a momentum factor of about 1.02 — so the momentum half of the entrance loss nearly vanishes in turbulent flow, and what remains is almost entirely the excess wall shear. The two halves that happen to be equal in the laminar case are not two halves of one thing, and they scale in completely different ways: one is decided by the shape a profile ends up with, the other by how hard the wall pulls while it is getting there.

One further consequence of the exponential approach is worth carrying to any measurement. Because the departure decays at a fixed rate in $x/(a\,Re)$, a station quoted in diameters is only meaningful alongside the Reynolds number it was measured at — and two laboratories reporting “thirty diameters” at Reynolds numbers a decade apart are reporting two states of development that differ by many decades of residual error.

What the picture cannot show #

The inlet is impossible. A slab of uniform flow with no slip at the wall is discontinuous there — the layer at $x = 0$ is infinitely thin — and the first few per cent of the march is resolving that rather than resolving a fluid. The momentum balance is twenty-one per cent out at the fourth station, five per cent at the eighth and two per cent from there on, and those first stations are excluded from the check with the reason stated.

A real entrance is not a slab either. Water entering a pipe from a plenum comes through a contraction with its own boundary layer and its own separation bubble if the lip is sharp, and the entrance loss of a real inlet — the one in the fittings tables — is that geometry’s loss plus the development cost computed here.

It is laminar and two-dimensional. A turbulent entrance develops in ten to twenty diameters almost regardless of Reynolds number, because turbulent transport is far more effective than diffusion, and the whole scaling above is replaced by a weak power law. A round pipe’s numbers differ from a channel’s by the factors quoted above and by an entrance constant of 0.0575 rather than 0.011.

And the parabolised equations cannot go backwards. Marching downstream assumes that nothing downstream affects anything upstream, which is exactly false wherever the flow separates — so this method is incapable of an entrance with a sharp lip, which is the entrance most pipes actually have.

What a designer does about it #

Three responses, in increasing order of effort, and the arithmetic above decides which is worth it.

Accept it and add the loss. The entrance costs 0.679 dynamic pressures once, whatever the pipe’s length, so on a long run it is negligible and on a short one it is not. The crossover is where the developed friction over the run equals two thirds of a dynamic pressure, which for laminar flow is around thirty diameters. Below that, the entrance is a substantial fraction of the whole pressure budget.

Shorten the development by starting closer to the answer. Nothing shortens the length itself — it is set by diffusion crossing the duct and by nothing a designer controls — but the profile at the inlet is controllable, and a contraction with screens and honeycomb delivers something far closer to the developed shape than a slab. Wind tunnels are built this way for the flow quality rather than for the pressure, and the pressure benefit is incidental.

Or avoid needing the developed profile at all. Most measurements that specify a straight run are specifying it because their instrument assumes a profile: an orifice plate, a turbine meter, a pitot traverse. An instrument that does not care — a Coriolis meter, a full-bore magnetic meter — removes the requirement rather than satisfying it, and in a plant where twenty diameters of straight pipe is a wall, that is the cheaper engineering.

Who measured it, and the connection worth keeping #

Hagenbach measured the extra pressure drop in 1860, forty years before anybody could have computed it, by noticing that Poiseuille’s law over-predicted the flow through short tubes. Boussinesq gave the first analysis in 1891, and Schlichting produced the profile-family solution in 1934 that the handbook numbers still trace back to. The value computed here by marching agrees with that lineage to under a per cent.

The connection worth keeping is with the essay beside this one. Both layers stop growing, and the reasons are entirely different. The Ekman layer stops because rotation supplies a frequency that diffusion has to compete with, so its depth contains no distance at all. This layer stops because it meets the one growing from the opposite wall, so its length contains the Reynolds number and nothing else. Two boundary layers, two terminations, and the same lesson: a layer’s thickness is never a property of viscosity alone — it is the outcome of a competition, and what it competes against comes from outside.

Where the ladder goes next #

The rung above is the turbulent entrance, where the development is much shorter and the mechanism is entrainment rather than diffusion, and where the useful length is set by how fast a turbulent layer grows rather than by how fast momentum diffuses.

The one beside it is the same march with a second field on it. The temperature entering a duct develops in its own length, which is the momentum length multiplied by the Prandtl number — so in a liquid metal the flow is developed long before the temperature is, and in an oil it is the other way about. That number, and the layer it decides, is the other layer.