The part of the closure a pipe cannot see

Stopping water costs more than moving it computes the Joukowsky rise: shut a valve on a moving column of water and the pressure at the valve steps up by the wave speed times the velocity that was destroyed, over gravity. That essay also records the standard remedy — close the valve slowly — and the standard caveat, which is that slowly has to mean slowly compared with the time it takes the news to reach the far end and come back.

This essay is about what “slowly” means when it is written into a specification, and it starts from an observation that is easy to make and rarely made: a closure time does not determine a peak.

What the gauge is actually integrating #

The pressure at the valve at any instant is set by one characteristic arriving from upstream and by the valve’s own law at that instant. The arriving characteristic left the reservoir half a round trip ago, and what it carries was decided by the valve half a round trip before that. So the head at the valve now is a function of the valve’s position now, and of its position one round trip ago, and two round trips ago, and so on back — with a sign that alternates and an amplitude that friction eats.

That is a convolution, and it has the shape every memory in this collection has: the recent past weighted heavily, the distant past weighted less, and a definite window beyond which nothing matters. The only unusual thing about this one is that the window has a hard edge rather than a soft one, because the information travels at a finite speed rather than diffusing.

The consequence is the thing a specification cannot express. A number — six seconds — is not an input to a convolution. A function is.

The line, and the two numbers it is made of #

Everything below is one pipeline: 1,200 metres of half-metre pipe carrying water at two metres a second from a reservoir at a hundred metres of head, with a wave speed of a thousand metres a second and a Darcy friction factor of 0.018.

Two numbers come out of that geometry and they are the only two that matter.

The Joukowsky rise is the wave speed times the velocity, over gravity: 203.9 metres of head, or about twenty bar. That is the most this line can do to itself.

The round trip is twice the length over the wave speed: 2.40 seconds. That is how long the valve has to wait before anything it does comes back to it.

The march used here is the method of characteristics on two hundred reaches, with the friction term carried on each characteristic and a valve law that closes the quadratic orifice relation at each step. Shutting the valve instantaneously gives a rise of 203.90 metres against the analytic 203.94 — four hundredths of a metre, which is the discretisation and not the physics.

Closing slowly buys nothing until the news gets back #

Run the same line with a linear closure at twelve different durations and the answer divides into two regimes with a sharp knee between them.

Below one round trip the peak does not move. A closure taking a twentieth of a round trip gives 203.7 metres; one taking a whole round trip gives 200.1. That is a factor of twenty in closure time for a two per cent change in peak. The reason is exactly the reason this essay exists: until the relief wave returns, the line has no way of knowing that the valve is moving rather than shut, so every history inside that window is the same history as far as the pressure is concerned.

Above one round trip the peak falls fast. One and a half round trips gives 116 metres, two gives 72, five gives 23. Once the relief wave is arriving while the valve is still moving, each further increment of closure time is being paid for at the full rate.

The handbook rule for the second regime is Allievi’s: the rise is approximately twice the length times the velocity, over gravity times the closure time. It is drawn on the figure and it is a reasonable envelope — within about fifteen per cent over the slow half of the range, and an overestimate at the knee, where it crosses the Joukowsky line at exactly one round trip because that is how it was constructed.

What the rule cannot do is tell two shapes apart, because the closure time is the only thing in it.

Which instants the peak is listening to #

The honest way to ask which part of the closure matters is not to reason about characteristics but to measure it, and the measurement is simple. Take a linear eight-second closure. Add a small bump to the valve’s opening at one instant. Recompute the peak. Divide the change by the bump. Move the bump and do it again — a hundred and sixty times.

What comes out is a sensitivity kernel, and it has four features worth reading off it.

It is zero after the peak. A bump applied at seven and a half seconds does nothing to a peak that happened at 7.2, which needs no fluid mechanics at all and is the check that the measurement is measuring what it claims to.

It is largest a little before the peak — at 5.6 seconds, which is 0.67 of a round trip earlier. That is where the valve’s own motion and the characteristic returning from its motion one round trip ago are both working in the same direction.

It is structured rather than smooth. There are steps in it at the reflection arrivals, because the contribution from one round trip back switches on at a definite instant rather than fading in.

And it has a front edge. Before 0.45 seconds the sensitivity is under a hundredth of its largest value. The first 5.6 per cent of the closure is not weakly important; it is measurably invisible.

The part the pipe cannot see #

That front edge is worth dwelling on, because it is the property of this system that a designer could actually use and it is not in any of the standard rules.

The peak occurred at 7.2 seconds. The valve’s behaviour before 0.45 seconds contributed less than one per cent as much as its behaviour at the most sensitive instant. Whatever the valve did in that first half second — whether it dawdled, jumped, or was mis-specified entirely — the gauge at the far end never found out.

The window is not one round trip. It is 6.75 seconds, which is 2.8 round trips, and it is not a fixed number of round trips either: changing the wave speed over a factor of six moves the window in seconds by less than a factor of two. The window is set jointly by the reflection train and by how much of the closure is doing significant throttling, and the second of those depends on the valve law rather than on the pipe. That is a less tidy answer than “the last two lengths of line” and it is what the measurement says.

The tidy claim — that only the final round trip matters — is the one this essay set out expecting to confirm, and it does not survive the kernel. What survives is weaker and more useful: there is a front edge, it is measurable, and it is a small fraction of the closure rather than most of it.

One closure time, three peaks #

Here are three closure laws. All three are open at zero, shut at six seconds, and monotone in between.

The linear one closes at a constant rate. The late finish loiters near open and then closes quickly at the end. The early finish does most of its closing in the first half and then dribbles shut.

A specification that says “close in six seconds” is satisfied by all three. So is a specification that adds “monotonically”. So is one that adds a maximum rate, provided the maximum is set by the fastest of them.

The peaks are 50.2, 84.4 and 99.2 metres of head. The largest is very nearly twice the smallest, and the spread is 97 per cent of the linear answer.

The ordering is not a surprise once the kernel is in view — both rearrangements move closing rate into the region the kernel weights most, and both raise the peak. What is worth noticing is that the linear closure is the mildest of the three, which is not something a rule stated in terms of duration could predict, and which is a good reason to specify the law rather than the time.

What the solver computed, and how it was checked #

The march is standard and the checks are the interesting part.

Against the analytic answer. An instantaneous closure must give the Joukowsky rise exactly, and it gives it to 0.02 per cent. That is a check on the characteristics, the friction term and the valve boundary all at once, because a mistake in any of them shows up here.

Against monotonicity. The peak must not rise as the closure is slowed. The twelve-point sweep is checked to be non-increasing within a two per cent tolerance, which catches the class of error where a resonance is being manufactured by the grid rather than by the pipe.

Against the future. The sensitivity kernel must be zero for bumps applied after the peak. It is, outside the width of the bump itself, and that is the check that the perturbation experiment is measuring a sensitivity and not an artefact of moving the peak around.

And against a null. The three-shape comparison is required to produce a spread, because a measurement that found all three shapes equivalent would mean the march had lost the shape somewhere — which is precisely what a scheme with too much numerical damping does.

What this makes of a specification #

The practical reading is short and it is not what is usually written.

A closure time is a bound, not a value. If the closure time is under one round trip, the peak is the Joukowsky rise and the shape is irrelevant — that regime is genuinely specified by a number, and it is the number to check first. If it is over, the shape decides and the duration only bounds how bad it can be.

The bound is loose by about a factor of two. On this line, six seconds admits everything from 50 to 99 metres, and a designer who computed the linear case and applied a factor of 1.5 would still be under-designed against the late-finish law.

The remedy is to specify the law. A valve characteristic — opening against stem position — and a stem motion together give a function, and that function is what the pipe integrates. This is routinely available and routinely not required.

And a butterfly valve is the case to worry about, because its opening area is strongly non-linear in stem angle: a constant-speed actuator produces something much closer to the late-finish law than to the linear one. The shape is a property of the hardware and it arrives whether or not anybody specified it.

Where the same shape appears elsewhere on this site #

This is a memory with a kernel, and the essays around it found several.

The kernel here is finite and hard-edged because the information travels at the wave speed. The kernel in the wall the fluid is listening to is infinite and heavy-tailed because it travels by diffusion, which is why that one has no mean delay at all. The kernel in the drag that integrates a whole history is the same diffusive shape acting on a particle.

The common structure is the point of every memory number is one time over another, which is counting what matters applied to this collection’s own results, and this line’s ratio is the closure time over the round trip — the number the plateau figure is plotted against. Below one, the memory is longer than the process and the flow behaves as though the valve shut instantly. Above one, the memory is being paid off while the process runs.

A third kind sits between them. A duct that forgets everything but one number has a memory that decays selectively — every mode but the slowest is gone after a stated distance, so what survives is one number rather than a window — and the pipeline’s equivalent would be a line long enough that only the fundamental of the reflection train is left. And where a kernel is not merely present but lagging, it can drive rather than damp: that is what two lifts at one incidence is about, and it is the reason surge analysis and flutter analysis are the same kind of calculation done by different departments.

The same arithmetic decides the design of a ram pump, which is a pump with no engine: that machine exists to harvest the surge rather than to survive it, and its waste valve is deliberately designed to close inside a round trip.

What friction does to the memory #

Friction is the only thing in this problem that makes the past fade, and it does two separate jobs that are worth separating.

It damps the reflection train. The square wave in the first figure decays, and the rate is set by the friction head, which is about four and a half metres on this line against a two-hundred-metre wave. That is why the kernel’s contributions from three and four round trips back are small: not because the information failed to arrive, but because it arrived attenuated.

And it packs the line. With flow established there is a head gradient along the pipe; when the flow stops, that gradient has to unwind, and it does so as a slow rise on top of the elastic wave. On a long pipeline with high friction this line packing can exceed the Joukowsky rise itself, which is why the short-line arithmetic here is not directly transferable to a cross-country main.

A frictionless line would have an infinite memory, in the sense that the kernel would never decay, and the peak would depend on the whole closure history however long. Nothing physical is frictionless, but a short steel line with clean water is close enough that the tail matters more than the intuition suggests.

What the picture cannot show #

The kernel figure is a first-order sensitivity: it says what a small perturbation does, and the three-shape comparison is not small. The two are consistent — the shapes with more closing rate late have higher peaks, as the kernel predicts — but the kernel cannot be integrated against a large rearrangement to get the peak, and it is not drawn as though it could be.

The plateau figure is drawn for one valve law. The knee sits at one round trip for any law, because the knee is about the wave and not the valve; the shape of the fall past it is not universal, and the Allievi line is drawn to make that comparison rather than to endorse it.

And every figure here is a head at the valve. The maximum head in the line is not always at the valve — for a slow closure it can be at an intermediate section — and a design has to check the profile rather than the endpoint.

Where the model stops #

One pipe, one reservoir, one valve. Real systems have branches, pumps, air vessels and check valves, and each of them is another boundary condition that reflects. A surge analysis of a network is a different computation with the same physics in it.

The wave speed is constant. It depends on the pipe’s elasticity and on how much free gas is in the water, and a few per cent of entrained air can halve it. Halving the wave speed halves the Joukowsky rise and doubles the round trip, which moves both axes of every figure here.

The same kernel width, in a machine that mixes rather than reflects, is the essay after this one.

Column separation is not modelled. The negative half of the wave is clipped at vapour pressure in reality, and the cavity’s collapse produces a second, higher spike that this march would not show. The first essay in this ladder computes where that clipping starts.

And the friction is quasi-steady. The Darcy factor used at each step is the one for a steady flow at the instantaneous velocity, and in a reversing transient the real wall shear carries its own history — which is the same unsteady-friction effect that the wall the fluid is listening to computes for a plate. Unsteady friction models exist, they add damping, and leaving them out makes the train here decay more slowly than a real one would.

Who found it, and when #

Joukowsky’s measurements on the Moscow mains are of 1897 and Allievi’s theory is of 1902; the characteristics method for pipelines is Bergeron’s, from the 1930s, and became the standard numerical treatment with Streeter and Wylie in the 1960s.

That the peak depends on the closure law rather than on its duration is not a new observation — it is implicit in every characteristics computation and is why surge software takes a valve curve as input. What is unusual here is measuring the sensitivity directly rather than inferring it, and the front edge is a consequence of the wave speed that the standard rules do not name.

Limits recorded rather than smoothed over #

The front edge is one line’s number. 0.45 seconds of an eight-second closure is a measurement on this geometry with this valve law, not a general fraction. What generalises is that a front edge exists.

The window is not a clean multiple of the round trip. It came out at 2.8 round trips here and between 0.9 and 5.8 across a sixfold change in wave speed at fixed closure time. The tidy claim was tested and did not hold, and it is recorded that way rather than being quietly narrowed until it did.

Two hundred reaches. The Joukowsky check passes at 0.02 per cent, which bounds the discretisation error on the peak; the kernel is computed on a hundred reaches for cost, and its front edge moves by less than one sample between the two resolutions.

And the three shapes are constructed rather than observed. They are chosen to have the same duration and different distributions of closing rate, which is what makes the point; whether a particular valve produces one of them is a question about hardware.