Stopping water costs more than moving it

A tap closed sharply on a domestic supply makes a bang that can be felt through the wall. In an industrial line the same event splits pipes, tears anchors out of concrete and destroys pumps, and it happens in a system whose steady operating pressure is a few bar.

The temptation is to look at the momentum in the pipe — a long line holds a lot of water, moving water carries momentum, and stopping momentum takes force. That account gets the size of the pressure wrong, gets its dependence on the pipe wrong, and gets the remedy wrong.

The right answer contains no pipe length at all.

The control volume that moves #

Consider the instant after the valve shuts. The water immediately at the valve has stopped. The water a hundred metres upstream has not — it cannot have, because nothing has told it yet.

What separates the two is a front, travelling upstream at the speed a pressure signal moves in the pipe. Ahead of the front the water is still running at V₀ at the original pressure; behind it the water is at rest at a higher pressure. That front is the whole phenomenon, and the way to analyse it is to draw a control volume that travels with it.

In the front’s frame the fluid arrives at a + V₀ and leaves at a. The momentum flux therefore changes by ρa(V₀) per unit area, and the pressure has to supply it:

$$\Delta p = \rho\, a\, \Delta V$$

That is Joukowsky’s equation, and it is worth reading what is absent from it. No pipe length. No time. No valve detail. No steady operating pressure. No diameter. Any sudden change of velocity costs ρaΔV, and for water in a steel pipe that is about 12 bar per metre per second — so a flow of three metres per second, which is unremarkable, carries a 36-bar surge behind it.

This is the applied field’s method in the one place it has to move: the box is drawn round the front rather than round the pipe, because that is where everything crossing the faces is known.

The speed the message travels at #

The wave speed is not the speed of sound in water, and the difference is a factor of four.

A pressure pulse in a pipe stores energy in two places: compressing the water, and stretching the pipe. The two compliances are in series, so the pipe’s elasticity slows the wave:

$$a = \frac{\sqrt{K/\rho}}{\sqrt{1 + KD/Ee}}$$

which is Korteweg’s result. For water in a thick steel pipe the correction is modest — 1,342 m/s against 1,481 in free water — and for plastic it is enormous.

That is a design lever rather than a curiosity: the surge pressure a system will see is set by the pipe’s stiffness, and can be reduced by a factor of several by changing the material — before any valve, any surge vessel and any control strategy is considered.

The wave, in space and time #

The formula gives the height of the step. What happens next needs the wave followed.

The method of characteristics is the natural tool, and on a frictionless pipe it is not an approximation at all. Along the lines dx/dt = ±a the combination H ± BQ is constant — those are the Riemann invariants, and they are exactly the same object as the ones that solve a shock tube. The scheme is a bookkeeping device for them, so refining the grid changes nothing: 32 reaches and 256 reaches give the same answer to five decimal places, and the site checks that they do.

The reservoir is what turns a step into a cycle. It holds the head fixed whatever the flow does, so the arriving high-pressure wave is reflected with its sign changed, and the pipe alternates between over-pressure and under-pressure with a period of 4L/a — 1.79 seconds for the pipe drawn here.

That is where the pipe’s length finally enters: not in the size of the pressure, but in how long it lasts and how often it repeats.

Why closing the valve slowly does not always help #

The obvious remedy is to close the valve more slowly. It works, and it works only after a threshold that the analysis above supplies exactly.

While the valve is still closing, the pressure at it is rising and a wave is running upstream. If the valve finishes before the first reflection returns — that is, in less than 2L/a — the pipe’s far end has not yet had any say, and the peak is the full Joukowsky value. Closing in a tenth of the return time and closing instantaneously produce identical pressures, which the site verifies to nine figures rather than asserting.

Past the threshold the peak falls roughly as 1/T, and the curve is where a real finding turned up.

The standard slow-closure estimate is Michaud’s rule, ΔH = 2LV₀/gT, which appears in every design handbook. The characteristics do not approach it. They approach half of it — the rigid-column value LV₀/gT, which is the pressure needed simply to decelerate the whole column at V₀/T with no wave in it anywhere. At sixteen return times the computed peak is 51 per cent of Michaud’s estimate, at thirty-two it is 51 per cent, and extrapolating the sequence gives 0.500.

The reason is that Michaud’s derivation assumes the velocity falls linearly, and a valve whose area falls linearly does not produce that: the flow through a closing valve holds up while the head rises behind it, and most of the deceleration happens at the end. Michaud is therefore a bound rather than an answer, and a conservative one by a factor of two — which is presumably why it has survived in the handbooks for a century.

The half of the wave that cannot happen #

The characteristics are symmetric, and the negative swing is as large as the positive one. That is where the model leaves physics.

A pipe cannot go below the vapour pressure of what is in it. When the low wave takes the head there, the column separates — a cavity of vapour opens at the valve or at a high point — and the two halves of the column then move independently until they meet again. What follows the collapse can be worse than the original hammer — not because the columns meet faster than the flow was moving, but because the collapse lands on waves the cavity left in the pipe, and the pulse that results can pass the closure’s rise by up to twice the margin the line had to vapour pressure. The collapse itself is essentially the bubble problem at pipe scale.

Nothing in the characteristics knows this. The limit is drawn on from outside, which is the honest way to show where a model has stopped applying: the solver reports where its own answer has left physics rather than quietly clamping it.

The same event in a softer pipe #

The wave speed enters the pressure linearly, so changing the pipe changes the whole picture rather than shading it.

That independence is the practically important half of the formula. A surge is not a percentage of the working pressure; it is an absolute quantity set by the velocity change, so the systems that suffer most are the ones that run fast at low pressure. Irrigation mains are the classic case, and the roughness that decides their friction has nothing to do with it.

There is a family resemblance to the compressible field worth naming. A valve closing quickly is a piston pushing into a gas at rest, and the wave it sends is the same object a shock tube’s diaphragm releases — a finite-amplitude disturbance whose speed is set by the medium’s stiffness. What water hammer lacks is the nonlinearity: the pressure changes are small enough compared with the bulk modulus that the wave speed does not depend on the amplitude, so the front stays a step and never steepens into the discontinuity a gas produces.

What is missing from the model, and what it changes #

Friction. There is none here, which is why the oscillation never decays. A real line damps a surge out in a few tens of cycles, faster in a rough pipe than a smooth one, and the damping is set by the friction factor. It affects the decay and not the first peak, which is why the frictionless model is the right one for design: the largest pressure the system will ever see is in the first cycle, before friction has had time to do anything.

Unsteady friction. Even where friction is included, a quasi-steady friction factor underestimates the damping of a rapid transient, because the velocity profile in the pipe is nothing like the fully-developed one the correlation was measured on. This is a live research subject and nothing here computes it.

Anchors and supports. A pipe that can move axially carries a second wave — a stress wave in the pipe wall — coupled to the fluid one, and the two exchange energy. The model has an infinitely rigid pipe in every direction except radially.

The valve’s own characteristic. The boundary condition used here is an orifice law with a linearly falling area, which is a reasonable model of a gate valve and a poor one of a ball valve, whose effective area collapses in the last few degrees of travel. Since the whole slow-closure argument is about how the area falls, this matters: a valve that does most of its closing at the end behaves like a fast valve however slowly the handle is turned.

The one place this is wanted rather than feared #

Everything above is written as a hazard, which is how the subject is usually taught. The next rung inverts it: a hydraulic ram is a machine whose entire mechanism is water hammer, deliberately provoked several times a minute, and used to lift a fraction of its own supply to a height the supply never reached. The pressure that bursts a pipe is the pressure that pumps the water, and the ceiling on what such a machine can do is exactly the aV/g of this essay.

Why this is the only unsteady thing in the field #

Every other essay in this field is a steady balance. A disc extracting power, a weir choking, a sailing polar, a bed floating: all of them are statements about a flow that has settled, and the control volume is drawn once and read off.

A surge cannot be treated that way, and the reason is worth naming because it separates two kinds of problem. In a steady flow the pressure adjusts everywhere at once — the incompressible pressure field is elliptic, and every point feels every other — so a control volume can be drawn anywhere and the answer does not depend on when. Allow the fluid the slightest compressibility and that stops being true: information travels at a finite speed, the equations become hyperbolic, and the answer at a point depends only on what has had time to reach it.

That is the same change of type the compressible field makes at Mach one, arriving in water at a millionth of the Mach number. Water is very nearly incompressible and very nearly is the operative phrase: the bulk modulus is finite, so the signal speed is finite, and a valve that shuts faster than the pipe can answer produces a pressure that has nothing to do with the pressure driving the flow.

The transient that actually breaks pipelines #

Every event in this essay begins with a valve being shut, and that is not the transient most systems are destroyed by. The worst case in a pumped main is a power failure.

When the supply is lost the pump does not stop instantly — it coasts down on the inertia of its own rotating parts — but it stops fast, and the head it was producing collapses with it. So the first wave is a down-surge, propagating from the pump into the line, and the pressure behind it falls by $\rho a \Delta V$ below the running value rather than rising above it.

That is much more dangerous than the valve case, for the reason the vapour-pressure figure shows: a system has a great deal of headroom above its working pressure and very little below it. A down-surge of twelve bar in a line running at four has taken the pressure below vacuum long before the arithmetic finishes, so column separation is the normal outcome of a pump trip rather than an exotic one — and the damaging event is the rejoining, when the two halves of the column come back together at whatever relative velocity nothing has damped.

And the obvious protection is frequently the cause. A non-return valve is fitted to stop the column running back through the pump. If it closes before reverse flow develops it does no harm; if it closes after, it is shutting on a flow that is already moving backwards, and the surge it generates is $\rho a$ times that reverse velocity. A check valve slamming on a returning column routinely produces a larger pressure than anything the forward flow could have made, and the failure is at the valve — which is why such valves are specified by their dynamic closing behaviour rather than by their leak-tightness.

The remedies map onto the ones already listed, with the clock running on a different quantity. A flywheel on the pump extends the speed decay past the return time $2L/a$, which is the closure-time remedy applied to a machine instead of a valve. An air vessel on the discharge gives the wave somewhere to reflect from before it can pull the line apart. Neither strengthens anything.

The four remedies, and which one the arithmetic prefers #

Every term in Joukowsky’s expression is a lever, and setting them out in order is the practical summary of the whole essay.

Reduce ΔV. The pressure is proportional to the velocity change, so a system designed to run at one metre per second rather than three has a third of the surge before anything else is done. Pipe velocity is chosen for friction and for capital cost, and this is the third consideration.

Reduce a. A more compliant pipe carries a slower wave. Plastic rather than steel is a factor of three or four; an air vessel or a length of flexible hose near the valve does the same thing locally.

Extend the closure past 2L/a. Effective only past the threshold, and then roughly as 1/T. This is the remedy everybody reaches for first and the one with a hard floor: below the return time it does nothing whatever.

Give the wave somewhere to go. A surge tank, an air vessel or a relief valve provides a reflecting boundary closer than the reservoir, which shortens the return time and caps the pressure. This is the remedy of choice for long lines, and it works by changing L rather than by changing anything about the water.

What the arithmetic will not support is the intuition that a stronger pipe is the answer. A pipe strong enough to hold 12 bar per metre per second is a pipe whose wall is thicker, and a thicker wall carries a faster wave — so the pressure it has to hold goes up as its ability to hold it does. The two effects do not cancel, but they are in the same direction, and a system that is designed to survive its surges rather than to avoid them is being designed against a moving target.

Who found it, and when #

Nikolai Joukowsky measured and derived the surge in Moscow’s water supply in 1898, in one of the most thorough experimental studies of its era — he shut valves on the city’s mains at night and recorded the pressures. The wave speed is Korteweg’s, from 1878. The method of characteristics was applied to pipe transients by Allievi in 1902, whose charts remained the design tool until computers made the characteristics themselves cheap enough to run directly.

The same Joukowsky appears on this site as the man whose transformation makes an aerofoil out of a circle and whose theorem gives its lift. It is the same habit in both places: find the control volume or the mapping in which the answer is exact, and refuse to approximate anything until it is.

Where the ladder goes next #

The slam has been characterised, feared, and shown to be independent of everything except the wave speed and the velocity change. What follows is the machine that lives on it.