Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Water hammer — where it appears

The pressure surge produced when a flow in a pipe is stopped quickly, equal to the density times the wave speed times the velocity change. It has nothing to do with the length of the pipe and can burst it in a fraction of a second.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

12 bar from stopping one metre per second. The head at the valve after it shuts, computed by the method of characteristics on a 600 m pipe. The rise is 122.4 m of water, which is ρaΔV/ρg to 1.4e-14 m — and the scheme was told neither ρaΔV nor anything else about the answer. The wave then runs to the reservoir and back every 2.000 s, and with no friction in the model it never decays: a real pipe damps this out in a few tens of cycles.

Stopping water costs more than moving it

Shut a valve on water running at one metre per second and the pressure that appears is twelve bar — not because the water was pushing hard, but because the only way to stop a column of fluid is to send a message back along it, and the message travels at the speed of sound in the pipe.

applied · Water hammer Nothing can deliver more than h/H, and here that is 10.0%. The fraction of the supply a ram can deliver, against the height it is asked to deliver to, from a supply falling 2 m. The upper curve is the exact ceiling h/H, which follows from the energy audit with every loss set to zero and can be reached by no real machine; the lower one is what a ram at 65% efficiency actually sends. Asking for twice the height halves the delivery, exactly, and there is no design that escapes it.

A pump with no engine

A hydraulic ram lifts water uphill using nothing but the water that is already falling. It has one moving part and no power supply, and everything it can and cannot do follows from an energy audit that fits on one line — including a ceiling nothing about its design can move.

applied · Water hammer How much the peak cares about the valve at each instant. A small bump is added to the valve's opening at one moment and the peak is recomputed; the difference, divided by the bump, is plotted against the moment. It is flat and zero for the first 0.45 seconds — that part of the closure is invisible to the peak — rises through the reflection arrivals, and falls to nothing after the peak has happened, which is the one part of the shape that needs no fluid mechanics.

The part of the closure a pipe cannot see

A surge specification says how long the valve takes to shut. The line does not integrate the duration, it integrates the shape — and it is measurably blind to the first half-second of an eight-second closure, while three closures of identical length give peaks 97 per cent apart.

applied · Water hammer The sound speed of a mixture, against how much of it is gas. Wood's formula for air in water. Both ends are the pure phases at 343 and 1,481 metres a second; in between the mixture takes the water's inertia and the air's springiness and the speed collapses to twenty-four metres a second — a fourteenth of the slower constituent.

Slower than either of them

Sound travels at 343 metres a second in air and 1,481 in water. In a mixture of the two it travels at twenty-four, because the mixture takes the water's inertia and the air's springiness — and one per cent of air by volume is enough to take water down to a twelfth of its own speed.

compressible · Speed of sound A 204 m hammer traded for a 8.57 m swing over 299 seconds. The water level in a 10 m surge tank at the end of a 2 km tunnel 3 m across, carrying 2 m/s, after the turbine is shut off at once, with the level measured from the reservoir's. Without friction it rises to V₀√(L Aₜ/g Aₛ) = 8.57 m and swings with a period 2π√(L Aₛ/g Aₜ) = 299.1 s, the integration agreeing with both closed forms. With the tunnel's 5 m of friction the level starts 5 m below the reservoir, peaks at 5.61 m after 98 s, falls to −3.70 m, and decays. The same tunnel shut at its end with no tank would take the Joukowsky rise of 204 m. The tank does not remove the column's momentum; it gives it a free surface to push against, slowly.

A tank that turns a hammer into a swing

Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.

applied · Water hammer Below Thoma's 6.07 m² the governed tank's swing grows; above it, it dies. The tank level after the turbine's power demand drops by two per cent, with a governor holding the power constant, for tanks of 0.7 and 1.3 times Thoma's area of 6.07 m² — a tank 2.78 m across. The smaller tank's swing grows by a factor of 1.47 every 74 s cycle and has reached −12.20 m by 427 s; the larger one's keeps 0.75 of itself every 100 s and is barely visible. Carried on, the smaller tank's run is refused at 794 s, where the head at the turbine has fallen below a quarter of its design value and the governor would be asking for a flow no turbine passes. The instability has nothing to do with the tank's height: it is the governor drawing more water as the level falls, which feeds the swing, against the tunnel's friction, which is the only thing damping it.

The better tunnel needs the bigger tank

A turbine governed to hold its power opens further when the head at it falls, and draws the tank down harder. That makes it a negative resistance, the tunnel's friction is the only thing damping the swing against it, and so the smallest stable tank grows as the friction shrinks — 2.78 metres across for five metres of friction, 6.09 for one.

applied · Water hammer 121 m after the cavity closes, against 69 m from the closure. The head at a valve shut instantly on water flowing at 0.36 m/s through 600 m of 100 mm pipe, a = 1200 m/s, with a steady head of 25 m. The closure raises it to 69.1 m, the Joukowsky head; the reflection returns at one round trip, 1.00 s, and takes the head down to the vapour head, −10.1 m, where a cavity opens (shaded). It closes 2.146 round trips after the closure, and the first pulse after it reaches 121.3 m — 52.3 m above the Joukowsky head — for 146 ms. The step line is the exact solution between events; the thin line is a 240-reach grid solver that was told nothing about it and agrees with its first pulse to better than a millimetre.

Twice the margin, on top of the hammer

Shut a valve on a line whose pressure is low and the returning wave boils the water beside it. When that cavity closes, the head at the valve can pass the Joukowsky rise — by up to twice the margin that let the water boil, in a sawtooth that jumps each time one more round trip fits into the cavity's life, and for a time that is shortest exactly when the pulse is tallest.

applied · Water hammer The throat holds the hammer back only while its cavity lasts. Left, pressure at the closing valve (red) and at the upstream face of the venturi (gold); right, the throat's cavity volume; after the valve shuts with a 5 mL cavity in the throat. The valve sees the full Joukowsky rise of 14.8 bar at once. The wave reaches the venturi 33.3 ms later, and for the next 7.9 ms the upstream pipe hears nothing: its pressure stays at 5 bar while the cavity is squeezed. When the cavity closes at 41.3 ms the surge passes into the upstream pipe at 13.8 bar above its steady pressure.

A choked throat buys time, not silence

A venturi whose throat has reached vapour pressure passes a flow the downstream pressure cannot change, and it is tempting to read that as isolation: whatever happens downstream, the upstream pipe will not hear it. Slam a valve downstream and it hears it. The cavity at the throat holds the surge back only for as long as it takes to fill, and then lets 93 per cent of it through.

misconceptions · Venturi

Named alongside it

The objects these essays reach for when they reach for this one.

Wave speedModel limitColumn separationMethod of characteristicsCavitationVapour pressureCompressibilityControl volumeFriction factorMeasurementOscillationSurge tank

All concepts