The floodgate that opens at a chosen depth

On many irrigation canals and some flood defences there are gates that nobody operates. They are flat plates hung on horizontal pivots, resting against a seat on their downstream side, and they stay shut while the water behind them rises, until at one level they swing open and let it through; when the level falls they swing shut again. On the spillways of some dams there are larger versions — concrete or steel blocks standing in a row along the crest — each designed to tip over at a different flood level and be washed away, opening the spillway in stages, and each set off by nothing but the water itself.

None of these has a float or a motor. What they have is a hinge, or a toe, put at a particular height, and the physics they use is the oldest result in hydrostatics after Archimedes’: where the water’s push on a submerged surface acts. The pressure that only knows depth found that the push on a dam acts two-thirds of the way down. That point moves when the water is deeper than the surface it pushes on, and a gate that opens itself is a gate whose hinge has been placed in its way.

Where the push acts #

Pressure in still water grows in proportion to depth, so the push on a vertical gate is not uniform: it is least at the top and greatest at the bottom. The whole push can be replaced by a single force of the same total, acting at one height — the centre of pressure — chosen so that it would turn the gate about any axis exactly as the distributed push does. The same push, further out made the point that a force’s turning effect depends on where it acts as much as on how big it is, and the centre of pressure is where a single force must act to have the right turning effect.

When the water stands no higher than the top of the gate, the pressure across the wetted part is a triangle, zero at the surface and greatest at the gate’s foot, and its centre of pressure is a third of the way up the wetted part — the two-thirds-down point of the dam. Raise the water above the gate’s top and the pressure on the gate becomes a trapezoid: the triangle of the water level with the gate’s top, plus a uniform pressure from the extra water above, which pushes equally on every part of the gate.

The uniform part acts at the gate’s middle. The triangular part acts a third of the way up. As the water deepens, the uniform part grows and the triangle stays the same size, so the resultant moves towards the middle. It never gets there: at any finite depth the water at the bottom of the gate is a little deeper than at the top, and the resultant always sits a little low.

The offset, and why depth erases it #

The distance by which the centre of pressure lies below the gate’s centroid has a compact general form:

$$y_{\text{cp}} - \bar y = \frac{I}{A\bar y},$$

where $\bar y$ is the depth of the centroid below the surface, $A$ the gate’s area and $I$ the second moment of its area about a horizontal line through the centroid. For a rectangle of height $H$, $I/A = H^2/12$; for a circle of diameter $H$, $H^2/16$. The offset falls as one over the depth.

The formula has the same shape as the parallel-axis rule for a body’s moment of inertia, and for the same reason: the pressure is linear in depth, so the moment of the push about any line is a first moment of area weighted by depth, and the shift from centroid to centre of pressure is the second moment divided by the first. The geometry of the gate enters only through $I/A$, which is why a circle, which has less of its area far from its middle than a square of the same height does, has a smaller offset.

The physical reading is simpler. The offset exists because the pressure varies across the submerged gate; it shrinks because, deep down, the variation across the gate is a small fraction of the pressure itself. A sluice gate ten of its own heights below the surface has a resultant within a per cent of its centre — uniform pressure, as the push that has no direction found, has no preference for any part of a surface, and so no lever to favour one edge. Engineers designing deep outlet gates take the force at the centroid and need no correction; designers of surface gates cannot.

A hinge put on the climb #

Now hang the gate on a horizontal hinge across its width, at some height $y_h$ above its bottom, and let it rest against a sill on the downstream side so that it can swing open but not back. The water’s moment about the hinge is the resultant force times its lever arm, and the lever arm is the distance from the hinge to the centre of pressure. While the centre of pressure is below the hinge, the water pushes the bottom of the gate harder than the top — about the hinge — and presses it into its seat. Once the centre of pressure rises above the hinge, the water’s moment reverses and swings the gate open.

The climb of the centre of pressure as the water rises therefore sets the opening level. A hinge at $0.3H$ is passed while the water is still below the top of the gate, at a depth of $0.90H$, on the triangular part of the climb. A hinge at $0.4H$ is passed at $1.33H$. A hinge at $0.45H$, closer to the middle, is passed only at $2.17H$. A hinge at or above the middle is never passed at all.

The opening depth is a design choice, made once, by placing a pivot. The steepness of the curve near the middle means the choice is sensitive there: moving the hinge from $0.40H$ to $0.45H$ raises the opening level from $1.33H$ to $2.17H$. Gates intended to open at a modest overtopping use a hinge well below the middle; gates intended to hold until a flood is deep use one close to it, and their opening level depends sensitively on getting the hinge exactly where it was designed.

The exact level is the one at which the centre of pressure reaches the hinge. For a hinge above a third of the height the gate is overtopped when it opens, and solving the trapezoid’s centre for the water depth $D$ gives

$$D = \frac{(2 - 3y_h/H)}{3 - 6y_h/H}\,H,$$

which runs away to infinity as $y_h$ approaches $H/2$.

A gate of a practical size makes the numbers concrete. A flap two metres high and three wide, hinged 0.8 metres above its sill — at $0.4H$ — stays shut until the water stands 2.67 metres deep, two-thirds of a metre over its top. At that moment the water pushes on it with about 98 kilonewtons, the weight of a ten-tonne lorry, and the whole of that push acts exactly on the line of the hinge, so that the hinge carries it and the gate is balanced on a knife-edge. A few centimetres more water and the push acts a few millimetres above the hinge; the moment that results, a few hundred newton-metres, is enough to start the gate moving, and the more it opens, the more the flow beneath it unloads the bottom of the flap and the further it swings.

Held harder, then let go #

The way the hinged gate opens has a feature that the climb alone does not show. Below the opening level the water’s moment presses the gate shut, but it does not do so more and more firmly all the way up.

As the water first rises, the total force grows faster than the lever arm shrinks, and the gate is pressed into its seat harder. Past $0.80H$ for this hinge, the lever arm shrinks faster than the force grows, the grip loosens, and at $1.33H$ it is zero. The same water that was holding the gate shut releases it. That matters for the seal — a self-acting gate leaks least when it is held hardest, at the levels it most often sees — and for the opening, which begins smoothly from zero moment rather than with a jolt. Once open, the water’s moment on the flap grows without limit and it swings fully open; in practice it is stopped by a buffer, and the water flowing under it changes the pressure from hydrostatic to something that has to be computed with the flow.

Real self-acting gates are seldom this simple. The flat plate described here would oscillate near its opening level, opening a crack, releasing water, the level dropping, the gate shutting, and a practical gate adds a counterweight, a float or a damper to make the action steady. But the principle that sets the level — the climb of the centre of pressure — is the same, and it is the part that needs no adjustment.

A block that tips on purpose #

The same arithmetic applied to something resting on a surface instead of hanging from a hinge gives the opposite design: a block that stays put through ordinary floods and tips over at a chosen extreme one. Slide or topple found that a block pushed sideways tips about its leading edge when the pushing force’s moment about that edge exceeds the weight’s restoring moment. On a spillway crest the push is the water’s, acting on the block’s upstream face at the centre of pressure, and the block tips about its downstream toe.

A solid concrete block whose length along the flow is 0.7 of its height would tip on its face pressure alone when the water stood $1.85H$ above the crest. That level depends on the block’s weight and on the climb of the centre of pressure, and the tip comes gradually as the moments approach each other, which makes the level imprecise. Add a chamber beneath the block, connected to the upstream water through an inlet at a set height, and something else happens when the water reaches the inlet: the chamber floods, and the water pushes up on the block’s base with the full pressure at the crest. The block the water does not lift found that buoyancy needs water underneath — a block sealed to the bottom feels no upward push at all. The chamber is a way of letting the water underneath at a chosen moment.

The uplift arrives as a step, and the step carries the tipping moment past the restoring one at once. The block tips at the inlet’s height, a level set by the position of a hole rather than by the block’s weight or the exact shape of the pressure. That is how the fusegates installed on dam spillways since the 1990s work: rows of blocks along the crest, each with a well and a chamber whose inlet is set at a different height, so that as a flood rises beyond what the spillway can pass, the blocks tip one after another and widen the spillway in steps. Each is lost when it tips — it is a fuse — but a dam that might otherwise be overtopped is saved by opening its spillway exactly when needed.

The oldest calculation in the subject #

The force on a submerged wall was one of the first problems hydrostatics solved quantitatively. Simon Stevin, the Flemish engineer who also worked out the balance of forces on an inclined plane, computed the force of water on a vertical wall in 1586 by cutting it into thin horizontal strips, bounding the pressure on each strip between its values at the strip’s top and bottom, and letting the strips become thin — an integral before integrals had a name. He found the force on a wall to be the weight of a column of water with the wall’s area as its base and half its height as its height, which is the statement that the force is the pressure at the centroid times the area. Stevin also stated the hydrostatic paradox that force multiplied, and nothing gained explored: that the force on a vessel’s bottom depends on the depth of water above it and not on the amount, so a narrow column can press as hard as a wide lake.

The centre of pressure — not just how much the water pushes but where — came later, with the mechanics of moments, and it is the part that the self-acting gate uses. A force calculation tells an engineer how strong the gate must be. Only the centre of pressure tells where to put its hinge, and it was the realisation that the centre moves with the water level that turned a calculation about strength into a design for a gate that measures the flood itself.

The centre of pressure and the metacentre #

The centre of pressure is one of a family of points in hydrostatics at which a distributed push can be replaced by a single force, and each has its own surprise. Why a ship comes back upright found that the buoyant force on a tilted hull acts through a point that moves as the hull tilts, and that a ship is stable if its centre of gravity is below the metacentre, the point where the line of the shifted buoyancy crosses the hull’s centreline. The log that floats on its corner found a square log choosing to float tilted because that point lies differently for different orientations. In each case, the question “does it turn?” is answered by where the water’s resultant acts relative to a pivot — a hinge, a toe, a centre of gravity — and in each case the answer is computed from a first and a second moment of a shape.

That is why these designs can dispense with any mechanism. A float, a sensor or a motor measures the water level and acts on it. A hinge placed on the centre of pressure’s climb does both at once: the climb is the measurement, the crossing is the action, and the only part that can wear is the pivot.

What the figures leave out #

The figures assume still water: the pressure everywhere is the hydrostatic $\rho g$ times depth. Once a gate opens and water flows under or over it, the pressure on the gate is no longer hydrostatic — water accelerating towards an opening has lower pressure than still water at the same depth — and the moment on an open gate must be computed with the flow. The gates are taken as rigid, flat and vertical; an inclined gate has the same formulas with distances measured along its slope, and the climb is stretched by the inclination. Water on the downstream side, which pushes back with its own centre of pressure, is taken to be absent. And the block on the weir crest is drawn without the friction at its base, which resists sliding rather than tipping, or the scour beneath it, which in a real flood can change its support before the water reaches the inlet.

The domain is static: a gate or block in still water whose level changes slowly enough that the pressure is hydrostatic at every moment. Within it the climb of the centre of pressure and the opening levels are exact.

Still open: how self-acting gates behave in a moving flood #

What is not fully settled is how self-acting gates behave dynamically, when the flood rises quickly and the water accelerates through the opening as the gate swings. Flap gates in rivers and tidal outlets have been observed to oscillate — slamming open and shut, or vibrating at a frequency set by the flap’s own inertia and the flow beneath it — and some such oscillations have damaged gates and their supports. Whether a given design will open smoothly or flutter depends on the coupling between the flap’s motion, the pressure of the flow through the gap, and the waves sent upstream and downstream when the flap moves, and the predictions rely on scale models and on simulations of the flow that are only now detailed enough to capture the gap. The static design, which sets the level, is exact; the dynamics that decide whether the flap opens cleanly are not yet reliably computed in advance.

The static result is short. The water’s push on a submerged gate acts below the gate’s centroid by $I/A\bar y$ — a third of the way up the wetted part while the gate is not overtopped, then climbing towards the middle as the water deepens — so a gate hinged at $0.4H$ is pressed shut, hardest at $0.80H$ of water, and swings open at $1.33H$, with no mechanism but the hinge’s height; a block on a weir tips at the height of an inlet that floods a chamber beneath it. The water measures its own depth by where it pushes.