The depth that costs least

Water in a channel carries its own energy budget. Some of it is height — the depth itself — and some is speed, and the two trade against each other because the discharge is fixed: deeper water is slower water.

That trade has a floor. There is a least amount of energy that will carry a given flow, no depth carries it for less, and a channel asked to deliver a flow with less energy than that does not comply. It backs up until it has enough.

The curve, and why it has the shape it has #

Measure the energy from the bed, per unit weight of fluid, as a length. It is the depth plus the velocity head, and with the discharge per unit width $q = Vh$ held fixed:

$$E = h + \frac{V^2}{2g} = h + \frac{q^2}{2gh^2}$$

Two terms with opposite behaviour. The first rises with depth linearly — deep water is high water. The second falls as the inverse square — deep water is slow water and slow water has little kinetic energy. At large depth the curve is asymptotic to the line $E = h$; at small depth it is asymptotic to the vertical axis, because a very thin sheet of water carrying a fixed discharge has to move very fast indeed.

Between two divergences there is a minimum, and differentiating gives it:

$$\frac{\mathrm{d}E}{\mathrm{d}h} = 1 - \frac{q^2}{gh^3} = 0 \quad\Longrightarrow\quad h_c = \left(\frac{q^2}{g}\right)^{1/3}$$

Substituting back, $E_{\min} = \tfrac{3}{2}h_c$ exactly, and the Froude number at that depth is

$$\mathrm{Fr} = \frac{q}{h\sqrt{gh}} = \sqrt{\frac{q^2}{gh^3}} = 1$$

exactly. Those two facts — that the energy minimum and the Froude number of one are the same depth, and that the minimum energy is three halves of it whatever the discharge — are the whole content of the curve, and neither is obvious from looking at it.

What was computed, and what the assertions catch #

The minimum is found by golden-section search on the computed curve, not by the formula, and then three things are asserted.

The search’s answer is the closed form. $h_c = 0.29427746$ from the search against $(q^2/g)^{1/3} = 0.29427746$, to eight figures.

The Froude number there is one, to six figures. That is the assertion worth having, because it is the one that would fail if the two definitions had drifted apart — an energy expression with a mistyped constant would still have a minimum somewhere, and the minimum would still look like a minimum, and only its Froude number would give it away.

The least energy is three halves of the critical depth, to six figures. This one catches a different failure: a curve computed with the wrong power of $h$ in the velocity head has a minimum, at a plausible depth, with a plausible Froude number, and the ratio comes out wrong.

The solver also refuses to compute a flow whose energy is below the floor, and the refusal is worth quoting because it says what the physics does instead:

Past it there is no depth over the crest that carries the discharge, and what happens is not a thinner sheet of water — it is the upstream depth rising until the arriving energy is enough. The channel chokes, exactly as a nozzle does.

Two branches, and the answer depends on which one #

The two depths at a given energy are the same flow rate arranged in two entirely different ways, and almost every counter-intuitive result in open-channel flow is a consequence of not knowing which branch is in play.

The names are borrowed wholesale from the gas, and the borrowing is exact rather than decorative: subcritical is subsonic and supercritical is supersonic, with the same statement about which way a signal can travel underneath both.

The upper branch is subcritical: deep, slow, $\mathrm{Fr} < 1$. Long waves can travel upstream faster than the water travels down, so the flow knows what is ahead of it and is controlled from downstream. A river approaching a weir is on this branch, and it starts responding to the weir long before it reaches it.

The lower branch is supercritical: shallow, fast, $\mathrm{Fr} > 1$. No signal can travel upstream, so the flow does not know what is ahead and is controlled from upstream. Water leaving the foot of a spillway is on this branch, and it arrives at whatever is downstream without having prepared for it — which is why it so often has to be dealt with by forcing a jump.

Now put a bump in the bed. Raising the bed by $\Delta z$ takes $\Delta z$ out of the specific energy, which is a step to the left along the curve. On the upper branch, moving left means moving down: less depth. On the lower branch, moving left means moving up: more depth.

At the case computed, a 0.60 m approach at Froude 0.34 drops by 58 mm over a 50 mm bump, and a 0.10 m approach at Froude 5.05 rises. The bump is identical. The fluid is identical. Which way the surface goes is a question about the Froude number.

Both behaviours are asserted together rather than one at a time, because either alone reads as an accident of the numbers chosen.

A family of curves, one per discharge #

There is not one specific-energy curve; there is one for every discharge, and the way they nest is the second half of the picture.

Since $h_c = (q^2/g)^{1/3}$, doubling the discharge raises the critical depth by a factor of $2^{2/3} = 1.587$ and the minimum energy with it. So the question “can this channel carry this flow” has an answer that depends on both, and a channel that copes with a river in summer can be entirely unable to pass the same section in flood — not because it has run out of room at the banks, but because the flow no longer has the energy to get through the narrowest place.

Read the other way round, the family is what makes a flume a measuring instrument. Force the flow through critical and the depth there is $h_c$, which depends on $q$ and on nothing else; measure it, invert the cube root, and the discharge follows. No velocity has been measured, no coefficient calibrated, and nothing downstream can affect the reading. It is the closest thing open-channel hydraulics has to the pitot tube — an instrument whose reading is a consequence of a theorem rather than of a calibration curve, and which is trusted for the same reason.

Choking, and the throat it is the same as #

Keep raising the bump. The flow steps further and further left along the curve, the depth over the crest approaches critical, and at some rise there is nothing left to give.

The largest bump this flow can climb is $E_1 - E_{\min}$, which comes out at 0.194 m for the case drawn. Ask for more and the calculation has no root, because the demanded energy is below the floor.

The physical resolution is that the upstream state was not fixed after all. The channel adjusts: the approach depth rises, which raises $E_1$, until the flow arrives with exactly enough energy to pass over the crest at critical depth. The crest is then a control — the depth there is $h_c$, fixed by the discharge alone, and knowing it is what lets a weir be used as a flow meter. That is why a measuring flume works: force the flow through critical, measure one depth, and the discharge follows without any calibration of a velocity.

This is the same statement as the throat that stops listening, with depth in the part pressure plays.

• A converging nozzle accelerates a subsonic gas; a rising bed decelerates a subcritical stream in energy terms and thins it.
• The throat is where the Mach number reaches one; the crest is where the Froude number reaches one.
• Past the choking condition, lowering the back pressure changes nothing upstream of a nozzle throat; past the choking bump, nothing downstream changes the discharge over a crest.
• The flow can pass through the sonic — or critical — condition once, at the narrowest section, and not twice.

The other way to choke a channel, which is the nozzle’s own #

A bump chokes a channel by taking energy away from it. There is a second route that leaves the energy entirely alone and chokes the flow anyway, and it is the one that makes the analogy with a nozzle exact rather than merely close — because it is a change of area.

Narrow the channel at constant bed level. The total discharge $Q$ is unchanged, so the discharge per unit width $q = Q/b$ rises, and the whole specific-energy curve is a different member of the family: critical depth $(q^2/g)^{1/3}$ deeper, minimum energy $\tfrac32 h_c$ larger. The flow has the same energy it always had and the floor has come up to meet it.

The arithmetic for the case drawn above is worth doing, because it puts the two routes on one scale. That flow arrives with $E_1 = 0.635$ m and $q = 0.5$ m²/s. Choking requires the crest depth to be two thirds of the arriving energy, $h_c = 0.424$ m, which the curve reaches at $q_{\max} = \sqrt{g h_c^3} = 0.864$ m²/s — so the channel may be squeezed to 58 per cent of its width before there is no depth that will pass the flow. Narrow it further and the same thing happens as with too tall a bump: the water upstream rises until it arrives carrying enough energy, and the narrowest section becomes a control at critical depth.

That is the nozzle, term for term. A converging duct chokes when its throat area falls below what the mass flow requires; a converging channel chokes when its width falls below what the discharge requires; and in both cases the response is not a thinner stream but a rearrangement of everything upstream. The bump, by contrast, corresponds to something a nozzle does not have — it removes energy the way friction or heat does, arriving at the same singularity by a different door.

Which is why the two are combined in practice. A critical-flow flume narrows the walls and raises the floor at once, so that the section is forced through critical with a margin, and then measures one depth upstream. The reading is a discharge because critical depth depends on discharge alone; the combination is used because either device on its own has to be large to guarantee the crossing, and the two together do it in a shorter structure with less head lost. A Parshall flume is exactly this, and it has been the standard instrument for measuring flow in irrigation canals and treatment works for a century.

The general statement is the one the curve has been making throughout. There is only one way for a channel to refuse, and the ways of arriving at the refusal — a rise in the bed, a narrowing of the walls, an increase in the discharge — are three routes to the same point on the same curve.

What the curve does not contain #

No bed friction. The specific-energy curve is a statement at one cross-section, and over a long channel friction removes energy continuously. The subject that results — gradually varied flow, and its dozen classified surface profiles — is entirely about that term, and none of it is here.

No slope. A channel on a slope has gravity feeding energy in as fast as friction takes it out, and the balance between them defines a normal depth which the flow tends towards. The relation between the normal depth and the critical depth is what makes a channel steep or mild, and it decides whether a jump forms at all.

One dimension, and a rectangle. The whole calculation assumes hydrostatic pressure, no vertical velocity and a flat wide channel. Over a sharp-crested weir none of those holds and the flow separates from the crest entirely; over a broad-crested weir they hold well and the critical-depth argument is used directly. The distinction between the two kinds of weir is exactly the question of whether this model applies.

No wave system. A channel with a genuinely two-dimensional disturbance in it — a pier, a bend, an obstruction — sends out oblique standing waves in supercritical flow, exactly as a supersonic body sends out oblique shocks, and the pattern is closely related to the wake a ship drags behind it. None of that is visible in a one-dimensional energy balance, which knows only about the mean depth at each station.

And no dispersion. Long waves in shallow water are non-dispersive in this model. Real ones are not, slightly, and near Froude one that slight dispersion is what produces the train of smooth standing waves — the undular jump — instead of a broken one. That is the case where the previous rung’s calculation is least honest, and it is the same missing term.

Two curves, and knowing which one to use #

The pair of essays in this anchor turns on a distinction that is easy to state and easy to forget: energy says what is possible, momentum says what happens.

A gradual transition — over a bump, through a contraction, along a gentle slope — is smooth, loses essentially nothing, and is governed by the specific-energy curve. Follow that curve and the depth follows continuously.

An abrupt transition — a jump — destroys energy, so the energy curve cannot connect its two ends. What survives is the specific force, and the conjugate-depth relation is the answer. Try to use the energy curve there and it says the transition is impossible; try to use the momentum curve on the gradual case and it gives the wrong depth, because momentum is not conserved along a channel with a sloping bed.

Getting that choice right is most of what open-channel hydraulics is, and it is a cleaner version of a decision that recurs everywhere in this subject: which conserved quantity survives the thing being crossed. It is the same judgement that decides whether Bernoulli’s equation may be carried across a fan and whether the total pressure survives a shock.

There is a third curve nobody draws and it is worth naming for completeness. Holding the energy fixed and varying the discharge gives a curve with a maximum rather than a minimum — the largest flow a channel of a given energy can pass — and its maximum is at the same critical depth. That coincidence is not luck: minimising energy at fixed discharge and maximising discharge at fixed energy are the same stationarity condition read along two different sections of the same surface. Critical flow is therefore the answer to two questions at once, which is why so many different arguments in this subject converge on Froude one.

Every one of these statements has a twin in the compressible field, and this is the one where the twin is most useful: the sonic throat is simultaneously the minimum area for a given mass flow and the maximum mass flow for a given area, and the two branches through it are the two branches here.

Who found it, and when #

Boris Bakhmeteff introduced specific energy as a concept in 1912, at St Petersburg and later Columbia, and the curve is usually named after him. The critical depth and its properties were implicit in Bélanger’s work of the 1820s and in Bresse’s of the 1860s; what Bakhmeteff added was the diagram, and the diagram is what makes the two branches a single object rather than two cases.

The choking argument arrived from the other direction and much later. Its gas-dynamics twin belongs to the 1930s and to the development of supersonic wind tunnels, and the recognition that the two are the same statement is largely due to the wartime generation who used water tables as analogue computers for compressible flow — a shallow layer of water over a shaped floor, photographed from above, standing in for a gas whose behaviour nobody could otherwise visualise.

Where the ladder goes #

Both open-channel rungs are conservation arguments with no material property in them, which is what they share with the actuator disc and the sudden expansion.

The field now turns to a case where that is no longer possible. A ball in flight is decided by where its boundary layer separates, that separation cannot be computed here at any Reynolds number, and everything that follows has to be built on a correlation with the borrowing marked. The result is one of the strangest facts in this subject: there is a band of speeds in which a ball’s drag force falls as it goes faster.