The number that really is one

This collection has spent an essay establishing that a dimensionless group is one where the two terms it compares are equal and that almost nothing happens there. What “of order one” is worth puts fourteen groups on an axis and finds their behavioural thresholds scattered from a five-hundred-and-ninety-fourth of one to seventeen hundred times it.

Two of the fourteen sit exactly on one. This essay is about why, and the answer is not that those two groups were formed more carefully.

The condition, which is not a comparison #

Shallow water carries surface waves at $\sqrt{gh}$ relative to the fluid. A disturbance in a stream moving at $U$ therefore travels at $U + \sqrt{gh}$ downstream and $U - \sqrt{gh}$ upstream, and the second of those changes sign when

$$U = \sqrt{gh}, \qquad\text{that is}\qquad \mathrm{Fr} = \frac{U}{\sqrt{gh}} = 1.$$

What happens at that point is not that one term becomes larger than another. It is that a signal stops being able to reach upstream at all. Below it, the flow downstream can influence the flow upstream, so a change made at a weir is felt above it. Above it, the flow has no way of knowing what lies ahead, and a change made downstream is simply swept away.

That is a change in the structure of the problem rather than in the relative size of two of its terms. Mathematically the equations change from elliptic to hyperbolic — the same equations, with the same coefficients, and a different classification because a discriminant has changed sign.

And nothing about it depends on how accurately anybody wants an answer. There is no observable whose error reaches a per cent somewhere; there is a signal that arrives or does not.

The second statement, which is about energy #

There is a completely independent route to the same number, and its independence is the interesting part.

For a channel carrying a flow rate $q$ per unit width, the specific energy — the head measured from the bed — is

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

The first term rises with depth and the second falls, so $E$ has a minimum. Differentiating and setting to zero gives $q^2 = gh^3$, which is $U = \sqrt{gh}$, which is $\mathrm{Fr} = 1$.

So the depth at which a channel carries a given flow with the least energy is exactly the depth at which a disturbance can no longer travel upstream. Nothing in the derivation mentions waves, and nothing in the wave argument mentions energy.

Two arguments, no shared assumptions, one number, exactly. That is not what a term ratio looks like; a term ratio’s threshold moves when the observable or the tolerance changes, and these two observables could not be less alike.

The consequence that looks like a paradox #

The two branches produce the single most counter-intuitive result in open-channel flow, and it follows from the energy curve without any further physics.

Put a small rise in the bed of a channel. The flow crossing it loses specific energy equal to the rise, so it moves along the curve towards the minimum.

On the subcritical branch, moving towards the minimum means moving to smaller depth: the surface goes down over the bump. On the supercritical branch it means moving to larger depth: the surface goes up. A rise in the bed lowers the water in a slow stream and raises it in a fast one, and the whole of it is which side of $\mathrm{Fr} = 1$ the flow started on.

If the bump is high enough the flow reaches the minimum and can go no further — the channel chokes, exactly as a nozzle does, with depth in the place of pressure and the Froude number in the place of the Mach number.

The same argument, with sound #

The parallel with compressible flow is exact rather than decorative, and it is the second of the two rows that sit on one.

A gas carries pressure signals at the speed of sound. A stream moving faster than sound cannot send a signal upstream, so it cannot be told about an obstruction, so the warning cannot arrive and the adjustment happens abruptly in a shock. The equations change type at $M = 1$ and the area–Mach relation has its minimum there, which is why a converging–diverging nozzle works at all.

Every statement in this essay has a compressible twin:

The analogy is close enough to be quantitative: a shallow-water flow with a free surface behaves like a gas with a ratio of specific heats of exactly two, which is a result this collection computes rather than asserts.

What makes a characteristic condition different #

The general point is worth isolating, because it is the one thing that separates these two groups from the other twelve.

A term ratio compares the size of two contributions to the same equation. Its value of one is a statement about magnitudes, the solution depends smoothly on it, and any threshold has to be extracted from the solution together with a tolerance.

A characteristic condition is a statement about the direction in which information travels. It does not compare contributions; it asks whether a family of characteristics points upstream or downstream, and the answer flips at a point. There is no small parameter, no expansion, no smooth observable and therefore no tolerance.

The test is one calculation: ask for the threshold at a different tolerance and see whether it moves. A term ratio’s moves proportionally. The Froude number’s does not move at all, because there is nothing in the statement for a tolerance to attach to.

The condition decides where it is allowed to happen #

There is a further consequence of the exactness that is stronger than anything about tolerances, and it is the one that makes the Froude number useful to somebody building a channel rather than analysing one.

Write the gradually varied flow equation for a frictionless channel with a bed elevation $z_b(x)$. Differentiating the specific energy along the channel gives

$$\left(1 - \mathrm{Fr}^2\right)\frac{dh}{dx} = -\frac{dz_b}{dx},$$

and the left-hand side has the critical condition sitting in it as a factor. At $\mathrm{Fr} = 1$ that factor vanishes, so unless the right-hand side vanishes at the same station the depth gradient would have to be infinite — which is to say the flow cannot be critical there at all.

So the critical condition is not merely a value the flow may take; it is a value it may take only where the bed has an extremum, or, in the varying-width version of the same algebra, only where the channel is narrowest. Everywhere else it is forbidden by the equation itself.

That is the shallow-water twin of a result this collection computes for a gas: sonic conditions occur only at a throat, and a throat that has gone sonic stops listening to anything downstream. The two derivations are the same three lines with the area in place of the bed, and both end in a factor of $(1 - \mathrm{Fr}^2)$ or $(1 - M^2)$ multiplying the gradient of the quantity that is being solved for.

And the consequence in both is control. A section where the flow passes through critical is a hydraulic control: it fixes the relation between depth and discharge at that station, so conditions upstream of it are decided by the control and not by whatever lies further down. That is why a weir works, why a flume measures without calibration, and why an engineer looking at a long river reach begins by finding the controls and integrating away from them in both directions. Between two controls the profile is determined; across one it is not.

The behaviour at the critical station itself is worth a sentence because it is where the exactness turns into an ambiguity of a familiar kind. With both sides of the equation vanishing, $dh/dx$ is $0/0$ there, and its value has to be recovered from a second-order expansion — which returns two roots. One continues on the same branch, the flow reaching critical at the crest and returning to subcritical beyond it; the other crosses, going supercritical downstream. Both are solutions, both satisfy every condition imposed so far, and which one occurs is settled by what is downstream — a tailwater level, a gate, a jump further along.

That is this collection’s recurring shape, arriving one last time and in the one place where the threshold itself was exact. The equations and the boundary conditions upstream are complete, they are sharp, they contain no tolerance whatever — and at the critical section they leave two answers, and something else has to say which. The nozzle does the identical thing: a converging–diverging duct at its throat admits a subsonic continuation and a supersonic one, and the back pressure chooses. A condition that determines the location of a transition with perfect precision still does not determine what the flow does when it gets there.

What the number is not exact about #

The exactness belongs to one statement and it is worth fencing it off, because the Froude number does several jobs and only one of them has a sharp threshold.

Wave resistance. A ship’s wave-making drag depends on the Froude number formed on its length, and the dependence is a set of humps and hollows produced by interference between the bow and stern wave systems. Nothing sharp happens at $\mathrm{Fr} = 1$; the humps are wherever the interference is constructive, and the design values that matter are 0.3, 0.4 and 0.5.

Hull speed. The folklore threshold for a displacement hull is $\mathrm{Fr} \approx 0.4$, and it is a rule of thumb with a tolerance in it exactly like the ones this collection keeps taking apart. There is no discontinuity at it.

Stratified flow. The stratified Froude number $U/NH$ compares a flow’s speed with a buoyancy frequency, and although its threshold at one is about whether flow goes over or round an obstacle — which is a blocking condition and has some of the same character — it is smeared over a range rather than sharp.

So the same group has a sharp threshold in one problem and a soft one in another, which settles the question of whether sharpness is a property of the group. It is not. It is a property of what the group is being asked.

Where the exactness runs out #

It would be wrong to leave the impression that everything about a critical flow is exact. The condition is; the flow near it is not, and the difficulty is severe in exactly the opposite way from a term ratio’s.

Near $\mathrm{Fr} = 1$ the specific-energy curve is flat — its second derivative is finite but its first vanishes — so a small change in energy produces a large change in depth. A channel near critical is therefore unstable in practice: small disturbances produce large surface excursions, the standing waves that appear are not a numerical artefact, and open-channel designers avoid running close to critical for exactly that reason.

So the number is exact and the neighbourhood of it is the least predictable part of the whole flow. That is the opposite of a term ratio, where the threshold is vague and the flow on either side of it is well behaved.

The one place the exactness is useful #

That flatness is exploited rather than avoided in one application. A critical-depth flume forces the flow through the critical condition by narrowing the channel, and because $q^2 = gh_c^3$ at critical, measuring the depth gives the flow rate directly.

The measurement is good precisely because the condition is exact. There is no coefficient to calibrate, no Reynolds-number correction and no tolerance: if the flow is critical at the throat, the flow rate follows from the depth by an identity. Every other flow meter in this collection — an orifice plate, a venturi, a weir — carries a discharge coefficient that has to be measured.

A threshold with no tolerance in it makes an instrument with no calibration in it, which is about as direct a consequence of this essay’s distinction as could be asked for.

What the picture cannot show #

Shallow water, and only shallow water. Everything here assumes the pressure is hydrostatic and the velocity uniform over the depth, which requires the wavelength to be long compared with the depth. Short waves are dispersive, their speed depends on wavelength, and there is no single wave speed for the flow speed to equal — so the sharp condition blurs.

No friction. A real channel has a bed, and a long channel’s depth is set by a balance between gravity and friction rather than by the energy curve. The critical condition is local and the normal depth is not.

One dimension. A wide channel with a variable bed has a critical line rather than a critical section, and the flow can be supercritical in some of the width and subcritical in the rest.

And the jump is not in these figures. The transition from supercritical to subcritical dissipates energy and cannot be described by the energy curve at all — the shock in a river computes it from momentum instead, which is the same reason a shock is computed from momentum rather than from Bernoulli.

An audit of the fourteen #

Setting this essay’s two rows beside the other twelve makes the rarity plain. Of the groups this collection computes thresholds for:

Eight are term ratios. Their onsets sit between seven and five hundred and ninety-four times below their balances, and every one of those factors is a tolerance divided by a slope.

Two are eigenvalues. Their thresholds are 3.657 and 1707.762, and the relation to one is that there is none.

Two are discriminants. An eighth and a quarter, exact, sharp and algebraic.

And two are characteristic conditions. Exactly one, in both cases, and this essay is about why.

Two out of fourteen is the reputation of the whole folklore. It is not a bad hit rate for a rule of thumb, and it is a very poor one for a rule that is stated without qualification in the first week of every course.

Who found it, and when #

Bélanger derived the jump conditions in 1828 and Bresse the backwater equations shortly after, both before anybody had a Froude number to write them in. William Froude’s own work in the 1860s and 1870s was about ship resistance and model testing, and the number was named for him afterwards; the open-channel usage came later still.

The gas-dynamic twin arrived from the other direction, with de Laval’s nozzle in the 1880s and the theory catching up over the following forty years. That the two subjects were the same subject was noticed by von Kármán in 1938, and the shallow-water analogy was used during the Second World War to study supersonic flows in a water table, because water tables were cheaper than supersonic tunnels.

The surprising connection is with the rest of this collection’s thresholds. The Froude number is not better formed than the Knudsen number; it is answering a different question. Both are ratios of two quantities, both are one where those quantities are equal, and only one of them is asking about the direction information travels. The folklore about groups of order one is right in precisely the cases where the group happens to encode a characteristic condition, and those are rare enough that the rule’s reputation rests on two examples out of fourteen.

Where the ladder goes next #

Above this rung is the two-dimensional problem, where critical flow becomes a line in a plane and the methods of characteristics that nozzle design uses become necessary. Beside it sits the shock in a river, which is what happens at the transition itself, and the warning that cannot arrive, which is the same argument in a gas.

Below it is the angle that does not care, where the Froude number governs a wave system rather than a characteristic condition — the same number doing an entirely different job, and having a threshold of the ordinary kind while doing it.