The depth at the edge is not the critical one

The specific-energy curve, the critical depth, the classification of gradually varied profiles and every backwater calculation rest on one assumption, and it is almost never stated where it is used. The pressure at every point is taken as hydrostatic — ρg times the depth below the surface — which is true when the streamlines are nearly straight and nearly horizontal, and is the reason a channel can be described by one number per station at all.

At a free overfall the streamlines are neither. The bed simply ends; the water falls off; and over the last few depths before the edge the streamlines curve sharply downwards, with a vertical acceleration of order $g$. The hydrostatic assumption is not approximately right there. It is entirely wrong, and what makes the overfall worth an essay is that the size of its wrongness can be measured with a ruler.

The momentum balance, which needs no assumption about the brink #

Take a control volume between two sections: an upstream one where the flow is critical and the pressure is still hydrostatic, and the brink itself. Momentum in a horizontal, frictionless channel gives

$$\frac{q^2}{h_c} + \tfrac12 g h_c^2 = \frac{q^2}{h_b} + \tfrac12 k\,g h_b^2,$$

where $k$ is the fraction of the hydrostatic pressure force the brink section retains. At the brink the nappe is falling freely with atmospheric air above and below it, so $k = 0$, and with the critical condition $q^2 = g h_c^3$ the whole thing collapses to

$$\tfrac32 h_c^2 = \frac{h_c^3}{h_b} \qquad\Longrightarrow\qquad h_b = \tfrac23\,h_c.$$

Two thirds, with no empirical content anywhere: a momentum balance, the definition of critical depth, and the observation that a free jet carries no pressure.

Three things about that derivation are worth marking, because each is a place where a different choice would have been made and would have been wrong.

It is a momentum balance and not an energy one. The energy between the two sections is not conserved — the flow is accelerating into a fall, there is a wake behind the nappe, and the specific energy at the brink is not the specific energy upstream. Momentum is conserved, because the only forces on the control volume are the pressures on its two faces and a bed shear that has been neglected. Every place in this field where the hydrostatic description fails is a place where momentum survives and energy does not, and the hydraulic jump is the other one.

The upstream section is taken at critical depth rather than anywhere convenient. That is what closes the problem: it supplies the relation $q^2 = gh_c^3$ and removes the discharge, so the answer is a pure ratio. Choosing a section further upstream would have left two unknowns and one equation.

And the brink’s pressure is taken as zero rather than small. That is the physical statement in the whole calculation, it is the one the measurement disagrees with, and it is the reason the answer comes out as a clean fraction rather than as a number.

Seven per cent, and what it is made of #

The measured end-depth ratio is 0.715, and it has been measured many times since Rouse in 1936. The momentum answer is seven per cent low.

That gap is a genuine result rather than experimental scatter, and the figure above says what it would take to close it: keeping about forty per cent of the hydrostatic pressure at the brink rather than none.

Forty per cent is a great deal more than a falling nappe looks as though it should have. So the honest reading is that the gap is not all pressure. The balance also assumes the velocity is uniform over the depth at both sections, and at the brink it is markedly not — the surface streamline has been accelerating downwards while the bed streamline has not — so the momentum flux $q^2/h_b$ understates the true one. Part of the seven per cent is pressure and part is a velocity distribution, and one measurement cannot separate them.

What the calculation does establish, firmly, is the bracket: a hydrostatic brink would give one, a free jet gives two thirds, and the truth is close to the free jet and not to the hydrostatic assumption every reach calculation makes.

The sign of the discrepancy is worth one more line, because it is informative and it is the direction a reader would not guess. Both of the neglected effects push the ratio up — a residual pressure at the brink holds the water back and deepens it, and a non-uniform velocity carries more momentum than the uniform estimate and therefore needs less depth to balance it, which also deepens the upstream side of the exchange. So the measured value has to exceed two thirds, and two thirds is a floor rather than an estimate.

That matters for how the constant is used. A designer who does not know the local value can use two thirds and know the direction of the error, which is worth more than a number of unknown sign: the discharge will be overstated by at most eleven per cent, and never understated.

Why the assumption fails here and nowhere else in the field #

It is worth asking why a description this successful should collapse at one particular place, because the answer says where else to be careful.

A hydrostatic pressure distribution follows from the vertical momentum equation with the vertical acceleration set to zero. That is legitimate when the vertical velocity is small and changing slowly, which in a channel means the surface slope is gentle and its curvature is gentler. The quantity that has to be small is not the slope but $h\,\mathrm{d}^2 z_s/\mathrm{d}x^2$ — the depth times the curvature of the surface — because that is what sets the centripetal acceleration of a fluid particle following the surface.

Along a reach that quantity is of order the square of the slope a uniform flow settles at and is utterly negligible. At a bump it is a few per cent and the theory is excellent. At a brink the surface turns through nearly a right angle in a distance of order the depth, so the quantity is of order one, and there is no small parameter left.

That criterion is the useful thing to carry away, because it names the other places in the subject where the same failure is waiting: the crest of a sharp-crested weir, the toe of a spillway, the lip of a sluice gate, and the face of a hydraulic jump. All four are treated in practice with measured coefficients rather than with the theory, and all four are treated that way for this reason rather than out of timidity.

A flow meter with nothing in the channel #

The reason anybody cares is that this makes an overfall into an instrument.

Measure one depth at the edge of a drop and the discharge follows: $q = \sqrt{g\,(h_b/0.715)^3}$. There is no weir to build, no approach-velocity correction, no calibration, and nothing in the channel to silt up or to be washed away. An overfall is the cheapest flow-measuring structure there is, and its accuracy is the accuracy of one constant.

The error a hydrostatic reading makes is the point of the whole exercise. Taking $h_b = h_c$ takes the critical depth thirty per cent too small; the discharge goes as the three-halves power of it; so the discharge comes out at $(0.715)^{3/2} = 0.605$ of the truth — understated by forty per cent.

That scale-freedom is worth noticing. The calculation contains a discharge, a depth and $g$, and those combine into exactly one dimensionless group — so everything here is a statement about a ratio, it is the same ratio in a laboratory flume and over a dam, and a forty per cent error is a forty per cent error at every size.

There is a second consequence of having only one group, and it is the reason this essay can be written at all. A problem with one dimensionless group cannot have a Reynolds number in it, so the ratio does not depend on the viscosity, on whether the flow is turbulent, or on the scale of the experiment — which is why a measurement made in 1936 in a small laboratory flume is quoted without qualification for a spillway a hundred times the size.

That is an unusually strong position for an empirical constant to be in. Most of the coefficients in hydraulics — a discharge coefficient, a loss coefficient, a friction factor — carry a Reynolds-number dependence that has to be quoted with them, and the quoting is where most of the error in practice comes from. This one does not, because gravity and inertia are the only two things in the problem and they make one group between them.

The same argument says what would bring a second group in: surface tension, which matters when the nappe is thin enough for the sheet to break up, and viscosity, which matters when the approach flow’s boundary layer occupies a substantial fraction of the depth. Both arrive at small scale, which is the one direction in which the constant is known to drift — end-depth measurements in very shallow flumes are the ones that disagree.

Where the critical depth actually is #

One more thing the arithmetic implies, and it is the part most often got wrong in practice.

Critical depth does not occur at the brink. It occurs upstream of it, at the last station where the flow is still hydrostatic enough for the specific-energy curve to mean anything — which measurement puts at three to four critical depths back from the edge. Between that station and the brink the depth falls from $h_c$ to $0.715 h_c$ along a curve that no one-dimensional theory describes, because the whole region is the one where the theory does not hold.

The reason that station exists at all is the reason every control section in this field exists. A subcritical reach is controlled from downstream: information travels upstream, and the reach adjusts to whatever it is told. A fall at the end is the strongest possible downstream condition, so the channel must pass through critical depth somewhere before it — and where it does is where the downstream influence stops being communicable, which is a few depths back.

That is the same statement as the choking condition in the essay that computes it: the flow cannot pass through the sonic point twice, so it passes once, and the passing is at the control. An overfall is a control section whose structure is the absence of a channel.

So a gauging station at an overfall measures at the brink and computes as though the critical section were there. That works because the ratio is constant and known — which is a substitution of one measured constant for a piece of theory, and is a perfectly respectable way to make an instrument.

It is also the reason the overfall is a better instrument than it has any right to be. A structure whose theory fails in the region being measured ought to be unreliable, and this one is not, because the failure is confined and its outcome is a single number that does not depend on anything an operator can get wrong. The discharge cannot drift with the channel’s roughness, with the approach velocity, or with the temperature of the water; it depends on one length and one constant.

Compare that with the structure it replaces. A sharp-crested weir needs its crest kept sharp, its nappe ventilated, its approach pool deep enough that the velocity of approach can be neglected, and a head measured far enough upstream to be clear of the drawdown — and its coefficient drifts with all of them. The overfall has no crest to maintain because the crest is the end of the channel.

What the picture cannot show #

The control volume is horizontal and frictionless. A real overfall sits at the end of a sloping reach with a bed shear on it, and both add momentum terms of the same order as the seven per cent being argued about — which is another reason the implied pressure of forty per cent should not be taken as a measurement of anything.

The nappe is taken as ventilated. If the air beneath the falling sheet cannot be replaced, the nappe entrains it, the pressure under the sheet falls below atmospheric, and the sheet is drawn towards the wall. A confined overfall is a different flow and the ratio is not 0.715.

The channel is rectangular and wide. The whole of the arithmetic above uses $q^2 = gh_c^3$, which is the wide-rectangular critical condition. A trapezoidal or circular section has a different relation between the area and the depth, so a different critical condition and a different end-depth ratio; the ratios for those sections are measured rather than derived.

And the flow arriving is subcritical. A supercritical approach flow does not pass through critical depth at all, the control is upstream rather than at the brink, and the depth at the edge is simply whatever the supercritical profile delivers there — which is a different problem with the same geometry.

And the bed is smooth right up to the edge. A real brink is a concrete lip or a rock sill with its own roughness and its own small step, and the last few centimetres of a channel are where the boundary layer is thickest relative to the depth. The ratio is quoted for a clean, sharp, horizontal lip, and an overfall that has been in service for twenty years does not have one.

Who found it, and when #

Rouse measured the end-depth ratio in 1936 and reported 0.715, which is the value still used. The momentum derivation of two thirds is older than that in form and was applied to the overfall afterwards; Hager’s analyses of the 1980s are the standard modern treatments, and extend the ratio to other section shapes and to sloping beds.

The surprising connection is with an instrument that measures the same quantity a completely different way, and pays a completely different price. A differential-pressure flow meter infers a discharge from a pressure drop across a constriction, and its cost is a permanent head loss — the meter destroys a fraction of the energy of the flow it is measuring, permanently, and the fraction is the price of the reading. An overfall costs nothing at all: the water was going to fall off the end anyway, and the measurement is made by looking. The two instruments bracket what a flow measurement can cost, and the reason the free one is not used everywhere is that it needs a place where the channel ends, which a pipeline does not have.

Still open: whether the two contributions can be separated #

The seven per cent is shared between a residual pressure at the brink and a non-uniform velocity there, and no measurement of the end-depth ratio can say in what proportion.

They are separable in principle, because they affect different things. The residual pressure contributes to the pressure-force term and the velocity distribution to the momentum-flux term, and the two respond differently to the one parameter an experimenter can vary without changing the geometry: the approach Froude number. A brink fed by a flow that is only just subcritical arrives with a flatter velocity profile than one fed from a deep pool, so the momentum-flux correction should move and the pressure correction should not.

The experiment is a sequence of end-depth measurements at a fixed geometry and a swept approach condition, with the ratio plotted against the approach Froude number rather than averaged over it — and a direct measurement of the pressure on the end wall beneath the nappe, which is a row of tappings and is easier than it sounds. Both have been done separately and, as far as the record shows, never together on the same channel; doing so would turn a single calibration constant into two measured quantities and would say which half of the hydrostatic assumption fails first.