Open-channel flow
A river has no wall on top. Nothing forces its cross-section to be any particular value, and nothing pumps it — gravity does the entire job, acting on a slope you can barely see by eye. Every equation in this article exists because a pipe's simplest assumption, a fixed flow area, is not available here.
Every result in this article traces back to one structural difference from pipe flow. A pipe fixes its own cross-section; fill it and the area the fluid occupies is simply the pipe's area, decided by whoever cast the pipe. A channel does no such thing. Its water is free to rise or fall, and the depth it settles at is not a given quantity but an outcome — something the flow rate, the slope and the channel shape jointly determine. Reading that depth back out is most of what open-channel hydraulics is for.
Free-surface flow
Two things distinguish an open channel from a pipe, and it is worth keeping them separate because they cause different kinds of trouble. The first is the free surface itself: the top of the flow sits at atmospheric pressure, not at whatever pressure a pump or an upstream reservoir has put behind it. There is no \(\Delta P\) to drive the flow forward the way there is in a pipe, because the pressure is fixed at both ends of any slice you care to take. What is left to do the driving is gravity, acting through the slope of the channel bed, and that is the second distinguishing feature: an open channel is not pushed, it is tipped.
The consequence that actually makes the mathematics different is the one that follows from both: the flow area is not a fixed property of the conduit, the way a pipe's bore is, because the depth is free to move. Raise the flow rate through a fixed pipe and the velocity alone has to rise to carry it, since the area cannot change. Raise the flow rate through a channel and both the depth and the velocity are free to adjust, in whatever combination the channel's shape and roughness allow. Every unknown in the rest of this article is ultimately this one: given a flow rate and a channel, what depth does the water actually run at.
That depth is not free to be anything at all — it settles at a value fixed by a balance between the gravity component pulling the water along the slope and the friction the bed and banks exert against it, and finding that balance is the subject of the next section.
A pipe is a closed conduit: fill it, and its area is fixed by the wall.
Uniform flow and the Manning equation
Run a channel long enough and steadily enough, and the depth stops changing along its length: the gravity component driving the flow forward is exactly balanced, at every station, by the friction resisting it. This is uniform flow, and the depth it settles at is called the normal depth. The friction side of that balance depends on how much of the cross-section is actually in contact with a solid boundary, which is measured by the wetted perimeter \(P\) — the length of bed and banks the water touches, deliberately excluding the free surface at the top, since nothing drags on the water there.
Divide the flow area by that wetted perimeter and the result is the hydraulic radius, the one length scale that captures how much boundary the flow has to fight per unit of area it is carrying:
The Manning equation is an empirical correlation, not a derivation, for the velocity a channel settles at once it knows \(R_h\), its bed slope \(S_0\), and a roughness coefficient \(n\) that summarises everything about the surface, from smooth concrete to a weed-choked earth ditch, in a single tabulated number:
It is worth flagging that this equation is dimensional rather than a clean ratio of dimensionless groups: the SI form above needs \(R_h\) in metres and returns \(V\) in metres per second, and a different constant appears if the equation is written in US customary units. Multiply through by the flow area and the result is the flow rate the channel actually carries at its normal depth:
A concrete rectangular channel four metres wide, laid on a slope of 0.001 with a Manning roughness of 0.015, running 1.2 metres deep, has \(A = 4.8\) m², \(P = 6.4\) m and so \(R_h = 0.75\) m. The equation returns \(V \approx 1.74\) m/s and a flow rate of about 8.35 cubic metres per second — one depth, one flow rate, out of the entire range the channel could in principle carry.
Run the channel at a steady, unchanging depth, and every slice looks like the last.
Specific energy and critical depth
Not every channel runs uniform, and even a channel that eventually does needs a way to describe the depth as it approaches that state. The natural quantity is specific energy, energy measured relative to the channel bed rather than to some fixed datum, so that it combines the depth itself with the kinetic energy of the flow moving through it:
For a fixed flow rate, plot \(E\) against \(y\) and the curve has a distinctive shape: it falls steeply as \(y\) shrinks from a large value, reaches a minimum, and then rises again as \(y\) shrinks further still. That shape is not a coincidence of the particular numbers chosen; it follows directly from the two terms fighting each other. Large \(y\) means small \(V\) for a fixed flow rate, so the kinetic term is small and \(E\) is dominated by \(y\) itself. Small \(y\) forces \(V\) to be large to carry the same flow through a smaller area, and the kinetic term, which grows with the square of \(V\), eventually swamps everything.
The minimum itself, found by setting \(dE/dy = 0\), defines a special depth called the critical depth. For a wide or rectangular channel, writing the flow rate per unit width as \(q = Q/b\), it works out to
and at that depth the velocity satisfies \(V_c^2 = g y_c\) exactly — a relation that will reappear as soon as the Froude number is introduced. Away from the minimum, the curve has two branches, and a horizontal line drawn across it at any energy above \(E_{\min}\) crosses both: the same specific energy is available at two different depths, one on each branch. Which of those two the flow actually takes up is not a question the energy equation alone can answer, and it is exactly the question the next section exists to settle.
Fix the flow rate and plot how much energy the water carries at every possible depth.
Subcritical and supercritical flow
A small disturbance on a free surface — a stone dropped in, a slight bump on the bed — sends out a ripple, and that ripple travels at a definite speed relative to the water it is moving through, set purely by the local depth:
Whether that ripple can fight its way back upstream against the current depends on a straightforward race between \(c\) and the flow's own velocity \(V\), and their ratio is the Froude number:
When \(Fr < 1\), the flow is subcritical: deep and comparatively slow, and a wave's own speed beats the current, so disturbances can and do propagate upstream — which is exactly why the depth downstream of a subcritical channel, at a gate or a weir, can influence conditions upstream of it. When \(Fr > 1\), the flow is supercritical: shallow and fast, and nothing generated downstream can ever climb back against it. A supercritical channel does not care what is waiting for it further along; information about that only ever travels one way.\(Fr = 1\) is precisely the critical condition from the previous section — the depth at which a disturbance's own speed exactly matches the flow's.
The analogy this sets up with compressible flow is not a loose one. The Froude number plays exactly the role the Mach number plays there, with a surface gravity wave standing in for a sound wave, and it carries through to a genuine geometric consequence: a disturbance in a supercritical stream is confined to a wedge downstream of its source, with a half-angle \(\beta\) given by \(\sin\beta = 1/Fr\), in exact correspondence with a Mach cone trailing a body moving faster than sound.
Drop a disturbance into slow, deep flow and its ripples can climb back upstream.
The hydraulic jump
A channel can carry supercritical flow into a stretch that, for whatever reason, only supports subcritical flow further on — a milder slope, a downstream obstruction, a wider section. The two states cannot connect smoothly. A gradual transition from supercritical to subcritical would have to pass continuously through the critical depth, and doing so would require the flow's energy to rise on its own, which nothing here supplies. Nature's actual answer is abrupt rather than gradual: a hydraulic jump, a short, violently turbulent roller in which the depth increases suddenly over a distance of only a few channel depths.
Energy is not conserved across the jump — the turbulence in the roller dissipates a substantial fraction of it — but momentum is, because the jump is short enough that the forces of bed friction and gravity along its length are negligible next to the momentum flux entering and leaving it. Applying the momentum equation across the jump, for a rectangular channel, gives the sequent-depth relation, connecting the depth after the jump to the Froude number before it:
A channel arriving at the jump with \(y_1 = 0.3\) m and \(Fr_1 = 3\) leaves it at \(y_2 \approx 1.13\) m — nearly quadruple the depth, from a jump only a few metres long. The energy lost in reaching that new depth follows from the specific-energy relation evaluated at both ends:
Far from being a nuisance, that dissipation is frequently the point. Water leaving a spillway arrives at the base of a dam supercritical and carrying a dangerous amount of kinetic energy; a stilling basin is built specifically to force a hydraulic jump inside it, so that energy is destroyed deliberately, in a structure engineered to survive it, rather than left to erode the riverbed immediately downstream.