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A hydraulic jump connects fast, shallow, supercritical flow to slower, deeper, subcritical flow. Mass and momentum constrain the depth change, while turbulence dissipates mechanical energy.
Contents
Understand the mechanism · Chapters 1–3
1. Recognise a change in flow regime
Work the model · Chapters 4–5
Test the explanation · Chapters 6–8
6. Downstream conditions decide where the jump can sit
Water emerging quickly beneath a gate can form a thin sheet, then suddenly thicken into a churning region. Downstream, the flow is deeper and slower. This abrupt transition is a hydraulic jump. Its explanation needs both the water arriving from upstream and the conditions that the downstream channel imposes.
The word jump describes the rise in the free surface. It does not mean every water parcel leaps through the air. Inside the transition there can be strong mixing, rollers and entrained air, while the overall discharge continues downstream.
This guide uses a straight, level, rectangular channel to make the reasoning explicit. Circular jumps around an impinging tap jet share a family resemblance, but radial spreading and surface tension can matter there. We should not insert a kitchen-sink measurement blindly into the rectangular-channel equations.
For long, small-amplitude gravity waves on otherwise still shallow water, wave speed is approximately c = √(gy). Here y is water depth and g is gravitational acceleration. Relative to the ground, a wave trying to travel upstream against a current of speed u moves at approximately u − c.
If u exceeds c, even that upstream-directed disturbance is swept downstream. If u is below c, information can travel upstream through these waves. The ratio Fr = u/√(gy) is the Froude number for this rectangular-channel model. Fr greater than one is supercritical; Fr less than one is subcritical.
The Bureau of Reclamation’s flow guidance places this distinction alongside critical-depth and specific-energy relationships. For nonrectangular channels the relevant hydraulic depth must be chosen correctly; it is not automatically the deepest point in the section.
The wave argument explains why two regimes behave differently. It does not, by itself, calculate the jump’s position. Downstream depth, channel geometry and the available discharge still have to fit together.
Original numerical model. Let the upstream depth be y₁ = 0.050 m and the average speed be u₁ = 2.00 m/s. Take a channel width b = 0.40 m and g = 9.81 m/s². The volume discharge is Q = by₁u₁ = 0.040 m³/s, or 40 litres per second.
Per unit channel width, q = Q/b = uy = 0.100 m²/s. If the downstream depth becomes larger, the downstream average speed must decrease so that u₂y₂ remains 0.100 m²/s. This statement assumes steady flow with no water entering or leaving through the sides.
At the upstream section, √(gy₁) = √(0.4905) ≈ 0.700 m/s. Therefore Fr₁ ≈ 2.86. The incoming flow is comfortably supercritical in this model.
Continuity alone does not determine y₂. Many depth-speed pairs have the same product. A 0.10 m depth with 1.00 m/s speed and a 0.20 m depth with 0.50 m/s speed both carry the required discharge. Momentum supplies the additional constraint.
Choose sections outside the violently changing part of the jump. Assume approximately hydrostatic pressure at those sections, nearly uniform section velocities, the same bed elevation and negligible integrated bed friction over the short control volume. The rectangular-channel momentum function per unit width is M = y²/2 + q²/(gy).
Setting M₁ = M₂ gives the conjugate-depth relation y₂/y₁ = [√(1 + 8Fr₁²) − 1]/2. This is a pressure-and-momentum result. It is not obtained by pretending that mechanical energy stays unchanged through the turbulent jump.
For our values, the depth ratio is approximately 3.57, so y₂ ≈ 0.1785 m and u₂ = 0.100/0.1785 ≈ 0.560 m/s. The downstream Froude number is about 0.423: the predicted outflow is subcritical.
The US Army Corps of Engineers’ open-channel manual sets out the momentum basis and this conjugate-depth relationship. Its use depends on the channel and flow assumptions, not merely on seeing white water.
Specific mechanical energy relative to the bed is E = y + u²/(2g) in this simplified model. Upstream, E₁ = 0.050 + 4/19.62 ≈ 0.2539 m. Downstream, E₂ ≈ 0.1785 + 0.560²/19.62 ≈ 0.1945 m. The mechanical head decrease is approximately 0.0594 m.
The same result follows from ΔE = (y₂ − y₁)³/(4y₁y₂). With a water density of 1000 kg/m³, the mechanical power dissipated is ρgQΔE ≈ 23.3 W. These are calculations for the invented example, not measurements from a real channel.
Energy has not ceased to exist. Organised flow energy is transferred into turbulent motion and ultimately internal energy, with other small pathways possible. The practical term energy loss refers to mechanical energy no longer available in the mean flow.
Notice that the water surface rises while total specific mechanical energy falls. Looking only at the height would miss the much larger reduction in velocity head. The Reclamation energy-balance guide helps distinguish water-surface level from the complete energy account.
| Quantity | Upstream | Downstream |
|---|---|---|
| Depth | 0.0500 m | 0.1785 m |
| Average speed | 2.00 m/s | 0.560 m/s |
| Froude number | 2.86 | 0.423 |
| Specific mechanical energy | 0.2539 m | 0.1945 m |
The calculated y₂ is the depth compatible with a particular upstream state under our assumptions. It is not a guarantee that a real channel will supply that depth at any desired location. The tailwater condition can move, submerge or destabilise the transition.
A Reclamation investigation of stilling basins describes unstable surging when tailwater is inadequate. This is why a basin is not designed by finding a depth ratio and stopping. The surrounding geometry, range of discharges and downstream conditions must also be considered.
For a thought experiment, hold the gate opening and discharge fixed while slowly changing downstream water level. Record the jump’s position and structure. If the jump migrates, the result demonstrates that the transition is a system response rather than an object fixed to one patch of bed.
This discussion explains the science; it is not a design specification. Hydraulic jumps and low-head structures can contain dangerous recirculating currents. Observations should come from controlled laboratory records or safe public viewing locations.
A useful dataset would include channel width, discharge, upstream and downstream depths, mean velocities and time variability. Measurements should be made at representative sections away from the rapidly varying core. Repeated readings help separate a stable average from a jump moving back and forth.
Start with continuity: do the measured products byu agree within uncertainty? Next test momentum using M. Then calculate the mechanical-energy difference. This order locates different failures. A poor water balance can indicate an unmeasured inflow or measurement error before any exotic jump physics is required.
If momentum agreement is weak, inspect pressure and velocity assumptions, bed slope, sidewall effects and the chosen section locations. Air entrainment and nonuniform velocity profiles may matter. The Reclamation study on applicability of the jump formula is a primary example of checking a theoretical relation against controlled observations.
White water alone does not establish a hydraulic jump. A waterfall, obstacle wake or breaking wave can also entrain air. The transition in depth, average velocity and flow regime gives the more discriminating description.
Question: Why can we conserve momentum in the model while mechanical energy decreases? Answer: Momentum and energy are different accounts. Forces determine the momentum balance; irreversible processes convert organised mechanical energy into other forms. One statement does not cancel the other.
Question: If the discharge doubles, can we keep the same two depths and simply double both speeds? Answer: That preserves continuity but changes the momentum terms and Froude numbers. The old conjugate-depth pair generally will not satisfy the new flow.
Question: Does a deeper downstream surface prove that total energy increased? Answer: No. Compare depth plus velocity head, using consistent reference elevations.
For the wider framework, continue to Fluid Dynamics. The central habit is to keep mass, momentum, mechanical energy and boundary conditions in the same explanation.