A hydraulic jump is an abrupt increase in the depth of a fast-moving liquid stream with a corresponding decrease of speed. When liquid at high velocity discharges into an area of lower velocity and greater depth, a wavy or turbulent zone is formed. Turbulence caused by the abrupt deceleration of the flow dissipates some of the initial kinetic energy to heat.
Hydraulic jumps are commonly seen in open channel flows, such as rivers and spillways, and these jumps are designed and managed in hydraulics and civil engineering. Hydraulic jumps are commonly introduced as energy dissipators downstream of dam spillways.
The jump phenomenon is dependent upon the initial fluid speed. If the initial speed of the fluid is below the critial speed (the speed of water waves), then no jump is possible. For initial flow speeds which are not significantly above the critical speed, the transition appears as an undulating wave. As the initial flow speed increases further, the transition becomes more abrupt, until at high enough speeds, the transition front will break and curl back upon itself. When this happens, the jump can be accompanied by violent turbulence, eddying, air entrainment, and surface undulations, or waves. These transitions are captured by the value of the upstream Froude number.
In the context of whitewater rafting, a hydraulic jump is called a stopper wave. Hydraulic jumps have some similarity to shock waves in supersonic flows.
Leonardo da Vinci described and sketched water flows involving jet impingement and recirculation in his Codex Leicester (c. 1504âÂÂ1510; e.g., fols. 4r, 5r, 33v), which are now interpreted as including hydraulic jumps. The mathematics were first described by Giorgio Bidone of Turin University when he published a paper in 1820 called "Expériences sur le remou et sur la propagation des ondes".
There are two main manifestations of hydraulic jumps, either stationary or moving upstream against the flow. Historically, different terminology has been used for each, but they are simply variations of each other seen from different frames of reference. The same physical processes and analytical techniques describe both types.
These phenomena are addressed in an extensive literature from a number of technical viewpoints.
The two types of hydraulic jump are:
A related case is a cascade â a wall or undulating wave of water moves downstream overtaking a shallower downstream flow of water as shown in Figure 5. If considered from a frame of reference which moves with the wave front, the behavior is that of a stationary jump.
A tidal bore is a hydraulic jump which occurs when the incoming tide forms a wave (or waves) of water that travel up a river or narrow bay against the direction of the current.
Bores take on various forms depending upon the difference in the water level upstream and down, ranging from an undular wavefront to a shock-wave-like wall of water.
Figure 3 shows a tidal bore with the characteristics common to shallow upstream water â a large elevation difference is observed. Figure 4 shows a tidal bore with the characteristics common to deep upstream water â a small elevation difference is observed and the wavefront undulates. In both cases the tidal wave moves at the speed characteristic of waves in water of the depth found immediately behind the wave front. A key feature of tidal bores and positive surges is the intense turbulent mixing induced by the passage of the bore front and by the following wave motion.
A moving hydraulic jump is called a surge. The travel of the wave is faster in the upper portion than in the lower portion of positive surges.
A stationary hydraulic jump is the type most frequently seen on rivers and on engineered features such as outfalls of dams and irrigation works. They occur when a flow of liquid at high velocity discharges into a zone of the river or engineered structure which can only sustain a lower velocity. When this occurs, the water slows in an abrupt step or standing wave on the liquid surface.
Comparing the characteristics before and after, one finds:
Another type of stationary hydraulic jump occurs when a rapid flow encounters a submerged object which throws the water upward. The mathematics behind this form is more complex and will need to take into account the shape of the object and the flow characteristics of the fluid around it.
In spite of the apparent complexity of the flow transition, a simple two-dimensional analysis yields results which closely parallel both field and laboratory measurements. The principles of conservation of mass, conservation of momentum, and conservation of energy lead to quantitative relationships for:
The height of the jump is derived from the application of the equations of conservation of mass and momentum. The flow upstream of the jump has a depth and an average speed . Downstream of the jump, the depth and average speed are and . The liquid has a density , and is gravitational body force. For a rectangular channel of constant width , mass conservation for the jump is
and momentum conservation is
Solving these equations, the upstream and downstream depth are related by
This is known as Bélanger equation. This result may be extended to an irregular cross-section.
The ratio is the dimensionless Froude number which relates inertial to gravitational forces in the upstream flow
The depth relationship is then
The solution has three cases:
Therefore, the hydraulic jump is possible only when . Since is the speed of a shallow gravity wave, the condition is equivalent an initial velocity greater than the wave speed, called supercritical flow. The downstream is subcritical flow ().
Engineers use hydraulic jumps is to dissipate mechanical energy (the sum of kinetic and potential energy) in channels, dam spillways, and similar structures. The aim is to prevent high kinetic energy from damaging these structures. The rate of mechanical energy dissipation or head loss across a hydraulic jump is a function of the hydraulic jump inflow Froude number and the height of the jump.
The steady flow energy equation can also be applied to a control volume around the jump, to assess the loss. The mechanical energy loss at a hydraulic jump expressed as loss of hydraulic head (in meters) is:
In the design of a dam, the kinetic energy of the fast-flowing stream over a spillway can cause erosion of the streambed downstream. This energy can be partially dissipated by causing a hydraulic jump at the base of the dam. To limit damage, this hydraulic jump normally occurs on an apron engineered to withstand hydraulic forces and to prevent local cavitation and other phenomena which accelerate erosion.
In the design of a spillway and apron, the engineers select the point at which a hydraulic jump will occur. Slope changes, or sometimes obstructions, are routinely designed into the apron to force a jump at a specific location. To trigger the hydraulic jump without obstacles, an apron is designed such that the flat slope of the apron retards the rapidly flowing water from the spillway. If the apron slope is insufficient to maintain the original high velocity, a jump will occur.
Two methods of designing an induced jump are common:
In both cases, the final depth of the water is determined Eq. (1)
NB: the above classification is very rough. Undular hydraulic jumps have been observed with inflow/prejump Froude numbers up to 3.5 to 4.
Practically this means that water accelerated by large drops can create stronger standing waves (undular bores) in the form of hydraulic jumps as it decelerates at the base of the drop. Such standing waves, when found downstream of a weir or natural rock ledge, can form an extremely dangerous "keeper" with a water wall that "keeps" floating objects (e.g., logs, kayaks, or kayakers) recirculating in the standing wave for extended periods.
The hydraulic jump is characterised by a highly turbulent flow. Macro-scale vortices develop in the jump roller and interact with the free surface leading to air bubble entrainment, splashes and droplets formation in the two-phase flow region. The airâÂÂwater flow is associated with turbulence, which can also lead to sediment transport. The turbulence may be strongly affected by the bubble dynamics. Physically, the mechanisms involved in these processes are complex.
The air entrainment occurs in the form of air bubbles and air packets entrapped at the impingement of the upstream jet flow with the roller. The air packets are broken up in very small air bubbles as they are entrained in the shear region, characterised by large air contents and maximum bubble count rates. Once the entrained bubbles are advected into regions of lesser shear, bubble collisions and coalescence lead to larger air entities that are driven toward the free-surface by a combination of buoyancy and turbulent advection.
A number of variations are amenable to similar analysis:
Figure 2 above illustrates an example of a hydraulic jump, often seen in a kitchen sink. Around the place where the tap water hits the sink, a smooth-looking flow pattern will occur. A little further away, a sudden "jump" in the water level will be present. This is a hydraulic jump.
A circular impinging jet creates a thin film of liquid that spreads radially, with a circular hydraulic jump occurring downstream. For laminar jets, the thin film and the hydraulic jump can be remarkably smooth and steady. In 1993, Liu and Lienhard demonstrated the role of surface tension in setting the structure of hydraulic jumps in these thin films. Many subsequent studies have explored surface tension and pattern formation is such jumps.
A 2018 study experimentally and theoretically investigated the relative contributions of surface tension and gravity to the circular hydraulic jump. The authors performed experiments on horizontal, vertical and inclined surfaces finding that irrespective of the orientation of the substrate, for same flow rate and physical properties of the liquid, the initial hydraulic jump happens at the same location. They proposed a general criterion for a thin film hydraulic jump to be
where is the local Weber number and is the local Froude number. For some kitchen-sink-scale hydraulic jumps, the Froude number is very high, therefore, the effective criteria for the thin film hydraulic jump is . In other words, a thin film hydraulic jump occurs when the liquid momentum per unit width equals the surface tension of the liquid. However, this claim is heavily contested.
Turbidity currents can result in internal hydraulic jumps (i.e., hydraulic jumps as internal waves in fluids of different density) in abyssal fan formation. The internal hydraulic jumps have been associated with salinity or temperature induced stratification as well as with density differences due to suspended materials. When the slope of the bed (over which the turbidity current flows) flattens, the slower rate of flow is mirrored by increased sediment deposition below the flow, producing a gradual backward slope. Where a hydraulic jump occurs, the signature is an abrupt backward slope, corresponding to the rapid reduction in the flow rate at the point of the jump.
Hydraulic jumps occur in the atmosphere in the air flowing over mountains. A hydraulic jump also occurs at the tropopause interface between the stratosphere and troposphere downwind of the overshooting top of very strong supercell thunderstorms. A related situation is the Morning Glory cloud observed, for example, in Northern Australia, sometimes called an undular jump.
The hydraulic jump is the most common choice of design engineers for energy dissipation below spillways and outlets. A properly designed hydraulic jump can provide for 60-70% energy dissipation of the energy in the basin itself, limiting the damage to structures and the streambed. Even with such efficient energy dissipation, stilling basins must be carefully designed to avoid serious damage due to uplift, vibration, cavitation, and abrasion. An extensive literature has been developed for this type of engineering.
While travelling down river, kayaking and canoeing paddlers will often stop and playboat in standing waves and hydraulic jumps. The hydraulic jumps and associated standing waves make for popular locations for such recreation.
Similarly, kayakers and surfers sometimes ride tidal bores up rivers.
Hydraulic jumps have been used by glider pilots in the Andes and Alps and to ride Morning Glory effects in Australia.