Ahydraulic jump is a phenomenon in the science ofhydraulics which is frequently observed inopen channel flow such asrivers andspillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface. The rapidly flowing liquid is abruptly slowed and increases in height, converting some of the flow's initial kinetic energy into an increase in potential energy, with some energy irreversibly lost through turbulence to heat. In an open channel flow, this manifests as the fast flow rapidly slowing and piling up on top of itself similar to how ashockwave forms.
It was first observed and documented byLeonardo da Vinci in the 1500s.[1] The mathematics were first described byGiorgio Bidone ofTurin University when he published a paper in 1820 calledExperiences sur le remou et sur la propagation des ondes.[2]
The phenomenon is dependent upon the initial fluid speed. If the initial speed of the fluid is below the critical speed, then no jump is possible. For initial flow speeds which are not significantly above thecritical 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, orwaves.
There are two main manifestations of hydraulic jumps and historically different terminology has been used for each. However, the mechanisms behind them are similar because they are simply variations of each other seen from different frames of reference, and so the physics and analysis techniques can be used for both types.
The different manifestations are:
A related case is a cascade – 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, this is amenable to the same analysis as a stationary jump.
These phenomena are addressed in an extensive literature from a number of technical viewpoints.[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]
Hydraulic Jump is used sometimes in mixing chemicals.[19]
Hydraulic jumps can be seen in both a stationary form, which is known as a "hydraulic jump", and a dynamic or moving form, which is known as a positive surge or "hydraulic jump in translation".[16] They can be described using the same analytic approaches and are simply variants of a single phenomenon.[15][16][18]
Atidal 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.[16] As is true for hydraulic jumps in general, bores take on various forms depending upon the difference in the waterlevel upstream and down, ranging from an undular wavefront to ashock-wave-like wall of water.[9] Figure 3 shows a tidal bore with the characteristics common to shallow upstream water – a large elevation difference is observed. Figure 4 shows a tidal bore with the characteristics common to deep upstream water – 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.[20]
Another variation of the moving hydraulic jump is the cascade. In the cascade, a series of roll waves or undulating waves of water moves downstream overtaking a shallower downstream flow of water.
A moving hydraulic jump is called a surge. The travel of wave is faster in the upper portion than in the lower portion in case 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 a rather abrupt rise (a step orstanding wave) on the liquid surface.[17]
Comparing the characteristics before and after, one finds:
| Characteristic | Before the jump | After the jump |
|---|---|---|
| fluid speed | supercritical (faster than the wave speed) also known as shooting or superundal | subcritical also known as tranquil or subundal |
| fluid height | low | high |
| flow | typically smooth turbulent | typicallyturbulent flow (rough and choppy) |
The other stationary hydraulic jump occurs when a rapid flow encounters a submerged object which throws the water upward. Themathematics 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, application of simple analytic tools to a two dimensional analysis is effective in providing analytic results which closely parallel both field and laboratory results. Analysis shows:
The height of the jump is derived from the application of the equations of conservation of mass and momentum.[18] There are several methods of predicting the height of a hydraulic jump.[3][4][5][6][10][15][18][21]
They all reach common conclusions that:
For a known flow rate as shown by the figure below, the approximation that the momentum flux is the same just up- and downstream of the energy principle yields an expression of the energy loss in the hydraulic jump. Hydraulic jumps are commonly used as energy dissipators downstream of dam spillways.
In fluid dynamics, theequation of continuity is effectively an equation ofconservation of mass. Considering any fixed closed surface within an incompressible moving fluid, the fluid flows into a given volume at some points and flows out at other points along the surface with no net change in mass within the space since the density is constant. In case of a rectangular channel, then the equality of mass flux upstream () and downstream () gives:
with the fluiddensity, and the depth-averaged flow velocities upstream and downstream, and and the corresponding water depths.
For a straight prismatic rectangular channel, the conservation of momentumflux across the jump, assuming constant density, can be expressed as:
In rectangular channel, such conservation equation can be further simplified todimensionless M-y equation form, which is widely used in hydraulic jump analysis in open channel flow.
Jump height in terms of flowDividing by constant and introducing the result from continuity gives
which, after some algebra, simplifies to:
where Here is thedimensionlessFroude number, and relates inertial to gravitational forces in the upstream flow. Solving this quadratic yields:
Negative answers do not yield meaningful physical solutions, so this reduces to:
known asBélanger equation. The result may be extended to an irregular cross-section.[18]
This produces three solution classes:
This is equivalent to the condition that. Since the is the speed of a shallowgravity wave, the condition that is equivalent to stating that the initial velocity representssupercritical flow (Froude number > 1) while the final velocity representssubcritical flow (Froude number < 1).
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 aweir 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.
One of the most important engineering applications of the hydraulic jump is to dissipate energy in channels, dam spillways, and similar structures so that the excess kinetic energy does not damage these structures. The rate of energy dissipation orhead loss across a hydraulic jump is a function of the hydraulic jump inflow Froude number and the height of the jump.[15]
The energy loss at a hydraulic jump expressed as a head loss is:
In the design of adam the energy of the fast-flowing stream over aspillway must be partially dissipated to preventerosion of the streambed downstream of the spillway, which could ultimately lead to failure of the dam. This can be done by arranging for the formation of a hydraulic jump to dissipate energy. To limit damage, this hydraulic jump normally occurs on an apron engineered to withstand hydraulic forces and to prevent localcavitation 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. Obstructions or slope changes are routinely designed into the apron to force a jump at a specific location. Obstructions are unnecessary, as the slope change alone is normally sufficient. 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 by the downstream characteristics. The jump will occur if and only if the level of inflowing (supercritical) water level () satisfies the condition:
The hydraulic jump is characterised by a highly turbulent flow. Macro-scale vortices develop in the jump roller and interactwith the free surface leading to air bubble entrainment, splashes and droplets formation in the two-phase flow region.[23][24] The air–water 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.[25] 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.
| Amount upstream flow is supercritical (i.e., prejump Froude Number) | Ratio of height after to height before jump | Descriptive characteristics of jump | Fraction of energy dissipated by jump[11] |
|---|---|---|---|
| ≤ 1.0 | 1.0 | No jump; flow must be supercritical for jump to occur | none |
| 1.0–1.7 | 1.0–2.0 | Standing or undulating wave | < 5% |
| 1.7–2.5 | 2.0–3.1 | Weak jump (series of small rollers) | 5% – 15% |
| 2.5–4.5 | 3.1–5.9 | Oscillating jump | 15% – 45% |
| 4.5–9.0 | 5.9–12.0 | Stable clearly defined well-balanced jump | 45% – 70% |
| > 9.0 | > 12.0 | Clearly defined, turbulent, strong jump | 70% – 85% |
NB: the above classification is very rough. Undular hydraulic jumps have been observed with inflow/prejump Froude numbers up to 3.5 to 4.[15][16]
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.[26] Many subsequent studies have explored surface tension and pattern formation is such jumps.[27]
A 2018 study[28] experimentally and theoretically investigated the relative contributions of surface tension and gravity to the circular hydraulic jump. To rule out the role of gravity in the formation of a 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 model for the phenomenon andfound the general criterion for a thin film hydraulic jump to be
where is the localWeber number and is the localFroude number. For kitchen sink scale hydraulic jumps, the Froude number remains 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.[28] However, this model stays heavily contested.[29]
Turbidity currents can result in internal hydraulic jumps (i.e., hydraulic jumps asinternal waves in fluids of different density) inabyssal fan formation. The internal hydraulic jumps have been associated with salinity or temperature inducedstratification 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.[30]
Hydraulic jumps occur in the atmosphere in the air flowing over mountains.[31] A hydraulic jump also occurs at thetropopause interface between the stratosphere and troposphere downwind of the overshooting top of very strongsupercell thunderstorms.[32] A related situation is theMorning Glory cloud observed, for example, in Northern Australia, sometimes called an undular jump.[16]
The hydraulic jump is the most commonly used 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.[7][8][13][15]
While travelling down river,kayaking andcanoeing paddlers will often stop andplayboat in standing waves and hydraulic jumps. The standing waves and shock fronts of hydraulic jumps make for popular locations for such recreation.
Similarly, kayakers andsurfers have been known to ridetidal bores up rivers.
Hydraulic jumps have been used byglider pilots in the Andes and Alps[31] and to rideMorning Glory effects in Australia.[33]