Hydraulic jump - AbsoluteAstronomy.com

A hydraulic jump is a phenomenon in the science of hydraulics

Hydraulics

Hydraulics is a topic in applied science and engineering dealing with the mechanical properties of liquids. Fluid mechanics provides the theoretical foundation for hydraulics, which focuses on the engineering uses of fluid properties. In fluid power, hydraulics is used for the generation, control,...

which is frequently observed in open channel flow

Open channel flow

Open-Channel Flow, a branch of Hydraulics, is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways, but differ in one important respect; the free surface...

such as rivers and spillways. 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 a shockwave forms.

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 the critical

Supercritical flow

A supercritical flow is when the flow velocity is larger than the wave velocity. The analogous condition in gas dynamics is supersonic....

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 wave

Wave

In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s.

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:

The stationary hydraulic jump – rapidly flowing water transitions in a stationary jump to slowly moving water as shown in Figures 1 and 2.

The tidal bore
Tidal bore
A tidal bore is a tidal phenomenon in which the leading edge of the incoming tide forms a wave of water that travel up a river or narrow bay against the direction of the river or bay's current...

– a wall or undulating wave of water moves upstream against water flowing downstream as shown in Figures 3 and 4. If considered from a frame of reference which moves with the wave front, you can see that this case is physically similar to a stationary jump.

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.

Classes of hydraulic jumps

Hydraulic jumps can be seen in both a stationary form, called a hydraulic jump, and a dynamic or moving form, called a positive surge or "hydraulic jump in translation". They can be described using the same analytic approaches and are simply variants of a single phenomenon.

Moving hydraulic jump

A tidal bore

Tidal bore

A tidal bore is a tidal phenomenon in which the leading edge of the incoming tide forms a wave of water that travel up a river or narrow bay against the direction of the river or bay's current...

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. 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 a shock-wave-like

Shock wave

A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

wall of water. 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.

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.

Stationary hydraulic jump

The stationary hydraulic jump is 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 or standing wave) on the liquid surface.

Comparing the characteristics before and after, one finds:

**Descriptive Hydraulic Jump Characteristics**
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	typically turbulent flow (rough and choppy)

The other stationary hydraulic jump occurs when a rapid flow encounters a submerged object which throws the water upwards. The mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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.

Analysis of the hydraulic jump on a liquid surface

In spite of the apparent complexity of the flow transition, application of simple analytic tools to a two dimensional analysis are effective in providing analytic results which closely parallel both field and laboratory results. Analysis shows:

Height of the jump: the relationship between the depths before and after the jump as a function of flow rate.
Energy loss in the jump
Location of the jump on a natural or an engineered structure
Character of the jump: undular or abrupt

Height of the jump

There are several methods of predicting the height of a hydraulic jump.

They all reach common conclusions that:

The ratio of the water depth before and after the jump depend solely on the ratio of velocity of the water entering the jump to the speed of the wave over-running the moving water.
The height of the jump can be many times the initial depth of the water.

Assuming a two-dimensional situation with 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 dissipaters downstream of dam spillways.

Applying the continuity principle
In fluid dynamics, the equation of continuity is effectively an equation of conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

. 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 fluid density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

and

the depth-averaged

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

flow velocities upstream and downstream, and

and

the corresponding water depths.

Conservation of momentum flux
The conservation of momentum flux

Flux

In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

across the jump, assuming constant density, can be expressed as:

Jump height in terms of flow
Dividing by constant

and introducing the result from continuity gives

Which, after some algebra, simplifies to:

where

Here

is the dimensionless Froude number

Froude number

The Froude number is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance of an...

, 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 as Bélanger

Jean-Baptiste-Charles-Joseph Bélanger

Jean-Baptiste Charles Joseph Bélanger was a French applied mathematician who worked in the areas of hydraulics and hydrodynamics. He was a professor at the Ecole Centrale des Arts et Manufactures, Ecole Polytechnique and École des Ponts et Chaussées in France...

equation.

This produces three solution classes:

When , then (i.e., there is no jump)
When , then (i.e., there is a negative jump – this can be shown as not conserving energy and is only physically possible if some force were to accelerate the fluid at that point)
When or , then (i.e., there is a positive jump)

This is equivalent to the condition that

. Since the

is the speed of a shallow gravity wave

Gravity wave

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....

, the condition that

is equivalent to stating that the initial velocity represents supercritical flow

Supercritical flow

A supercritical flow is when the flow velocity is larger than the wave velocity. The analogous condition in gas dynamics is supersonic....

(Froude number > 1) while the final velocity represents subcritical flow (Froude number < 1).

Undulations downstream of the jump
Practically this means that water accelerated by large drops can create stronger standing waves (undular bore

Undular bore

In meteorology, an undular bore is a wave disturbance in the Earth's atmosphere and can be seen through unique cloud formations. They normally occur within an area of the atmosphere which is stable in the low levels after an outflow boundary or a cold front moves through.In hydraulics, an undular...

s) in the form of hydraulic jumps as it decelerates at the base of the drop. Such standing waves, when found downstream of a weir

Weir

A weir is a small overflow dam used to alter the flow characteristics of a river or stream. In most cases weirs take the form of a barrier across the river that causes water to pool behind the structure , but allows water to flow over the top...

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.

Energy dissipation by a hydraulic jump

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 or head loss across a hydraulic jump is a function of the hydraulic jump inflow Froude number. The larger the jump, as expressed in terms of its inflow Froude number, the greater the head loss.

Analytically, the fractional energy loss (FEL) can be expressed in terms of the Froude number (

) for the incident flow as:

Since

this is equivalent to concluding the energy loss can be predicted by predicting or measuring the speed and depth of the entering water.

Location of hydraulic jump in a streambed or an engineered structure

In the design of a dam

Dam

A dam is a barrier that impounds water or underground streams. Dams generally serve the primary purpose of retaining water, while other structures such as floodgates or levees are used to manage or prevent water flow into specific land regions. Hydropower and pumped-storage hydroelectricity are...

the energy of the fast-flowing stream over a spillway

Spillway

A spillway is a structure used to provide the controlled release of flows from a dam or levee into a downstream area, typically being the river that was dammed. In the UK they may be known as overflow channels. Spillways release floods so that the water does not overtop and damage or even destroy...

must be partially dissipated to prevent erosion

Erosion

Erosion is when materials are removed from the surface and changed into something else. It only works by hydraulic actions and transport of solids in the natural environment, and leads to the deposition of these materials elsewhere...

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 local cavitation

Cavitation

Cavitation is the formation and then immediate implosion of cavities in a liquidi.e. small liquid-free zones that are the consequence of forces acting upon the liquid...

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:

If the downstream flow is restricted by the down-stream channel such that water backs up onto the foot of the spillway, that downstream water level can be used to identify the location of the jump.

If the spillway continues to drop for some distance, but the slope changes such that it will no longer support supercritical flow, the depth in the lower subcritical flow region is sufficient to determine the location of the jump.

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:

= Upstream Froude Number

g = acceleration due to gravity

G-force

The g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...

(essentially constant for this case)

h = height

Height

Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example "The height of the building is 50 m" or "The height of the airplane is 10,000 m"...

of the fluid (

= initial height while

= final downstream height)

Air entrainment in hydraulic jumps

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 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. Once the entrained bubbles are advected into regions of lesser shear, bubble collisions and coalescence lead to larger air entities that are driven towards the free-surface by a combination of buoyancy and turbulent advection.

Tabular summary of the analytic conclusions

**Hydraulic Jump Characteristics**
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
≤ 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.

Shallow fluid hydraulic jumps

The hydraulic jump in your sink
Figure 2 above illustrates a daily example of a hydraulic jump can be seen in the sink. Around the place where the tap water hits the sink, you will see a smooth-looking flow pattern. A little further away, you will see a sudden "jump" in the water level. This is a hydraulic jump.

The nature of this jump differs from those previously discussed in the following ways:

The water is flowing radially. As a result it continuously grows shallower and slows due to friction (the Froude number drops) up to the point where the jump occurs.
The flow depth is thin enough that the surface tension can no longer be neglected, changing the wave solution conclusions. The higher speed of the surface tension waves bleed off the high frequency component, making an undular jump the dominant form.

Changes in the behavior of the jump can be observed by changing the flow rate.

Hydraulic jumps in abyssal fan formation

Turbidity current

A turbidity current is a current of rapidly moving, sediment-laden water moving down a slope through water, or another fluid. The current moves because it has a higher density and turbidity than the fluid through which it flows...

s can result in internal hydraulic jumps (i.e., hydraulic jumps as internal wave

Internal wave

Internal waves are gravity waves that oscillate within, rather than on the surface of, a fluid medium. They are one of many types of wave motion in stratified fluids . A simple example is a wave propagating on the interface between two fluids of different densities, such as oil and water...

s in fluids of different density) in abyssal fan

Abyssal fan

Abyssal Fans, also known as deep-sea fans, underwater deltas, and submarine fans, are underwater structures that look like deltas formed at the end of many large rivers, such as the Nile or Mississippi Rivers. Abyssal fans are also thought of as an underwater version of alluvial fans.- Formation...

formation. The internal hydraulic jumps have been associated with salinity or temperature induced stratification

Stratification (water)

Water stratification occurs when water masses with different properties - salinity , oxygenation , density , temperature - form layers that act as barriers to water mixing...

as well as with density differences due to suspended materials. When the bed slope over which the turbidity current 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.

Atmospheric hydraulic jumps

A related situation is the Morning Glory cloud

Morning glory cloud

The Morning Glory cloud is a rare meteorological phenomenon occasionally observed in different locations around the world. The southern part of Northern Australia's Gulf of Carpentaria is the only known location where it can be predicted and observed on a more or less regular basis. The settlement...

observed, for example, in Northern Australia, sometimes called an undular jump.

Industrial and recreational applications for hydraulic jumps

Industrial

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

Cavitation

Cavitation is the formation and then immediate implosion of cavities in a liquidi.e. small liquid-free zones that are the consequence of forces acting upon the liquid...

, and abrasion. An extensive literature has been developed for this type of engineering.

Recreational

While travelling down river, kayaking

Kayaking

Kayaking is the use of a kayak for moving across water. Kayaking and canoeing are also known as paddling. Kayaking is distinguished from canoeing by the sitting position of the paddler and the number of blades on the paddle...

and canoeing

Canoeing

Canoeing is an outdoor activity that involves a special kind of canoe.Open canoes may be 'poled' , sailed, 'lined and tracked' or even 'gunnel-bobbed'....

paddlers will often stop and playboat

Playboating

Playboating is a discipline of whitewater kayaking or canoeing where the paddler performs various technical moves in one place , as opposed to downriver whitewater canoeing or kayaking where the objective is to travel the length of a section of river...

in standing waves and hydraulic jumps. The standing waves and shock fronts of hydraulic jumps make for popular locations for such recreation.

Similarly, kayakers and surfers

River surfing

River surfing is the sport of surfing either standing waves or tidal bores in rivers. Claims for its origins include a 1955 ride of 1.5 miles along the tidal bore of the River Severn...

have been known to ride tidal bore