Physics:Hydraulic jump

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Short description: Discharge of high velocity liquid into lower velocity area
Figure 1: A common example of a hydraulic jump is the roughly circular stationary wave that forms around the central stream of water from a tap when it hits the flat surface of a sink. The jump is at the transition between the area where the circle appears still and where the turbulence is visible.

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. 1504–1510; e.g., fols. 4r, 5r, 33v), which are now interpreted as including hydraulic jumps.[1][2] 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".[3]

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.[4][5][6][7][8][9][10][11][12][excessive citations]

Classes of hydraulic jumps

Figure 2: A raft encountering a hydraulic jump on Canolfan Tryweryn in Wales
Figure 3: A tidal bore in Alaska showing a turbulent shock-wave-like front. At this point the water is relatively shallow and the fractional change in elevation is large.

The two types of hydraulic jump are:[12][13]

  • The stationary hydraulic jump – an abrupt transition of a rapidly moving stream to slowly moving water, as shown in Figures 1 and 2.
  • The tidal bore – a wall or undulating wave of water that moves upstream against water flowing downstream as shown in Figures 3 and 4. If one considers a frame of reference which moves along with the wave front, then the wave front is stationary relative to the frame and has the same essential behavior as the 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, the behavior is that of a stationary jump.

Moving hydraulic jump

Figure 4: An undular front on a tidal bore. At this point the water is relatively deep and the fractional change in elevation is small.

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.[13]

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.[14]

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.[15]

Figure 5: Series of roll waves moving down a spillway, where they terminate in a stationary hydraulic jump

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.

Stationary hydraulic jump

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.[16]

Comparing the characteristics before and after, one finds:

Descriptive Hydraulic Jump Characteristics[7][8][11][12]
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 typically turbulent flow (rough and choppy)

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.

Analysis of the hydraulic jump on a liquid surface

Burdekin Dam on the Burdekin River in Queensland, Australia showing pronounced hydraulic jump induced by down-stream obstructions and a gradient change

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.[17] The principles of conservation of mass, conservation of momentum, and conservation of energy lead to quantitative relationships for:

  • Height of the jump: the relationship between the depths before and after the jump as a function of flow rate
  • Mechanical energy dissipation in the jump
  • Location of the jump on a natural or an engineered structure

Height of the jump

The height of the jump is derived from the application of the equations of conservation of mass and momentum.[6][10][18] The flow upstream of the jump has a depth h0 and an average speed v0. Downstream of the jump, the depth and average speed are h1 and v1. The liquid has a density ρ, and g is gravitational body force. For a rectangular channel of constant width w, mass conservation for the jump is

ρv0wh0=ρv1wh1

and momentum conservation is

ρwv02h0+12ρwgh02=ρwv12h1+12ρwgh12

Solving these equations, the upstream and downstream depth are related by

h1h0=1+8v02gh0−12

This is known as Bélanger equation.[19] This result may be extended to an irregular cross-section.[20]

The ratio v0/gh0 is the dimensionless Froude number which relates inertial to gravitational forces in the upstream flow

Fr0=v0gh0

The depth relationship is then[18]

h1h0=1+8Fr02−12   (1)

The solution has three cases:

  • When Fr0>1, then h1h0>1 (i.e., depth increases at the jump)
  • When Fr0=1, then h1h0=1 (i.e., there is no jump)
  • When Fr0<1, then h1h0<1 (i.e., the depth would decrease. However, this solution does not conserve energy. It would only physically possible if some force were to accelerate the fluid at that point)

Therefore, the hydraulic jump is possible only when Fr0>1. Since gh0 is the speed of a shallow gravity wave, the condition Fr0>1 is equivalent an initial velocity greater than the wave speed, called supercritical flow. The downstream is subcritical flow (Fr1<1).

Mechanical energy dissipation by a hydraulic jump

Saint Anthony Falls on the Mississippi River showing a pronounced hydraulic jump

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.[12]

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:[18]

Head Loss=(h1−h0)34h0h1

Location of hydraulic jump in a streambed or an engineered structure

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.

Supercritical flow down the Cleveland Dam spillway at the head of the Capilano River in North Vancouver, British Columbia, Canada

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 Eq. (1)

Naturally occurring hydraulic jump observed on the Upper Spokane Falls north channel

Characteristics of the hydraulic jump as a function of Froude number

Hydraulic jump characteristics[10][17]
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.[12][13]

==Undulations downstream of the jump == 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.

Air entrainment in hydraulic jumps[non-primary source needed]

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.[21][22] 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.[23] 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.

Hydraulic jump variations

A number of variations are amenable to similar analysis:

Shallow fluid hydraulic jumps

The hydraulic jump in a sink

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.[24] Many subsequent studies have explored surface tension and pattern formation is such jumps.[25]

A 2018 study[26] 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

1We+1Fr2=1

where We is the local Weber number and Fr is the local Froude number.[clarification needed] For some kitchen-sink-scale hydraulic jumps, the Froude number is very high, therefore, the effective criteria for the thin film hydraulic jump is We=1. In other words, a thin film hydraulic jump occurs when the liquid momentum per unit width equals the surface tension of the liquid.[26] However, this claim is heavily contested.[27][disputed – discuss]

Internal wave hydraulic jumps

Hydraulic jumps in abyssal fan formation

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.[28]

Atmospheric hydraulic jumps

Hydraulic jumps occur in the atmosphere in the air flowing over mountains.[29] A hydraulic jump also occurs at the tropopause interface between the stratosphere and troposphere downwind of the overshooting top of very strong supercell thunderstorms.[30] A related situation is the Morning Glory cloud observed, for example, in Northern Australia, sometimes called an undular jump.[13]

Industrial and recreational applications for hydraulic jumps

Energy dissipation using hydraulic jump

Industrial

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.[7][8][11][12]

Kayak playing on the transition between the turbulent flow and the recirculation region in a pier wake

Recreational

Hydraulic jumps have been used by glider pilots in the Andes and Alps[29] and to ride Morning Glory effects in Australia.[31]

See also

References and notes

  1. ↑ Marusic, Ivan; Broomhall, Susan (2021). "Leonardo da Vinci and Fluid Mechanics". Annual Review of Fluid Mechanics 53: 1–25. doi:10.1146/annurev-fluid-022620-122816. 
  2. ↑ Leonardo da Vinci (c. 1504). Codex Leicester. pp. fols. 4r, 5r, 33v. 
  3. ↑ Cabrera, Enrique (2010). Water Engineering and Management through Time: Learning from History. CRC Press. ISBN 978-0415480024. 
  4. ↑ Douglas, J.F.; Gasiorek, J.M.; Swaffield, J.A. (2001). Fluid Mechanics (4th ed.). Essex: Prentice Hall. ISBN 978-0-582-41476-1. 
  5. ↑ Faulkner, L.L. (2000). Practical Fluid Mechanics for Engineering Applications. Basil, Switzerland: Marcel Dekker AG. ISBN 978-0-8247-9575-7. 
  6. ↑ 6.0 6.1 Fox, R.W.; McDonald, A.T. (1985). Introduction to Fluid Mechanics. John Wiley & Sons. ISBN 978-0-471-88598-6. 
  7. ↑ 7.0 7.1 7.2 Hager, Willi H. (1995). Energy Dissipaters and Hydraulic Jump. Dordrecht: Kluwer Academic Publishers. ISBN 978-90-5410-198-7. 
  8. ↑ 8.0 8.1 8.2 Khatsuria, R.M. (2005). Hydraulics of Spillways and Energy Dissipaters. New York: Marcel Dekker. ISBN 978-0-8247-5789-2. 
  9. ↑ Roberson, J.A.; Crowe, C.T (1990). Engineering Fluid Mechanics. Boston: Houghton Mifflin Company. ISBN 978-0-395-38124-3. 
  10. ↑ 10.0 10.1 10.2 Streeter, V.L.; Wylie, E.B. (1979). Fluid Mechanics. New York: McGraw-Hill Book Company. ISBN 978-0-07-062232-6. https://archive.org/details/fluidmechanic00stre. 
  11. ↑ 11.0 11.1 11.2 Vischer, D.L.; Hager, W.H. (1995). Energy Dissipaters. Rotterdam: A.A. Balkema. ISBN 978-0-8247-5789-2. 
  12. ↑ 12.0 12.1 12.2 12.3 12.4 12.5 Chanson, H. (2004). The Hydraulic of Open Channel Flow: an Introduction (2nd ed.). Butterworth-Heinemann. ISBN 978-0-7506-5978-9. 
  13. ↑ 13.0 13.1 13.2 13.3 Chanson, H. (2009). "Current Knowledge In Hydraulic Jumps And Related Phenomena. A Survey of Experimental Results". European Journal of Mechanics B 28 (2): 191–210. doi:10.1016/j.euromechflu.2008.06.004. Bibcode: 2009EuJMB..28..191C. https://espace.library.uq.edu.au/view/UQ:162239/ejmbf_09.pdf. 
  14. ↑ Lighthill, James (1978). Waves in Fluids. Cambridge: Cambridge University Press. ISBN 978-0-521-29233-7. 
  15. ↑ Koch, C.; Chanson, H. (2009). "Turbulence Measurements in Positive Surges and Bores". Journal of Hydraulic Research 47 (1): 29–40. doi:10.3826/jhr.2009.2954. Bibcode: 2009JHydR..47...29K. https://espace.library.uq.edu.au/view/UQ:164015/jhr09_01a.pdf. 
  16. ↑ Murzyn, F.; Chanson, H. (2009). "Free-Surface Fluctuations in Hydraulic Jumps: Experimental Observations". Experimental Thermal and Fluid Science 33 (7): 1055–1064. doi:10.1016/j.expthermflusci.2009.06.003. Bibcode: 2009ETFS...33.1055M. http://espace.library.uq.edu.au/view/UQ:179807. 
  17. ↑ 17.0 17.1 United States Bureau of Reclamation (1 June 1955). Research studies on stilling basins, energy dissipators, and associated appurtenances (Report). Hydraulic Laboratory Report. Denver, Colorado: U.S. Bureau of Reclamation. pp. 20-23. https://www.usbr.gov/tsc/techreferences/hydraulics_lab/pubs/HYD/HYD-399.pdf. Retrieved 30 March 2026. 
  18. ↑ 18.0 18.1 18.2 White, Frank M. (2008). Fluid Mechanics (6th ed.). New York: McGraw-Hill. ISBN 978-0-07-128645-9. 
  19. ↑ Chanson, Hubert (2009). "Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger (1828)". Journal of Hydraulic Engineering 135 (3): 159–63. doi:10.1061/(ASCE)0733-9429(2009)135:3(159). http://espace.library.uq.edu.au/view/UQ:165002/jhe09_03.pdf. 
  20. ↑ Chanson, Hubert (2012). "Momentum Considerations in Hydraulic Jumps and Bores". Journal of Irrigation and Drainage Engineering 138 (4): 382–385. doi:10.1061/(ASCE)IR.1943-4774.0000409. Bibcode: 2012JIDE..138..382C. 
  21. ↑ Chanson, H.; Brattberg, T. (2000). "Experimental Study of the Air-Water Shear Flow in a Hydraulic Jump". International Journal of Multiphase Flow 26 (4): 583–607. doi:10.1016/S0301-9322(99)00016-6. Bibcode: 2000IJMF...26..583C. https://espace.library.uq.edu.au/view/UQ:9346/ARTI_77E.pdf. 
  22. ↑ Murzyn, F.; Chanson, H. (2009). "Two-phase gas-liquid flow properties in the hydraulic jump: Review and perspectives". in S. Martin and J.R. Williams. Multiphase Flow Research. Hauppauge NY, USA: Nova Science Publishers. pp. Chapter 9, pp. 497–542. ISBN 978-1-60692-448-8. https://espace.library.uq.edu.au/view/UQ:179852/nova_murzynchanson_2009.pdf. 
  23. ↑ Chanson, H. (2007). "Bubbly Flow Structure in Hydraulic Jump". European Journal of Mechanics B 26 (3): 367–384. doi:10.1016/j.euromechflu.2006.08.001. Bibcode: 2007EuJMB..26..367C. https://espace.library.uq.edu.au/view/UQ:12836/ejmbf_07.pdf. 
  24. ↑ Liu, Xin; Lienhard, John H. (July 1993). "The hydraulic jump in circular jet impingement and in other thin liquid films". Experiments in Fluids 15 (2): 108–116. doi:10.1007/BF00190950. Bibcode: 1993ExFl...15..108L. 
  25. ↑ Bush, John W. M.; Aristoff, Jeffery M. (2003). "The influence of surface tension on the circular hydraulic jump". Journal of Fluid Mechanics 489: 229–238. doi:10.1017/S0022112003005159. Bibcode: 2003JFM...489..229B. 
  26. ↑ 26.0 26.1 Bhagat, R.K.; Jha, N.K.; Linden, P.F.; Wilson, D.I. (2018). "On the origin of the circular hydraulic jump in a thin liquid film". Journal of Fluid Mechanics 851. doi:10.1017/jfm.2018.558. Bibcode: 2018JFM...851R...5B. 
  27. ↑ Duchesne, Alexis; Limat, Laurent (2022-02-28). "Circular hydraulic jumps: where does surface tension matter?". Journal of Fluid Mechanics 937. doi:10.1017/jfm.2022.136. ISSN 0022-1120. Bibcode: 2022JFM...937R...2D. http://dx.doi.org/10.1017/jfm.2022.136. 
  28. ↑ Kostic, Svetlana; Parker, Gary (2006). "The Response of Turbidity Currents to a Canyon-Fan Transition: Internal Hydraulic Jumps and Depositional Signatures". Journal of Hydraulic Research 44 (5): 631–653. doi:10.1080/00221686.2006.9521713. Bibcode: 2006JHydR..44..631K. 
  29. ↑ 29.0 29.1 Clément, Jean Marie (2015). Dancing with the wind. Pivetta Partners. ISBN 978-8890343247. 
  30. ↑ "Hydraulic jump dynamics above supercell thunderstorms", Science, O'Neill et al, Vol. 373, Issue 6560, September 10, 2021
  31. ↑ "Cloud-surfers ride Morning Glory in north Queensland". ABC News. 3 October 2017. http://www.abc.net.au/news/2017-10-04/cloud-surfers-ride-morning-glory-in-north-queensland/9010504. 



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