Physics:Hydraulic jump

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Short description: Abrupt increase in depth and decrease in speed of a stream

The roughly circular hydraulic jump is around the central stream of water from a tap when it hits the flat surface of a sink. The thin region in the center has high speed and the thick region beyond has low speed.
The roughly circular hydraulic jump is around the central stream of water from a tap when it hits the flat surface of a sink. The thin region in the center has high speed and the thick region beyond has low speed.

A hydraulic jump is an abrupt increase in the depth of a fast-moving liquid stream in an open channel, which is accompanied by a decrease in speed. The jump appears as a wavy or turbulent region between the high-speed upstream flow and the slower downstream flow. A common example is the circular jump formed when a tap runs into a kitchen sink.

Hydraulic jumps occur below dam spillways and in rivers. Hydraulic jumps may be stationary, as below a dam, or they may propagate as surges along a stream, as in a tidal bore. Civil engineers design spillways and stilling basins to create hydraulic jumps that dissipate the mechanical energy of water flowing over dams.

A hydraulic jump can form only when the upstream flow moves faster than shallow-water waves, so that small disturbances to the flow cannot travel upstream. For speeds only slightly above the wave speed, the transition is a rolling wave. As the flow speed increases, the transition becomes more abrupt, until at high enough speeds the front breaks and curls upstream. These regimes are characterized by the ratio of upstream speed to wave speed, which is called the Froude number.

Hydraulic jumps also occur in stratified flows, including in the atmosphere and the oceans. In rivers, they can create both recreational whitewater features and dangerous recirculating currents.

History

Photo of the overflow spillway and jump at Lower St. Anthony Falls, Minneapolis
Concrete overflow spillway at Lower St. Anthony Falls on the Mississippi River, showing a pronounced hydraulic jump at its base.

As early as 1504, Leonardo da Vinci described and sketched water flows now understood as hydraulic jumps in his Codex Leicester.[1][2] The first experimental investigations of hydraulic jumps were published by Giorgio Bidone in 1820.[3][4][5] Jean-Baptiste Bélanger formulated the first modern theory of the hydraulic jump in 1841.[6][7]

Experimental and theoretical studies of hydraulic jumps continued during the second half of the 19th century, but Hager described Safranez’s 1927 work as the first systematic experimental investigation of the phenomenon.[8][9] Research in the 1930s established the importance of the Froude number for characterizing the flow in hydraulic jumps.[10][11]

Hydraulic works had used devices such as stepped cascades to reduce the energy of flowing water since antiquity.[12] In the early 20th century, hydraulic jumps, energy dissipators, and stilling basins became subjects of intensive study, and by the mid-20th century standard stilling-basin design guidance had been codified.[5][13]

Stationary and moving hydraulic jumps

Photo of a tidal bore in Alaska
A tidal bore in Alaska showing a steep, turbulent front. The upstream water is relatively shallow and the fractional change in elevation is large.

Hydraulic jumps may also be classified according to whether the transition is stationary or propagates as a surge.

A stationary hydraulic jump occurs at a fixed location. Upstream of the jump, the flow is fast and shallow; downstream, it is slow and deep. In the transition zone, the water slows and deepens in an abrupt step or standing wave. Downstream of the jump, the flow is typically turbulent and choppy.[14]

Photo of an undulating tidal bore
An undular front on a tidal bore. The upstream water is relatively deep and the fractional change in elevation is small.

A moving hydraulic jump, or surge, is a steep or undulating wavefront that propagates along the stream. A positive surge is a sudden increase in water depth that propagates as a wave either upstream or downstream. For example, when a dam breaks, a steep wall of water rushes downstream, and in tidal bores, a surge propagates upstream as the tide comes in.[15]

Tidal bores occur in rivers or narrow bays when the incoming tide travels upstream against the current. A tidal bore advancing into shallow upstream water typically shows a large and steep elevation difference, whereas a tidal bore entering deep upstream water may have a small elevation difference and an undulating wavefront. In both cases, the bore moves at the speed characteristic of waves in water of the depth immediately behind the wavefront.[16][17][18]

In a frame of reference moving with a surge, the surge is equivalent to a stationary jump.[19]

The Bélanger equation and the Froude number

The principles of conservation of mass and conservation of momentum lead to an equation relating the depths downstream and upstream of the jump.[20][21][22] The equation, known as the Bélanger equation,[7] agrees closely with both field and laboratory measurements.[13]

Line drawing of a control volume around a hydraulic jump
A hydraulic jump surrounded by an imaginary control surface (dotted green line) showing upstream and downstream hydrostatic forces

The Bélanger equation describes a hydraulic jump in a rectangular channel of uniform width, under idealized assumptions. The flow upstream of the jump has depth

h1

and an average speed

v1

. Downstream of the jump, the depth and average speed are

h2

and

v2

(see figure). Drag forces from the surface below the jump are presumed to be negligible. The liquid has a density

ρ

, and

g

is the gravitational acceleration.

If the flow is steady, the mass flow rate into the jump equals the mass flow rate out of the jump. Per unit width, this gives:

ρv1h1=ρv2h2

The momentum inflow and hydrostatic pressure force upstream must equal the momentum outflow and hydrostatic pressure force downstream, so that:[23]

ρv12h1+12ρgh12=ρv22h2+12ρgh22

Combining these expressions gives the ratio h2/h1:

h2h1=1+8v12gh1−12

This result is called the Bélanger equation.[7]

The change in depth for non-rectangular cross-sections has also been studied.[8]

The Froude number

The ratio v1/gh1 that appears in the Bélanger equation is called the Froude number of the upstream flow:

Fr1=v1gh1

The Froude number is dimensionless. In terms of the Froude number, the Bélanger equation is:[21]

h2h1=1+8Fr12−12


If Fr1>1, the equation shows that h2>h1, corresponding to an increase in depth across the jump. For Fr1<1, the equation gives h2<h1, implying a decrease in depth, but this solution is not physically admissible because it would require an increase in mechanical energy across the jump, violating the second law of thermodynamics.[21] For Fr1=1, the depth does not change.

Thus, the hydraulic jump can occur only when Fr1>1.

The speed of a shallow-water gravity wave is gh1, so the condition Fr1>1 is equivalent to an upstream speed greater than the wave speed. This situation is called supercritical flow.[24] The flow downstream of the jump moves slower than water waves (Fr2<1) and is thus subcritical flow. A hydraulic jump is always a transition from supercritical to subcritical flow.

Analogy to shock waves in compressible flow

In compressible-fluid dynamics, flow with speed greater than the speed of sound is termed supersonic, while flow with speed less than the speed of sound is subsonic. The ratio of flow speed to sound speed is the Mach number, Ma. Across a shock wave, a supersonic flow (Ma > 1) abruptly changes to subsonic flow (Ma < 1). A hydraulic jump is analogous to a shock wave in that it marks a transition from supercritical flow (Fr > 1) to subcritical flow (Fr < 1), with the Froude number playing a role analogous to the Mach number.[25]

Hydraulic jumps in civil engineering

Photo of Burdekin Dam in Queensland, Australia
Burdekin Dam on the Burdekin River in Queensland, Australia showing a hydraulic jump induced by both downstream obstructions and a change of slope
Series of roll waves moving down a spillway, where they terminate in a stationary hydraulic jump

The high kinetic energy of water flowing down a dam spillway can cause erosion of the streambed downstream, potentially undermining the structure.[26] A hydraulic jump can dissipate much of this energy. To limit damage, this hydraulic jump should normally occur on an apron engineered to withstand hydraulic forces[26] and to resist local cavitation and other erosion-causing phenomena.

In the design of a spillway and apron, engineers control the location of the hydraulic jump. Structural features such as slope changes or obstructions are often incorporated into the apron to induce a jump at a specific location. Alternatively, a horizontal apron may be designed so that the supercritical flow from the spillway transitions to subcritical flow as it encounters deeper downstream water. In practice, the position of the hydraulic jump is governed by both the apron geometry and the downstream water depth (tailwater), which together determine whether the flow can remain supercritical.[8][27][28]

Engineers often use hydraulic jumps for energy dissipation below spillways and outlets. A properly designed hydraulic jump can dissipate on the order of 60–70% of the flow’s mechanical energy within the stilling basin,[8] 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.[27][28]

Conservation of energy can be applied across the jump to calculate the dissipation of mechanical energy (the sum of kinetic and potential energy per unit mass of water). Expressed as loss of hydraulic head (in meters), the dissipation is:[21]

head loss=(h2−h1)34h1h2

This equation also agrees closely with measurements.[13] The head loss increases with the difference in downstream and upstream depth and therefore rises with the Froude number.

The following table summarizes the trend between depth change, energy dissipation, and jump characteristics with increasing Froude number. The ranges given for the jump characteristics are approximate and can overlap. The length of the hydraulic jump ranges from approximately 4 times the downstream depth for weak jumps to approximately 6.2 times the downstream depth for steady and strong jumps.[13][14]

Depth change, features, and energy dissipation for rising Froude number[13][14]
Froude number upstream of jump Ratio of depth after to depth before jump Descriptive characteristics of jump Fraction of mechanical energy dissipated by jump
≤ 1.0 1.0 No hydraulic jump forms none
1.0–1.7 1.0–2.0 Water surface shows undulations (“undular jump”) < 5%
1.7–2.5 2.0–3.1 Small rollers develop on the surface, but downstream water is smooth (a “weak jump”) 5% – 15%
2.5–4.5 3.1–5.9 Oscillating jet moves up and down in jump and large waves can travel far downstream (“oscillating jump”) 15% – 45%
4.5–9.0 5.9–12.0 Stable jump; position is less sensitive to downstream depth and energy dissipation is high (“steady jump”) 45% – 70%
> 9.0 > 12.0 Rough surface and downstream waves, high dissipation (“strong jump”) 70% – 85%

Internal hydraulic jumps

Most hydraulic jumps discussed in open-channel flow occur at a free surface. In stratified fluids, analogous jumps can occur at an internal density interface rather than at the air–water boundary. These internal hydraulic jumps are rapid transitions between internally supercritical and internally subcritical flow, meaning that the flow changes from faster than internal waves along the interface to slower than them. Internal jumps are often associated with strong mixing of the fluid layers.[29][30]

Internal hydraulic jumps occur in a variety of geophysical settings, including two-layer flows over underwater sills and stratified atmospheric flow over mountains.[30] In the atmosphere, they may form on the lee side of topography and are often discussed alongside mountain lee waves.[31] A well-documented example occurs in the lee of the Sierra Nevada in California, where cloud formations can make the jump visible.[32]

Recreation and hazards

A raft encountering a hydraulic jump on a river in Wales
A raft encountering a hydraulic jump on Canolfan Tryweryn in Wales

Hydraulic jumps and the standing waves associated with them are used in whitewater recreation, including kayaking, canoeing, rafting, and river surfing. In artificial whitewater parks, in-stream structures are often designed to create hydraulic jumps for kayakers and other boaters.[33] Kayakers and surfers sometimes ride tidal bores up rivers.[17]

Hydraulic jumps can also pose serious hazards to recreational river users. Below sharp drops, and especially below low-head dams, a submerged hydraulic roller may form in which recirculating flow can trap boats, swimmers, and debris for extended periods.[34] In whitewater paddling, highly retentive holes are often called “keepers”.[35][36] Low-head dams are sometimes described in safety literature as “drowning machines”.[37]

Surface tension and pattern formation in thin-film jumps

A liquid jet striking a surface, as in a kitchen sink, creates a thin liquid film that spreads radially before undergoing a hydraulic jump. For laminar jets, the thin film and the hydraulic jump can be entirely smooth and steady. In 1993, Liu and Lienhard showed that surface tension sets the shape of these jumps.[38] Many subsequent studies have explored the role of surface tension in such jumps.[39][40] When capillary instability appears, a jump can adopt complex non-circular patterns, including polygons, three- and four-leaf clovers, bow ties, and cat’s eyes.[41]

See also

References

  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. Bibcode: 2021AnRFM..53....1M. 
  2. ↑ Codex Leicester. c. 1504. pp. fols. 4r, 5r, 33v. 
  3. ↑ "Expériences sur le remou et sur la propagation des ondes". Memorie della Reale Accademia delle Scienze di Torino (Imprimerie Royale) 25: 21–112. 1820. https://archive.org/details/TO0E040768_TO0324_PNI-2246_000002. 
  4. ↑ Cabrera, Enrique (2010). Water Engineering and Management through Time: Learning from History. CRC Press. ISBN 978-0415480024. 
  5. ↑ 5.0 5.1 De Padova, Diana; Mossa, Michele (2021). "Hydraulic Jump: A Brief History and Research Challenges". Water 13 (13): 1733. doi:10.3390/w13131733. Bibcode: 2021Water..13.1733D. 
  6. ↑ Bélanger, J. B. (1841) (in fr). Notes sur le cours d'hydraulique. 1841 à 1842 : récapitulation préliminaire des notions essentielles de la dynamique. Paris: École Royale des Ponts et Chaussées. https://gallica.bnf.fr/ark:/12148/bd6t54644538r. 
  7. ↑ 7.0 7.1 7.2 Chanson, Hubert (2010). "Jean-Baptiste Bélanger: hydraulic engineer and academic". Proceedings of the Institution of Civil Engineers - Engineering and Computational Mechanics 163 (4): 227–233. doi:10.1680/eacm.2010.163.4.227. 
  8. ↑ 8.0 8.1 8.2 8.3 Hager, Willi H. (1995). Energy Dissipators and Hydraulic Jump. Dordrecht: Kluwer Academic Publishers. ISBN 978-90-5410-198-7. 
  9. ↑ Safranez, K. (1927). "Wechselsprung und die Energievernichtung des Wassers" (in de). Bauingenieur 8: 898–905. 
  10. ↑ Rouse, H. (1934). "On the Use of Dimensionless Numbers". Civil Engineering 4: 563–568. 
  11. ↑ Bakhmeteff, B. A.; Matzke, A. E. (1936). "The hydraulic jump in terms of dynamic similarity". Transactions of the American Society of Civil Engineers 101: 630–680. doi:10.1061/TACEAT.0004708. 
  12. ↑ Chanson, Hubert (2000–2001). "Historical Development of Stepped Cascades for the Dissipation of Hydraulic Energy". Transactions of the Newcomen Society 72 (2): 295–318. doi:10.1080/03720187.2000.12023617. https://staff.civil.uq.edu.au/h.chanson/reprints/newcom01.pdf. 
  13. ↑ 13.0 13.1 13.2 13.3 13.4 Peterka, A. J. (1958). Hydraulic Design of Stilling Basins and Energy Dissipators. Engineering Monograph. 25 (Revised January 1978 ed.). Denver, Colorado: U.S. Department of the Interior, Bureau of Reclamation. https://www.usbr.gov/tsc/techreferences/hydraulics_lab/pubs/EM/EM25.pdf. 
  14. ↑ 14.0 14.1 14.2 Chow, Ven Te (1959). "Chpt. 15: Hydraulic jump and its use as energy dissipator". Open-Channel Hydraulics. New York: McGraw-Hill. https://archive.org/details/openchannelhydra0000chow/mode/2up. 
  15. ↑ Chow, Ven Te (1959). "Chpt. 19: Rapidly varied unsteady flow". Open-Channel Hydraulics. New York: McGraw-Hill. https://archive.org/details/openchannelhydra0000chow/page/554/mode/2up. 
  16. ↑ Lighthill, James (1978). Waves in Fluids. Cambridge: Cambridge University Press. ISBN 978-0-521-29233-7. 
  17. ↑ 17.0 17.1 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. 
  18. ↑ Chanson, Hubert (2011). Tidal Bores, Aegir, Eagre, Mascaret, Pororoca: Theory and Observations. Singapore: World Scientific. doi:10.1142/8035. ISBN 978-981-4335-41-6. https://www.worldscientific.com/worldscibooks/10.1142/8035. 
  19. ↑ Chanson, H. (2004). The Hydraulics of Open Channel Flow: An Introduction (2nd ed.). Butterworth-Heinemann. ISBN 978-0-7506-5978-9. 
  20. ↑ Streeter, V.L.; Wylie, E.B. (1979). Fluid Mechanics (7 ed.). New York: McGraw-Hill Book Company. ISBN 978-0-07-062232-6. https://archive.org/details/fluidmechanic00stre. 
  21. ↑ 21.0 21.1 21.2 21.3 White, Frank M. (2008). Fluid Mechanics (6th ed.). New York: McGraw-Hill. ISBN 978-0-07-128645-9. 
  22. ↑ Munson, Bruce R.; Young, Donald F.; Okiishi, Theodore H. (1990). Fundamentals of Fluid Mechanics (1st ed.). New York: Wiley. pp. 688–694. ISBN 0-471-85526-X. 
  23. ↑ Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1987). Fluid Mechanics. Course of Theoretical Physics. 6 (2nd ed.). Oxford: Pergamon Press. pp. 412–413. ISBN 978-0750627672. 
  24. ↑ Chow, Ven Te (1959). "Chpt. 2: Open channels and their properties". Open-Channel Hydraulics. New York: McGraw-Hill. https://archive.org/details/openchannelhydra0000chow/mode/2up. 
  25. ↑ Thompson, Philip A. (1972). "Chpt. 11: Analogs in compressible flow". Compressible-fluid dynamics. New York: McGraw-Hill Book Company. pp. 517–533. https://archive.org/details/compressibleflui0000phil. "Discusses stationary jumps, tidal bores, and dam break problems using ideas from compressible flow theory." 
  26. ↑ 26.0 26.1 Davis, Calvin Victor, ed (1942). "Chpt. 7: Spillway and stream-bed protection works". Handbook of Applied Hydraulics. New York: McGraw-Hill Book Company. pp. 333–357. 
  27. ↑ 27.0 27.1 Khatsuria, R.M. (2005). Hydraulics of Spillways and Energy Dissipators. New York: Marcel Dekker. ISBN 978-0-8247-5789-2. 
  28. ↑ 28.0 28.1 Vischer, D.L.; Hager, W.H. (1995). Energy Dissipators. Rotterdam: A.A. Balkema. ISBN 978-0-8247-5789-2. 
  29. ↑ Turner, J. S. (1973). Buoyancy Effects in Fluids. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press. ISBN 978-0-521-08623-3. https://archive.org/details/buoyancyeffectsi0000turn/page/n5/mode/2up. 
  30. ↑ 30.0 30.1 Lawrence, Gregory A.; Armi, Laurence (2022). "Stationary internal hydraulic jumps". Journal of Fluid Mechanics 936. doi:10.1017/jfm.2022.74. Bibcode: 2022JFM...936A..25L. 
  31. ↑ Peña, Alfredo; Santos, Pedro (2021). "Lidar observations and numerical simulations of an atmospheric hydraulic jump and mountain waves". Journal of Geophysical Research: Atmospheres 126 (4). doi:10.1029/2020JD033744. Bibcode: 2021JGRD..12633744P. 
  32. ↑ Armi, Laurence; Mayr, Georg J. (2011). "The Descending Stratified Flow and Internal Hydraulic Jump in the Lee of the Sierras". Journal of Applied Meteorology and Climatology 50 (10): 1995–2011. doi:10.1175/JAMC-D-10-05005.1. Bibcode: 2011JApMC..50.1995A. 
  33. ↑ Stephens, T. A.; Bledsoe, B. P.; Fox, B. D.; Kolden, E.; Kondratieff, M. C. (2015). "Effects of whitewater parks on fish passage: a spatially explicit hydraulic analysis". Ecological Engineering 83: 305–318. doi:10.1016/j.ecoleng.2015.06.032. 
  34. ↑ Wahl, Tony L.; Svoboda, Connie D. (2020). "Defining Dangerous Flow Ranges of Low-Head Dams". https://www.usbr.gov/tsc/techreferences/hydraulics_lab/pubs/PAP/PAP-1205.pdf. 
  35. ↑ Kauffman, Robert B. (2016). "River Dynamics". Frostburg State University. https://www.frostburg.edu/faculty/rkauffman/_files/images_swr/Ch04_RiverDynamics_v2.pdf. 
  36. ↑ "River Levels". National Park Service. 2020. https://www.nps.gov/biso/planyourvisit/riverlevels.htm. 
  37. ↑ Tschantz, Bruce A.; Wright, Kenneth R. (2011). "Hidden Dangers and Public Safety at Low-Head Dams". The Journal of Dam Safety (Association of State Dam Safety Officials) 9 (1). https://damsafety.org/sites/default/files/TschantzWright_PublicSftyLowDams_JDS2011_1.pdf. 
  38. ↑ Liu, Xin; Lienhard, John H. (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. 
  39. ↑ 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. 
  40. ↑ Duchesne, Alexis; Limat, Laurent (2022). "Circular hydraulic jumps: where does surface tension matter?". Journal of Fluid Mechanics 937. doi:10.1017/jfm.2022.136. Bibcode: 2022JFM...937R...2D. 
  41. ↑ Bush, J. W. M.; Aristoff, J. M.; Hosoi, A. E. (2006). "An experimental investigation of the stability of the circular hydraulic jump". Journal of Fluid Mechanics 558: 33–52. doi:10.1017/S0022112006009839. Bibcode: 2006JFM...558...33B. 



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