Hydraulic jump

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

thumb|upright=2|Concrete overflow spillway at Lower [[St. Anthony Falls on the Mississippi River, showing a pronounced hydraulic jump at its base.|alt=Photo of the overflow spillway and jump at Lower St. Anthony Falls, Minneapolis]]

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

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. Research in the 1930s established the importance of the Froude number for characterizing the flow in hydraulic jumps.

Hydraulic works had used devices such as stepped cascades to reduce the energy of flowing water since antiquity. 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.

Stationary and moving hydraulic jumps

thumb|upright=1.7|A tidal bore in Alaska showing a steep, turbulent front. The upstream water is relatively shallow and the fractional change in elevation is large.|alt=Photo of a tidal bore in Alaska

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.

upright=1.25|thumb|An undular front on a tidal bore. The upstream water is relatively deep and the fractional change in elevation is small.|alt=Photo of an undulating tidal bore

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.

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.

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

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. The equation, known as the Bélanger equation,

:<math> \rho v_1^2h_1 + {1 \over 2} \rho gh_1^2 = \rho v_2^2h_2 + {1 \over 2} \rho gh_2^2 </math>

Combining these expressions gives the ratio <math>h_2/h_1</math>:

:<math> {h_2 \over h_1} =\frac{ \sqrt{1+{\frac{8v_1^2}{gh_1}-1}{2} </math>

This result is called the Bélanger equation. The flow downstream of the jump moves slower than water waves (<math>\text{Fr}_2 < 1</math>) 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.

Hydraulic jumps in civil engineering

upright=1.5|thumb|[[Burdekin Dam on the Burdekin River in Queensland, Australia showing a hydraulic jump induced by both downstream obstructions and a change of slope|alt=Photo of Burdekin Dam in Queensland, Australia]]

upright=1.1|thumb|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. 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

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,

Internal hydraulic jumps occur in a variety of geophysical settings, including two-layer flows over underwater sills and stratified atmospheric flow over mountains. A well-documented example occurs in the lee of the Sierra Nevada in California, where cloud formations can make the jump visible.

Recreation and hazards

thumb|upright=1.2|A raft encountering a hydraulic jump on [[Canolfan Tryweryn in Wales|alt=A raft encountering a hydraulic jump on a river 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. Kayakers and surfers sometimes ride tidal bores up rivers. In whitewater paddling, highly retentive holes are often called “keepers”. Low-head dams are sometimes described in safety literature as “drowning machines”.

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. Many subsequent studies have explored the role of surface tension in such jumps. 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.