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
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
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.
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, agrees closely with both field and laboratory measurements.
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 h 1 {\displaystyle h_{1}} and an average speed v 1 {\displaystyle v_{1}} . Downstream of the jump, the depth and average speed are h 2 {\displaystyle h_{2}} and v 2 {\displaystyle v_{2}} (see figure). Drag forces from the surface below the jump are presumed to be negligible. The liquid has a density ρ {\displaystyle \rho } , and g {\displaystyle 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:
ρ v 1 h 1 = ρ v 2 h 2 {\displaystyle \rho v_{1}h_{1}=\rho v_{2}h_{2}}
The momentum inflow and hydrostatic pressure force upstream must equal the momentum outflow and hydrostatic pressure force downstream, so that:
… excerpt ends here. Continue reading the full article.
