Hydraulic jump in a rectangular channel, also known as classical jump, is a natural phenomenon that occurs whenever flow changes from supercritical to subcritical flow. In this transition, the water surface rises abruptly, surface rollers are formed, intense mixing occurs, air is entrained, and often a large amount of energy is dissipated. Numeric models created using the standard step method or HEC-RAS are used to track supercritical and subcritical flows to determine where in a specific reach a hydraulic jump will form. There are common hydraulic jumps that occur in everyday situations such as during the use of a household sink. There are also man-made hydraulic jumps created by devices like weirs or sluice gates. In general, a hydraulic jump may be used to dissipate energy, to mix chemicals, or to act as an aeration device. To produce equations describing the jump, since there is an unknown energy loss, there is a need to apply conservation of momentum. To develop this equation, a general situation in which there may or may not be an energy loss between upstream and downstream, and there may or may not be some obstacle on which there is a drag force Pf is considered. However, for a simple or classic hydraulic jump the force per unit width (Pf) equals 0. From there the momentum equation, and the conjugate depths equation, can be derived.
About hydraulic jumps The depth of supercritical flow, y1, ‘jumps’ up to its subcritical conjugate depth, y2, and the result of this abrupt change in flow conditions is considerable turbulence and Energy Loss, EL. Figure 1 shows a schematic of typical jump characteristics where E1 is the energy of the upstream flow, E2 is the energy of the downstream flow and Lj is the length of the hydraulic jump. A series of small surface rollers are formed in a standing wave like the one shown in Figure 1.
Figure 1. Hydraulic Jump Overall Schematic
Common hydraulic jumps Hydraulic jumps occur commonly in everyday situations such as during the use of any household sink. The jump can be seen in the form of a circular, stationary wave surrounding the inflow of water. The hydraulic jump occurs at the point where the seemingly still water becomes turbulent. As water hits the sink, it disperses, increasing in depth to a critical radius where the flow (supercritical with low depth, high velocity, and a Froude number greater than 1) must suddenly jump to a greater, subcritical depth (high depth, low velocity, and a Froude number less than 1) that is known to conserve momentum. Figure 2. Turbulent hydraulic jump can be created in sink (left), viscous hydraulic jump can create advanced shapes (right) (Images courtesy of John Bush, MIT)
Man-made hydraulic jumps Hydraulic jumps may also be manmade; as seen in Figure 2, scientists have been experimenting with the effects of viscosity on the hydraulic jump and have been able to create steady asymmetrical forms. In more practical applications, jumps are created in the environment with specific purposes such as erosion prevention. Erosion in stream beds is often caused by a high velocity water flow which leads to sediment transport. This process can be prevented by decreasing the velocity of the flow into the stream bed with the introduction of a hydraulic jump. Often in these cases, a hydraulic jump is created by devices such as a weir or sluice gate where the turbulent flow enters the stream. The mixture of chemical constituents in a solution is another practical use for hydraulic jumps. Introducing a hydraulic jump rapidly increases the turbulence of the flow, allowing sufficient constituent mixing without the use of any additional mechanisms. The wastewater industry sometimes uses hydraulic jumps as a way to mix solutions, minimizing the need to implement more expensive mechanical mixing systems. Figure 3. Weir in Riverfront Park (left) and Hydraulic Jump in Coagulation Chamber (right) Still another use for manmade hydraulic jumps is energy dissipation. One example of an energy dissipating use is a hydraulic jump stilling basin. In these basins, horizontal and sloping aprons are used to dissipate up to 60% of the energy of incoming flow; the basins implement devices such as chute blocks, baffle piers, and dentated ends whose effectiveness in energy dissipation is dependent on the Froude number of the incoming flow. ‘Hydraulic jump stilling basins are not typically suggested for use when dealing with heads greater than 100 meters due to complications caused by turbulences like intermittent cavitation, vibration, uplift, and hydrodynamic loading.’ Other hydraulic structures such as dams and weirs also use these same energy dissipating principles to reduce the incoming force from turbulent flows that tend to scour or erode downstream areas. Figure 4. Stilling Basin On Oker River in the Harz-Mointains at Opened Scour Outlet (left) and Stilling Basin for Griggs Dam in Columbus, OH (right)
Derivation of formula for simple, momentum conserving hydraulic jump in rectangular channel
Definitions of momentum Momentum is defined as the product of mass times velocity, and like velocity, it is a vector. French Scientist and Philosopher of the early 1600s René Descartes first discovered the concept of momentum but got stuck on the amount of motion (speed) which was not being conserved. Christiaan Huygens, a Dutch Scientist, pointed out that the "quantity of motion" did not need to be a positive value; a negative value meant that it was moving in the opposite direction.
Definition of variables mv = momentum = mass x velocity [=] MLT−1 ρ = density [=] ML−3 q = Q''/w = flow rate per unit width [=] L2T−1 Fd = dynamic force due to frictional resistance [=] MLT−2 P1 = upstream pressure [=] ML−1T−2 P2 = downstream pressure force [=] ML−1T−2 y1 = upstream depth [=] L y2 = downstream depth [=] L Fr = Froude number [dimensionless] [=] L2T−1 hj = height of hydraulic jump [=] L M = momentum function (specific force + momentum) [=] L2 γ = specific weight of water (9810 N/m3) [=] ML−2T−2 The basic principles behind the momentum function are:
Conservation of momentum which "states that the total momentum of a closed system of objects (which has no interactions with external agents) is constant" and Newton's laws of motion stating that the sum of the forces in a particular direction is equal to the mass times acceleration in that direction.
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