Hydraulic jump

This equation is cubic in hquid depth. Below a minimum value of Ejp there are no real positive roots above the minimum value there are two positive real roots. At this minimum value of Ejp the flow is critical that is, Fr = 1, V= V, and Ejp = (3/2)h. Near critical flow conditions, wave motion ana sudden depth changes called hydraulic jumps are hkely. Chow (Open Channel Hydraulics, McGraw-Hill, New York, 1959), discusses the numerous surface profile shapes which may exist in nommiform open channel flows. 

Additional theoretical bac-kground can be obtained from Preiswerk, Application of the Methods of Gas Dynamics to Water Flows with Free Suiface, part 1 Flows with No Energy Dissipation, NACA Tech. Mem. 934, 1940 part 11 Flows with Momentum Discontinuities Hydraulic Jumps), NACA Tech. Mem. 935, 1940. 

If a liquid is flowing in a rectangular channel in which a hydraulic jump occurs between sections 1 and 2, as shown in Figure 3.23, then the conditions after the jump can be determined by equating the net force acting on the liquid between the sections to the rate of change of momentum, if the frictional forces at the walls of the channel may be neglected. 

This expression gives D2 as a function of the conditions at the upstream side of the hydraulic jump. The corresponding velocity u2 is obtained by substituting in the equation ... 

The minimum depth at which a hydraulic jump can occur is found by putting Dx = D2 = D in equation 3.112 giving ... 

Thus a hydraulic jump can occur provided that the depth of the liquid is less than the critical depth. After the jump the depth will be greater than the critical depth, the flow having changed from rapid to tranquil. 

The energy dissipated in the hydraulic jump is now calculated. For a small change in the flow of a fluid in an open channel ... 

The hydraulic jump may be compared with the shock wave for the flow of a compressible fluid, discussed in Chapter 4. 

A hydraulic jump occurs during the flow of a liquid discharging from a tank into an open channel under a gate so that the liquid is initially travelling at a velocity of 1.5 m/s with a depth of 75 mm. Calculate the corresponding velocity and the liquid depth after the jump. 

Plug flow. Plug in horizontal flow, as shown in Figure 3.16, consists of enlongated gas bubbles. Although the name is sometimes used interchangeably with slug flow, it is differentiable in horizontal flow by the shape of the gas cavity or the appearance of staircase hydraulic jumps at the tails of air pockets (Ruder and Hanratty, 1990). 

The cross-sectional area available for flow in the collector ring should match that for the exit port or interstage line. This condition allows a smooth transition, as the liquid leaves the collector ring with no hydraulic jumps that would cause the flow to... 

Many of the examples show a rapid change from a depth below the critical to a depth above the critical. This is a local phenomenon, known as the hydraulic jump, which will be discussed in detail following the examples of gradually varied flow. 

The M3 scenario. This occurs because of an upstream control, as by the sluice gate. The bed slope is not sufficient to sustain lower-stage flow, and, at a certain point determined by energy and momentum relations, the water surface will pass through a hydraulic jump to upper-stage flow unless this is made unnecessary by the existence of a free overfall before the M3 crest reaches critical depth. 

The S scenarios. These are steep slope cases. They may be analyzed in much the same fashion as the M scenarios, having due regard for downstream control in the case of upper-stage flow and upstream control for lower-stage flow. Thus a dam or an obstruction on a steep slope produces an Si scenario, which approaches the horizontal asymptotically but cannot so approach the uniform depth line, which lies below the critical depth. Therefore this curve must be preceded by a hydraulic jump. The S2 scenario shows accelerated lower-stage flow, smoothly... 

The H and A scenarios. These scenarios have in common the fact that there is no condition of uniform flow possible. The smooth-plane fall drop-down curves are similar to the M2 scenario, but even more noticeable. The value of ye given in Eq. (10.129) applies strictly only to the smooth-plane scenario, but is approximately true for the M2 scenario also. The sluice gate on the horizontal and adverse slopes produces H3 and As scenarios that are like the sluice gate scenario but cannot exist for as long as the M3 scenario before a hydraulic jump must occur. Of course it is not possible to have a channel of any appreciable length carry water on a horizontal or an adverse grade. 

By far the most important of the local nonuniform flow phenomena is that which occurs when supercritical flow has its velocity reduced to subcritical. We have seen in these example scenarios that there is no ordinary means of changing from lower- to upper-stage flow with a smooth transition, because the theory calls for a vertical slope of the water surface. The result, then, is a marked discontinuity in the surface, characterized by a steep upward slope of the profile, broken throughout with violent turbulence, and known universally as the hydraulic jump. 

In the case of a hydraulic jump on a sloping channel, it is simply necessary to add the sine component of the weight of the water to the general momentum equation. Although there exist more refined methods for doing this [41], a good approximation may be obtained by assuming the jump section to be a trapezoid with bases jq and y2 and altitude of about 6Y2. 

The problem of determining where a hydraulic jump will occur is a combined application. In the case of supercritical flow on a mild slope, for instance, the tail water depth y2 is determined by the uniform flow depth jo for that slope. The rate of flow and the application of Eq. (10.133) then fix yu and the length of the M3 curve required to reach this depth from the upstream control may be computed from Eq. (10.123). Similarly, in the case of subcritical flow on a steep slope, the initial depth is equal to y0, the tail water depth is given by Eq. (10.133), and the length of the Si curve to the jump from the downstream control is computed from Eq. (10.123). For application of the hydraulic jump to design problems, and for analysis of the jump in circular and other nonrectangular sections, the reader is referred to more extensive treatises on the subject [42],... 

Example 10.9 A wide rectangular channel, of slope S = 0.0003 and roughness n = 0.020, carries a steady flow of 50 ft3/s per foot of width. If a sluice gate is so adjusted as to produce a depth of 1.5 ft in this channel, determine whether a hydraulic jump will form downstream, and, if so, find the distance if from the gate to the jump. 

If the channel is constricted laterally, the specific energy remains constant but the discharge q per unit width increases. An important application of the foregoing theories is the Parshall flume [46], which involves a constriction in the sides and a depression in the floor as well. This construction causes critical depth to occur near the beginning of the drop in floor level. When the flume operates submerged, a hydraulic jump forms downstream of the sloping section but this does not affect conditions upstream, and the flume will measure flow with an error of 3% or less for submergence ratios of up to 0.77. 

Ellms, R. W., Computation of the Trailwater Depth of the Hydraulic Jump in Sloping Flumes, cited in R. L. Daugherty and A. C. Ingersoll, Fluid Mechanics, McGraw-Hill, New York, 1954. 

Explain the phenomenon of hydraulic jump which occurs during the flow of a liquid in an open channel. 

The hydraulic jump as a mixing device." J. Am. Water Works Assoc., 17 1-23. 

Figure 6.9 shows a hydraulic jump and its schematic. By some suitable design, the chemicals to be mixed may be introduced at the point indicated by 1 in the figure. Hydraulic-jump mixers are designed as rectangular in cross section. 

---