M1Q9 PART #4 (METHOD)

The jet emerging from an underflow vertical sluice gate contracts at the "vena contracta" just downstream of the gate, where the depth is 0.7 m. Downstream of the gate, where the flow is unaffected, water depth coincides with the normal depth dn = 3.2 m. The channel has got a rectangular cross section 8 m wide, n = 0.03 and slope S0 = 0.002. PART #4 (method): Which method could be used to determine the hydraulic jump location?

Question

M1Q9 PART #4 (METHOD)

The jet emerging from an underflow vertical sluice gate contracts at the "vena contracta" just downstream of the gate, where the depth is 0.7 m. Downstream of the gate, where the flow is unaffected, water depth coincides with the normal depth dn = 3.2 m. The channel has got a rectangular cross section 8 m wide, n = 0.03 and slope S0 = 0.002. PART #4 (method): Which method could be used to determine the hydraulic jump location?

BlackTom AI · Accepted Answer

First, let's restate the question and the provided options to ensure clarity of what is being evaluated. The problem asks which method could be used to determine the hydraulic jump location given a supercritical jet that contracts at the vena contracta downstream of an underflow vertical sluice gate (depth 0.7 m), with downstream depth dn = 3.2 m, in a rectangular channel 8 m wide, with Manning’s n = 0.03 and slope S0 = 0.002.

Option 1: The upstream depth of the jump is the vena contracta; hence, the jump occurs immediately downstream of the gate without any jet adjustment.
- This is not correct because the jet contracts at the vena contracta, but after leaving the gate the flow undergoes gradual adjustment before a hydraulic jump can form. The jump does not occur immediately at the gate; the flow typically transitions from supercritical to subcritical over a finite distance, with depth changes occurring downstream as the jet adjusts. Thus, claiming the jump is at the vena contracta depth and immediately downstream ignores the necessary progression of the flow and is inconsistent with how hydraulic jumps form in real channels.

Option 2: Consider how the supercritical jet gradually adjusts and identify where flow conditions satisfy jump formation.
- This approach aligns with the physical nature of hydraulic jumps. A supercritical jet that contracts will undergo gradual deceleration and depth increase as it interacts with the downstream region, and a hydraulic jump forms where the flow transitions from supercritical to subcritical under appropriate energy and momentum considerations. Evaluating the gradual adjustment and locating the point where flow criteria for a jump are met (e.g., Froude number dropping below a critical value, continuity of momentum, depth change) is a sound method.

Option 3: Assume jet adjustment before the jump ends at the section where velocity is maximum and depth is minimum.
- This is problematic because the location of a hydraulic jump is not defined by the point of maximum velocity or minimum depth in isolation. The jump forms where the flow cannot sustain a supercritical condition and must transition to subcritical, which depends on global energy and momentum balances, not simply on a single point of maximum velocity or minimum depth. Therefore, this criterion is not a reliable method to locate the jump.

Option 4: Apply the momentum equation directly between the gate and a downstream section, assuming uniform depth and ignoring gradual depth variations.
- Using the momentum equation across a jump is a valid technique, but applying it directly between the gate and a downstream section while assuming uniform depth and ignoring gradual depth variations is inconsistent with the actual physics. Between the gate and the downstream section, the depth is not uniform due to the gradual jet adjustment and the development of the hydraulic jump. Ignoring depth variation would lead to an inaccurate determination of the jump location. This makes the approach flawed for locating a hydraulic jump in this setup.

In summary, the reasoning for each option hinges on how well the method reflects the real flow behavior: hydraulic jumps arise from a gradual adjustment of a supercritical jet and occur where flow conditions permit a transition to subcritical, not instantaneously at the vena contracta, nor at a single peculiar point of velocity/depth, and not by assuming uniform depth with a simplistic momentum balance over a non-uniform region. Option 2 most accurately captures this, while the other options rely on incorrect or oversimplified assumptions about where and how a hydraulic jump forms.

类似问题

M1Q10 PART #2 (METHOD) Water discharging from a spillway into a rectangular horizontal (wide) channel goes through a hydraulic jump, before returning to a river (see schematic below). PART #2 (method): Which method could be used to determine the height of this step?

M1Q8 Which of these actions would make a hydraulic jump taller (increase the difference between upstream and downstream depths)?

M1Q6 Which factors control the length of a hydraulic jump?

With regard to a hydraulic jump, which of the following is true?

A rectangular channel 10 m wide form part of a dam spillway. The discharge is 36.5 m3/s when the depth of flow is 0.43 m. At the foot of the spillway the channel is almost horizontal with an hydraulic jump. Calculate the energy loss at the jump.

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