page_title: Dimensionless momentum-depth relationship in open-channel flow
section_title: Solution of hydraulic jump with sluice gate
hierarchical_section_title: Dimensionless momentum-depth relationship in open-channel flow / Solution of hydraulic jump with sluice gate
caption_attribution_description: English: Hydraulic Jump with Sluice Gate
context_page_description: Momentum for one-dimensional flow in a channel can be given by the expression:
  where:
  M is the momentum [L³]
  Q is the rate of flow [L³/s]
  g is the acceleration due to gravity [L/T²]
  A is the cross sectional area of flow [L²]
  ȳ is the distance from the centroid of A to the water surface [L]
  For open channel flow calculations where momentum can be assumed to be conserved, such as in a hydraulic jump, we can equate the Momentum at an upstream location, M₁, to that at a downstream location, M₂, such that:
context_section_description: The following example of a hydraulic jump at a sluice gate outlet will give a clear idea about how conservation of energy and conservation of momentum apply in open channel flow.
  As shown in the middle panel in schematic plot, in a rectangular channel, deep upstream flow (position 1) encounters a sluice gate in front of position 2. A sluice gate imposes adecrease in flow depth at position 2, and a hydraulic jump is formed between position 2 and far downstream where the flow depth increases again (position 3). The left panel in Figure 2 shows the M-y diagram of these 3 positions (momentum is also referred to as other definitions in different references, e.g. “Specific Force” in (Chaudhry 2008)), while the right panel in schematic plot shows the E-y diagram for these 3 positions. Energy loss can be neglected between position 1 and 2 (e.g. assuming conservation of energy), but the external thrust on the gate causes significant momentum loss. By contrast, between positions 2 and 3, turbulence in the hydraulic jump dissipates energy, while the momentum can be assumed to be conserved. If we know the unit discharge as q = 10 ft²/s and the flow depth at position 1 as y₁ = 8.0 ft, by applying energy conservation between position 1 & 2 and momentum conservation between 2 & 3, the flow depths at position 2 (y₂) and 3 (y₃) can be computed.
  Applying conservation of energy between position 1 & 2:
  Applying conservation of momentum between position 2 & 3:
  In addition, we can obtain the thrust on the sluice gate as well:
  (The example above comes from Dr. Moglen’s “Open Channel Flow” course (CEE5384) in Virginia Tech, U.S.)
caption: Hydraulic Jump with Sluice Gate
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