page_title: Momentum–depth relationship in a rectangular channel
section_title: Hydraulic jumps and conservation of momentum
hierarchical_section_title: Momentum–depth relationship in a rectangular channel / Hydraulic jumps and conservation of momentum
caption_attribution_description: English: Conservation of Momentum – Forces applying to a control volume surrounding a hydraulic jump
context_page_description: In classical physics, momentum is the product of mass and velocity and is a vector quantity, but in fluid mechanics it is treated as a longitudinal quantity evaluated in the direction of flow. Additionally, it is evaluated as momentum per unit time, corresponding to the product of mass flow rate and velocity, and therefore it has units of force. The momentum forces considered in open channel flow are dynamic force – dependent of depth and flow rate – and static force – dependent of depth – both affected by gravity.
  The principle of conservation of momentum in open channel flow is applied in terms of specific force, or the momentum function; which has units of length cubed for any cross sectional shape, or can be treated as length squared in the case of rectangular channels. Although not being technically correct, the term momentum will be used to replace the concept of the momentum function. The conjugate depth equation, which describes the depths on either side of a hydraulic jump, can be derived from the conservation of momentum in rectangular channels, based upon the relationship between momentum and depth of flow.
context_section_description: Figure 2 depicts a hydraulic jump. A hydraulic jump is a region of rapidly varied flow and is formed in a channel when a supercritical flow transitions into a subcritical flow. This change in flow type is manifested as an abrupt change in the flow depth from the shallower, faster-moving supercritical flow to the deeper, slower-moving subcritical flow. Assuming no additional drag forces, momentum is conserved.
  A jump causes the water surface to rise abruptly, and as a result, surface rollers are formed, intense mixing occurs, air is entrained, and usually a large amount of energy is dissipated. For these reasons, in engineered systems a hydraulic jump is sometimes forced in an attempt to dissipate flow energy, to mix chemicals, or to act as an aeration device.
  The law of conservation of momentum states that the total momentum of a closed system of objects (which has no interactions with external agents) is constant. Despite the fact that there is an energy loss, momentum across a hydraulic jump is still conserved. This means that the flow depth on either side of the jump will have the same momentum, and in this way, if the momentum and flow depth on either side of the jump is known, it is possible to determine the depth on the other side of the jump. These paired depths are known as sequent depths, or conjugate depths. The latter is valid unless the jump is caused by an external force or outside influence.
  The green box in Figure 2 represents the control volume enclosing the jump system and shows the major pressure forces on the system (FP1 and FP2). As this system is considered to be horizontal (or nearly horizontal) and frictionless, the horizontal components of force that normally exist due to friction (Ff) and the weight of water from a sloping channel (Fw) are neglected. It is worth to noting that the slope of the triangular hydrostatic pressure distributions on each location corresponds to the specific weight of water (γ), which has units of (m/L²t²)
caption: Figure 2: Conservation of Momentum – Hydraulic Jump
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