Momentum-depth relationship in a rectangular channel

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 (i.e. one dimension) 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.In fluid dynamics, the momentum-force balance over a control volume is given by: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.An M-y diagram is a plot of the depth of flow (y) versus momentum (M). It should be noted that in this case M does not refer to momentum (M/Lt2), but to the momentum function (L3 or L2). This produces a specific momentum curve that is generated by calculating momentum for a range of depth values and graphing the results. Each M-y curve is unique for a specific flowrate, Q, or unit discharge, q. The momentum on the x-axis of the plot can either have units of length3 (when using the general momentum function equation) or units of length2 (when using the rectangular form Munit equation). In a rectangular channel of unit width, an M-y curve is plotted using:A flow is termed critical if the bulk velocity of the flow V {displaystyle V}   is equal to the propagation velocity of a shallow gravity wave g y {displaystyle {sqrt {gy}}}  . At critical flow, the specific energy and the specific momentum (force) are at a minimum for a given discharge. Figure 4 shows this relationship by showing a specific energy curve (E-y diagram) side-by-side to its corresponding specific momentum curve (M-y diagram) for a unit discharge q = 10 ft2/s. The green line on these figures intersects the curves at the minimum x-axis value that each curve exhibits. As noted, both of these intersections occur at a depth of approximately 1.46 ft, which is the critical flow depth for the specific conditions in the given channel. This critical depth represents the transition depth in the channel where the flow switches from supercritical flow to subcritical flow or vice versa.As mentioned before, an M-y diagram can provide an indication of flow classification for a given depth and discharge. When flow is not critical it is classified as either subcritical or supercritical. This distinction is based on the Froude number of the flow, which is the ratio of the bulk velocity (V) to the propagation velocity of a shallow wave : ( g y ) {displaystyle left({sqrt {gy}} ight)}  . The generic equation of the Froude number is expressed in terms of gravity (g), the flow’s velocity (V) and the hydraulic depth (A/B), where (A) represents the cross sectional area and (B) the top width. For rectangular channels, this ratio is equal to the depth of flow (y).Conjugate, or sequent, depths are the paired depths that result upstream and downstream of a hydraulic jump, with the upstream flow being supercritical and downstream flow being subcritical. Conjugate depths can be found either graphically using a specific momentum curve or algebraically with a set of equations. Because momentum is conserved over a hydraulic jump conjugate depths have equivalent momentum, and given a discharge, the conjugate to any flow depth can be determined with an M-y diagram (Figure 6).Start with the conservation of momentum function M 1 = M 2 {displaystyle M_{1}=M_{2}}  , for rectangular channels:It is important not to confuse conjugate depths (between which momentum is conserved) with alternate depths (between which energy is conserved). In the case of a hydraulic jump, the flow experiences a certain amount of energy headloss so that the subcritical flow downstream of the jump contains less energy than the supercritical flow upstream of the jump. Alternate depths are valid over energy conserving devices such as sluice gates and conjugate depths are valid over momentum conserving devices such as hydraulic jumps.The momentum equation can be applied to determine the force exerted by water on a sluice gate (Figure 7). Contrary to the conservation of fluid energy when a flow encounters a sluice gate, the momentum upstream and downstream of the gate is not conserved. The thrust force exerted by water on a gate placed in a rectangular channel can be obtained from the following equation, which can be derived in the same way as the conservation of momentum equation for rectangular channels: