2.6 Energy and Momentum Concepts

Section 2.3 developed the four fundamental equations of open channel hydraulics. These equations can now be applied to further the understanding of open channel phenomena, including the determination of alternate and sequent depths, the computation of normal and critical depths, and the hydraulic jump. This section develops these concepts by incorporating example computations for open channel flow analysis.

Specific Energy and Alternate Depths

The concept of specific energy is a modification of the total energy equation by dropping the datum energy term (z). Specific energy is then equal to the total energy as referenced to the channel invert. The equation for the specific energy at a selected location is


where	E	=	the specific energy at a point on the channel, as measured from the channel invert (ft, m)

Using the specific energy equation allows depth and velocity changes to be easily estimated over a short distance for a simple channel.

Example 2.6 Applying specific energy to analyze the effect of an obstruction to flow.

For the short channel reach shown in the following figure, compute the depth on the step of the channel bottom. Assume no losses between points 1 and 2. The width of the channel, T, is 10 ft and the discharge, Q, is 1000 ft³/s.

Solution

The specific energy equation (assuming a = 1) between the two points may be written as

Because depth and velocity on the step are unknown, a second equation is required to relate depth and velocity at this location to substitute into the preceding equation. The continuity equation can be employed because it is known that the discharge is constant between the two points, which gives

An iterative solution yields 9.24 ft for y1. Substituting for y1 gives V1 = 10.82 ft/s.

These are reasonable estimates because depth and velocity are expected to change by a small amount at a small obstruction in gradually varied flow; therefore, the values at location 1 should not be too different from those at location 2. However, solving the previous equation for y1 yields multiple roots, as illustrated in the following figure. That is, depths of 5.12 and -3.29 ft both satisfy the equation. Certainly, a negative depth is impossible, so this answer can be discarded. However, the other positive depth is possible since the energy balance is satisfied. How can two different depths satisfy the same energy criterion? The answer lies in evaluating the condition of flow, or flow regime for each. Computing the Froude number for the two possible depths gives Fr = 0.63 for the 9.24 ft depth and Fr = 1.52 for the 5.12 ft depth. Because changes in depth and velocity should be gradual to apply the energy equation, the subcritical solution is adopted for the depth on the step. If the depth on the step were supercritical, the basic assumption of zero losses between points 1 and 2 would be invalid.

From the data in Example 2.6, assume that the flow of 1000 ft³/s (28.3 m³/s) is held constant to calculate the specific energy for various depths. A curve with a somewhat parabolic shape can be plotted, as shown in Figure 2.18. The upper half of the curve is asymptotic to the flow depth (y), and the lower half is asymptotic to depth = 0. This specific energy curve shows that there are two possible depths for any given energy. These depths are referred to as alternate depths, one subcritical and one supercritical for the same energy value. As shown in the figure, the alternate depths are 9.24 ft and 5.12 ft (2.82 and 1.56 m) for the same specific energy of 11.05 ft (3.37 m). Only one value of depth is possible when the specific energy is a minimum. Figure 2.18 shows this value to be about 6.6-6.8 ft (2.0-2.1 m). This value is referred to as critical depth (y_c), the subject of the next section.

Critical Depth

The depth at the point of minimum energy is referred to as the critical depth (y_c). Depths greater than the critical depth indicate subcritical flow and depths less than the critical depth indicate supercritical flow. Critical depth can be determined graphically, as shown in Figure 2.18, where y_c is estimated from the graph to be about 6.6 to 6.8 ft. However, plotting this curve to compute critical depth is laborious and the accuracy of the critical depth calculation may be unacceptable. A direct solution is preferable. An equation can be developed by differentiating Equation 2.28 and setting the result equal to zero, then solving for the minimum value. Algebraic manipulation leads to the general equation for critical depth for any cross-section shape as


where	A_c	=	the cross-sectional channel area at the critical depth (ft², m²)
	T_c	=	the top width of flow at the critical depth (ft, m)

This equation holds only for the critical depth. For the special case of a rectangular channel, it is usually convenient to work with a unit discharge, given by


where	q	=	the unit discharge (ft³/s/ft, m³/s/m)
	T	=	the top width of the flow in a rectangular channel (ft, m)

The unit discharge, q, is applicable only for a rectangular channel. Substituting Equation 2.30 into Equation 2.29 and rearranging terms gives

Note that in both Equations 2.28 and 2.30 for critical depth, only discharge and cross-section shape are needed to solve for y_c. A change in channel slope or channel roughness has no effect on the solution for critical depth.

For Example 2.6, y_c can now be obtained directly with Equation 2.31 (y_c = 6.77 ft, or 2.10 m), because it is known that the channel is rectangular, making the unit discharge equal to 100 ft³/s/ft (9.29 m³/s/m). Critical depth is an important parameter that allows for the analysis of profile shape when the normal depth (the subject of the next section) and the actual water depth are known. The presence of critical depth is also considered a control on the flow regime, with subcritical flow upstream of critical depth and supercritical flow downstream. Significant obstructions, such as a dam or spillway, or a large slope change, such as a waterfall or the start of rapids, are typical instances of where critical depth will occur. Critical depth can be measured as the flow regime changes from subcritical to supercritical; however, it cannot be measured or located for the reverse situation, from supercritical to subcritical. The hydraulic jump that occurs for this latter condition contains significant turbulence and does not present a smooth profile, precluding a precise determination and location of critical depth. Laboratory measurements have found that the critical depth occurs about four times the critical depth upstream of the weir or channel dropoff. Discharge can be measured quite accurately at locations where critical depth occurs, such as at a dam or weir, and because critical depth is a function of only discharge and geometry, it is easy to calculate. Agencies such as the U.S. Geological Survey install special weirs on small streams to cause critical depth for low streamflow conditions. Determining the depth (head) on the weir can be used to easily determine the discharge over the weir from the weir equation (discussed in Chapter 6).

Example 2.7 Computing the critical depth.

For the channel conditions of Example 2.2, determine the critical depth for a flow rate of 400 ft³/s. The trapezoidal channel has a 10-ft bottom width and 1V:3H side slopes.

Solution

Because the channel is nonrectangular, the Equation 2.29 for critical depth is appropriate. From Table 2.1, the area at critical depth for a trapezoidal shape is 10y_c + 3y_c².

Normal Depth

When Chézy conducted his observations of flow in prismatic canals, he noted that the depth and velocity appeared to be uniform with distance. Normal depth, y_n, is the depth in the channel for uniform flow. Depth and velocity do not vary with distance; therefore, the channel invert profile, the water surface profile, and the energy grade line are all parallel, as shown in Figure 2.19. Uniform flow or normal depth rarely occurs and only in a long reach of a prismatic channel with constant slope and roughness coefficient.

Normal depths for any arbitrary shape can be computed by applying the Manning equation for discharge (Equation 2.27), knowing the discharge, channel slope, and an appropriate value for Manning's n. The area and hydraulic radius terms are both functions of depth; therefore, a depth found using the channel invert slope is normal depth. Normal depth computations usually require an iterative solution, as shown in Example 2.8. Various parameters (discharge, depth, slope, n) for a normal depth condition may be computed with HEC-RAS (see Chapter 11, (see page 401)).

Example 2.8 Computing the normal depth.

For the channel of Example 2.2, compute normal depth. The slope of the channel invert is 0.004 and Manning's n is 0.03. The discharge is 400 ft³/s.

Solution

For the 10-ft bottom width trapezoidal channel with 1V:3H side slopes, apply Manning's equation for discharge. From Table 2.1, the area of a trapezoid at the normal depth is

The Hydraulic Jump

When the flow regime changes from supercritical flow to subcritical flow, a sudden and abrupt rise in the water surface forms, known as a hydraulic jump. The hydraulic jump can occur only when the specific forces are equal on both the supercritical and subcritical sides of the jump: Equation 2.22 must be satisfied. Hydraulic jumps are often used in stilling basin design to intentionally reduce the energy of the flow.

For a rectangular channel and with the substitutions q = Q/T, y/2 for the centroid of a rectangular shape, and y = q/V, Equation 2.22 can be modified to yield


where	y1	=	the supercritical depth immediately before the initiation of the jump (ft, m)
	y2	=	the subcritical depth immediately after completion of the jump (ft, m)

These depths upstream and downstream of the jump are called sequent depths or conjugate depths and are illustrated in Figure 2.20.

It should be noted that the subscripts increase in the upstream direction in open channel hydraulic analysis and for the equations and examples in this book. The equations for hydraulic jumps are an exception to this rule, however. The smaller subscript is normally used for the supercritical side of the jump and the larger subscript represents the subcritical side. Equation 2.32 requires an iterative solution to determine y2, given y1. It is thus preferable to develop a direct (non-iterative) solution to determine this unknown depth. Through substitution and algebraic manipulation of Equation 2.32, a quadratic equation for the depth in a rectangular cross section is derived as


where	Fr₁	=	the supercritical Froude number immediately upstream of the initiation of the jump

Equation 2.33 can be used to compute the subcritical sequent depth knowing the supercritical depth and Froude number. For the reverse calculation, the equation is


where	Fr₂	=	the subcritical Froude number immediately downstream of the end of the jump

Example 2.9 Computing the sequent depth in a hydraulic jump.

A hydraulic jump occurs in a rectangular channel of horizontal slope. The depth immediately prior to the jump (y1) is 1 m and the unit discharge is 20 m³/s/m. Compute the sequent depth downstream of the jump and the percentage of energy lost in the hydraulic jump.

Solution

For the depth and unit discharge given, the velocity immediately before the jump is

Because the actual depth and hydraulic depth are the same in a rectangular channel, the Froude number before the jump is

The specific energy before and following the hydraulic jump can be used to estimate the energy loss as