A channel invert slope is designated as mild if the normal depth is greater than critical depth (that is, subcritical flow is the expected condition). Figure 2.22 shows a pris-matic channel with a mild slope and the calculated normal and critical depths for a known discharge. The water surface profiles for this channel carry an M classification because of the mild slope. Also shown on Figure 2.22 are three zones: Zone 1 for actual depths greater than both normal and critical depth, Zone 2 for actual depths greater than critical but less than normal depth, and Zone 3 for actual depths less than both normal and critical depths. Therefore, water surface profiles in Zone 1 are classi-fied as M1, in Zone 2 as M2, and in Zone 3 as M3. In gradually varied flow, there will be only one profile shape possible for a known depth in a particular zone. If the actual depth for a known discharge at a point along the channel is greater than both normal and critical depth, it would be useful to be able to sketch the expected water surface profile. The sketch would not reflect a precise elevation of the profile, but rather the expected profile shape. Equation 2.39 can aid in sketching the resulting water surface profile.
With no need to compute numerical values for dy/dx, Equation 2.39 can be applied to determine the sign of dy/dx. If the sign of dy/dx is positive, the depth increases in the downstream direction. If the sign of dy/dx is negative, the depth decreases in the downstream direction. Four additional rules are also needed for profile classification, as follows:
1. Profiles approach normal depth asymptotically.
2. Profiles intersect critical depth at a sharp angle.
Figure 2.21 Development of the profile shape equation.
Section 2.5 Profile Shapes 47
3. Obstructions or changes in subcritical flow affect the profile upstream, but not downstream.
4. Obstructions or changes in supercritical flow affect the profile downstream, but not upstream.
In using Equation 2.39 to determine the sign of dy/dx, the most difficult part is the determination of the magnitude of the energy grade line, or friction slope (sf), com-pared to the invert slope (so). The slope of the energy grade line generally follows the slope of the water surface profile; therefore, the faster the velocity, the steeper the water surface and energy grade line profiles. Also, for normal depth the velocity is constant. Therefore, for depths greater than normal depth, the velocity is less than normal velocity (same flow, but more cross-sectional area). Similarly, for depths less than normal depth, the velocities are greater than normal velocity (same flow, but less cross-sectional area). Thus, for depths exceeding normal depth, the energy grade line slope is flatter (smaller) than the energy grade line slope for normal depth. For known depths less than normal depth, the energy grade line slope is steeper (larger) than that for normal depth.
Inspection of Figure 2.22a shows that the known depth exceeds both normal and criti-cal depths. Thus, for this depth, the velocity is less than normal velocity, causing the friction slope (sf) to be flatter (smaller) than so. For so greater than sf, the numerator in Equation 2.39 is positive. Because the known depth is greater than critical, the regime is subcritical; therefore, the Froude number is less than 1. For Fr < 1, the denominator
Figure 2.22 Profile classifications for mild slopes.
in Equation 2.39 is also positive, giving a positive dy/dx term. As indicated earlier in this section, a positive dy/dx means that depth is increasing in the downstream direc-tion. Thus, rule (1) above states that flow will eventually approach normal depth (upstream for subcritical flow) in a prismatic channel of constant invert slope, if the channel is long enough. With this knowledge, the resulting profile is sketched as shown in Figure 2.22a.
The profile shown is Figure 2.22a is classified as an M1 shape, the most common of all profile shapes. It occurs when a downstream obstruction forces an upstream depth increase. This “backup” caused by the obstruction, or backwater effect, gives the typi-cal backwater curve, or M1 shape. Narrowing of a channel, a reduced flow area caused by a bridge or culvert, or a flattening of the downstream channel slope all result in a backwater condition and an M1 shape. The first and third rules, along with the knowledge that dy/dx is positive, allow the resulting profile to be sketched in Figure 2.22a.
Equation 2.39 and the preceding classification rules allow examination of the situation in which the known depth is greater than critical but less than normal (Zone 2). In this case, the Froude number is again less than one and the denominator of Equation 2.39 is still positive. For depths in Zone 2, the velocity is greater than normal velocity, thus the friction slope (sf) is steeper (greater) than the channel slope (so). This situation gives a negative numerator for Equation 2.39.
Thus, the dy/dx term is negative for depths between critical and normal on a mild slope. A negative value of dy/dx indicates that the depth decreases in the downstream direction. With this knowledge, and applying the first through third rules for profile classification, the M2 shape can be sketched, as shown in Figure 2.22b. As the profile approaches critical depth, depth and velocity change abruptly with distance; thus, the M2 shape stops a short distance upstream of the location of critical depth. The depth is thus rapidly varied for a relatively short distance downstream of that point.
Figure 2.14 shows a water surface profile having an M2 shape. A mild profile that passes through critical depth is often referred to as a drawdown curve. An M2 profile ending at a waterfall, a spillway, a supercritical length of channel, or a sudden enlargement of the channel geometry results in a drawdown curve. Figure 2.14 and Figure 2.15a show the water surface profile of a drawdown curve.
The third mild shape exists for a known depth that is less than both normal and criti-cal. A depth less than critical means that the flow is supercritical and the Froude num-ber exceeds 1. For Fr > 1, the denominator of Equation 2.39 is negative. Supercritical velocities for this situation far exceed the velocity at normal depth, thus requiring an sf term much larger (steeper) than so. For Equation 2.39, a negative numerator and denominator result in dy/dx being positive and the depth increasing in the down-stream direction. The second and fourth rules in the preceding section are used to sketch an M3 profile, as shown in Figure 2.22c.
How can supercritical flow occur on a mild slope? Obviously, some manipulation of upstream conditions is required. An example is the presence of a sluice gate upstream, which can restrict depth of flow under the gate to less than critical depth, causing the flow to become supercritical for a short distance downstream of the gate.
However, supercritical flow cannot be sustained for long on a mild slope before a hydraulic jump returns the flow to subcritical. Figure 2.15b illustrates a profile possi-bly having an M3 shape.
Section 2.5 Profile Shapes 49
Example 2.10 Profile classification.
For the depth given in Example 2.2 and the critical and normal depths found in Exam-ple 2.7 and ExamExam-ple 2.8, classify the profile and suggest a cause for the resulting pro-file shape.
Solution
From Example 2.2, the known depth is 4 ft. Critical and normal depths computed from Example 2.7 and Example 2.8 are 2.78 and 3.57 ft, respectively. Since normal depth is greater than critical depth, the slope classification is mild. Also, since the known depth exceeds both normal and critical depths, the known depth is in Zone 1. Therefore, the profile is classified as an M1 shape, a typical backwater curve.
The M1 shape is caused by downstream conditions. Potential causes could be a nar-rowing of the downstream channel, a reduced channel slope downstream, an increase in the downstream Manning’s n, or an obstruction in the downstream channel, such as a low dam or weir, that causes an increase in upstream depth.
In open channel hydraulics, the most common profile shapes encountered are M1 and M2. However, there are other profiles: three classifications (S1, S2, and S3) for steep slopes (yn< yc), two classifications (H2 and H3) for horizontal slopes (so = 0), three classifications (C1, C2, and C3) for critical slopes (yn = yc), and two classifications (A2 and A3) for adverse slopes (so < 0). Supercritical flow is expected on steep slopes. A horizontal slope is most common for energy dissipation structures (for instance, a stilling basin) to control a hydraulic jump. Critical and adverse slopes are less com-mon than mild, steep, or horizontal slopes. For horizontal and adverse slopes, there is no Zone 1 because normal depth is undefined (it would be equal to infinity for hori-zontal slopes and negative for adverse slopes). Consequently, there are only two zones for horizontal and adverse classifications. Figure 2.23 shows the 13 possible profile classification shapes.
Example 2.11 Profile analysis for gradually varied flow.
For the channel system shown in the following figure, sketch and label the likely water surface profiles.
Solution
Gradually varied profile analysis involves determining in what zone the actual water surface elevations or depths will fall throughout each subreach of the channel system.
The water surface elevations at the boundary must be given, as they are in this exam-ple, or, if not specified, would possibly be assumed to be at normal depth at the boundary. In comparing normal and critical depth for a certain discharge on each of the three channel segments, it is seen that the upper segment is steep (normal depth less than critical depth), the middle segment is horizontal (no real normal depth for a zero slope), and the lower segment is mild (normal depth greater than critical depth).
Figure 2.23 Examples of flow profiles.
Section 2.5 Profile Shapes 51
Each of the three segments may be evaluated separately, with the profiles sketched during the analysis.
Steep slope: Flow passes under the sluice gate, with the water surface elevation at the lip of the gate less than both normal and critical depths. Because this corresponds to Zone 3 on a steep slope, an S3 curve is drawn from the lip of the gate and transitions into normal depth. Presumably, the depth will approach or reach normal depth on the steep slope. The profile would then overlay normal depth for the balance of the steep slope, until the channel slope changes to horizontal. Because supercritical flow cannot be sustained for a significant distance on a zero slope, a hydraulic jump will occur. The jump could initiate on either the steep or horizontal slope. There is insufficient data given in this example to determine on which slope the jump will commence, because a momentum balance between the depths just prior to and following the hydraulic jump is required (using Equation 2.22), along with knowing the discharge. Because this information is not furnished, the hydraulic jump could initiate on the steep slope, with an S1 curve following the jump, or the jump could begin on the horizontal slope, with an H3 shape before the hydraulic jump. For this example, the jump is assumed to begin on the steep slope with an S1 classification, as shown in the following figure. It is equally correct for this example to sketch the profile at normal depth to the intersec-tion of the steep and horizontal slope, and then show an H3 shape for a short distance prior to a hydraulic jump on the horizontal slope, as shown with the alternate shape on the figure.
Horizontal slope: With the hydraulic jump assumed to occur on the steep slope, flow on the horizontal slope must be subcritical, with depths greater than critical depth. This situation defines an H2 classification and shape, as shown on the following figure. The depth on the horizontal slope is expected to exceed the depth on the downstream mild slope (a flatter slope means lower velocity for the same discharge and therefore a greater depth) and this depth on the horizontal slope would decrease as the flow approaches the mild slope. The depth at the junction of the horizontal and mild slopes would be equal to or less than normal depth on the mild slope.
Mild slope: At the end of the mild slope, the known depth is less than critical depth.
Consequently, the profile along the mild reach must transition from nearly normal depth at the upstream end to less than critical depth at the downstream end. As the profile is in Zone 2, an M2 classification and profile are apparent. The profile becomes rapidly varied as it passes through critical depth a short distance upstream of the channel terminus at the dropoff, or free overfall. No classification is appropriate for the short reach of mild channel between critical depth and the downstream water sur-face, as this reach contains rapidly varied flow. The entire profile through the channel system is shown in the following figure. The profile will either reflect an S3, then H3, then H2 then M2 shape, or it could reflect an S3, S1, H2 and M3 shape, depending on where the hydraulic jump is initiated.
2.6 Computational Methods
The ultimate goal of most open channel hydraulics computations is the depth, or water surface elevation, at all desired locations along a length, or reach, of channel or river. Several different graphical and analytical techniques have been developed since the early 1900s, but only two are still applied on a regular basis: the direct step method and the standard step method.