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3.1 ENERGY PRINCIPLE
Under steady flow, mass and energy are conserved in openchannel flow.
On the other hand, under unsteady flow, mass and momentum are conserved.
Energy is expressed in FL units With reference to the channel of large slope of Fig. 31, the total head H at a section 0 containing point A on a streamline may be written as follows:
in which z_{A} = elevation of point A above a datum plane, d_{A} = depth of point A below the water surface, θ = slope angle of the channel bottom, and V_{A}^{2}/(2g) = velocity head of the flow in the streamline passing through point A.
Due to crosssectional nonuniformity, the velocity head is likely to vary along the flow depth and width. In practice, at a given cross section, the velocity head is based on the mean velocity V for that cross section, and the Coriolis coefficient (Section 2.2) is used to account for the nonuniformity in the velocity distribution. This leads to the total energy for the channel section:
For a channel of small slope, cos θ ≅ 1. Thus, the total energy reduces to:
The statement of conservation of energy between cross sections 1 and 2 leads to (Fig. 31):
in which h_{f} = energy loss. The line representing the total head is the energy grade line (EGL) or energy line. The slope of the line is the energy gradient, or S_{f}. The slope of the water surface is denoted by S_{w} and the slope of the channel bottom is S_{o}, where S_{o} = tan θ. Under uniform flow conditions, all three slopes are the same: S_{f} = S_{w} = S_{o}. For a channel of small slope, Eq. 34 reduces to:
When α_{1} = α_{2} ≅ 1 (a hydraulically wide channel), and h_{f} = 0, Eq. 35 reduces to the statement of conservation of energy, or Bernoulli equation:
3.2 SPECIFIC ENERGY
Equation 36 is applicable to a channel of small slope. In the limit, when the channel is horizontal, or else, as an approximation, when the channel is sufficiently short, Eq. 36 simplifies to a statement of conservation of specific energy. In effect, for z_{1} ≅ z_{2}, Eq. 36 reduces to:
Equation 37 states the constancy of specific energy in a hydraulically wide horizontal channel. The specific energy is:
where E is the energy per (unit of) weight, measured with respect to the channel bottom, defined in terms of the flow depth y and the velocity head V^{ 2}/(2g). Since V = Q /A, the specific energy in terms of discharge is:
Since Q is a constant under steady flow, and flow area A is always a unique function of flow depth y, the specific energy for a given Q is only a function of y. For a given cross section and discharge Q, a plot of flow depth y versus specific energy E leads to the specific energy curve. A typical curve is shown in Fig. 32. For a given Q, the specific energy curve is a type of hyperbolic function, featuring a minimum value of E and two limbs, a lower limb AC and an upper limb BC.
The minimum value of E (at point C) characterizes the critical state of flow, with critical depth. The lower limb AC, with smaller depths, describes supercritical flow. This limb is asymptotic to the horizontal axis. The upper limb BC, with larger depths, describes subcritical flow. This limb is asymptotic to a 45° line OD that originates at the origin (O). At any point P on the curve, the abscissa represents the specific energy E and the ordinate the flow depth y.
For a given specific energy, there are two points on the curve, one corresponding to low stage (small depth) and another corresponding to high stage (large depth). These depths are referred to as alternate depths. At point C, the alternate depths coalesce into one depth, termed the critical depth; at that point, the specific energy is a minimum. When the depth of flow is greater than the critical depth, the flow is termed subcritical. When the depth of flow is smaller than the critical depth, the flow is termed supercritical. For a discharge Q, the specific energy curve is curve AB in Fig. 32. For a smaller discharge, it is A'B'; for a larger discharge, it is A"B". Critical flow criterion From Fig. 32, critical depth (critical flow) corresponds to minimum specific energy. To prove this assertion mathematically, differentiate Eq. 39 with respect to y and equate to zero, to yield:
By definition, the differential change in flow area dA is equal to the top width T times the differential change in flow depth dy (Fig. 32):
Thus, Eq. 310 reduces to:
Since D_{c} = A_{c} /T_{c} , and V_{c} = Q /A_{c} , Eq. 312 reduces to:
The lefthand side of Eq. 313 is effectively the square of the Froude number (Section 13). Thus, the condition F^{ 2} = 1, or better yet, F = 1, properly describes the condition of critical flow, for which the specific energy is a minimum. Alternatively, Eq. 313 may be expressed as follows:
which states that, under critical flow, the velocity head is equal to onehalf of the hydraulic depth. Figure 33 shows a channel operating at near critical flow.
Equation 314 describes the critical flow condition for a channel of small slope that is hydraulically wide (α = 1). In general, for a channel of large slope and arbitrary crosssectional shape, the critical flow condition is:
in which D_{c} is the hydraulic depth corresponding to critical flow, measured normal to the channel bottom. Thus, the general critical flow condition is:
and the general definition for Froude number, applicable to any channel, regardless of slope and crosssectional shape, is:
3.3 LOCAL PHENOMENA
Changes from supercritical to subcritical flow, or from subcritical to supercritical flow, occur frequently in openchannel flow, depending on the prevailing channel slope and cross section. If the changes occur within a relatively short distance, they constitutes local phenomena. The hydraulic drop and the hydraulic jump are examples of local phenomena. The hydraulic drop The hydraulic drop is triggered by a sharp depression in the water surface, usually caused by an abrupt change in bottom elevation. The free overfall shown in Fig. 34 is an example of a hydraulic drop.
Flow in the proximity of a free overfall is usually rapidly varied; therefore,
the brink depth is somewhat smaller than the critical depth
computed by the theory of parallel flow. Figure 35 shows a schematic of the flow near the brink.
The actual drawdown curve is shown with a solid line,
while the theoretical watersurface curve, assuming parallel flow,
is shown with a dashed line.
For channels of small slope, the computed critical depth is about 1.4 times the brink depth;
i.e.,
The hydraulic jump The hydraulic jump is triggered by an abrupt rise in the water surface as the flow progresses downstream. In a hydraulic jump the flow changes from supercritical upstream to subcritical downstream, accompanied by an appreciable loss of energy. The amount of energy loss depends on the upstream and downstream flow conditions. The jump occurs frequently under one of the following situations:
The flow depth before the jump is called the initial depth y_{1}, while the flow after the jump is called the sequent depth y_{2}. The initial and sequent depths y_{1} and y_{2} are shown on the specific energy curve of Fig. 37. They should be distinguished from the alternate depths y_{1} and y_{2}', which are two possible depths for the same specific energy. The specific energy E_{1} at the initial depth y_{1} is greater than the specific energy E_{2} at the sequent depth y_{2}. The energy loss due to the hydraulic jump is the difference between specific energies for initial and sequent depths:
Also shown, to the right of Fig. 37, is the specific force curve (Section 3.5), where the sequent depths y_{1} and y_{2} have the same specific force: F_{1} = F_{2}.
3.4 MOMENTUM PRINCIPLE
Momentum M is equal to a force integrated over a period of time, or mass times velocity, Eq. 227, repeated here for convenience:
The momentum flux, or force F, of a flow with velocity V through a cross section of area A, Eq. 230, repeated here as Eq. 320, is:
Since Q = VA, the momentum flux, or force F, exerted by a flow of discharge Q and velocity V is:
According to Newton's second law of motion, the change ΔF in momentum flux through a control volume is equal to the resultant of all the external forces acting on the control volume. The external forces are: (1) body force, and (2) surface forces. The body force is the gravitational force, resolved along the direction of motion (the force labeled W sinθ in Fig. 38). In general, there is a nonzero channel bottom slope θ; otherwise, the channel bottom would be horizontal and the gravitational component along the direction of motion would vanish. The surface forces on the control volume are of three kinds: (1) on the bottom, (2) on the sides, and (3) on the top. The bottom surface force is due to friction, which is always acting in the direction opposite to the flow (the force labeled F_{f} in Fig. 38). There is no such thing as zero friction; under certain conditions, friction may be neglected, but it is never zero.
The side surface forces are two:
one upstream, the force labeled P_{1} in Fig. 38,
and another downstream, the force labeled P_{2}. These forces are
due to the water pressure, which is
either hydrostatic under parallel flow, or nonhydrostatic under convex or concave
curvilinear flow
( The top surface force is due to wind. Under the scales normally considered in openchannel flow, wind forces are small and are usually neglected. However, wind forces may not be negligible in cases on freesurface flow in reservoirs or flow in a wide open space such as the ocean. The statement of momentum (flux) conservation in a control volume is (Fig. 38):
Or, in terms of unit weight:
Following the usual convention of mechanics, the momentum flux difference is equal to the flux at the downstream section 2 minus the flux at the upstream section 1. The forces acting on the control volume are positive in the flow direction and negative against it. Equation 323 is known as the momentum flux balance equation or, for short, the momentum equation. For parallel flow in a rectangular channel of small slope and width b, the force P_{1} is:
Similarly, the force P_{2} is:
The weight W of the control volume (Fig. 38) is:
The weight of the control volume, resolved along the direction of motion (Fig. 38), is:
Generally, under typical gradually varied flow conditions, the friction force F_{f} along the channel bottom is about equal and opposite in sign to the gravitational force W sin θ (Eq. 327). Thus, the friction force may be expressed as follows:
in which h_{f}' = head loss due to friction. The discharge Q is:
Substituting Eqs. 324 to 329 into Eq. 323 and simplifying terms, the following equation is obtained:
Equation 330 differs from Eq. 35 in several important respects:
Thus, while Eqs. 35 and 330 look similar, they are not equivalent.
Equation 35 applies to steady gradually varied flow, while
Eq. 330 applies for unsteady gradually varied flow.
In other words, In practice, the momentum principle applies to problems where forces can be shown to play a significant role. Typically, problems of steady gradually varied flow use conservation of energy, while problems of unsteady gradually varied flow use conservation of momentum. The exception is the hydraulic jump, which is rapidly varied. Both energy and momentum principles are used in the solution of the hydraulic jump. 3.5 SPECIFIC FORCE
In horizontal channels, the gravitational force resolved along the direction of motion is effectively zero. As a convenient approximation, for nearly horizontal channels, the gravitational force may be considered small and be neglected on practical grounds. The frictional force develops along the channel bottom; the longer the channel, the greater the frictional force. Thus, for a short channel, the frictional force may be taken as sufficiently small and neglected on practical grounds. Note that the frictional force is never zero; its neglect is only justified as an approximation, when compared with the other forces that are present in openchannel flow. The neglect or elimination of the gravitational and frictional forces in the momentum flux balance reduces it to:
Assuming β_{1} = β_{2} = 1, Eq. 331 reduces to:
Since V_{1} = Q / A_{1}, and V_{2} = Q / A_{2}:
The pressure force P acting on a crosssection of area A and distance z̄ from the centroid of the area to the water surface [Fig. 39 (b)] is:
Thus:
Substituting Eqs. 335 and 336 into Eq. 333, and dividing by unit weight γ :
In general, the specific force is defined as follows:
Equation 337 states that specific force is conserved in openchannel flow in a short horizontal channel, i.e., F_{1} = F_{2}. Note that specific force is a force per unit of γ, the weight per unit of volume; therefore, specific force has the units of volume [L^{3}]. The specific force curve shown in Fig. 39 (c) is obtained by plotting F in the abscissas and y in the ordinates. This curve is similar to the specific energy curve [Fig. 39 (a)], but with significant differences. The limb AC approaches the horizontal axis asymptotically toward the right. The limb BC rises upward and extends without limit to the right. For a given value of specific force F_{1}, the curve has two possible depths: y_{1} and y_{2}. These are the initial and sequent depths of a hydraulic jump, respectively. At point C [Fig. 39 (c)], the sequent depths coalesce into one depth, termed the critical depth; at that point, the specific force is a minimum. Note that this is the same critical depth obtained by specific energy considerations; see Fig. 39 (a). Critical flow criterion for specific force As in the case of specific energy, to prove that minimum specific force corresponds to the critical flow criterion, differentiate Eq. 338 with respect to y to yield:
For a change in depth dy, the corresponding change d(z̄A) in the static moment of the water area about the free surface is equal to:
As usual in differential calculus, the secondorder term in Eq. 340 is neglected, to yield:
Therefore, Eq. 339 reduces to:
Simplifying Eq. 342:
Since dA /dy = T, Q /A = V, and A /T = D, Eq. 343 reduces to:
which is the square of the Froude number:
Equation 345 is the criterion for critical flow, applicable to both specific energy and specific force (specific momentum) curves. Note that the sequent depth y_{2} is always smaller than the high alternate depth y_{2}' (Fig. 37). Furthermore, the energy E_{2} is always smaller than the energy E_{1}, while the specific force F_{2} remains equal to the specific force F_{1} [Fig. 37 and Fig. 39 (c)]. In order to maintain a constant specific force, the flow depth must increase from y_{1} to y_{2} at the cost of losing a certain amount of energy. The energy loss is equal to: ΔE = E_{1}  E_{2}. This situation occurs in the hydraulic jump, where the specific forces before and after the jump are equal, but with a consequent loss of energy (Fig. 310).
Specific force per unit of channel width In a hydraulically wide channel, the specific force per unit of channel width b is:
where q = Q /b. In terms of mean velocity V = q/y, the specific force per unit of channel width b is:
Specific force in units of force In units of force, the specific force is:
In units of force, the specific force per unit of channel width b is:
In units of force, and in terms of mean velocity V, the specific force per unit of channel width b is:
3.6 MOMENTUM PRINCIPLE IN NONPRISMATIC CHANNELS
When the external forces are negligible or known in advance, the momentum principle may be applied to nonprismatic channels. The momentum principle is particularly applicable to the hydraulic jump, where the high internal losses that occur cannot be fully evaluated using the energy principle alone. The following example illustrates how the momentum principle may be applied to the design of a channel transition (Chow, 1959). In this case, the upstream flow is supercritical and the downstream flow subcritical, opening up the possibility that a hydraulic jump may form somewhere within the transition. The latter may be avoided by modifying the shape of the channel cross section in the transition; for example, raising the bed elevation as shown in part (b) below. Example 32: Design of a channel transition, from supercritical to subcritical flow
Assume that a rectangular channel 8ft wide carries 100 cfs at a depth of 0.5 ft.
The channel is connected through a 50ft long straightline transition
to a 10ft wide channel flowing at a depth of 4 ft.
It is desired to design the transition, including the calculation of the flow profile. For simplicity, neglect friction and other minor losses
Solution.
Use Eq. 39 to calculate
the total energy at the upstream end of the transition: The difference between these two values is the energy loss, or energy drop: ΔE = E_{u}  E_{d} = 6.117 ft. This energy must be dissipated through the transition by some means. The following two cases are considered:
A. Abrupt energy dissipation.
The upstream and downstream Froude numbers (Eq. 19) are
F_{u} = 6.23 and
Table 31 shows the computations of the channel transition, with hydraulic jump. Note the following:
Table 31 shows that the specific force for the low stages (Col. 5) varies little
along the transition, having a mean value of about 78.8 ft^{3}.
On the other hand, the specific force for the high stages By varying the shape (width) of the cross sections in the transition, the location of the intersection of the specific force lines may be changed, i.e., the position of the jump. Changing the flow depth in the downstream channel will also change the position of the jump. Generally, an increase in the downstream depth will move the jump upstream; conversely, a decrease in the downstream depth will move the jump downstream. B. Gradual energy dissipation. The hydraulic jump may be avoided by providing a way to gradually dissipate the energy. This can be accomplished by varying the width or raising the bottom of the transition in such a way that the actual energy grade line (EGL) is a straight line joining the total heads at the two end sections [Fig. 312 (b)]. In an actual design, first assume the flow profile and then proportion the dimensions of the raised bottom following applicable hydraulic laws. The following steps are recommended:
Table 32 shows the computations of the channel transition, without hydraulic jump. Note the following:
Figure 312 (b) shows the shape of the hump necessary to provide a gradual drop in the energy gradeline (EGL), thus avoiding the hydraulic jump. Note that the specific force lines intersect at the critical flow section, located to the right of Section 9 (Table 32).
QUESTIONS
PROBLEMS
REFERENCES
Chow, V. T. 1959. Openchannel Hydraulics. McGraw Hill, New York. Henderson, F. M. 1966. Open channel flow. Macmillan, New York.

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