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4.1 CRITICAL FLOW
In openchannel flow, two characteristic thresholds describe the state of flow (Section 1.3):
The ratio V/F embodies two important properties:
The dischargeflow area rating, Eq. 14, is repeated here for convenience:
in which α = coefficient of the rating, and β = exponent. The latter is defined as follows (Section 1.3):
Values of Froude number F corresponding to values of Vedernikov number V = 1, i.e., values of F_{ns}, are listed in Table 11.
Critical flow occurs under the following conditions (Fig. 41):
The condition of critical flow represents a threshold between subcritical flow, for which F < 1, and supercritical flow, for which F > 1. Dynamic waves in openchannel flow have two components: (1) primary, and (2) secondary (Ponce and Simons, 1977). Primary waves travel with absolute velocity:
Secondary wave travels with absolute velocity:
While primary waves always travel downstream, secondary waves may travel upstream or downstream, depending on the flow conditions. In subcritical flow, w > u, and secondary waves are able to travel upstream. In supercritical flow, w < u, and secondary waves cannot travel upstream, instead traveling downstream. In practice, this means that subcritical flow is controlled from downstream, because surface perturbations are able to travel upstream. Conversely, supercritical flow cannot be controlled from downstream, because surface perturbations are not able to travel upstream. Instead, supercritical flow can be controlled only from upstream. Critical flow may occur in one of two distinct modes:
The DarcyWeisbach equation for open channel flow The DarcyWeisbach equation for open channel flow, Eq. 132, is repeated here for convenience (Section 1.4):
in which the Froude number is:
For a hydraulically wide channel, for which D ≅ R, Eq. 45 reduces to:
For application to openchannel flow, a modified DarcyWeisbach friction factor f, equal to 1/8 of the usual DarcyWeisbach friction factor f is applicable. The modified DarcyWeisbach equation for openchannel flow is:
Physical meaning of the critical slope The critical slope is that for which F = 1. In Eq. 45, for F = 1:
in which S_{c} = critical slope. In Eq. 47, for F = 1:
Furthermore, in Eq. 48, for F = 1:
Equations 49 to 411 confirm that the friction factor and the critical slope are indeed closely related.
For a channel of arbitrary cross section, In general:
in which S_{c} is appropriately defined by either Eqs. 49, 410, or 411. Equation 412 constitutes a type of DarcyWeisbach equation applicable to openchannel flow. In uniform flow, this equation provides an enhanced physical meaning to the concept of critical slope. In gradually varied flow, Eq. 412 enables an increased understanding of the asymptotic limits to the water surface profiles (Section 7.3). The following conclusions are drawn from Eq. 412:
As Eq. 412 indicates, critical flow occurs when the discharge is such that the critical slope (friction slope) equals the bottom slope. This is possible in a lined channel of fixed geometry, where as discharge increases, the friction slope eventually decreases to match the bottom slope. Thus, in an artificial prismatic channel, it is possible to achieve critical, and even supercritical, flow depths. Supercritical flows are rare, but not unheard of. For example, under Chezy friction in hydraulically wide channels, roll waves form under Vedernikov number V = 1, which corresponds to Froude number F = 2 (Fig. 16 and Table 11). The situation is quite different in a natural channel, where the flow is able to freely interact with the boundary, increasing the "actual" or effective friction slope. In practice, the friction slope is not able to decrease to match the bottom slope, with the actual flow remaining subcritical, at least through most of its domain. Thus, as argued by Jarrett (1982), it is extremely rare to have critical or supercritical flow in a natural channel (Fig. 43).
4.2 COMPUTATION OF CRITICAL FLOW
From Eq. 46, the square of the Froude number is:
In Eq. 413, replacing V = Q /A and D = A /T leads to:
At critical flow, F = 1, and Eq. 414 is conveniently expressed in the following form:
With reference to Fig 44, the top width T is:
The flow area A is:
Replacing Eqs. 416 and 417 into Eq. 415:
Given g = gravitational acceleration, and input data consisting of discharge Q, bottom width b,
and side slope z [z:H to 1:V, Fig. 43], Eq. 418 is solved for the critical depth
Equation 418 may be expressed as follows:
Equation 423 is the general formula for critical flow, applicable to trapezoidal channels.
For a rectangular channel: In Eq. 423, making the change of variable x = y_{c} for simplicity:
The solution of the cubic Eq. 424 may be accomplished by a trialanderror procedure. A timetested algorithm for the approximation to the first root of the equation is described below.
Critical depth in a hydraulically wide channel
For a hydraulically wide channel, or else, a rectangular channel: Q = qb, and z = 1. Replacing these values is Eq. 418 leads to an expression for
critical depth y_{c} in terms only of unitwidth discharge
4.3 CRITICAL FLOW CONTROL
In Eq. 423, the critical flow depth is a function only of Q, b, and z.
Thus, the critical flow depth, and for that matter, critical flow, is
independent of the bottom slope and channel friction.
Note that
in The unique dischargeflow area (and unique dischargestage rating), i.e., only one flow depth for every discharge, and vice versa, is referred to as control. Control can be either of types: (1) section control (transversal control), acting at a cross section, or (2) channel control (longitudinal control), acting along the channel. Figures 45 to 47 shows three cases of critical flow control in a prismatic channel, under the following slopes: (1) subcritical, (2) critical, and (3) supercritical. The subcritical flow condition depicted on Fig. 45 shows the existence of critical section control only at the downstream dam crest. Thus, the flow is controlled at the downstream end.
The critical flow condition of Fig. 46 shows critical control in two places: (1) section control at the downstream dam crest, and (2) channel control along the upstream critical slope channel. As shown, a backwater profile, which is a nearly horizontal pool level, connects the two flow control sections. The flow is controlled at the downstream end.
The supercritical flow condition of Fig. 47 shows critical control in two places: (1) section control at the downstream dam crest, and (2) section control at the very upstream section of the supercritical channel. The flow is controlled at both downstream and upstream ends, with a hydraulic jump occurring somewhere in the middle.
The concept of uniqueness of the rating qualifies critical flow as a (section or channel) control. This provides an expedient way of determining discharge from stage, or stage from discharge, if one or the other is known. This property of critical flow is useful in flow measurements. In practice, critical control for flow measurement is accomplished in two ways: (1) weir flow (section control), and (2) critical flow flume (channel control). Weir flow is described in Section 4.4. A typical example of a critical flow flume is the Parshall flume (Fig. 48). Under freeflowing conditions (low tailwater depth), only one gage measurement is required to determine the discharge. However, under submerged conditions (with high tailwater depth), two gage measurements are required (note the two measuring tubes in Fig. 48).
In a broadcrested weir, critical flow occurs in the vicinity of the crest. The discharge per unit of width is:
By definition, the critical depth is 2/3 of the total head H measured above the weir crest:
Replacing Eq. 429 in Eq. 428:
in which C is a discharge coefficient defined as follows:
In SI units, C = 1.704; in U.S. Customary units, C = 3.087. For various reasons, an actual design value C_{d} may differ from the theoretical value C. Experience has shown that the approximate range is: 0.8 ≤ C_{d} /C ≤ 1.3. Figure 49 shows a broadcrested weir for which C_{d} = 1.45 (SI units). In practice, H is taken as the elevation of the water surface above the weir crest. This assumes that the approach velocity V_{a} at a section sufficiently upstream from the weir is zero: V_{a} = 0.
4.4 SHARPCRESTED WEIRS
Sharpcrested weirs are used to force section control in an open channel, for purposes of flow measurement. In practice, sharpcrested weirs have been built using: (1) triangular, (2) trapezoidal, or (3) rectangular sections.
Triangular weirs can be of two types: (a) fully contracted, or (b) partially contracted. Contraction refers to the size of the weir flow area as compared to the size of the approach channel flow area. Depending on size and design, a weir may be subject to both vertical and horizontal contraction. For a weir to be fully contracted, the ends of the weir should be sufficiently far from the sides and bottom of the approach channel (Fig. 410). Full contraction increases the measurement accuracy by providing a more precise channel control (a unique dischargestage relation) in the immediate vinicity of the weir.
Fully contracted Vnotch weir. The Vnotch weir is a commonly used type of triangular sharpcrested weir.
For Vnotch weirs, full contraction is produced when the distance
b from each side of the weir notch to each side of
the weir pool is greater than 2H. For a 90° Vnotch weir,
the flow width at head level is equal to 2H. Therefore,
the weir may be considered to be fully contracted when
the ratio B/H > 6, i.e., for A weir not satisfying the above criterion is partially contracted, i.e., the approach channel width B is too narrow relative to the head H. In USBR practice, the practical criterion for a partially contracted Vnotch weir is: H/B ≤ 0.4.
The fully contracted Vnotch weir formula, in U.S. Customary units (Q in cfs, H in ft), is (USBR Water Manual):
In Eq. 432, the discharge Q is a function of hydraulic head H and angle θ. The discharge coefficient C_{e} and head correction coefficient k are a funtion of θ. The width of the approach channel B is used to check to see if the weir is fully contracted: H/B ≤ 0.2. The formula (polinomial fit) for C_{e}, with θ in degrees, is:
The formula (polinomial fit) for k, with θ in degrees, is (LMNO Engineering):
The fully contracted Vnotch weir is restricted to the following conditions:
Partially contracted Vnotch weir. For the partially contracted 90° Vnotch weir, Eq. 432 reduces to:
The discharge coefficient C_{e} is a function of the ratios H/P and P/B, as shown in Fig. 411.
The formula for the partially contracted 90° Vnotch weir is subject to the following restrictions:
A standard Cipolletti weir is trapezoidal in shape (Fig. 412). The crest and sides of the weir plate are placed far enough from the bottom and sides of the approach channel to produce full contraction. The sides incline outwardly at a slope of 1 horizontal to 4 vertical. The computation procedure follows Section 12 of Chapter 7 of the USBR Water Measurement Manual.
The formula for the Cipolletti weir, in U.S. Customary units, is:
in which L = length of the weir crest, in ft, and H = head on the weir crest, in ft.
The accuracy of measurements obtained by Eq. 436
is considerably less than that obtainable with Vnotch weirs. The accuracy of the discharge coefficient is The Cipolletti weir is subject to the following restrictions:
The head H is measured at a distance of at least 4H upstream from the crest.
A rectangular weir has a rectangular shape, as shown in Fig. 413. To produce full contraction, the crest and sides of the weir plate are placed sufficiently far enough from the bottom and sides of the approach channel. The computation procedure follows Section 6 of Chapter 7 of the USBR Water Measurement Manual.
The KindsvaterCarter formula for a rectangular weir, in U.S. Customary units, is:
in which C_{e} = effective coefficient of discharge; L = length of the weir crest, in ft, k_{b} = a correction factor to obtain effective weir length, in ft; H = head measured above the weir crest, in ft; and Q = discharge, in cfs. The value B is the average width of the approach channel. The correction factor k_{b} is a function of the ratio L/B, as shown in Fig. 414.
The effective coefficient of discharge C_{e} includes effects of relative depth and relative width of the approach channel. It is a function of H/P and L/B, as shown in Fig. 415.
Given H, L, B and P, and the ratios H/P and L/B, the computation proceeds with the following steps:
The rectangular weir equation (Eq. 437) is subject to the following restrictions:
Standard contracted rectangular weir.
The standard contracted rectangular weir is shown in
The formula for the standard contracted rectangular weir is the Francis equation. In U.S. Customary units, this equation is:
in which Q = discharge, in cfs, L = length of the weir crest, in ft, and H = head on the weir crest, in ft.
The accuracy of measurements obtained by Eq. 438
is considerably less than that obtainable with Vnotch weirs. The accuracy of the discharge coefficient is Equation 438 has a constant discharge coefficient (3.33), which facilitates computations. However, the coefficient does not remain constant for a ratio of headtocrest H/L > 1/3, and the actual discharge exceeds that given by the equation. Francis' experiments were made on comparatively long weirs, most of them with crest length L = 10 ft and heads in the range 0.4 ft ≤ H ≤ 1.6 ft. Thus, the equation applies particularly to such conditions. USBR experiments on 6in, 1ft, and 2ft weirs has shown that the Francis equation also applies fairly well to shorter crest lengths L, provided H/L ≤ 1/3. Equation 438 is subject to the following restrictions:
Standard suppressed rectangular weir. A standard suppressed rectangular weir has a horizontal crest that crosses the full channel width (Fig. 418). The elevation of the crest is high enough to assure full bottom crest contraction of the nappe. The vertical sidewalls of the approach channel continue downstream past the weir plate, to prevent side contraction or lateral expansion of the overflow jet. The computation procedure follows Section 10 of Chapter 7 of the USBR Water Measurement Manual. Special care must be taken to secure proper aeration beneath the overflowing sheet at the crest. Aeration is usually accomplished by placing vents on both sides of the weir box under the nappe. Other conditions for accuracy of measurement for this type of weir are generally the same as for those of the contracted rectangular weir, except for the absence of side contraction. However, the crest height P should be at least 3H.
The formula for the standard suppressed rectangular weir is the Francis equation. In U.S. Customary units, this equation is:
in which Q = discharge, in cfs, L = length of the weir crest, in ft, and H = head on the weir crest, in ft.
The accuracy of the discharge coefficient is Equation 439 is subject to the following restrictions:
The head H is measured at an upstream distance of at least 4H from the weir. The sidewalls must extend a distance of at least 0.3H downstream from the crest. The overflow jet must be adequately ventilated to the atmosphere. A tabular comparison of sharpcrested weirs for discharge measurement in openchannel flow is given in Ponce (2013).
QUESTIONS
PROBLEMS
REFERENCES
Chow, V. T. 1959. Openchannel Hydraulics. McGraw Hill, New York. Jarrett, R. D., 1984. Hydraulics of HighGradient Streams. ASCE Journal of Hydraulic Engineering, Vol. 110, No. 11, November, 15191539. Ponce, V, M., and D. B. Simons. 1977. Shallow wave propagation in openchannel flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 14611476.

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