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Full Equations Utilities (FEQUTL) Model for the Approximation of Hydraulic Characteristics of Open Channels and Control Structures During Unsteady Flow

# 4.3 Embankments and Weirs

Flow over embankment-shaped weirs as well as weirs of other shapes is computed in the FEQUTL command EMBANKQ (section 5.6). The computations follow the principles developed and outlined in Hulsing (1967, p. 26-27). The principles described in Hulsing are based on research reported by Kindsvater (1964). The procedure applied in FEQUTL divides embankment-shaped weirs into two classes: weirs with paved surfaces and weirs with gravel surfaces. The variation of the weir coefficient with flow conditions in each of these classes is further subdivided into high-head and low-head cases. The boundary between high-head and low-head cases was set at 0.15 of the crest width of the embankment. The high-head weir coefficients vary with the ratio of piezometric head to crest width, whereas the low-head weir coefficients vary with the piezometric head. In both cases, the piezometric head is used in the weir equation. The work by Kindsvater (1964) indicated that the effect of embankment height was rendered negligible if the piezometric head was used in the weir equation. Submergence-reduction factors for both paved- and gravel-surface weirs are applied in FEQUTL. These factors are applied as functions of the ratio of the piezometric tail-water head to the piezometric head on the crest.

Six tables, which may be electronically retrieved with FEQUTL as described in section 1.1, are included in a file named EMBWEIR.TAB to represent embankment-shaped weirs. These tables contain the weir coefficients for high-head and low-head flow and the submergence-reduction factors for weirs with paved surfaces and with gravel surfaces. The values included in these tables were derived from the information in Hulsing (1967) and Kindsvater (1964). The physical dimensions of the embankment-shaped weir are input by the user, and the appropriate coefficient tables are accessed to compute 2-D tables of types 6 and 13 for a user-specified range of upstream heads and partial free drops.

The six tables described are listed as default table numbers in the input description for the EMBANKQ command (section 5.6). If 2-D tables for other types of weirs (sharp crested, broad crested, or ogee shaped) are to be computed, the user must input tables containing the weir coefficients for high-head and low-head flow and the submergence-reduction factor using the FTABIN command (section 5.13). These tables are referenced in the EMBANKQ command (section 5.6).

The weir crest is assumed to be level in the typical weir equation. This is rarely true for the profile of a roadway crest as illustrated in figure 18 (in section 5.6). Integration of the unit-width weir equation is applied in EMBANKQ to account for the possibility that the weir crest is not horizontal. Horton (1907, p. 57), in his summary of weir formulas and coefficients, recommended this procedure. Brater and King (1976, p. 5-16) showed that the equation resulting from the integration of the unit-width weir equation gives a close approximation to the flow for a triangular, sharp-crested weir (V-notch weir). A single equation represents a wide range of experiments and angles of notch within 5 percent. Therefore, integration along an embankment crest much less sharply inclined than a V-notch weir should produce useful results.

The user must specify the elevation of the crest, the width of the crest in the direction of flow, the elevation of the approach surface to the weir, and the nature of the crest surface (PAVED or GRAVEL) for each of a series of locations along the weir crest perpendicular to the flow as shown in figure 18 and described in detail in sections 5.5 and 5.6. These values are then a function of the offset distance, s , along the weir crest measured from some convenient reference point. The elevation of the approach surface is needed to define the height of the weir crest to compute the velocity head of approach.

The unit-width discharge over a weir, q w, is computed in the EMBANKQ command as

(97)

for the high-head range, and

(98)

for the low-head range,

where

 W(s) is the crest width represented at the local offset s; fCL is the function that gives the weir discharge coefficient for low-head flow; fCH() is the function that gives the weir discharge coefficient for high-head flow; H is the piezometric head on the weir; hw is the piezometric head on the weir; ht is the downstream head causing flow submergence at a weir; and fs() is the submergence function.

The total flow at a given upstream water-surface elevation, Q w ( z w ), is

(99)

,

where sL and sR give the limits of integration that depend on the water-surface elevation, zw. Equation 99 does not include the case of two or more separate paths for flow over the roadway, but the generalization to that case is straightforward. Equation 99 can be applied to each path and the results summed to obtain the total flow. Hulsing (1967) indicated that the width to use for defining flow over the roadway is difficult to define and recommended that five-sixths of the maximum piezometric head be used to define the level that establishes the limits of flow. This approximation is not applied in the EMBANKQ command. Instead, the water-surface elevation is projected to the weir crest to define the limits of integration. This approach produces good results for V-notch weirs up to a central angle of 120 degrees. No data are available to check the results for smaller slopes. It is clear that the wetted width of the roadway will be narrower than the width obtained by projecting the upstream water-surface elevation to the crest. However, it is not clear what the limits of integration should be. Careful measurements on large-scale inclined weirs are needed to draw conclusions on this matter. The typical slopes along roads are small. Although the difference in integration widths obtained from the two assumptions may be substantial, the differences in flow may not be substaintial. The piezometric head on the weir crest in the region where the integration limits differ will be relatively small. Thus, the total contribution of flow from these regions of the crest to total flow will be small.

The integral in equation 99 is approximated on the basis of Simpson's Rule. Simpson's Rule is applied to each line segment defined by the successive points along the weir crest. The weir crest is assumed to vary linearly between adjacent points. Thus, for each line segment, the flow per unit width is computed at three points: at each end and at the midpoint. The velocity head of approach is computed for each point and results from the unit width of flow over the weir at that point alone. This deviates from the typical practice for flow over weirs wherein a global velocity head is used. This means that the velocity head is not a constant along the weir crest and may vary from point to point. This gives a more realistic representation of the flow field approaching the weir when flow over the flood plain is present than the constant-velocity-head approximation. If a crest segment is very long with a large change in elevation between the two ends, it may be necessary to subdivide this segment for computational accuracy in Simpson's Rule.

The free flow and a range of submerged flows are computed for a series of upstream maximum piezometric heads defined by the user. The maximum piezometric head is computed from the water-surface elevation to the point of minimum crest elevation. Given these values and the six function tables defining the weir characteristics, the following steps are applied to compute the values.

1. The upstream water-surface elevation is computed for the current maximum piezometric head.
2. The line segments that compose the weir crest are scanned.
3. For each line segment completely submerged, the unit-width flow rate is computed as defined above for the particular head range and surface class for each end and for the midpoint of the segment. If culvert computations are involved, the various flux terms also are computed applying the same rule for integration.
4. For each partly submerged line segment, the intersection point with the water surface is located, and the flux terms defined in step 3 are computed for the truncated line segment.
5. Simpson's Rule is applied to approximate the integral in equation 99 over the line segment.
6. The line-segment values are summed to obtain the free-flow values at the given upstream water-surface elevation.
7. The drop to the downstream water-surface elevation at the free-flow limit is computed if the submergence functions have been given. If not, the free-flow result is placed in a function table of type 2, and the next maximum upstream piezometric head given by the user is evaluated.
8. For each of a series of partial free drops defined by user-input values, the submerged flows over the weir are computed in a manner analogous to that for free flow.
9. The results are placed in a 2-D table of type 13, and the next maximum upstream piezometric head given by the user is evaluated.

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