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Enhancements and Modifications to the Full Equations Utilities (FEQUTL) Model, March 1995 to August 1999.
Note: This document is separate from the U.S. Geological Survey report by Franz and Melching (1997). This description of enhancements and modifications to the Full Equations Utilities Model has not been approved by the Director of the U.S. Geological Survey.

New subsection for section 4. Approximation of Hydraulic Characteristics of Control Structures, Franz and Melching (1997b), p. 70

Section 4.7a Flow in Orifices

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Orifices are sometimes used to slowly release water from a detention reservoir to lower the water after a flood event to prepare for the next flood event. The orifice is sized so that the flow will not exceed a limit value for the design condition and to also permit some minimum dry-weather flow to pass through the structure. Orifice plates are also sometimes used as a retrofit on culverts whose capacity proves to be too large for some change in conditions.
We define four flow conditions for an orifice: free orifice flow (FO) in which the opening is filled with water and the tailwater level is below the orifice invert; submerged orifice flow (SO) in which the opening is filled with water but the tailwater level is above the orifice invert; free weir flow (FW) in which the orifice opening is only partially filled with water and the tailwater level is below the invert; and submerged weir flow (SW) in which the orifice opening is only partially filled with water but the tailwater is at or above the invert of the orifice.
We do not assume any particular shape to the orifice and the equations developed below apply to a wide range of shapes. Figure 4-11 shows a sketch of an orifice with some of the defining dimensions. The greatest horizontal width of the orifice, called its horizontal diameter, is denoted by W and its greatest vertical height, called its vertical diameter, is denoted by D. The piezometric head upstream of the orifice is Symbol and the piezometric head downstream will be denoted by Symbol. Following Rouse(1946) and Bos(1989) we estimate the incremental flow, dQ, through the incremental strip of height dz and width, T(z) shown in Figure 4-11 as

Equation

where c = an incremental discharge coefficient that will vary with z, and Symbol as well as the shape of the orifice and the nature of the edges of the opening; and g = acceleration due to gravity; and z =distance of the strip from the orifice invert. We integrate this incremental flow value from z = 0 to z = D, assuming that the discharge coefficient will allow this to be done with sufficient accuracy. We then get

Equation

where Qf = the free orifice flow. We do not know the value of Symbol with any precision and the function would prove difficult to measure in any case, so we make a further approximation, justified in part by the first mean-value theorem for definite integrals, and use an average discharge coefficient, Symbol, outside the integral. With this change and with the transfer of the constant Symbol outside the integral, we get

Equation

as the defining equation for free-orifice flow. Note that the integrand in Eq. 4-31 is a function of the shape of the orifice and the current upstream head value. This equation applies to all shapes of orificesassuming a suitable value of discharge coefficient can be found for the shapes.

If the upstream head falls below the top of the orifice opening, we get the defining equation for flow over a sharp-crested weir. In this case, the upper limit of the integral becomes the upstream head and not the vertical diameter. If we assume that we can find a suitable discharge coefficient we can generalize Eq. 4-31 to include free-weir flow also by changing the upper limit to include the two options. That is, the upper limit will be the smaller of either the upstream head, Symbol, or the vertical diameter of the opening, D. We then get

Equation

as the defining equation for both free orifice and free weir flow.

Figure

Figure

Figure 4-12 shows a sketch of our prototypical orifice with a tailwater level added. There is no data in the literature on orifice flow with the orifice partially under tailwater. However, such a condition can exist, if only for a short time, during an unsteady flow simulation and we must be able to describe it. The approach used here is simple: we assume that the flow in the orifice below the tailwater level is affected by the tailwater but that the flow in the orifice above the tailwater level is not affected by the tailwater. Thus the submerged flow is represented in two parts: one part is computed as if it were free and the other as if it were completely submerged.
The free part is defined by changing the lower limit on the integral in Eq. 4-32 to reflect that the flow may be affected by tailwater. When the tailwater reaches the top of the orifice the free-flow integral must vanish. Also the flow should be free of tailwater if the head downstream is below the invert of the orifice. Thus the lower limit becomes Equation to describe all cases of interest. We then get the free part of the submerged flow, Symbol, for both orifice and weir flow as

Equation

Eq. 4-33 also defines free flow if Symbol.
The net head on the submerged part of the flow in the orifice is just the difference between the upstream head and the downstream head, that is, Equation. This head difference is a constant. If the orifice is not submerged at all, we want the integral to be zero, and this gives an upper limit of zero. If the orifice is partially submerged the upper limit should be Symbol. Finally if the orifice is totally submerged, the upper limit should be D. This then gives

Equation

as the defining equation for the submerged part of submerged flow for both weir and orifice flow if such flow exists. If Symbol then Symbol". Note that the integral gives the area of the orifice that is below the tailwater level.
With this careful definition of integration limits, the flow is given by Equation for all four flow conditions. Thus Eq. 4-33 and 4-34 are the defining equations for all flow conditions for orifices in FEQUTL. Note that these equations apply to all shapes for which it is meaningful to define a discharge coefficient.
We could specialize to particular shapes and develop equations for each shape. However, we choose not to do so. Instead, we will evaluate the defining integrals numerically so that all shapes of interest are computed in the same way and with the same equations. This will be done even in those cases were the integrals for a particular shape can be computed in closed analytical form. The computational cost of doing so is nil in comparison to the possibility of error in transcribing the sometimes complex expressions for the integrals. By using the basic definitions for the flows we also include the small effect of head variation across the orifice opening when the upstream heads are small.
It proves convenient to express the integrals in dimensionless form. To this end we will normalize all relevant values with respect to the vertical diameter, D, of the orifice opening. That is all linear dimensions are expressed relative to D. We place a tilde over the symbol to indicate that it is relative to the vertical diameter. For example, the width function, T(z) in dimensionless form becomes Equation.
The defining equation for the free part of the flow becomes

Equation

and the equation for the submerged part of the flow becomes

Equation

For some shapes, such as a circle, the integrand will have an unbounded slope at one or both limits of integration. These unbounded derivatives make numerical integration difficult. However, Davis and Rabinowitz(1984, p. 441) report on a transformation designed by W. Kahan that alleviates this problem for integrals like those here. This approach gives the result that

Equation

where Equation, for any integral with finite limits. FEQUTL then uses a 10-point Gaussian rule to compute the value of the transformed integral. Testing shows that the errors are small with a relative error on the order of Equation or less.
This approach is used for all flows through the orifice. Submerged weir flow is one of these flow conditions. Values for submergence effects on weir flow are available even if partial submergence effects on orifices are not. Therefore a comparison was made between the weir submergence predicted by this approach with the much used Villemonte formula as reported in Brater and King (1967). This comparison was done for both triangular and rectangular weirs. The submergence relationship implied by the above integrals agreed within 5 percent of the Villemonte formula for all submergence ratios less than 0.9 for the rectangular weir and within 3 percent for the triangular weir. Thus this approach to representing both weir and orifice flow through an orifice is of suitable accuracy. We retain it because the above equations will provide a smooth transition between weir and orifice flow if the coefficient for both is the same or nearly so.
For a wide range of sharp or square edged orifices and weirs the dimensionless discharge coefficient as defined above has a value close to 0.6. This is true for rectangular and circular shapes in both weir and orifice flow. Weir flow for triangular weirs also falls close to 0.6. This near constancy is not only convenient but also serves to lend support to the assumptions used in developing the equations above. Since the overall discharge coefficient is nearly constant we can infer that the incremental coefficient is also. Thus moving the unknown incremental coefficient from under the integral in Eq. 4-30 was valid. If the orifice is larger than about 0.075 meter(3 inches), then the values given in Bos(1989) for weir and orifice flow, both free and submerged, deviate from 0.6 no more than 1.2 percent for all heads greater than 0.25 of the vertical diameter of the opening. The deviation increases to about 8 per cent at a relative head of 0.1, the lower limit recommended for accurate measurement. Thus a constant value of 0.6 can be used as a reasonable value for all but the smaller circular orifices. The coefficient for a V-notch weir is about 0.58 and about 0.59 for a rectangular weir with negligible velocity of approach. No values for orifice flow through triangular openings were found. However, the coefficient for a submerged rectangular orifice was given as 0.61 if it is fully contracted. Again in these cases a fixed value of 0.6 is in error by about 3 per cent. These values assume that the edge is sharp and distinct. Rounding of the edges of the opening will require an increased value for the coefficient. Little data seems to be known for this case. Consult Bos(1989) for more details and more accurate values of the discharge coefficient.
Approach velocity heads have not been included in the above analysis because in most cases the flow through an orifice is small enough that the velocity head in the approaching flow is negligible. However, the user can request inclusion of velocity head by providing an approach section for the orifice. FEQUTL then computes the flow ignoring velocity head as a first estimate and then uses the estimated flow to compute an estimated velocity head. This estimated velocity head is then used to convert the piezometric head to the total head before evaluating the integrals for the final time. Only one correction is made, again with the assumption that the velocity heads will be small.

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