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

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 97-4037

2.1 One-Dimensional Function Tables


Six of the seven types of 1-D function tables computed in FEQUTL list characteristics of stream cross sections, whereas the seventh type lists flow-rating curves. Complete descriptions of these function tables are given in the following sections.

2.1.1 Type 2 Tables

Table type 2 lists a function of one argument. It is applied for simple rating curves for a variety of structures. Because there is only one argument, the rating is assumed to be unaffected by tail-water variations. This assumption must be considered when tables of type 2 are applied. In FEQUTL and FEQ computations, the variation of values between tabulated arguments is assumed to be linear. Thus, a piecewise-linear approximation of a function of one variable is represented in these tables. The piecewise-linear approximation is computed in FEQ as

(1)

Equation,

where

Equation(yt) = the function value estimated from the table for argument value, yt, of interest,

Equation = the ith function value tabulated,
Equation = the corresponding argument value, and Equation.

Enough data points should be included in the table so that straight-line interpolation between data points results in negligible error.

By proper selection of the tabulation interval and function values, the model user can represent a wide variety of functions with this type of table. These functions include rating curves, variation of spillway coefficients with head, and energy-loss coefficients. The following is an example of a portion of a type 2 table representing a rating curve at a gaging station as output from FEQUTL for input to FEQ.

Table

2.1.2 Cross-Section Function Tables

Six cross-section function tables are computed in FEQUTL and used in FEQ. The type numbers assigned are from 20 to 25. The cross-sectional characteristics are tabulated as a function of the water-surface height (equal to maximum depth) in the cross section in all tables. Therefore, these function tables are 1-D because only one argument is present. However, these function tables contain more than one function value for each argument value. The equations used to compute the cross-sectional characteristics listed in the various function tables are given in sections 3.1.1, 3.1.2, and 3.1.3.

All of the six table types list water-surface height, y; top width, T; cross-sectional area, A; square root of conveyance, Equation; and the momentum-flux correction coefficient, This is the Greek letter Beta (sections 3.1.1 and 3.1.2). The first moment of area about the water surface, J (section 3.1.1), is added to table type 21. The first moment of area about the water surface, the energy-flux correction coefficient, This is the Greek letter Alpha, and critical flow rate, Qc (section 3.1.2), are added to table type 22. Table type 23 is similar to table type 20 with the addition of the weight factors to convert area to volume per unit length of channel, MA, and to convert flow rate to momentum content per unit length of channel, M Q, for curvilinear channels (section 3.1.3). In the same manner, table type 24 is similar to table type 21, and table type 25 is similar to table type 22. In terms of the defined symbols, the table types contain the cross-section characteristics as follows:

Type 20: y, T, A, Equation, This is the Greek letter Beta
Type 21: y, T, A, Equation, This is the Greek letter Beta, J
Type 22: y, T, A, Equation, This is the Greek letter Beta, J, This is the Greek letter Alpha, Qc
Type 23: y, T, A, Equation, This is the Greek letter Beta, MA, MQ
Type 24: y, T, A, Equation, This is the Greek letter Beta, J, MA, MQ
Type 25: y, T, A, Equation, This is the Greek letter Beta, J, This is the Greek letter Alpha, Qc, MA, MQ

Any cross-section table that contains the proper information can supply that information in FEQ simulation. Thus, tables of type 25 are applicable in all contexts. On the other hand, tables of type 20 are applicable in a more limited context. In FEQ simulation an error message results if a cross-section table does not contain the needed information.

The interpolation method for each of the cross-section characteristics is listed in table 2. In "integrated linear" interpolation, a linear function is integrated to find the interpolated value. Area is given by the integral of the top width. The top width is interpolated linearly between tabulated argument values. Therefore, the linear top-width variation is integrated to interpolate between the tabulated argument values. In the same way, the first moment of area is the integral of the area. The area is piecewise quadratic (an integral of a piecewise linear function). Thus, "integrated quadratic" interpolation is applied wherein the piecewise quadratic variation of area is integrated to interpolate for the first moment of area between tabulated argument values. Interpolation for the critical flow rate is done linearly for logarithmic transformed values ("linear in logarithms") because this yields exact values for many standard geometric shapes (Franz and Melching, 1997). Additional details are provided in sections FEQ: 3, 4, and 11 in the documentation report for the Full Equations model (Franz and Melching, 1997).

Table

The following is an example of a cross-section table of type 20 as output from FEQUTL for input to FEQ, where SQRT (CONV) is the square root of conveyance

Table
2.1.1 Type 2 Tables

2.1.2 Cross-Section Function Tables

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