Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations of Motion for One-Dimensional Unsteady Flow in Open Channels and Through
Control Structures
By proper choice of the tabulation interval and the choice of the function values, many functions can be represented with these table types. These include rating curves, hydrographs, variation of spillway coefficients with head, and any other function for which a single variable is a function of a single argument.
(142)
(143)
where the argument y is contained within the interval ( yi, yi+1 ). The trapezoidal rule is used for integration in equation 143 because it is exact for linear functions. For this table type, values must be computed such that equation 143 is correct. This means that the function value is computed by integration of the piecewise linear derivative.
No formal name has been applied for the interpolation defined by these equations. It will be called integrated linear interpolation for subsequent reference. The function, f T, is, at most, a piecewise quadratic function of its argument. If the integrated function is quadratic, then the interpolation will be called integrated quadratic interpolation.
Function table types 3 and 8 are ideal for representing the area and storage capacity of a level-pool reservoir because the integral of the surface area is the storage. The storage must be computed by trapezoidal-rule integration of the surface area.
(144)
(145)
Function tables of types 4 and 9 are useful whenever a function must be represented smoothly. However, the user must be aware that no checking on the validity of the relation between the function and its derivative is done in FEQ computations. Improper selection of the function and its derivative for tabulation when the function is close to zero can result in an interpolated value that is negative when such a function value is invalid. Many functions of interest in unsteady-flow analysis are not only positive but are always increasing. Linear interpolation of functional results will yield values that are always increasing. Linear integrated interpolation will yield values that are always increasing if the derivative, as tabulated, is always positive. To avoid nonincreasing interpolated values when piecewise cubic interpolation is applied, all tabulated derivatives must be nonnegative and the absolute value of the derivative at either end of the piecewise cubic part of the function must never exceed three times the absolute value of the straight line slope between the function values; that is, for all i and for j = 0,1 for each interpolation interval.
Depth, top width, area, square root of conveyance, and the momentum-flux correction coefficient are tabulated in all six table types. The first moment of area about the water surface is added in type 21. The first moment of area about the water surface, the energy-flux correction coefficient, and critical flow rate are added in type 22. Types 23 through 25 are similar to types 20 through 22 with the addition of the correction factors for channel curvilinearity. Table contents are summarized symbolically as follows:
Type 20: y, T, A, ,
Type 21: y, T, A, , , J
Type 22: y, T, A, , , J, , Qc
Type 23: y, T, A, , , MA, MQ
Type 24: y, T, A, , , J, MA, MQ
Type 25: y, T, A, , , J, , Qc, MA, MQ
The interpolation mode for each of the cross-sectional
characteristics is listed in table 5.
Integrated quadratic interpolation follows the concept of integrated linear interpolation. The first moment of area about the water surface is given by the integral of the area. This results in
(146)
Equation 146 is the trapezoidal rule with end corrections and is exact for polynomials of third order (cubic) or less. Therefore, the area is integrated exactly because the area as defined by integrated linear interpolation is a quadratic polynomial.
Linear interpolation of the critical flow in logarithms of flow and depth implies that Qc is a piecewise power function of water-surface height; that is, for each tabulation interval ( yi, yi+1 ),
(147)
where cai and cbi are the coefficient and power, respectively, of the power function for interval i. The power and the coefficient will vary between intervals. This interpolation for critical flow is exact for rectangular, triangular, and parabolic cross sections. It also is exact for cross sections where the top width varies as a power function of the water-surface height in the section. For other shapes of cross section, it is a good approximation over moderate ranges of water-surface height, which is acceptable for the purposes of table look up.
Cross-section function tables are input in program unit XSECIN. Table lookup is done in the subroutines XLKT nn, where nn gives the type number for the table. A function table can be applied in any context for which the table contains the required information; for example, tables of type 25 can be used even though all the characteristics are not needed. If sufficient information for a given context is not available in the selected cross-section table type, an error message is issued and processing stops.
Some obsolete table types are supported in FEQ so that streams simulated with earlier versions of FEQ can be easily adapted to the latest version. These types are types 1 and 12 for cross-section tables. They are exactly the same as the current types 21 and 22. These table types for cross-section function tables are no longer computed in FEQUTL, but they can be used in FEQ simulation. The type designation is converted on input to FEQ. A table of type 5 is computed in FEQ for internal use in abrupt expansions to extract the relation between water-surface height and critical flow from a cross-section table.