Full Equations Utilities (FEQUTL) Model for the Approximation of Hydraulic Characteristics of Open Channels and Control Structures During Unsteady Flow
A look-up table in the context of FEQUTL computation is an organized collection of information that defines the important hydraulic characteristics of a cross section or control structure in the stream system simulated in FEQ. Look-up tables have been utilized for mathematical functions, such as the square root and the logarithm, for many years. Various forms of interpolation are utilized to determine values of these functions. The ready availability of these standard functions on calculators and in computer languages have made the use of table look-up procedures for standard mathematical functions rare.
For more complex mathematical functions, such as critical values for statistical tests, table look-up procedures are commonly included in computer programs. The shape and size of a stream cross section is normally not described by some simple, standard function. Approximation using polynomials or some other standard function is not practical because stream shapes greatly vary. Thus, carefully defined look-up tables are an efficient means for defining nonstandard functions. Utilization of look-up tables in FEQ simulation results in the generality of description needed in an unsteady-flow modeling system. In addition to the generality of description, look-up tables make computations in FEQ more efficient. Millions of calculations are involved in an unsteady-flow simulation. Therefore, superfluous computations must be eliminated for efficient simulation.
1.2.1 Look-up Tables for Channel Cross SectionsCross sections of a stream usually are given in a series of horizontal distances from reference points, called offsets, and the elevation of points on the bed of the stream. The area of flow, the top width of the flow, the wetted perimeter of the flow, conveyance, and other characteristics can be computed given an elevation of the water surface. However, a review of the governing equations for unsteady flow (Franz and Melching, 1997) indicates that the details of which side is the right bank of a cross section and which side is the left bank of a cross section do not affect the computations if the left bank and right bank are defined consistently for curvilinear channels. All that is required to solve the governing equations is information on the variation with water-surface elevation of the key characteristics of the cross section, such as flow top width, area, square root of conveyance, correction coefficients for nonuniform flow, first moment of area about the water surface, and correction coefficients for channel curvilinearity. These characteristics can be computed from the basic cross-section description and stored in a look-up table. The hydraulic characteristics of open channels needed for flow simulation in FEQ are then readily available.
The application of look-up tables to describe the hydraulics of a variety of control structures eliminates superfluous computations of hydraulic characteristics of these structures during unsteady-flow simulation. Steady-flow relations are applied at these structures. Application of steady-flow relations is feasible because the change in the volume of water and in the momentum of the water in the control structure is small relative to the changes in the stream channels between structures. At a control structure, information on the relation between the flow through the structure and the water-surface elevation upstream and in some cases downstream from the structure are needed in flow simulation in FEQ, or any other similar model for unsteady-flow analysis. The detailed nature of the type of structure is irrelevant to the computations done in FEQ. Thus, in flow simulations done in FEQ, it is not important whether the structure is a bridge, a culvert, or a spillway. All that is needed for flow simulation is an adequate description of the relation between the flow through the structure, the water-surface elevation upstream from the structure, and the water-surface elevation downstream from the structure.
The description of the hydraulics of control structures is suited to application of look-up tables. The performance of the structure for a meaningful range of flows and water-surface elevations can be computed and the results placed in a look-up table. The look-up table is then accessed in FEQ simulations to complete the unsteady-flow computations. Thus, the computational burden of defining the flow through the structure need not be done in FEQ simulations. A major additional benefit of applying a look-up table is that the transitions in flow, such as the transition between partial and full flow in culverts, can be isolated and approximated before the unsteady-flow computations are done. This eliminates the difficulty of making a smooth transition between the flow patterns that can result in the structure as the computations proceed. Calculation of such transitions during the unsteady-flow computations is always difficult and sometimes nearly impossible. Computation of the transition flow conditions with steady-flow calculations in a utility program is feasible but may still be difficult.
A look-up table contains one or more input arguments, the values utilized to do the look up, and one or more output items used in simulation. For a cross section the argument in FEQ is the maximum depth found in the section. The output items of interest are the cross-sectional characteristics, which are all functions of the maximum depth, such as top width, area, and conveyance. These are discussed at length in section 3. For flow through a culvert, two input arguments are utilized--the piezometric head upstream from the culvert and the piezometric head downstream from the culvert measured from the same datum--and the item of interest is the flow through the culvert.
Several components are needed in the development of a useful table look-up procedure. The tables must be patterned for efficient look up, an efficient searching scheme must be applied, and a set of rules must be established for defining values intermediate to those that are tabulated (rules for interpolation). The pattern of the tables is a programming detail that is outlined in the source code of FEQ and FEQUTL. The rules of interpolation are discussed in section 2. The search technique used for all tables in FEQ and FEQUTL is the simple linear search. This technique proves to be most efficient because the table look ups are most often repetitive at nearly the same value of the argument. The value determined in the last table look up and its location in the table are stored in FEQ and FEQUTL simulations, and the search for the function value at the next look-up time in the simulation begins at that point. Therefore, in most computations, the number of comparisons needed to find the point of interpolation for the current argument is minimal. Only occasionally is a long search required.