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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

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 96-4240

2.1 Channel Segmentation

In steady-flow analysis, a governing equation is given that describes flow variation. This is most often written as an energy-conservation equation but a momentum-conservation equation also can be used. In either case, this equation is in differential or integral form; and the solution cannot be determined without application of numerical methods. Thus, only an approximate solution to the governing equation can be determined. To find this solution, the channel is subdivided into short pieces called computational elements. Then, for each computational element, the differential or integral terms in the governing equations are approximated algebraically to yield an algebraic equation that approximates the governing equation for that element. From these computational elements, the whole solution scheme proceeds.

The ends of the computational element, called nodes, are defined by cross sections either measured or estimated from field measurements. The cross section is at right angles to the direction of flow as best as can be determined. One way to visualize a computational element is as a slice of the channel whose ends are at right angles to the longitudinal axis (fig. 1). Adjacent computational elements have a cross section and, therefore, a node in common; thus, there will always be one more node (cross section) than the number of computational elements.

The values of interest in the cross sections are the width of the water surface, the flow area, the first moment of area, the conveyance, and, perhaps, an energy- or momentum-flux coefficient, all computed at any given elevation. The shape of the cross section does not appear explicitly in the governing equations, but only implicitly through these cross-sectional characteristics.

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