and the conservation of momentum equation :
in which t is time, x is distance along the longitudinal axis of the waterway, A is cross-sectional area, V is velocity, g is the gravity acceleration constant, and h is the water-surface elevation above a datum; Sf is the friction slope, which may now be
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evaluated using a steady flow empirical formula such as the Manning equation (Manning, 1889; Chow, 1959), i.e., in which Q = AV is discharge or flow, R = A/P is the hydraulic radius, and P is the wetted perimeter of the cross section, S0 is the channel bottom slope (dimension-less), p is a units conversion factor, i.e., 1.49 for U.S. units or 1.0 for SI, and n is the Manning roughness (friction) coefficient. Equations (1) and (2) are quasi-linear hyperbolic partial differential equations with two dependent parameters (V and h) and two independent parameters (x and t); A is a known function of h, and Sj is a known function of V and h. Derivations of the Saint-Venant equations can be found in the following references: Stoker (1957), Henderson (1966), StrelkofF (1969), and Liggett (1975).
Due to the mathematical complexity of the Saint-Venant equations (no analytical solution is known), simplifications were necessary to obtain feasible solutions for the salient characteristics of a propagating flood wave. This approach produced a profusion of simplified flow routing models. The simplified flow routing models may be categorized as: (I) purely empirical, (II) storage routing, based on the conservation of mass and an approximate relation between flow and storage, and (III) hydraulic, i.e., based on the conservation of mass and a simplified form of the conservation of momentum equation (2).
Categories I and II are further classified as lumped flow routing techniques in which the flow is computed as a function of time, only at the most downstream location of routing reaches along the waterway. Category III can be classified as distributed flow routing techniques in which flow and depth or water-surface elevation are computed as a function of time at frequent locations within routing reaches along the waterway. During the last two decades dynamic hydraulic distributed flow routing methods based on numerical solutions of the complete Saint-Venant equations have become economically feasible as a result of advances in computing equipment and improved numerical solution techniques. Following is a brief description of some of the more popular storage routing models as well as both simplified and dynamic hydraulic flow routing models.
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