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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

7.1 Approximation of the Equations of Motion in a Dummy Branch

The equations of motion for a dummy branch are simple because the storage and momentum content of a dummy branch are negligible, so flows and elevations at the two flow-path end nodes defining the dummy branch are virtually identical. The dummy branch is treated as a reservoir with constant surface area, no lateral inflow, and a linear friction-loss relation between flow and elevation difference. Therefore, the conservation of mass equation becomes

(86)

where A_S is a constant surface area and the weight for integration with respect to time has been set to 1. The left-hand node is taken to be the upstream node for the dummy branch. The default surface area applied in FEQ is 2 but this value may be overridden by the user.

The relation between the flow and the elevation difference is

(87)

where the node on the left is taken to be the upstream node. The friction loss coefficient, K_F, is set by default to
2 x 10 ^-8, but different values can be input. The average water-surface elevation is used in equation 86 so that the partial derivatives of this equation will be nonzero in the nonlinear solution for the unknowns.

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