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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

1.4 Major Assumptions in Unsteady-Flow Analysis

Analysis of 1-D unsteady flow in open channels requires many assumptions. The major assumptions are the following:

  1. The wavelength of the disturbance of the flow is very long relative to the depth of the flow. This "shallow-water wave assumption" implies that the flow is principally 1-D and basically parallel to the walls and bottom forming the channel. Thus, streamline curvature is small; lateral and vertical accelerations are negligible relative to the longitudinal accelerations; and, therefore, the pressure distribution is hydrostatic.

  2. The channel geometry is fixed so that the effect of deposition or scour of sediment is small.

  3. The bed of the channel has a shallow slope so that (a) the tangent and sine of the angle that the bottom makes with the horizontal have nearly the same value as the angle and (b) the cosine of the angle is approximately 1.
  4. The effect of boundary friction force can be estimated with a relation derived from steady uniform flow. Nonuniformity and unsteadiness are assumed to have only a small effect on the frictional losses.

  5. Channel alignment with respect to the effect of directional changes on the conservation of momentum principle may be treated as if it were rectilinear even though the channel is curvilinear. Thus, the water surface in any cross section of the stream is assumed to be horizontal. Super-elevation effects on the water surface in channel bends are not considered in the analysis and are assumed to have a small effect on the results.

  6. The fluxes of momentum and energy along the cross section resulting from nonuniform velocity distribution may be estimated by means of average velocities and flux-correction coefficients that are functions of location along the stream and water-surface elevation.

  7. The flowing fluid is homogeneous (constant density).
From these assumptions, formal statements of the conservation of water volume (mass) and conservation of water momentum can be developed. The conservation of volume (mass) principle relates to flows and changes in the quantity of water stored in the channels and reservoirs. No forces of any kind are considered in the conservation of mass. Forces, momentum fluxes, and the momentum of water in storage are related in the conservation of momentum principle. The factors involved in this equation are
  1. gravity force on the water in the channel,

  2. friction force on the wetted perimeter of the channel,

  3. pressure force on the boundaries,

  4. wind force on the water surface, and
  5. inertia of the water.
Some of these factors can be omitted to simplify the unsteady-flow computations. If all these factors are included in the analysis, the equations are referred to as the complete, full, dynamic, Saint-Venant, or shallow-water equations. If the inertia of the water is ignored, the zero-inertia form of the motion equation is obtained. If, in addition, the variations of pressure force along the channel are ignored because they are thought to be small, the kinematic form of the motion equation is obtained. Reservoir routing also is a form of unsteady-flow analysis in which the motion equation is simplified to a relation between water-surface elevation and the flow. In a certain sense, reservoir routing ignores all four factors although some or all are implicit in the relation between flow and water surface elevation. In each case, at least one of the factors is dropped from the motion equation. FEQ includes three of the four forms of motion equations for unsteady flow: (1) the full-equation form, including all four factors, (2) the zero-inertia form, in which the inertia of the water is omitted, and (3) the reservoir- routing form, in which the motion equation is reduced to a relation between water surface elevation and flow.

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