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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

1.3 Selection of Conservation Principles


Three conservation principles--conservation of water mass, conservation of the mechanical-energy content of the water, conservation of the momentum content of the water--are available for analysis of 1-D unsteady flow. Conservation of thermal energy is not considered because temperature-change and heat-transfer effects do not affect flow depth and discharge.

 The first principle selected is the conservation of water mass, which becomes the conservation of water volume if the density is constant. Equations derived from application of the conservation of mass principle are often referred to as "continuity equations."

 The choice of conservation of momentum instead of conservation of mechanical energy of water for FEQ was based on how well the various flow parameters and variables can be approximated and how well each particular principle works when only approximations to physical reality are possible. Both principles are exact given precise knowledge of all the flow parameters and variables; however, precise knowledge of these is never possible. Yen (1973) provides a detailed list of differences between the energy and momentum approaches. Many researchers, including Abbott (1974), Cunge and others (1980), and Liggett (1975), argue for combined application of the conservation of mass and conservation of momentum principles as the equations of motion because this combination gives the correct wave speed and height should abrupt waves (hydraulic bores) form during the modeling of rapidly increasing or decreasing flow. If the conservation of momentum principle is used with the continuity equation and the equations are properly approximated, then the correct wave speed and height will be computed. In contrast, application of the conservation of energy principle provides no simple approximation that can be applied to yield the correct wave speed and height.

 In many applications, the flow in the stream channels is derived from runoff entering the channels either overland or from storm sewers, drainage ditches, and streams too small to be explicitly represented in the model. These flows generally enter approximately at right angles to the main-channel flow, and complex interaction between these flows involves considerable turbulence. Application of the energy-conservation principle would require that the kinetic and potential energy of lateral flows be estimated, and such estimates are nearly impossible to make accurately. The turbulence results in an unknown increase in energy dissipation. Therefore, lateral inflows are better approximated by use of the conservation of momentum principle. Because these flows enter approximately at right angles to the main-channel flow direction, the effect is approximated in the momentum equation without an additional requirement for estimated losses. The applicability of the conservation of momentum principle to the solution of lateral inflow problems has been demonstrated in modeling of side-channel spillways and wash-water troughs (Henderson, 1966, p. 268), both of which cause much greater turbulence than normally results in unsteady flow.

Yen and others (1972) give further evidence for the choice of the momentum-conservation principle. Using artificial rainfall on a sloping glass flume, they computed resistance coefficients for steady, spatially varied flow for both the energy and momentum conservation principles and found that the resistance coefficient from the momentum principle was always closer to the coefficient estimated from steady flow without lateral inflow. Because use of Manning's equation for resistance losses yields a better estimate of the resistance coefficient for the momentum principle than for the energy principle, methods based on momentum conservation yield better estimates of the water-surface profile than do methods based on energy conservation, especially if Manning's n is calibrated to measured water-surface profiles or historic high water marks. In addition, the resistance coefficient estimated from the momentum principle was insensitive to variations in the velocity of lateral inflow (many applications of unsteady flow involve a wide range of lateral inflow rates).

 Finally, the equation obtained with the conservation of momentum principle is simpler than the equation obtained with the conservation of energy principle. The simplicity of the equation obtained with the conservation of momentum principle is twofold; the equation includes fewer terms, and less information is needed for each cross section.


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