Simplifications to the momentum equation

Depending on the relative importance of the various terms of the Momentum equation, it can be simplified for different applications as follow:

Since simplification means that some aspect is being ignored, it is important for a modeler to understand the basis of the model being applied to answer a hydraulic or hydrologic question. Further discussion on application and limitations of some of routing approaches that are used in many computer programs follows.

• Kinematic wave approximation—The kinematic wave approximation assumes that the gravitational and frictional forces are in balance. The kinematic wave approximation works best when applied to steep (0.0019, 10 ft/mi or greater), well-defined channels, where the floodwave is gradually varied. Changes in depth and velocity with respect to time and distance are small in magnitude when compared to the bed slope of the channel. The approach is often applied in urban areas because the routing reaches are generally short and well defined (circular pipes, concrete lined channels). However, the equations do not allow for hydrograph diffusion, but only simple translation of the hydrograph in time.
  The application of the kinematic wave equation is limited to flow conditions that do not demonstrate appreciable hydrograph attenuation. This may be an issue in wide channels, since attenuation increases with valley storage. The kinematic wave equations cannot handle backwater effects, since with a kinematic model flow disturbances can only propagate in the downstream direction.
• Modified Puls reservoir routing—This approach accounts for the difference of inflow as storage over some defined time period. This method is appropriate if lateral storage is the primary physical mechanism that affects the flood routing. This method disregards the equation of motion by focusing on continuity. It is closely related to level pool reservoir routing.
• Muskingum river routing—The Muskingum river routing method is based on two equations. The first is the continuity equation, and the second is a relationship of storage, inflow, and outflow of the reach. This method is based on a weighted function of the difference of inflow as storage over some defined time period. Typically, the coefficients of the Muskingum method are not directly related to physical channel properties and can only be determined from stream gage data.
• Diffusive wave approximation—The diffusion wave model is a significant improvement over the kinematic wave model because of the inclusion of the pressure differential term in the momentum equation. This term allows the diffusion model to describe the attenuation (diffusion effect) of the floodwave. It also allows the specification of a boundary condition at the downstream extremity of the routing reach to account for backwater effects. It also allows the specification of a boundary condition at the downstream extremity of the routing reach to account for backwater effects. Since it does not use the inertial terms (last two terms) from the full momentum equation, it is limited to slowly to moderately rising floodwaves in flat channels (Fread 1982). However, most natural floodwaves can be described with the diffusion form of the equations.
• Muskingum-Cunge—The theoretical development of the Muskingum-Cunge routing equation is based on the simplification of the convective diffusion equation. In the Muskingum-Cunge formulation, the amount of diffusion is controlled by forcing the numerical diffusion to match the physical diffusion represented by the convective diffusion equation. This approach accounts for hydrograph diffusion based on physical channel properties and the inflowing hydrograph. The method includes the continuity equation and a relationship of storage, inflow, and outflow of the reach. The solution is independent of the user-specified computation interval. The coefficients of the Muskingum- Cunge method are based on data such as cross section and estimated Manning’s n and are more physically based than the Muskingum method. Therefore, the Muskingum-Cunge method can be applied to ungaged streams. However, it cannot account for backwater effects, and the method begins to diverge from the full unsteady flow solution when very rapidly rising hydrographs are routed through flat channel sections.
• Quasi-steady dynamic wave approximation—The third simplification of the full dynamic wave equations is the quasi-steady dynamic wave approximation. In the case of flood routing, the last two terms in the momentum equation are often opposite in sign and tend to counteract each other. By including the convective acceleration term and not the local acceleration term, an error is introduced. This error is of greater magnitude than the error that results when both terms are excluded, as in the diffusion wave model. This approach is not often used in flood routing.
• Dynamic wave equations—The dynamic wave equations can be applied to a wide range of one-dimensional flow problems, such as dam break flood wave routing, tidal fluctuations, canal distribution, and forecasting water surface elevations and velocities in a river system during a flood. Solution of the full equations is normally accomplished with an explicit or implicit finite difference technique. The equations are solved for incremental times (dt) and incremental distances (dx) along the waterway.