External Boundaries

The characteristic form of the equations can be used to show that under sub-critical conditions, disturbances to the flow are propagated in both directions whereas under super-critical conditions disturbances can only propagate downstream. This implies that for wholly sub-critical flow, an independent boundary condition is required at both upstream and downstream ends of the model whereas for wholly super-critical flow two independent boundary conditions (usually flow and stage) are required at the upstream extent of the model.

For sub-critical flow, which is of primary interest, these boundary conditions are specified as discharge-time, stage-time or stage-discharge (rating curve) relationships. The following are known to lead to a properly posed system of equations:

• discharge hydrograph upstream and stage hydrograph downstream
• discharge hydrograph upstream and rating curve downstream

although other combinations may work in certain circumstances. The specification of a discharge hydrograph at the downstream end may lead to problems and Cunge J A et al (1980) noted that this can cause instabilities even though Amien M. and Fang C.S. (1970) claim to have implemented a model specifying discharge hydrographs at both ends.

In general the boundary conditions must be correctly specified since the uniqueness of the solution depends upon them. Irregularities in data can cause flow oscillations and it is often advisable to use a smoothed hydrograph, particularly at downstream boundaries.