Hydrodynamic Channel Flows

The motion of a body of water flowing in open channels can be described by the so called shallow water or St Venant equations, which express conservation of mass and momentum. Conservation of mass leads to the continuity equation which establishes a balance between the rate of rise of water level and the net inflow. Conservation of momentum leads to the dynamic equation which establishes a balance between inertia, diffusion, gravity and friction forces. Some other forces, such as the effect of wind or meanders, may also be included but usually these are small.

The governing equations are the continuity equation:

and the momentum equation:

The equations are described in mathematical terms as a pair of one-dimensional non-linear hyperbolic partial differential equations. In general, the solution of any system of differential equations depends on the existence, uniqueness and stability conditions of that solution. By transforming these equations into characteristic form, Courant R. and Lax P. (1959) have shown that they satisfy the Lipschitz condition ensuring the existence of a unique solution. Stability of the solutions has also been widely investigated. The interested reader is referred to standard texts for further details, such as Mahmood and Yevjevich (1975) and Cunge J A et al (1980).