The general hydrodynamic equations for a spatially varied unsteady flow in a prismatic open channel having an arbitrary cross-sectional shape can be derived from the equations of continuity and momentum. The assumptions based on the general concept of hydrodynamics and the theory of shallow water is introduced. The mathematical models in the surface irrigation can be formulated by these equations of motion with the appropriate initial and boundary conditions prescribed at the singularity point (the origin in the x, t-plane) and at x = 0. Therefore, the flow in the surface irrigation must be described by solving the boundary-value problem for the velocity and the depth of flow. In the two-dimensional flow, the discontinuity at the origin in the x, t-plane is overcome by imposing a critical velocity and correspondingly, a critical depth at the initial state. Without considering all the channel slope, friction, and infiltration terms, the mathematical model becomes the model for a “centered simple wave,” in which an exact solution, the Ritter solution, in the dam-breaking problem is already well-known. The Ritter solution satisfies the boundary condition at x = 0, where the discharge is always constant. However, even though an additional term, the channel slope, is considered, the modified Ritter solution no longer satisfies the same boundary condition. No exact solution seems possible with the present knowledge of mathematical techniques. The theory of characteristics is presented and the method of finite-difference based on this basic theory with the fixed-theme interval is introduced for the basic equations of motion in the surface irrigation. The trajectory of the wavefront is usually defined as a locus of zero celerity (c = 0) and forming an envelope of two different families of characteristics. Such an extremely complicated situation at the wavefront results in the possible failure of the present technique by simply using the finite-difference method unless some additional judicious assumptions at the wavefront, some type of the boundary-layer technique must be developed. This is the present status of the finite-difference method in the surface irrigation. The more simplified approximation by the kinematic-wave method is presented, based on an additional assumption analogous to the Dupit-Forchheimer theory in the groundwater flow. The method possesses a potential applicability in the surface irrigation. Inasmuch as no results are yet available, this study will be limited to a description of what is hoped to be accomplished and how it will be done.
Chen, Cheng-Lung, "Mathematical hydraulics of surface irrigation" (1966). Reports. Paper 563.