Governing Equations For One-Dimensional Flow

Three conservation laws – mass, momentum, and energy – are used to describe open-channel flows. Two flow variables, such as the flow depth and velocity, or the flow depth and rate of discharge, are sufficient to define the flow conditions at a channel cross section. Therefore, two governing equations may be used to analyze a typical flow situation. The continuity equation and the momentum or energy equation are used for this purpose. Except for the velocity head coefficient, α, and the momentum coefficient, β, the momentum and energy equations are equivalent [Cunge, et al., 1980] provided the flow depth and velocity are continuous, i.e., there are no discontinuities, such as a jump or a bore. However, the momentum equation should be used if the flow has discontinuities, since, unlike the energy equation, it is not necessary to know the amount of losses in the discontinuities in the application of the momentum equation.

In this chapter, we will derive the continuity and momentum equations, usually referred to as de Saint Venant equations. Several investigators [Stoker, 1957; Chow, 1959; Dronkers, 1964; Henderson, 1966; Strelkoff, 1969; Yen, 1973; Liggett, 1975; Cunge, et al., 1980; Lai, 1986; and Abbott and Basco, 1990] derived these equations by using different procedures. For illustration purposes, we will use two different procedures in our derivations. We will use the Reynolds transport theorem for the prismatic channels having lateral inflows or outflows. The type of the governing equations is then discussed. The equations describing flows having non-hydrostatic pressure distribution are derived by integrating the continuity and momentum equations for twodimensional flows. The chapter concludes by presenting integral forms of the governing equations.