L is a characteristic dimension, usually the depth (or the hydraulic radius) in open channel flow. In
laminar flow, viscous forces are dominant and Re is relatively small. In turbulent flow, Re is
large; that is, inertial forces are very much greater than viscous forces. Turbulent flows are
predominant in nature. Laminar flow occurs very infrequently in open channel flow.
Tranquil flow: In open channel flow, the free surface configuration, in response to changes in
channel geometry depends on the Froude number (Fr = V / gL ) , which is the ratio of inertial
forces to gravitational forces. The Froude number is also the ratio of the flow velocity V to the
celerity (c = gL ) of a small gravity wave in the flow (this concept is detailed in Section 2.5).
When Fr < 1, the flow is subcritical (or tranquil), and surface waves propagate upstream as well
as downstream. The boundary condition that controls the tranquil flow depth is always located at
the downstream end of the subcritical reach.
Rapid flow: When Fr > 1, the flow is supercritical (or rapid) and surface disturbances can
propagate only in the downstream direction. The control section of rapid flow depth is always at
the upstream end of the rapid flow region. When Fr = 1.0, the flow is critical and surface
disturbances remain stationary in the flow.
2.2 THREE BASIC EQUATIONS
The basic equations of flow in open channels are derived from the three conservation laws.
These are: (1) the conservation of mass; (2) the conservation of linear momentum; and (3) the
conservation of energy. The conservation of mass is another way of stating that (except for
mass-energy interchange) matter can neither be created nor destroyed. The principle of
conservation of linear momentum is based on Newton's second law of motion which states that a
mass (of fluid) accelerates in the direction of and in proportion to the applied forces on the mass.
In the analysis of flow problems, much simplification can result if there is no acceleration of the
flow or if the acceleration is primarily in one direction, the accelerations in other directions being
negligible. However, a very inaccurate analysis may occur if one assumes accelerations are
small or zero when in fact they are not. The concepts explained in this chapter assume
one-dimensional flow and the derivations of the equations utilize a control volume. A control
volume is an isolated volume in the body of the fluid, through which mass, momentum, and
energy can be convected. The control volume may be assumed fixed in space or moving with
2.2.1 Conservation of Mass
Consider a short reach of river shown in Figure 2.1 as a control volume. The boundaries of the
control volume are the upstream cross-section, designated section 1, the downstream
cross-section, designated section 2, the free surface of the water between sections 1 and 2, and
the interface between the water and the wetted perimeter (banks and bed).