Water in an open channel is effectively an incompressible fluid that is contained but can change its form according to the shape of the container. In nature, the bulk of fresh surface water either occupies hollows in the ground, as lakes, or flows in well-defined channels. Open channel flow also occurs in more regular man-made sewers and pipes as long as there is a free water surface and gravity flow.
The hydrologist is interested primarily in discharge of a river in terms of cubic metres per second (m3 s-1), but in the study of open channel flow, although the complexity of the cross-sectional area of the channel may be readily determined, the velocity of the water in metres per second (ms-1) is also a characteristic of prime importance. The variations of velocity both in space and in time provide bases for the standard classifications of flow.
In practice, uniform flow usually means that the velocity pattern within a constant cross-section does not change in the direction of the flow. Thus in Fig. 7.1, the flow shown is uniform from A to B in which the depth of flow, yo, called the normal depth, is constant. The values of velocity, v, remain the same at equivalent depths. Between B and C, the flow shown is non-uniform; both the depth of flow and the velocity pattern have changed. In Fig. 7.1, the depth is shown as decreasing in the direction of flow (y1 < yo). A flow with depth increasing (y1 > yo) with distance would also be non-uniform.
Over the cross-section of an open channel, the velocity distribution depends on the character of the river banks and of the bed and on the shape of the channel.
Velocity profile (b)
Velocity profile (b)
The maximum velocities tend to be found just below the water surface and away from the retarding friction of the banks. In Fig. 7.2a, lines of equal velocity show the velocity pattern across a stream with the deepest part and the maximum velocities typical of conditions on the outside bend of a river. A plot of the velocities in the vertical section at depth y is shown in Fig. 7.2b. The average velocity of such a profile is often assumed to occur at or near 0.6 depth.
When fluid particles move in smooth paths without lateral mixing, the flow is said to be laminar. Viscous forces dominate other forces in laminar flow and it occurs only at very small depths and low velocities. It is seen in thin films over smooth paved surfaces. Laminar flow is identified by the Reynolds number Re = pwuy/p, where pw is the water density and p the dynamic viscosity. (For laminar flow in open channels, Re is less than about 500). As the velocity and depth increase, Re increases and the flow becomes turbulent, with considerable mixing laterally and vertically in the channel. Nearly all open channel flows are turbulent.
7.1.4 Critical, slow and fast flow
Flow in an open channel is also classified according to an energy criterion. For a given discharge, the energy of flow is a function of its depth and velocity, and this energy is a minimum at one particular depth, the critical depth, yc (Fig. 7.3). It can be shown (Akan, 2006) that the flow is characterized by the dimensionless Froude number:
where u is the velocity, g is gravitational acceleration and y is the depth of flow. For Fr < 1, flow is said to be subcritical (slow, gentle or tranquil). For Fr = 1, flow is critical, with depth equal to yc the critical depth. For Fr > 1, flow is supercritical (fast or shooting) (Fig. 7.3). Larger flows have larger values of uc and yc. The occurrence of critical flow is very important in the measurement of river discharge because, at the point of critical flow for a given discharge, there is a unique relationship between the velocity and the discharge as u = -J(gy). Thus only depth has to be measured to calculate velocity. Elsewhere, the flow might be either a subcritical or supercritical state, and both velocity and depth would have to be measured to derive discharge. As discussed later in Section 7.6, if a cross-section where critical flow occurs naturally cannot be found, the installation of a weir or flume can force the flow to become critical.
7.1.5 Steady flow
This occurs when the velocity at any point does not change with time. Flow is unsteady in surges and flood waves in open channels. The analytical equations of unsteady flow are complex and difficult to solve (see Chapter 14) but the hydrologist is most often concerned with these unsteady flow conditions. With the more simple conditions of steady flow, some open channel flow problems can be solved using the principles of continuity, conservation of energy and conservation of momentum.
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