###### II. OPEN CHANNEL FLOWS THEORY

II.1 Specific energy

Specific energy (specific charge) in a cross section of a channel or a river is:

the average value of the energy of the molecules of the liquid of this section, by unit of weight of this liquid, compared to the

*   horizontal one passing by the point low of this section;
*   the average load of the section compared to a privileged datum-line;
*  the distance enters the line of load and the bottom a given section.

From Bernoulli's equation, the total energy per unit mass H in a turbulent open channel flow is given by where y is the flow depth, z the bed height above datum, g the acceleration due to gravity, and V the average velocity. In this equation, the velocity coefficient associated with the term V2/2g is set to unity, as usual for uniform flow. Without losses, H = constant, and flow over an elevated section of bed of height D z is described by where the specific energy E is given by in which subscripts 1 and 2 refer to locations upstream and downstream of the change in bed elevation, and q is the flow rate per metre width of channel (Figure 1). Figure 1

Given upstream conditions, the value of E downstream is obtained, and consequently the downstream flow depth. For a particular vale of E, there are two possible values of y, as shown in Figure 2.

II.2 Froude number :

The appropriate value of y is determined by the upstream value of the Froude number  Figure 2

For Fr > 1, the flow is supercritical and is fast and shallow (the lower arm of Figure 2), and for Fr < 1, the flow is subcritical and is slow and deep. Thus if the upstream flow is subcritical (E1, y1 in Figure 2), flow over the obstacle will be subcritical at reduced E, corresponding to reduced y (E2, y2 in Figure 2); the flow decreases over the raised bed. If D z is sufficiently large to attempt to force E2 to become less than the minimum possible value, the upstream flow will adjust, increasing the upstream flow depth so that E2 is the minimum value. In this case the structure is a control, and the flow depth yc, velocity Vc, and flowrate q at the minimum energy (critical) point are related by and .

If the bed level subsequently decreases, the critical point must occur at the maximum elevation, beyond which the flow becomes upercritical. A decrease in bed level corresponds to an increase in E, and consequently a decrease in y as the E, y relationship moves along the lower arm of Figure 2. Further downstream the imposed head may force a flow depth to be above that possible for a supercritical flow, in which case theflow must change back to subcritical. This is done by means of a hydraulic jump.

Thus for a given q, in the absence of losses the flowing depth over the maximum elevation is given by the critical condition. Conversely, a measurement of the flow depth at the critical point, which must be the crest, provides a measure of q. Downstream of the critical point, the flow is supercritical and is controlled only by the structure itself. If the downstream head condition is sufficiently high, the downstream jump may drown the obstacle, in which case the flow remains subcritical throughout.