chapter 3 hydraulics of open channel flow
chapter 3 hydraulics of open channel flow
chapter 3 hydraulics of open channel flow
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3.26 Chapter Three<br />
HYDRAULICS OF OPEN CHANNEL FLOW<br />
3.6.2 Rapidly Varied Unsteady Flow<br />
The terminology “rapidly varied unsteady <strong>flow</strong>” refers to <strong>flow</strong>s in which the curvature <strong>of</strong> the<br />
wave pr<strong>of</strong>ile is large, the change <strong>of</strong> the depth <strong>of</strong> <strong>flow</strong> with time is rapid, the vertical acceleration<br />
<strong>of</strong> the water particles is significant relative to the total acceleration, and the effect <strong>of</strong><br />
boundary friction can be ignored. Examples <strong>of</strong> rapidly varied unsteady <strong>flow</strong> include the catastrophic<br />
failure <strong>of</strong> dams, tidal bores, and surges that result from the quick operation <strong>of</strong> control<br />
structures such as sluice gates. A surge producing an increase in depth is termed a positive<br />
surge, and one that causes a decrease in depth is termed a negative surge. Furthermore,<br />
surges can go either upstream or downstream, thus giving rise to four basic types (Fig. 3.8).<br />
Positive surges generally have steep fronts, <strong>of</strong>ten with rollers, and are stable. In contrast,<br />
negative surges are unstable, and their form changes with the advance <strong>of</strong> the wave.<br />
Consider the case <strong>of</strong> a positive surge (or wave) traveling at a constant velocity (wave<br />
celerity) c up a horizontal <strong>channel</strong> <strong>of</strong> arbitrary shape (Fig. 3.8b). Such a situation can<br />
result from the rapid closure <strong>of</strong> a downstream sluice gate. This unsteady situation is converted<br />
to a steady situation by applying a velocity c to all sections; that is, the coordinate<br />
system is moving at the velocity <strong>of</strong> the wave. Applying the continuity equation between<br />
Sections 1 and 2<br />
(V 1 � c)A 1 � (V 2 � c)A 2<br />
(3.77)<br />
Since there are unknown losses associated with the wave, the momentum equation<br />
rather than the energy equation is applied between Sections 1 and 2 or<br />
γ<br />
γA1z�1 � γA2z�2 � �� y1 (V1 � c)(V2 � c � V1 � c) (3.78)<br />
g<br />
y 1<br />
y 1<br />
FIGURE 3.8 Definition <strong>of</strong> variables for simple surges moving in<br />
an <strong>open</strong> <strong>channel</strong>.<br />
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y 2<br />
y 2
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