Numerical Simulation of Positive Surge Moving Upstream
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predictions. Zheng et al. [20] experimentally studied various free surface characteristics of positive surge in an open channel flow and find a close agreement of maximum
wave height at the first wave crest with McCowan theory.
2 Unsteady Flow
In unsteady flow, “flow waves” are generated. A wave may be defined as spatial (i.e.,
with respect to distance) and temporal (i.e., with respect to time). A wave is named
as positive wave if water depth is higher behind the wave than the undisturbed flow
depth and as negative wave if the lower water flow depth behind the wave than the
undisturbed flow depth [4, 15]. The wave celerity is defined as relative wave velocity
with respect to a flowing fluid, whereas absolute wave velocity with respect to a fixed
frame of reference [4, 15].
2.1 Governing Equations for One-Dimensional Unsteady
Flow
The two governing equations used to analysis unsteady, one dimensional, open
channel flow are called the St. Venant equations. They are based on the principles of
conservation of mass and momentum [4, 5, 12].
The conservative form of the continuity equation
∂ A
∂t
+
∂ Q
∂ x
= q l
(1)
The conservative form of the momentum equation
∂ Q
∂t
+
∂
∂ x
(Q A + g A ¯
y) = g A
S 0 − S f
+ V x q l
(2)
The conservation forms of the St. Venant equations for one-dimensional unsteady
flow in vector form are
∂U
∂t
+
∂ F
∂ x
+ S = 0
( 3 )
where U =
A
V A
, F =
V A
V
2 A + g Ay
and S =
−q l
−g A(S 0 − S F ) − V x q l