240
P. Liu
using mass conservation and momentum theorem and analyzed the solution
methods of these equations.
(1) Continuous equation
As shown in Fig. 3.64, for prismatic channel without side flow, the differential
equation of unsteady gradient flow in open channel can be obtained by taking
the microelement control volume between section 1-2 and the conservation
of mass theorem, that is
ρ Q +
∂(ρ Q)
∂s
ds - ρ Q
dt = -
∂(ρ A)
∂t
dsdt
After simplification
∂ A
∂t
+
∂ Q
∂s
= 0
For rectangular channels, substituting A = bh and q = bhV into the above
equation, we can get
∂h
∂t
+ V
∂h
∂s
+ h
∂ V
∂s
= 0
This is the continuous equation of unsteady gradually varied flow in open
channel. The equation shows that in DT micro segment, when the mass of
the inflow micro control volume is greater than the outflow mass, the water
Wave
bottom
t time water surface
ds
dt
A
∂
t
∂
ds
( )
∂
s
∂
ρQ
ρQ+
ρ Q
1
b
2
a
b
a
t+dt time water surface
Fig. 3.64 Microelement control volume of unsteady gradient flow in open channel