3 Hydrodynamics
241
level in the open channel will rise, that is,
∂ Q
∂s < 0,
∂ A
∂t > 0; otherwise, when
the mass of the inflow micro control volume is less than the outflow mass,
the water level in the open channel will fall, that is,
∂ Q
∂s > 0,
∂ A
∂t < 0.
(2) Equation of motion
In the gradually varied flow channel, any segment of the microelement with
the length of ds is taken, as shown in Fig. 3.65. The cross-sectional area of the
microelement is A, and the angle between the channel bottom wall and the
horizontal direction is θ (the downward inclination along the flow direction is
positive). According to Newton’s second law, the motion equation established
along the flow direction is
∂ V
∂t
+ V
∂ V
∂s
+ g
∂(z b + h)
∂s
+ g
τ w
γ R
= 0
where z b is the elevation of the canal bottom, i = −
∂z b
∂s ; the shear stress of
the canal wall is treated according to the uniform flow, that is τ w = γ R J f =
γ
V 2
C 2 , R is the hydraulic radius and C is the chezy coefficient. Put it into the
above equation and get
∂ V
∂t
+ V
∂ V
∂s
+ g
∂h
∂s
= g
i −
V 2
C 2 R
This equation is the energy equation of unsteady gradually varied flow in
open channels. The Saint Venant equations are constructed together with the
1
2
t+dt
t
P 1
w
h 1
z r1
z r2
W
0
V 1
h 2
V 2
P 2
0
Wsinθ
θ
τ
Fig. 3.65 Microelement of unsteady gradually varied flow in open channel