236
P. Liu
If the resistance is ignored and the partial derivative of (h, V ) convection path s is far less than that of time, the following wave equation can be
obtained after simplification.
∂h
∂t
+
C 2
g
∂ V
∂s
= 0
∂ V
∂t
+ g
∂h
∂s
= 0
The wave equations of H or P are obtained by solving the above equations,
namely
∂ 2 h
∂t 2 = C
2 ∂ 2 h
∂s 2 ,
∂ 2 p
∂t 2 = C
2 ∂ 2 p
∂s 2
The general solution of this wave equation is
h − h 0 = F(s − Ct) + f (s + Ct)
V − V 0 =
g
C
(F(s − Ct) − f (s + Ct))
F (s-Ct ) is the wave function of the forward water hammer wave and f (s
+ Ct ) is the wave function of the reverse water hammer wave. There is an
analytic method, graphic method, and characteristic line method to solve the
water hammer problem.
3.6.4 Water Oscillating Flow
For the fluctuation of U-tube water level and water level in surge shaft of
hydropower station, the key point is to give the law of water level fluctuation, especially the surge height caused by turbine shutdown is an important
parameter to determine the geometric size of surge shaft. Different from the
water hammer problem, the compressibility of the water and the elasticity
of the pipe wall are not considered here. The water is incompressible and
the pipe is treated as rigid. The continuity obtained is VA = f (t ), and the
differential equation of motion is
1
g
∂ V
∂t
+
V
g
∂ V
∂s
+
∂h
∂s
+ λ
V |V |
2g D
= 0
1
g
dV
dt
+
∂h
∂s
+ λ
V |V |
2g D
= 0