3 Hydrodynamics
235
Substituting into the continuous differential equation can be
1
ρ
dρ
∂t
+
1
A
d A
∂t
+
∂ V
∂s
= 0
Bring relation
1
ρ
dρ
∂t
=
1
K w
d p
dt
,
1
A
d A
∂t
=
d
Eδ
d p
dt
Substituting into the above formula, we get
1
ρ
∂ p
∂t
+ V
∂ p
∂s
+ C
2 ∂ V
∂s
= 0
Combined with the equations of motion, the chain differential equations
(wave equations of water hammer, first-order nonlinear partial differential
equations) representing water hammer are formed, namely
1
ρ
∂ p
∂t
+ V
∂ p
∂s
+ C
2 ∂ V
∂s
= 0
1
g
∂ V
∂t
+
∂z
∂s
+
1
γ
∂ p
∂s
+
1
g
V
∂ V
∂s
+
4τ w
γ D
= 0
In the calculation of water hammer, the piezometric head h(= z + p/ γ )
and pipeline velocity V are often used as independent variables, and the above
equations are changed into
∂h
∂t
+ V
∂h
∂s
+ V sin θ +
C 2
g
∂ V
∂s
= 0
∂ V
∂t
+ V
∂ V
∂s
+ g
∂h
∂s
+ λ
V |V |
2D
= 0
The wall shear stress of the pipeline is replaced by the following formula,
where λ is the resistance coefficient along the pipeline.
τ w = λρ
V 2
8