3 Hydrodynamics
233
Compared with the energy equation of the constant total flow, there is one
more term of water head due to the acceleration of the fluid in the right term
of the equation, which is called inertia water head, expressed by hi. Namely
hi =
2
1
1
g
∂ V
∂t
ds
In a word, the differential equations of unsteady flow in pressurized pipes
can be obtained. Namely
∂(ρ A)
∂t
+
∂(ρ AV )
∂s
= 0
1
g
∂ V
∂t
+
∂z
∂s
+
1
γ
∂ p
∂s
+
1
g
V
∂ V
∂s
+
4τ w
γ D
= 0
The equations are suitable for the unsteady flow in a gradual pipeline,
including water hammer and oscillatory flow.
3.6.3 Water Hammer and Its Governing Equations
As early as 1898, the Russian scientist Jukowsky (as shown in Fig. 2.6)
established the relationship between the pressure increment and the velocity
change in elastic wave in the pipeline propagation based on the continuity equation and momentum theorem. For a simple pipe, the inlet end
is connected with the water tank, and the outlet end is provided with a
valve to regulate the flow, and the pipe length is L. In order to simplify the
problem, it is assumed that the pipeline is horizontal and the influence of
friction resistance is ignored. At this time, the piezometric headline under
the condition of the constant flow of pipeline is a horizontal line. When the
valve is fully opened, the flow in the pipe is constant, the flow rate is V 0 , and
the total head of the corresponding section is H 0 . Now the valve is suddenly
closed completely, resulting in water hammer wave (pressure wave) and wave
velocity C from the valve to the upstream, where the wave will cause the flow
rate to decrease and the pressure to increase. If the reference system is built
on the wave crest, the relationship between the hydraulic elements before
and after the liquid flow passing through the wave crest can be established
according to the continuity equation and momentum theorem. The relation
233
Compared with the energy equation of the constant total flow, there is one
more term of water head due to the acceleration of the fluid in the right term
of the equation, which is called inertia water head, expressed by hi. Namely
hi =
2
1
1
g
∂ V
∂t
ds
In a word, the differential equations of unsteady flow in pressurized pipes
can be obtained. Namely
∂(ρ A)
∂t
+
∂(ρ AV )
∂s
= 0
1
g
∂ V
∂t
+
∂z
∂s
+
1
γ
∂ p
∂s
+
1
g
V
∂ V
∂s
+
4τ w
γ D
= 0
The equations are suitable for the unsteady flow in a gradual pipeline,
including water hammer and oscillatory flow.
3.6.3 Water Hammer and Its Governing Equations
As early as 1898, the Russian scientist Jukowsky (as shown in Fig. 2.6)
established the relationship between the pressure increment and the velocity
change in elastic wave in the pipeline propagation based on the continuity equation and momentum theorem. For a simple pipe, the inlet end
is connected with the water tank, and the outlet end is provided with a
valve to regulate the flow, and the pipe length is L. In order to simplify the
problem, it is assumed that the pipeline is horizontal and the influence of
friction resistance is ignored. At this time, the piezometric headline under
the condition of the constant flow of pipeline is a horizontal line. When the
valve is fully opened, the flow in the pipe is constant, the flow rate is V 0 , and
the total head of the corresponding section is H 0 . Now the valve is suddenly
closed completely, resulting in water hammer wave (pressure wave) and wave
velocity C from the valve to the upstream, where the wave will cause the flow
rate to decrease and the pressure to increase. If the reference system is built
on the wave crest, the relationship between the hydraulic elements before
and after the liquid flow passing through the wave crest can be established
according to the continuity equation and momentum theorem. The relation