Flow in pressurized conduits 71
4.3.3 Linearization of water hammer equations
By neglecting advection and frictional effects, the system of Equations 4.21
and 4.24 can be linearized to give
∂
∂
+
∂
∂
=
h
t
c
g
u
x
2
0
(4.25)
∂
∂
+
∂
∂
=
u
t
g
h
x
0
(4.26)
Multiplying Equation 4.26 by c 2 /g and taking the derivative with respect
to x, and by taking the derivative of Equation 4.25 with respect to t, one of
the two variables (u, h) can be eliminated, leading to either of the following
equations:
∂
∂
−
∂
∂
=
2
2
2
2
2
0
h
t
c
h
x
(4.27)
∂
∂
−
∂
∂
=
2
2
2
2
2
0
u
t
c
u
x
(4.28)
Equations 4.27 and 4.28 are linear hyperbolic equations verifying the fact
that the ‘signals’ of either the velocity or the pressure head propagate as
waves with speed c, equal to the celerity of elastic waves. For metal ducts
carrying water, this celerity is of the order of 10 3 m/s, suggesting that the
water hammer phenomenon develops in a matter of a few seconds, a time
frame sufficient for the development of high, sometimes destructive, pressures within the pipe.
4.3.4 Numerical treatment of water
hammer equations
In most operational applications, the computer simulation model is comprised of the linearized continuity equation and the momentum equation,
where advection is neglected but the nonlinear term of wall friction is
accounted for. The frictional term does not introduce instabilities but
allows the simulation of energy losses, leading to a final equilibrium condition. For the problem to be well-posed, the appropriate initial and boundary conditions are provided.
4.3.3 Linearization of water hammer equations
By neglecting advection and frictional effects, the system of Equations 4.21
and 4.24 can be linearized to give
∂
∂
+
∂
∂
=
h
t
c
g
u
x
2
0
(4.25)
∂
∂
+
∂
∂
=
u
t
g
h
x
0
(4.26)
Multiplying Equation 4.26 by c 2 /g and taking the derivative with respect
to x, and by taking the derivative of Equation 4.25 with respect to t, one of
the two variables (u, h) can be eliminated, leading to either of the following
equations:
∂
∂
−
∂
∂
=
2
2
2
2
2
0
h
t
c
h
x
(4.27)
∂
∂
−
∂
∂
=
2
2
2
2
2
0
u
t
c
u
x
(4.28)
Equations 4.27 and 4.28 are linear hyperbolic equations verifying the fact
that the ‘signals’ of either the velocity or the pressure head propagate as
waves with speed c, equal to the celerity of elastic waves. For metal ducts
carrying water, this celerity is of the order of 10 3 m/s, suggesting that the
water hammer phenomenon develops in a matter of a few seconds, a time
frame sufficient for the development of high, sometimes destructive, pressures within the pipe.
4.3.4 Numerical treatment of water
hammer equations
In most operational applications, the computer simulation model is comprised of the linearized continuity equation and the momentum equation,
where advection is neglected but the nonlinear term of wall friction is
accounted for. The frictional term does not introduce instabilities but
allows the simulation of energy losses, leading to a final equilibrium condition. For the problem to be well-posed, the appropriate initial and boundary conditions are provided.