(18)
where
Integration of the nonlinear equations (18) we’ll execute by numerical
(19)
taking place along
Numerical integration of the equations (19) by method of characteristics is carried out as follows. On plane
Fig.4. Construction of direct and return characteristics
Let's consider the problem (18) – (19) solution by method of characteristics at the certain boundary conditions. We’ll enter in nodes of grid
of characteristics (Fig.4) the following designations:
t
)
=
_{
k
}
p
;
_{
ik
}
u
(
x
,
_{
i
}
t
) =
_{
k
}
u
;
_{
ik
}
λ
(
x
,
_{
i
}
t
) =
_{
k
}
λ
_{
}
. Then in view of boundary conditions
_{
ik
}
u
=
_{
ok
}
f
(
t
),
_{
k
}
u
=
_{
nk
}
g
(
t
) after exchanging in equations (19) derivatives by
finite-difference relations, and variables
_{
k
}
p, u
and
λ
– their average values in the neighbor nodes of grid on direct and return
characteristics [accordingly (
x
,
_{
i
}
t
), (
_{
k
}
x
_{
i
}
_{
-1
}
,
t
_{
k
}
_{
-1
}
) and (
x
,
_{
i
}
t
), (
_{
k
}
x
_{
i
}
_{
+1
}
,
t
_{
k
}
_{
-1
}
)], we’ll receive the following system of nonlinear algebraic equations:
(20) where
f
(
t
) and
_{
k
}
g
(
t
) – functions of time in boundary nodes of pipeline, defining
external influences acting on a liquid flow (for example, pulsation of flow because of the pump kinematics, volumetric or throttle regulation,
etc.);
_{
k
}
λ
_{
}
– coefficient of hydraulic resistance in node (
_{
ik
}
x
,
_{
i
}
t
):
_{
k
}
(21)
where
The received system of equations (20) – (21) is solved by an iterative method. |

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