Civil Engineering Reference
In-Depth Information
The method of characteristics is a finite difference technique where pressures were
computed along the pipe for each time step. Calculation automatically sub-divided the
pipe into sections (intervals) and selected a time interval for computations.
(
)
(
) (
)(
)
dp
/
dt
=∂∂+∂∂
p
/
t
p
/
s
ds
/
dt
,
(4)
(
)
(
) (
)(
)
(5)
dv
/
dt
=∂∂+∂∂
v
/
t
v
/
s
ds
/
dt
,
P
and
V
changes due to time are high and due to coordination are low then it can be
neglected for coordination differentiation:
(
) (
)(
)
(
) (
)
∂∂+ ρ∂ ∂+
v
/
t
1/
p
/
s
g dz
/
ds
+
f
/ 2
D v v
= (Euler equation)
0,
(6)
(
) (
)(
)
2
∂∂+ ∂∂= (Continuity equation), (7)
By linear combination of Euler and continuity equations in characteristic solution
Method:
(
C vs
/
1/
P Pt
/
0,
) (
)(
)
(
) (
)
(
) (
)(
)
2
λ∂∂+ ρ∂∂+
v
/
t
1/
p
/
s
g dz
/
ds
+
f
/ 2
D v v
+ ∂∂+ ∂∂=
C
v
/
s
1/
p
p
/
t
0,
(8)
+
−
λ=
c
&
λ=
c
(
) (
)(
)
(
) (
)
dv
/
dt
+
1/
cp
dp
/
ds
+
g dz
/
ds
+
f
/ 2
D v v
=
0,
(9)
(
) (
)(
)
(
) (
)
=
(10)
dv
/
dt
− ∂∂+
1/
cp
p
/
s
g dz
/
ds
+
f
/ 2
D v v
0,
Method of characteristics drawing in (s-t) coordination:
(
) (
)(
)
(11)
dv
/
dt
−
g
/
c
dH
/
dt
=
0,
(
)
dH
=
c
/
g dv
,
(Joukowski Formula),
(12)
By Finite Difference method:
(
)
(
)
(
)
(
=
(13)
)
(
)
(
)
c
+ − −+ − −+
:(
vp
v
Tp
0) (
g
/
c
Hp
H
/
Tp
0) (
fv
v
/2 )
D
0
Le
Le
Le
L e
=
(14)
(
)(
)
(
)(
) (
)
(
)
c
− − −+ − −+
:(
vp
vRi
Tp
0) (
g
/
c
Hp
HRi
/
Tp
0) (
fvR i vR i
/2 )
D
0
)
(
)
(
)
(
)
(
)
(
c
+ − + − +∆
:
vp
v
g
/
c
Hp
H
f
t
fv
v
/2
D
)
=
0
(15)
Le
Le
Le
Le
)
(
)
(
) (
)(
) (
c
− − + − +∆
:
vp
vRi
g
/
c
Hp
HRi
f
t
v
v
/2
D
)
=
0
(16)
Ri
Ri
(
) (
)(
)
V
++
V
gc H
/
-
H
Le
ri
Le
ri
V
=
1/ 2
,
(17)
)
(
)
(
P
-
f t DV
∆
/2
V
+
VV
Le
Le
ri
ri
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