Civil Engineering Reference
In-Depth Information
(
)
(
)
(
)
x xxxx
=
,,
,
yyxxx
=
,,
and
zzxxx
=
,,
. The metric
123
123
123
coefficients are
2
2
2
⎛
⎞
⎛
⎞
⎛
⎞
∂
x
∂
y
∂
z
[1.58]
h
=
+
+
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
i
∂
x
∂
x
∂
x
i
i
i
for
i
=
1, 2
and 3. The gradient is expressed by
G
G
G
G
e e
hxhxhx
∂
∂
e
∂
[1.59]
∇=
1
+
2
+
3
∂
∂
∂
1
1
2
2
3
3
G GG
(
)
The rotational (and the vorticity) in the system
eee
,,
123
is
G G G
he he he
11
2 2
3 3
G
G
G G G
G
∂
∂
∂
[1.60]
ωω
=+ + =∇∧=
e
ω
e
ω
e
u
11
2 2
3 3
∂
x x x
hu hu hu
∂
∂
1
2
3
11
2 2
3 3
The operator
2
is expressed by
∇
G
G
G
⎡
⎤
G
⎛
hh
⎞
⎛
hh
⎞
⎛
⎞
1
∂
∂ω
∂
∂ω
∂
hh
∂ω
2
[1.61]
∇=
ω
23
+
13
+
12
⎢
⎜
⎟
⎜
⎟
⎜
⎟
⎥
hhh
∂
x h
∂
x
∂
x h
∂
x
∂
x h
∂
x
⎢
⎥
⎝
⎠
⎝
⎠
⎝
⎠
⎣
⎦
123
1
1
1
2
2
2
3
3
3
Let us supplement these relations with the expression of
the divergence, which could be applied, for example, to the
velocity vector
G
(
)
uxxxt
,,;
123
G
G
⎡
⎤
1
∂
hhu
∂
hhu
∂
hhu
[1.62]
∇•
u
=
231
+
132
+
123
⎢
⎥
hhh
∂
x
∂
x
∂
x
⎣
⎦
123
1
2
3
The vorticity transport equation in the curvilinear
coordinates will be used (although not frequently) in
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