Civil Engineering Reference
In-Depth Information
matrix. The characteristic equation for a 3D flow can be
expressed in the following form ([CHO 90]):
3
2
[3.27]
λλλ
+++=
PQR
0
The parameters of the latter relation are
(
)
( )
Pa
=−
+
a
+
a tr
aa aa
=−
J
11
22
33
aa
1
2
(
)
()
11
12
11
13
22
23
2
2
Q
=
+
+
=
P
−
tr
J
aa
aa aa
21
22
31
33
32
33
1
1
2
[3.28]
=
Pa a
−
ik
ki
2
2
aaa
Raaa
11
12
13
1
(
)
()
()
3
3
=−
=−
det
J
=
−
P
+
3
Qtr
−
J
21
22
23
3
aaa
31
32
33
1
(
)
3
=−+
P
3
Qaaa
−
ik
kn
ni
3
By decomposing the tensor
into a symmetrical part
J
(
)
S
≡=
S
∂∂ ∂ ∂
Ux
+
Ux
2
ij
i
j
j
i
and an antisymmetrical part
(
)
,
A
≡=
AUx
∂∂ ∂ ∂
−
Ux
2
ij
i
j
j
i
by algebraic calculus, we find
PS
=−
ii
1
1
(
)
(
)
2
2
[3.29]
QPAAS S
=
−
−
=
PAAS S
+
−
ij
ji
ij
ji
ij
ij
ij
ij
2
2
1
(
)
R
=−+
P
3
3
PQS S
−
S
−
3
A A S
ij
jk
ki
ij
jk
ki
3
Search WWH ::
Custom Search