Civil Engineering Reference
In-Depth Information
where, as usual, Einsteinian convention of summation of
repeated indices is respected. The classification of the local
topology in the space was analyzed by Chong
et al.
[CHO 90]. The nature of the eigenvalue solution to equation
[3.27] indicates that the surface
(
)
P
,
Q
,
R
(
)
(
)
[3.30]
2
3
3
2
2
27
R
+
4
P
−
18
PQR
+
4
QP Q
−
=
0
divides the invariants space into two zones: in the first zone,
the characteristic equation [3.27] accepts one real root and
two complex conjugate roots, while in the second, there are
three distinct real roots. If, in addition, we use the notations
, and to represent the eigenvalues of the velocity
gradient tensor, i.e. the roots of equation [3.27], we can
express the invariants in the form
λ
1
λ
λ
2
3
(
)
P
Q
R
=− + +
=++
=−
λλ λ
λλ
1
2
3
[3.31]
λλ
λ λ
12
13
23
λλ λ
123
In the case of incompressible flows that we are dealing
with in this topic, for
reasons of continuity. The problem is, therefore, limited to
the plane , and equation [3.30] is considerably
simplified in this case. The discriminant
(
)
= 0
P
=−∂
U
1
∂
x
1
+ ∂
U
2
∂
x
2
+ ∂
U
3
∂
x
3
()
Q
,
R
(
)
2
3
[3.32]
Δ=
27 4
R
+
Q
divides the plane into two parts. We obtain one real
eigenvalue and two complex conjugate eigenvalues for
and three distinct real eigenvalues at . The existence of
the conjugate eigenvalues indicates a focal-type topology
similar to that of the two-dimensional (2D) dynamic system
discussed in the previous section. The discriminant
()
Q
,
R
Δ>
0
Δ≤
0
Δ=
0
(
)
(
)
3/2
corresponds to the row
R
=±
239
−
Q
,
which also
Search WWH ::
Custom Search