Civil Engineering Reference
In-Depth Information
(
)
.
The coupling term
L
is responsible for the non-orthogonality
of the eigenvectors [BUT 92, RED 93]. All the stable modes
can cause a primary transitory amplification before the
viscosity attenuates them, under the influence of the
coupling
L
. These modes are
subcritical
in relation to the
Tollmien-Schlichting modes. The transitory increase is
essentially algebraic, and appears to be a potential candidate
for bypass transition, provided the initial disturbance is
sufficiently intense and can survive the exponential decrease
stage. In summary, the mechanism of non-modal (non-
orthogonal) increase is linked to the non-normality of the
equations arising from the Orr-Sommerfeld and Squire
operators. This combination gives rise to a relation which
can formally be written as:
The operator
corresponds to
Δ≡
ddy
2
+
2
−
α
+
2
−
β
+
2
Δ
du
[]
dt
=
Lu
[5.80]
l
,
Consider the eigenvectors on the right
r
and left
j
[ ]
and
[ ]
*
defined, respectively, by
*
T
T
, where
λ
r
=
Lr
λ
l
=
l
L
ii
i
j
j
j
()
*
()
T
denote the conjugate and the transpose of the
vector. By multiplying the first equation by
and
l
on the left-
hand side and the second equation by
r
on the right-hand
side, and subtracting, we find that the eigenvectors on the
left and right are orthogonal, with
T
j
for
i
, and
rl
T
ij
=
0
≠
j
be the upper bound of the norm of
temporal variation. Look again at equation [5.80], with
T
jj
. Let
rl
≠
0
uut
=∂ ∂
max
du
N
N
∑
∑
[5.81]
=
β
r
u
=
α
r
ii
ii
dt
i
=
1
i
=
1
Equation [5.80] can be rewritten as
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