Environmental Engineering Reference
In-Depth Information
is the flow map that associates times
t
0
and
t
with the positions of fluid particles.
(Dependencies on
t
0
and
t
will herein be omitted for notational simplicity.) For any
smooth
v(
x
,
t
)
,
F
(
x
0
)
represents a diffeomorphism, which ensures invertibility of
D
F
(
x
0
)
and thus positive definiteness of
C
(
x
0
)
. Furthermore, incompressibility of
v(
x
,
t
)
implies det
C
(
x
0
)
=
1. Consequently, eigenvalues and normalized eigenvec-
tors of
C
(
x
0
)
satisfy:
1
ʻ
1
(
0
< ʻ
1
(
x
0
)<ʻ
2
(
x
0
)
≡
x
0
)
,
ʾ
1
(
x
0
)
↥
ʾ
2
(
x
0
).
(4)
Normal repulsion to a material line
ʳ
0
at time
t
0
with unit normal
n
0
is measured
by the local normal growth of
n
0
(Haller
2011
):
1
ˁ(
x
0
,
n
)
:=
,
(5)
x
0
)
−
1
n
0
n
0
,
C
(
x
0
))
≡
√
ʻ
2
(
where
·
,
·
is the Eucl
idean in
ner product. Note that
ˁ(
x
0
, ʾ
2
(
x
0
)>
1
√
ʻ
1
(
and
ˁ(
x
0
, ʾ
1
(
x
0
))
≡
x
0
)<
1. Accordingly, a material line everywhere tan-
gent to
ʾ
1
(
]isreferredtoasa
squeezeline
(resp.,
stretchline
).
Squeezelines obtained from backward (resp., forward) time integration and stretch-
lines obtained from forward (resp., backward) time integration attract (resp., repel)
nearby fluid trajectories. These material lines satisfy the following duality property
(Farazmand and Haller
2013
): squeezelines (resp., stretchlines) at
t
0
=
x
0
)
[resp.,
ʾ
2
(
x
0
)
t
+
obtained
from forward integration out to
t
=
t
−
map to backward stretchlines (resp., squeeze-
lines) at
t
0
=
=
t
+
.
Finally, material shear across
ʳ
0
is measured by the local tangential growth of
n
(Haller and Beron-Vera
2012
):
t
−
obtained from backward integration out to
t
0
−
1
10
x
0
)
−
1
˃(
x
0
,
n
0
)
:=
n
0
,
C
(
ʩ
n
0
ˁ(
x
0
,
n
0
),
ʩ
:=
.
(6)
Geometrical representations of the normal repulsion and Lagrangian shear are given
in Fig.
1
.
t
0
,t
ˁ
n
0
n
t
D
F
t,t
0
(
x
0
)
n
0
x
t
e
0
x
0
ʳ
t
ʳ
0
t
0
,t
˃
e
t
F
t
0
,t
Fig. 1
A material curve
ʳ
0
at time
t
0
is advected by the flow
F
t
0
,
t
to a curve
ʳ
t
at time
t
. At time
t
0
,
t
t
0
,
t
t
0
, normal repulsion,
ˁ
(
x
0
,
n
0
)
, over the interval
[
t
0
,
t
]
are
normal and tangential projections onto
ʳ
t
of the linearly advected normal to
ʳ
0
,D
F
t
0
,
t
(
x
0
,
n
0
)
, and Lagrangian shear,
˃
(
x
0
)
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