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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