Advanced Huid Dynamics
College 1
Part 4
Vorticity Dynamics
Vorticity and circulatie
Definition
•
Consider a flow ,
character ized b III. t ) =
( v.v ,
w )
the work
City as the Carl of the
Velocity
ad ( E. t) × ICE t ) ,
In cartesion coordinator
÷:
÷
Hou is a related to it ?
↳ ( Io t ) =L ( Xo ) t
É ( Ik -
Ik ) DX t É ( It If
-
it ) DX
try
-
translation
¥ rate of Strain tensor
= deflatie
= Rotaties
÷
÷÷÷÷: ÷
÷
÷
.
In -
DI = É e X EX
-
Rotatie where
rotatie vector is te
ij -
,-
couette flow : d- '
µ)
â
#â
€
→
←
IEË H -
÷÷÷÷÷÷÷÷÷÷÷:*
F-
ZD
-
:
§ tal
Congothe
C-
flow
-
ytly
# g
ctfdlt
t to
- -
④
g- y
- -
¥ XÉIX
T
§ Edf ! kaft ! Edf
dftsudf
=
=
-
Lidl dealt
= dx Î
=D
g
ÏÏ . . .
ydx
"
t dig "
t .
. .
VJDX
0
.jp#nseIEE
Stokester ( correctie between
workcity
& circulatie )
,
#
F-
§ tot
=
notie
vorti
Cityfield e
( Aftands )
.
ijij IJ
, t.us#F
Examt ê
=
§ Vêêdxdy
-
} Vdxdy =
✓ DX
B
-
= -
Irrotational flow :
A flow such that
{ If ¥] ( Carl of vetocity )
( no
vorticity)
ij
, Advanced Huid Dynamics
College 2
Vorti
City equator
•
assumep =
constant (incompressie flow)
•
assume inertie frame of reference
•
bodyforce f a Conservative :
f =
-
I 4
( mits)
µ dynamic viscose ( kg 1ms) ✓ %
=
:
•
,
The is defined ( I. t) I X UCI t)
vorticity as : =
,
px (Pl # te Ilt ) = -
IP thee -14 )
① ② ③ ④ ⑤
① px (E) =p Elixir ) =p # e
②I X (PHIL ) =P ( t.hu ) -
text) Ik HIHI -
)
xd
incontest
te te
µ
A- Ik
-
-
Peek I. e- o (
⇐÷÷÷÷÷÷÷÷÷
⑤ Extra ) 0
1=0
=
§ Idf →
[ ( Ixftnds
I XD
③ Ix (IP ) = 0
④ Extra ) =D (Ixus )
'
'
e
p 3¥ PEI =p ⇐ Ik twee
IE t
the =
elk twee
DIE =
E- Ik
Source
twee
(
term
/ mits =
diffusie constant
adrector of
vorticity
(
vorticity transported
with the flow)
College 1
Part 4
Vorticity Dynamics
Vorticity and circulatie
Definition
•
Consider a flow ,
character ized b III. t ) =
( v.v ,
w )
the work
City as the Carl of the
Velocity
ad ( E. t) × ICE t ) ,
In cartesion coordinator
÷:
÷
Hou is a related to it ?
↳ ( Io t ) =L ( Xo ) t
É ( Ik -
Ik ) DX t É ( It If
-
it ) DX
try
-
translation
¥ rate of Strain tensor
= deflatie
= Rotaties
÷
÷÷÷÷: ÷
÷
÷
.
In -
DI = É e X EX
-
Rotatie where
rotatie vector is te
ij -
,-
couette flow : d- '
µ)
â
#â
€
→
←
IEË H -
÷÷÷÷÷÷÷÷÷÷÷:*
F-
ZD
-
:
§ tal
Congothe
C-
flow
-
ytly
# g
ctfdlt
t to
- -
④
g- y
- -
¥ XÉIX
T
§ Edf ! kaft ! Edf
dftsudf
=
=
-
Lidl dealt
= dx Î
=D
g
ÏÏ . . .
ydx
"
t dig "
t .
. .
VJDX
0
.jp#nseIEE
Stokester ( correctie between
workcity
& circulatie )
,
#
F-
§ tot
=
notie
vorti
Cityfield e
( Aftands )
.
ijij IJ
, t.us#F
Examt ê
=
§ Vêêdxdy
-
} Vdxdy =
✓ DX
B
-
= -
Irrotational flow :
A flow such that
{ If ¥] ( Carl of vetocity )
( no
vorticity)
ij
, Advanced Huid Dynamics
College 2
Vorti
City equator
•
assumep =
constant (incompressie flow)
•
assume inertie frame of reference
•
bodyforce f a Conservative :
f =
-
I 4
( mits)
µ dynamic viscose ( kg 1ms) ✓ %
=
:
•
,
The is defined ( I. t) I X UCI t)
vorticity as : =
,
px (Pl # te Ilt ) = -
IP thee -14 )
① ② ③ ④ ⑤
① px (E) =p Elixir ) =p # e
②I X (PHIL ) =P ( t.hu ) -
text) Ik HIHI -
)
xd
incontest
te te
µ
A- Ik
-
-
Peek I. e- o (
⇐÷÷÷÷÷÷÷÷÷
⑤ Extra ) 0
1=0
=
§ Idf →
[ ( Ixftnds
I XD
③ Ix (IP ) = 0
④ Extra ) =D (Ixus )
'
'
e
p 3¥ PEI =p ⇐ Ik twee
IE t
the =
elk twee
DIE =
E- Ik
Source
twee
(
term
/ mits =
diffusie constant
adrector of
vorticity
(
vorticity transported
with the flow)
