Some simple immersed Some simple immersed boundary techniques for boundary techniques for simulating complex simulating complex flows with rigid flows with rigid boundary boundary Ming-Chih Lai Department of Applied Mathematics National Chiao Tung University 1001, Ta Hsueh Road, Hsinchu 30050 Taiwan [email protected]
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Some simple immersed boundary techniques for simulating complex flows with rigid boundary
Some simple immersed boundary techniques for simulating complex flows with rigid boundary. Ming-Chih Lai Department of Applied Mathematics National Chiao Tung University 1001, Ta Hsueh Road, Hsinchu 30050 Taiwan [email protected]. Outline of the talk :. - PowerPoint PPT Presentation
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Some simple immersed boundary Some simple immersed boundary techniques for simulating complex techniques for simulating complex
flows with rigid boundaryflows with rigid boundary
Update the velocity by the influence of the body force
,
nn n n
t Re
t
Step1:
u uu u u
Step2 :
u uf
,
, , ,
, ,
,
(1 ) ,
.
where is the volume fraction of the solid object in the ( , ) cell.
Projection s
i j
i j i j i j
i j i j
i j
ti j
u u
u f
Step3 :1 1
1
tep
,
0.
n n
n
t p
u u
u
,
,
,
,
Define a volume fraction field as
( ) ,
( )
1 if cell ( , ) is inside the object,
0 if cell ( , ) is o
i j
i j
i j
i j
vol solid part
vol cell
i j
i j
1 12 2
,
, 1, , , 1, ,
utside the object,
(0,1) if the object boundary cuts through the cell ( , ).
Furthermore, we define
0.5( ), 0.5( ).
i j
x yi j i j i j i ji j i j
i j
12, 1,, x
i j i ji j
12
, 1
,
,
i j
yi j
i j
* 11 1 *
Prediction step
3 4 1 2( ) ( ) ,
2Modification step
n nn n n n np
t Re
Step1:
u u uu u u u u
Step2 :
Second -order projection + VOF approach
,
, , ,
, ,
,2 3
(1 ) ,
3 .
2 Projection step
i j
i j i j i j
i j i j
t
t
u uf
u u
u f
Step3 :
1
1
1 1
,2 3
1 ( ).
nn
n n n
t
p pRe
u u
u
To compute the force ( ) accurately in the modification step,
so the prescribed boundary velocit
f x
Interpolating forcing approach : Joint work with C.- A. Lin and S.- W. Su, 2004.
Idea :y can be achieved.
Denote ( , ), Lagrangian markers
( , ), Cartesian grid points
( ) ( ) Let ( ), we need
k k k
i j
X Y
x y
t
X
x
u x u xf x ( ) ( ).k b k
u X u X
h
Interpolating forcing procedures:
(1) Find the boundary force ( ), 1, 2,..., .
(2) Distribute the force to the grid by the discrete delta function
( ) ( ) (
k
j j
k M
F
f x F X x X
1
2
2
1
) .
( ) ( )(3) ( ) (Thus, ( ) ( ).)
( ) ( )( ) ( )
( ( ) ( ) ) ( )
M
j
k b k
b k kh k
M
j h j h kj
s
t
ht
s h
x
x
u x u x f x u X u X
u uf x x
F x x
2
1
( ( ) ( ) ( ).M
h j h k jj
sh
x
x x )F
h1
2 2
Why don't we just use the marker forcing directly?
( ) ( )(1) ( )
(2) ( ) ( ) ( )
( ) ( )(3) ( )
( ) ( ) ( ) ( )(
b k kk
M
j jj
h k h k
st
s
t
h h
t
x x
Q :
u uF
f x F X x X
u x u xf x
Interpretation :
u x x u x xf
2
2
1
2 2
1
) ( )
( ) ( ) ( ( ) ( ) ) ( )
( ) ( ( ) ( )
( )
h k
Mk k
j h j h kj
Mj
h j h kj
k
h
s ht
s hs
s
x
x
x
x x
u uF x x
Fx x )
F
( ) ( ).k b k u u
Numerical Results
• Decaying vortex problem• Lid-driven cavity problem• A cylinder in lid-driven cavity• Flow around a circular cylinder• The flow past an in-line oscillating
cylinder
2
2
2
2 Re
2 Re
4 Re
( , , ) cos( )sin( ) ,
( , , ) sin( )cos( ) ,
1 ( , , ) (cos(2 ) cos(2 )) .
4
An immersed boundary virtually e
Decaying vortex.
t
t
t
u x y t x y e
v x y t x y e
p x y t x y e
Example 1:
2 2
xists in a form of the unit
circle ( 0.25) in [ 0.5,0.5] [ 0.5,0.5], such
that the velocity is prescribed.
(CFL 0.5, 1 N, 0.5 2 , = 4, Re=100).
x y
h s N s h
[ 1,1] [ 1,1].
1 Cavity position .
2
Lid -driven cavity,
x y
Example 2 :
Example 3 : A cylinder in the driven cavity.
1The figure of quiver with 100, .101Re h
1The figure of quiver with 1000, 100Re h
0, 0, 0y yu v p
0
0
0
x
x
x
u
v
p
13.4 16.5
0, 0, 0y y
D D
u v p
8.35
8.35
1
0
0x
D
D
u
v
p
X
Y D
Flow around a circular cylinderExample 4 :
A non-uniform grid (250 160) is adopted in .
A uniform grid (60 60) is in the region near the cylinder.
12
2
Drag coefficient:
, where x.2
Lift coefficient:
, where 2
DD D
LL
FC F f d
U D
FC
U D
Interesting quantities
2 x.
Strouhal number:
, where is the vortex shedding frequency.
D
qt q
F f d
f DS f
U
We consider the in-line oscillating cylinder in uniform flow at
Re 100 and the cylinder is now oscillating parallel to the free
stream at a fre
.
Example 5 : The flow past an in - line oscillating cylinder
quency 1.89 , where is the natural vortex
shedding frequency. The motion of the cylinder is prescirbed by
setting the horizontal velocities on the Lagrangian markers to