Some simple immersed boundary techniques for simulating complex flows with rigid boundary
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Transcript of Some simple immersed boundary techniques for simulating complex flows with rigid boundary
Some simple immersed boundary Some simple immersed boundary techniques for simulating complex techniques for simulating complex
flows with rigid boundaryflows with rigid boundary
Ming-Chih Lai
Department of Applied Mathematics
National Chiao Tung University
1001, Ta Hsueh Road, Hsinchu 30050
Taiwan
Outline of the talk:
• Review of the Immersed Boundary Method (IB method)
• Immersed boundary formulation for the flow around a solid body
• Feedback forcing + IB method
• Direct forcing approach
• Volume-of-Fluid approach
• Interpolating forcing approach
• Numerical results
Review of the IB method:
A general numerical method for simulations of biological systems
interacting with fluids (fluid interacts with elastic fiber).
Typical example: blood interacts with valve leaflet (Charles S.
Peskin, 1972, flow patterns around heart valves)
Applications:
• computer-assisted design of prothetic valve (Peskin & McQueecomputer-assisted design of prothetic valve (Peskin & McQueen)n)
• Platelet aggregation during blood clotting (Fogelson, Fausi)(Fogelson, Fausi)
• flow of particle suspensions (Fogelson & Peskin, Sulsky & Brac(Fogelson & Peskin, Sulsky & Brackbill)kbill)
• wave propagation in the cochlea (Beyer)(Beyer)
• swimming organism (Fausi)(Fausi)
• arteriolar flow (Arthurs, et. al.) (Arthurs, et. al.)
• cell and tissue deformation under shear flow (Bottino, Stockie &(Bottino, Stockie &
Green, Eggleton & Popel)Green, Eggleton & Popel)
• flow around a circular cylinder (Lai & Peskin)(Lai & Peskin)
• valveless pumping (Jung & Peskin)(Jung & Peskin)
• flapping filament in a flowing soap film (Zhu & Peskin)(Zhu & Peskin)
• falling papers, sails, parachutes, insect flight, ……
Recent review : C.S. Peskin, Acta Numerica, pp 1-39, (2002). Recent review : C.S. Peskin, Acta Numerica, pp 1-39, (2002).
Idea:
Mathematical formulation:• Treat the elastic material as a part of fluid.
• The material acts force into the fluid.
• The material moves along with the fluid.
Numerical method:• Finite difference discretization.
• Eulerian grid points for the fluid variables.
• Lagrangian markers for the immersed boundary.
• The fluid-boundary are linked by a smooth version of Dirac d
elta function.
Consider a massless elastic membrane immersed in viscous
incompressible fluid domain ,
: ( , ), 0
: unstressed lengthb
b
s t s L
L
Mathematical formulation
Equations of motion:
( )
0
( , ) ( , ) ( ( , ))
( , ) ( ( , ), ) ( , ) ( ( , ))
pt
t s t s t ds
s ts t t t s t
t
uu u u f
u =
f x F x
u u x x
( , ) ( ), ( ; , ),
( , ) : velocity ( , ) : boundary configuration
( ,
d
ss t T T s t
s s s
t s t
p
x
F
FLUID BOUNDARY
u x
x
) : pressure ( , ) : boundary force
: density : tension
: viscosity : unit tangent
t s t
T
F
( , ) ( , ) ( ( , ))
behaves like a one-dimensional delta function.
, ( , ) ( , )
( , ) ( ( , )) ( , )
t s t s t ds
t t d
s t s t ds t d
f x F x
f
f w f x w x x
F x w x x
( , ) ( , ) ( ( , ))
( , ) ( ( , ), )
If ( , ) ( , ), then
the total work done by the boundary the total work done on the fluid.
Thus, the solutio
s t t s t d ds
s t s t t ds
t t
F w x x x
F w
w x u x
n is NOT smooth. In fact, the pressure and velocity
derivatives are discontinuous across the boundary.
The force density is singular !
The pressure and the velocity normal derivative across the
boundary satisfy
,
.
T
ps
s
Theorem 1 :
F n
u F
n
Theorem 2 :
he normal derivative of the pressure across the boundary
satisfies
( )
.s sp
s
F
n
* Physical meaning of the pressure jump.
1 1
1 2 1 2 1 2
1 2 1 2
How to march ( , ) to ( , ) ?
Compute the boundary force
( ), , ( ) ,
where and are both defined on ( 1 2
n n n n
nn n n n ns kk s k k k s kn
s k
n nk k
DT D D T
D
T s k
u u
Step1:
F
) , and is defined on .
Apply the boundary force to the fluid
( ) ( ) .
Solve the Navier-Stokes equations with the f
nk
n n nk h k
k
s s k s
s
F
Step2 :
f x F x
Step3 :
1 2 20 1 1
1 1
0 1
1 2 1 2
orce to update the velocity
( ) ,
0.
where and are both defined on
n nn n n n ni i i i
i i
n
n nk k
u D D p D Dt
D
T
u uu u f
u
( 1 2) , and is defined on . nks k s s k s F
Numerical algorithm
1 1 2
Interpolate the new velocity on the lattice into the boundary points and
move the boundary points to new positions.
( )
n n nk h k h
x
Step4 :
U u x
1 1 .n n nk k kt U
( ) ( ) ( )
1. is a positive and continuous function.
2. ( ) 0, for 2
1 3. ( ) 1, for all ( ( ) ( ) )
2
4. ( ) ( ) 0, for all
5
h h h
h
h
h j h j h jjj even j odd
j h jj
x y
x x h
x h x h x h
x x h
x
22
2
2
. [ ( ) ] , for all ( ( ) ( ) )
3Uniquely determined:
81
(3 2 1 4 4( ) ) ,81
(5 2 7 12 4( ) ) 2 ,( )8
0
h j h j h jj j
h
x h C x x h C
C
x h x h x h x hh
x h x h x h h x hxh
otherwise.
hDiscrete delta function
Numerical issues of IB method:
• simple and easy to implement
• first-order accurate
• numerical smearing near the immersed boundary
• high-order discrete delta function
• numerical stability, semi-implicit method
IB formulation for the flow around a solid body
( ) : 0 bs s L
u
the fluid feels the force along the body surface to stop it !
1 ( )
0
( , ) ( ( ), ) ( ( ))
pt Re
t s t s
uu u u f
u =
f x F x 0
0 ( ( ), ) ( , ) ( ( ))
( , ) as
bL
b
ds
s t t s d
t
u u x x x
u x u x
11 1 1
1
1 1h
1
1
1 ( ) ,
0,
( ) ( ) ( ) , for all
( )
n nn n n n n
n
Mn n
j jj
nb k
pt Re
s
u uu u u f
u
f x F X x X x
U X 1 2h ( ) ( ) , 1, 2,... .n
k h k M x
u x x X
The boundary force ( ( ), ) is unknown and it must
be applied exactly to the fluid so the no-slip condition is satisfied.
Goldstein, Handler and Sirovich, 1993
Saiki and Biringe
s t
Main difficulty : F
n, 1996
Ye, Mittal, Udaykumar and Shyy,1999
Fadlum, Verzicco, Orlandi and Mohd-Yusof, 2000
Lai and Peskin, 2001
Kim, Kim and Choi, 2001
Lima E Silva, Silveira-Neto and Damasceno, 2003
Ravoux, Nadim and Haj-Hariri, 2003
Su, Lai and Lin, 2004
Denote ( , ) : Lagrangian markers ( : marker spacing)
( , ) : Cartesian grid points ( : grid spacing)
( ) : Boundary force at
k k k
i j
k
X Y s
x y h
X
x
F X
h1
2h
marker
Force distribution : ( ) ( ) ( )
Velocity interpolation : ( ) ( ) ( )
( ) ( ) ( )
with
k
M
j jj
k k
h k h i k h j k
s
h
d x X d y Y
x
X
f x F X x X
u X u x x X
x X
(1- ) ( )
0 otherwise.h
r h h r hd r
Feedback forcing + IB method : Goldstein, et. al., 1993. Saiki and Biringen, 1996.
La1
( ) ( , ) ( ( , )) ,
0,
( , ) ( ( ) ( , )), 1
( ,
e
p s t s t dst Re
s t s s t
s
u
u u u F x
u
F
i & Peskin, 2000.
)
( ( , ), ) ( , ) ( ( , )) ,
( , ) as .
Treat the body surface as a nearly rigid boundary (stiffness is large)
embedded in the fluid.
ts t t t s t d
tt
u u x x x
u x u x
Allow the boundary to move a little but the force will bring it back to
the desired location.
Simple but very small time step is needed!
Direct forcing approach : Mohd - Yusof, 1997. Fadlum, et. al., 2000. Lima E Silva et. al.,
Compute the boundary force at marker directly from the Navier - stokes
equations.
( ) 1 ( ) ( ) ( ) ( ) ( )
A complicated i
n nn n n nk
k k k kps t Re
Step1 :
F uu u u
2003.Lima E Silva et. al., 2003.
h
nterpolation procedure must be employed!
Distribute the boundary force at marker ( ) into the Eulerian grid
via the discrete delta function.
( ) ( ) ( )
nk
n nj j
Step2 : F
f x F X x X
1
Solve the Navier -Stokes equations on the Eulerian grid with the Eulerian
force obtained from Step2.
M
j
s
Step3 :
first - order projection method
staggered grid
Time discretization :
Spatial discretization :
Volume-of -Fluid (VOF) approach : Ravoux et. al., 2003.
Prediction step
1 ( ) .
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
( ) 0.24 cos(2 ).
c q q
b k c
f f f
U D f t