some simple immersed boundary techniques for simulating complex flows with rigid boundary
DESCRIPTION
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 PresentationTRANSCRIPT
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