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

mclai@math.nctu.edu.tw

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