A Boundary Condition Capturing Immersed Interface Method

Sheng Xu

Math Department

Southern Methodist University

Taiwan-NCTS Workshop on FSI, May 2011

Motivations

• Applications: Insect flight, …

• Accuracy: Second-order

• Efficiency: O(NlnN)+O(M)

• Stability: A wide range of Re

qXxFvpvvt

v

B

JdRe

1)(

An immersed boundary/interface formulation of FSI:

Fixed Cartesian grids (Peskin, JCP 1972)

• Flow around a solid → Flow around a fluid-fluid interface

• BCs on a solid → Singular force on a fluid-fluid interface

Solid

Fluid

Fluid

Fluid

Mathematical Formulation: (Peskin, JCP 1972)

• Jump conditions induced by the singular force:

...,][,][,][,][... pvpv

• Generalized finite difference/interpolation:

2

1

2

0

)(

1

2

0

)(

11

!!2

1

2d

dhOs

n

gs

n

g

hh

sgsg

s

sg n

i

n

nn

i

n

n

iii

1isis 1is

h h

The immersed interface method for the force singularity:

2nd order accuracy (LeVeque & Li, SINUM 1994)

vv ct

v

r

pph cp r

• Explicit RK4:

• FFT Poisson solver:

An efficient explicit implementation:

O(NlnN)+O(M) (Xu & Wang, JCP 2006)

• Solids change only the right hand side of the linear

system from discretization.

• The coefficient matrix of the linear system is the same

as that for a lid-driven cavity flow.

F

F

Body-fitted grid methods BC-capturing IIM

nap

BB

Ren

(1) Dirichlet velocity BC

(2) Neumann pressure BC

(1) Rigid motion of enclosed

fluid & proper jump of shear

(2) Proper jumps of pressure &

normal derivative of pressure

SF

Boundary condition capturing for the force density:

Stability at high Re (Xu, JCP 2008 & AML 2009)

• Rigid motion of enclosed fluid by piecewise continuous body force:

analytical pressure and velocity inside

x

xxxtq c

,0

,d

d

• Proper jump of shear: tangential force density

nvv

nFFF

B

nb

nn Re

1

Re

1

Dirichlet velocity BC

• Proper jump of pressure: normal force density

nn

bqr

FFB

nn

,],[,2

2

2

2

1

2

Jd][Re

1nn qF

nIn 2D:

- Predictor:

• Proper jump of normal derivative of pressure:

][

~~

21

211

nqFF

Jp

n

- Corrector (optional):

LP ppp

Neumann pressure BC

nP

P

P

Fp

pp

pp

][nn

B

P

BB

L

L

ppp

p

nnn

0

(a) Poisson part by IIM:

(b) ND map by BEM/FMM:

)||(|BPBBLn pppFCorrection:

Corrector in 2D: ND map

(c) Laplace part by IIM:

nL

L

L

Fp

p

p

0

0

n

Pp

nF

Lp

Corrector in 3D: Augmented-variable approach (Li et al, CCCP 2006 )

(a) Discrete Laplace equation

grid vector Lp

marker vector

CpL

(b) Interior surface pressure

ECpEp LLs

(c) Linear system for the desirable correction

PPLs ppECppp

(d) GMRES to solve the linear system

• Right-hand side: (c)

• Matrix-vector product: (b)

Validation of boundary condition capturing

• 2D flapper at Re=157

• 3D flapper at Re=157

• 2D flapper at Re=1000

- w. corrector

-. w/o corrector

-- feedback control

1:0.25 2D sinusoidal flapper at Re=157:

Effect of the corrector and comparison with feedback control

fTt 10

p

fTt 10

6:1:0.25 3D sinusoidal flapper at Re=157:

Comparison with 2D

2D

(t=0.8)3D

(t=0.8)

1:0.25 2D sinusoidal flapper at Re=1000:

Stable at standard CFL conditions

- Re=1000

-- Re=157

angular accelerationppressure

• Discrete pressure Poisson equation

Csp p

][

~~

21

211

nqFF

Jp

n

• Angular momentum balance

0pE

• Linear system solved by GMRES

0psEECI

Coupling fluid & Newtonian dynamics for free moving solids:

Augmented-variable approach

Aspect ratio β = 1:0.2

Reduced moment of inertia I*=0.48

Reynolds number Re=737

Test of the coupling: Tumbling of a falling plate

- simulation

-- experiment

Experiment (Andersen et al, JFM

2005)

Simulation

<u> 0.60 0.58

<v> -0.34 -0.37

<ω> 0.88 0.99

Decent slope 29.2 29.6

Short glide 1.1 1.2

Long glide 3.2 3.5

• Accuracy: 2nd for the velocity, 1st /2nd for the pressure w./wo. corrector

• Efficiency: O(N ln N)+O(M) without the corrector

• Stability: Stable at high Re with standard CFL conditions

• Coupling aerodynamics and Newtonian dynamics: Augmented-variable

approach with GMRES

Summary

Thanks

• The organizers for this wonderful workshop which gives me the

opportunity to present my work and visit Taiwan for the first time.

• Many of you in the audience for your help with my research and career.

• NSF for the financial support of this work.

• All of you for listening to my talk.