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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.