Simulating FSI Problems via the Curvilinear gImmersed Boundary Method
From Biofluids to Wind TurbinesFrom Biofluids to Wind Turbines
Iman Borazjani, Trung Le, SeokKoo Kang, Suresh
B h d F ti S ti lBehara, and Fotis Sotiropoulos
Computational Hydrodynamics & Biofluids GroupSaint Anthony Falls Laboratory
University of MinnesotaMinneapolis, MN
Immersed boundary Colloquium, AmsterdamJune 16, 2009
Sharp-Interface Cartesian/Immersed Boundary Approach
Gilmanov and Sotiropoulos, JCP, 2005
Arbitrarily complex 3D bodies are discretized with unstructured triangular mesh immersed in a gCartesian mesh
Immersed boundary treated as a h i t fFluid
Cartesian fluid mesh
sharp interface
Efficient node classification Borazjani et al JCP 2008
Fluid
Boundary
Solid
Unstructured
fluid mesh Borazjani et al., JCP, 2008
Velocities at the boundary are reconstructed via quadratic
Solid
body mesh interpolation in the normal direction2nd-order accurate - Gilmanov & Sotiropoulos, JCP, 2005
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
p , ,
Curvilinear Immersed Boundary FSI yMethod (CURVIB-FSI)
Moving immeresed boundaries over Body-fitted background mesh
Sharp interface treatment via 2nd
For details see:
Gilmanov and Sotiropoulos, A Hybrid Cartesian/Immersed Boundary Sharp interface treatment via 2nd-order accurate reconstruction
Fully-curvilinear, Cristoffel-symbol-free
p , y yMethod for Simulating Flows with 3D Geometrically Complex Moving Bodies J. Comp. Physics, 207 (2): 457-492, 2005
y , ynon-staggered grid fractional step method
Effi i t K l b d l
Ge and Sotiropoulos, A Numerical method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, J. Comp. Physics, Efficient Krylov-based solvers
MPI implementation
p , p y ,225(2), 1782-1809, 2007
Borazjani, Ge, and Sotiropoulos, Curvilinear Immersed Boundary Partitioned FSI approach: Strong coupling with dynamic under-relaxation
Convergence rate for discrete divergence of the velocity field (107
grid nodes)
j yMethod for Simulating Fluid Structure Interaction with Complex 3D Rigid Bodies, forthcoming, J. Comp. Physics, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Bi-leaflet Mechanical Heart Valves
Physiologic upstream flow waveform
Peak systole Re = 6000y
Plug inflow conditions
10×106 grid nodes, 2500 time steps per cyclecycle
PIV measurements obtained at the Georgia Tech CFM laboratory
FSI simulation
2 θ
θmin
Mdtdc
dtdI =+
θθ2
2 θ
θmax
I = 0.001 and c = 0
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Stability and Robustness of FSI SolveryBorazjani, PhD Thesis, 2008; Borazjani, Ge & Sotiropoulos, JCP, 2008
Due to small inertia even the strong-coupling might not converge.
Under-relaxation to stabilize:11 ~)1( ++ +−= lll QQQ αα
α = 0.7
Aitken acceleration to find stR+
<<1
20 α
α = 0.5I
Rs
fst ρ
ρ H=
under-relaxation coefficient:Aitken
1
11 ~
+
++
Δ
−=Δl
lll
QQQQ
1
1
11
1
)1(
+
+
++
−=
Δ−ΔΔ
−+=
l
ll
llll
QQQ
λα
λλλ
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
FSI Simulations Borazjani, PhD Thesis, 2008; Borazjani, Ge & Sotiropoulos, JCP, 2008
Q Criteria: ½(||Ω||2-||S||2) (H t 1988)
Out of plane vorticity
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
(Hunt 1988)
Out of Plane Vorticity Comparisons:Out of Plane Vorticity Comparisons:FSI vs. PIV
t=240 msect=240 msec
Borazjani, PhD ThesisPhD Thesis, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Out of Plane Vorticity Comparisons:Out of Plane Vorticity Comparisons:FSI vs. PIV
t=260 msect=260 msec
Borazjani, PhD ThesisPhD Thesis, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Out of Plane Vorticity Comparisons:Out of Plane Vorticity Comparisons:FSI vs. PIV
t=280 msect=280 msec
Borazjani, PhD ThesisPhD Thesis, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Out of Plane Vorticity Comparisons:Out of Plane Vorticity Comparisons:FSI vs. PIV
t 300t=300 msecBorazjani, PhD ThesisPhD Thesis, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Out of Plane Vorticity Comparisons:Out of Plane Vorticity Comparisons:FSI vs. PIV
t 440t=440 msecBorazjani, PhD ThesisPhD Thesis, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Out of Plane Vorticity Comparisons:Out of Plane Vorticity Comparisons:FSI vs. PIV
t 520t=520 msecBorazjani, PhD ThesisPhD Thesis, 2008
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
FSI Validation with In Vitro MeasurementsBorazjani PhD Thesis 2008; Borazjani et al JCP 2008Borazjani, PhD Thesis, 2008; Borazjani et al, JCP, 2008
Comparison of calculated leaflet kinematics with measurement in a straight aorta
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
measurement in a straight aorta
Anatomic AortaBorazjani et al, Ann. of Biomed Eng, Tentatively AcceptedSotiropoulos & Borazjani, Med. & Bio. Eng & Comp, 2009, review article
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Overset-CURVIB method
Overset gridOverset grid
Body-fitted grid for aorta and branches
Cartesian grid toCartesian grid to contain left ventricle
Left ventricle and the valve asthe valve as immersed boundary
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
BMHV Driven by the Ventricle MotionBMHV Driven by the Ventricle Motion
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Aquatic Swimming:Aquatic Swimming:Tethered Virtual Swimmers
Tethered LampreyBorazjani I and Sotiropoulos F
Tethered MackerelBorazjani I and Sotiropoulos F (2008) Borazjani, I. and Sotiropoulos, F.
(2009). Journal of Experimental Biology, 212, 576-592.
Borazjani, I. and Sotiropoulos, F. (2008). Journal of Experimental Biology 211, 1541-1558.
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Major Conclusions of Numerical Experiments with Tethered Swimmers
For each Reynolds number there is unique Strouhal number (St*) that self-propelled swimming is possible
fASTail-beat frequency f
St* is decreasing as Re increases
Lamprey (anguilliform) has higher efficiency in the transitional regime
UfSt = Tail-beat Amplitude A
Lamprey (anguilliform) has higher efficiency in the transitional regime (Re=4000) while mackerel (carangiform) in the inertial regime (Re=∞)
Lamprey (anguilliform) requires less power than mackerel (carangiform)Lamprey (anguilliform) requires less power than mackerel (carangiform) across all flow regimes!
Body shape or kinematics?
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
How to study effects of body shape and kinematics?kinematics?
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Body Shape and Kinematics Effects
KinematicsAnguilliform Carangiform
Mac
kere
l
pe
M
M k E l Mackerel
ody
Sha
p Macker-Eel Mackerel
Bo
yLa
mpr
ey
Lamprey Lamp-Rel
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Self-propelled Swimming H b id N i ti l F /I d B dHybrid Non-inertial Frame/Immersed Boundary
Borazjani, PhD Thesis, 2008
B d ki ti t il b t f d i it fBody kinematics, tail beat frequency f and viscosity of fluid are fixed
Th i i l it i d t i d b d th f th
ν
The swimming velocity is determined based on the forces on the fish body
Fdt
dumcm
=
Non-inertial frame of reference Conservative formulation -Developed by Beddhu et al (1996) and used by Kim & Choi (2006):
u 1⎞⎛ ∂ Ω
dt
uuwuvuu 2
Re1])[( ∇+−∇=+−⋅∇+⎟
⎠⎞
⎜⎝⎛∂∂ p
t r
kT tt uQuvu =+= )()(rr
Ω
r
rcm
ar tt
rw
ruv
uQuvu
×Ω=
×Ω+=
=+= )()(
Inertial frame
Non-inertial frame
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
r Inertial frame
Effect of Kinematics on Swimming Velocity (Re 4000)Effect of Kinematics on Swimming Velocity (Re~4000)
Borazjani & Sotiropoulos, Gallery of Fluid Motion winner, Phys. Fluids, 2009
The Camera move with the slower swimmerAnguilliform kinematics vs. carangiform kinematicsThe dots show the swimming speed of each swimmerThe dots show the swimming speed of each swimmer
Lamprey Body Mackerel Body
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Effect of Kinematics on Swimming Velocity (Inviscid)Effect of Kinematics on Swimming Velocity (Inviscid)
Borazjani & Sotiropoulos, Gallery of Fluid Motion winner, Phys. Fluids, 2009
The Camera move with the slower swimmerAnguilliform kinematics vs. carangiform kinematicsThe dots show the swimming speed of each swimmerThe dots show the swimming speed of each swimmer
Lamprey Body Mackerel Body
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
The Next Challenge: Swimming in Turbulent Flows
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Aquatic Plankton like PropulsionAquatic Plankton-like Propulsion
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Is the antenna deformation important?Is the antenna deformation important? Speed effect vs. Orientation effect
Rigid AntennaDeformable AntennaSt. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Rigid AntennaDeformable Antenna
Antenna Force ContributionBorazjani, PhD Thesis, 2008
Borazjani, Sotiropouols, Malkeil & Katz, J Exp Bio, under review
Th t
Drag
Phase Phase Ph PhaseThrust Phase 1
Phase 2
Phase3
Phase4
(CF)Rigid= -0.182 (CF)Deformable = -0.342St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
( F)Rigid ( F)Deformable
Stream Restoration
Cross vane Stone deflectors
Pictures from “Habitat improvement for trout streams”, Fish & boat commission PA 2007
Purpose: control of sedimentation and aquatic environments
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Fish & boat commission, PA, 2007
LES of Natural Meandering Streams g(Seokkoo Kang, PhD student, SAFL)
with in-stream structures no structures
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
AcknowledgementsAcknowledgements
Funding from NIH NCED NSF and resources formFunding from NIH, NCED, NSF and resources form Minnesota Supercomputing InstitutePofessor Sotiropoulos Trung Le SeokKoo KangPofessor Sotiropoulos, Trung Le, SeokKoo Kang, and Suresh BeharaExperimental data of BMHV flows provided by DrExperimental data of BMHV flows provided by Dr. Yoganathan at CFM lab Georgia TechExperimental data for Copepods provided by Dr. Yen p p p p yfrom Georgia Tech and Dr. Katz from Johns HopkinsLamprey CT scan was Dr. Fish, and Dr. Smits from p yPrinceton university
St. Anthony Falls LaboratoryUNIVERSITY OF MINNESOTA
Thank you!Thank you!