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The Fast Immersed Boundary Projection MethodThe Fast Immersed Boundary Projection Method
Kunihiko Taira, Clarence Rowley P i t U i itPrinceton University
Tim ColoniusCalifornia Institute of Technology
1Supported by US Air Force Office of Scientific Research
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The Immersed Boundary Method1
Momentum
Continuityboundary force
No-slip
• Flow field: Eularian (Cartesian grid)
• Body surf: Lagrangian
• Boundary force: enforce BC
• Delta function: regularization
21 Peskin (1972) – originally used for hemodynamics in heart
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Previous Force Calculations
Hooke’s law1 Feedback control2
Direct forcing3
Observations:Observations:• Need to tune ad hoc parameters (κ, α, β >> 1, stiffness)• Temporal offset in continuity & no-slip
31 Goldstein et al (1993), 2 Beyer & LeVeque (1998), 3 Mohd-Yusof (1997)
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Review of Projection Method
• Discretization of the NS eqs1
(staggered grid)(staggered grid)
• Projection/Fractional-Step Method2
momentum eq
pressure Poisson eqpressure Poisson eq
projection
41 Perot (1993), 2 Chorin (1968), Temam (1969)
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Review of Projection Method
• Observations
– Projection of u* to satisfy incompressibility – Pressure acts as a Lagrange multiplier
Constrained optimization problem– Constrained optimization problem (Karush-Kuhn-Tucker system)
Solution space with incompressibility satisfiedincompressibility satisfied
subject to
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Discrete Navier-Stokes Eqs with Immersed Boundary
Discrete Continuous
• Algebraically identical to traditional NS discretization, if E = HT
• Regularize Dirac delta function1g
61 Roma et al (1993)
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Immersed Boundary Projection Method
• Extend the traditional projection (fractional-step) method
LU decomposition
Immersed Boundary Projection Method1Immersed Boundary Projection Method1
Momentum equation
Modified Poisson equation
Projection
71 Taira & Colonius (2007)
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Comparison with Previous Methods
Hooke’s Law1
Artificial compressibility method2
Present approach requires no ad hoc parameters
(stiffness removed)(stiffness removed)
81 Beyer & LeVeque (1992), Lai & Peskin (2000); 2 Chorin (1967), Peyret (1976)
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Accuracy Assessment
2-D rotating cylinderstemporal spatial
• temporalAB2 + Crank-Nicolson
• spatial2nd finite vol + discrete delta fnc
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Validation
• 3D Flow around Low AR Wing (AR = 2)
3D simulation DPIV
Sim - 125x55x80- 150x66x96
250x88x128- 250x88x128
Exp -Thanks: Will Dickson
10Isosurface: ||ω|| = 3 with vortex core highlighted by Q = 3 1 Taira & Colonius (2009)
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Nullspace Approach
• Use of curl operator (in nullspace of div)
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Fast Immersed Boundary Projection Method
• Simplification can be made when grid is uniform• Eigenvalues of CTAC known for sine transformEigenvalues of C AC known for sine transform
Fast Immersed Boundary Projection Method1Fast Immersed Boundary Projection Method1
• Momentum eq solved• Small dimension (force eq)• Discrete streamfnc IBM
solution space ith li
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with no-slip satisfied
1 Colonius & Taira (2008)
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Far-Field BC with Multi Domains
• Computational domain size restricted to uniform grid• Remedy: multi domains1• Remedy: multi domains1
– cf. multi grid methods
• Coarsification– Larger domain to get accurate far field BC
• Interpolation– Use improved BC to find accurate inner soln– Use improved BC to find accurate inner soln
131 Colonius & Taira (2008)
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Speed-Up with the Fast IBPM
• Ex: flow over a cylinder (Re = 200)
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Fluid-Structure Interaction
• Couple of structural dynamics with predictor-corrector scheme• Ex: Vortex-induced vibration (Re = 250)Ex: Vortex-induced vibration (Re = 250)
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Summary
• Established IBM as a projection method B d f i d L lti li• Boundary force viewed as Lagrange multiplier
• No constitutive relations / incomp & no-slip enforced simultaneously• Nullspace approach in Fast IBPMNullspace approach in Fast IBPM• Multi-domain technique used for far-field BC• Fluid-structure interaction also possible• Fluidica© - Matlab toolbox
163D flapping wing
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References
T i & C l i (2007)• Taira & Colonius (2007)Journal of Computational PhysicsThe Immersed Boundary Method: A Projection ApproachThe Immersed Boundary Method: A Projection Approach
• Colonius & Taira (2008)Computer Methods in Applied Mechanics and EngineeringA Fast Immersed Boundary Method Using a Nullspace Approach and Multi-Domain Far-Field Boundary Conditionsy
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