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

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

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)

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)

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

5

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)

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)

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)

Accuracy Assessment

2-D rotating cylinderstemporal spatial

• temporalAB2 + Crank-Nicolson

• spatial2nd finite vol + discrete delta fnc

9

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)

Nullspace Approach

• Use of curl operator (in nullspace of div)

11

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

12

with no-slip satisfied

1 Colonius & Taira (2008)

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)

Speed-Up with the Fast IBPM

• Ex: flow over a cylinder (Re = 200)

14

Fluid-Structure Interaction

• Couple of structural dynamics with predictor-corrector scheme• Ex: Vortex-induced vibration (Re = 250)Ex: Vortex-induced vibration (Re = 250)

15

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

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

17