An Augmented IIM for Interface Problems in an Incompressible fluid

B. C. Khoo

Department of Mechanical Engineering Singapore-MIT Alliance National University of Singapore Collaborators: Zhijun Tan, D. V. Le, K.M. Lim

Outline

 Introduction  IIM for simulation of incompressible two-fluid interface • Jump Conditions across the Interface • • Numerical Algorithm Numerical Results  IIM for the dynamics of inextensible interfaces • • Numerical Algorithm Numerical Results  Conclusions

Introduction

 Peskin’s Immersed Boundary Method (IBM) • • • Fluid dynamics of blood flow in human heart Biological flows: platelet aggregation, bacterial organisms Rigid boundaries  Immersed Interface Method (IIM by LeVeque and Li) • • • • Elliptic equations, PDEs Stokes flows with elastic boundaries Navier-Stokes equations with flexible boundaries Streamfunction-vorticity equations on irregular domains

Peskin’s Immersed Boundary Method

• • • Use a discrete delta function to spread the force density to nearby Cartesian grid points.

F x

D h



k N

  1  

h

f

k



h



h

    4 1 0,

h

 

h



x



X

k

 

r

2

h

   , otherwise

r

 

s

 2

h

X

(s,t) Ω Γ(t) δΩ Ω + s Smearing out sharp interface of

O

(h).

First-order accurate for problems with non-smooth solutions.

Immersed Interface Method

• Incorporate the jumps in the solutions and their derivatives into the finite difference scheme near the interface

v x

 

v i

 1 

v i

 1 2

h

 { }

x i

 [ ],[ ],[

v xx

],   • • Avoid smearing out sharp interface Maintain second-order accuracy

(I) Navier-Stokes flows with discontinuous viscosity

• Incompressible Navier-Stokes Equations • The interface exerts singular force on the fluid  

s

• The motion of the moving interface satisfies

y x

    

X



n

      

Coupled Jump Conditions

Decoupled Jump Conditions

Numerical Algorithm:

Projection Method

A pressure-increment projection algorithm is employed on a MAC staggered grid

k

 1

u p v

 1

k u i

 1,

j p i

 1,

j v k

 1

u

-mesh point

v p

-mesh point -mesh point control point No need for pressure boundary conditions dealing with 

p

Projection Method: 3 steps

Correction terms in the projection method

Determination of q at control points

Matrix-vector multiplication

Interface Evolution: Moving Interface

Moving Interface: Implementation

Numerical Results: Exact solution

Numerical Results: Rotational Flow

• Ω = [-1, 1] × [-1, 1] • Interface: circle r = 0.5, located at (0, 0) • Force strength:

f

1  0,

f

2  0.1

• Viscosities:    0.01,    0.1

Numerical Results: Rotational Flow

Numerical Results: Elastic Membrane

• Ω = [-1.5, 1.5] × [-1.5, 1.5] • Semi-major axis: 0.75; semi-minor axis: 0.5

• Unstretched state: 0.5

• Elastic force:

f

  

s

 

T

0   

X



s

0

τ

 1   

Velocity and Pressure

Evolutions of semi-major and semi-minor axises   ( fixed)

Volume Conservation    0.1

   0.1,    0.01

Iterations in BFGS and GMRES

(II) Inextensible interface in Stoks Flows

• Incompressible Stokes Equations • The interface exerts singular force on the fluid

s

• The motion of the moving interface satisfies

y x

   

X



n

     

Interface constraint and singular force

• The inextensibility constraint for an evolving interface : • The force strength f exerted on the fluid: • An equivalent form: Schematic illustration of a 2D interface in shear flow

Finite difference MAC scheme with correction terms

(#) (#) Solved by FFT, Multigrid, PCG, etc

Determination of q at control points

Assuming that the tension q at the interface is known The velocity at the control points The surface divergenc at the control points: The surface divergence of the velocity at the interface can be written as

Matrix-vector multiplication

(*)

Evolving Inextensible Interface: Implementation

Numerical Results:

initially elliptical interface • Ω = [-3, 3] × [-1.5, 1.5] • Semi-major axis: 0.75; semi-minor axis: 0.5

• Initial orientation angle:   0 Pressure profile at steady state Streamline pattern at steady state

Shapes of deformed interface at different times

Initial (left) and final (right) shape of interface with different initial incidences.

Temporal evolution of orientation angle of interfaces with different initial incidences

Area conservation and arc length conservation

Numerical Results: initially concave interface

Initial orientation angle   0

Shapes of deformed interface at different times

Conclusions

 A second order accurate IIM for solving viscous incompressible flows with discontinuous viscosity is presented.

 An IIM is developed to simulate the dynamics of inextensible interface in a viscous fluid.

 Extend our IIM code to 3D problems.