An Augmented IIM for Interface Problems in an Incompressible fluid

1. 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

2. 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

3. 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

4. Peskin’s Immersed Boundary Method • Use a discrete delta function to spread the force density to nearby Cartesian grid points. • Smearing out sharp interface of O(h). • First-order accurate for problems with non-smooth solutions. Ω+ X(s,t) s Ω- Γ(t) δΩ

5. Immersed Interface Method • Incorporate the jumps in the solutions and their derivatives into the finite difference scheme near the interface • Avoid smearing out sharp interface • Maintain second-order accuracy

6. (I) Navier-Stokes flows with discontinuous viscosity • Incompressible Navier-Stokes Equations • The interface exerts singular force on the fluid • The motion of the moving interface satisfies

7. Coupled Jump Conditions

8. Decoupled Jump Conditions

9. -mesh point -mesh point -mesh point control point Numerical Algorithm:Projection Method A pressure-increment projection algorithm is employed on a MAC staggered grid No need for pressure boundary conditions dealing with

10. Projection Method: 3 steps

12. Correction terms in the projection method

13. Determination of q at control points

14. Matrix-vector multiplication

15. Interface Evolution: Moving Interface

16. Moving Interface: Implementation

17. Numerical Results: Exact solution

18. Numerical Results: Rotational Flow • Ω = [-1, 1]×[-1, 1] • Interface: circle r = 0.5, located at (0, 0) • Force strength: • Viscosities:

19. Numerical Results: Rotational Flow

20. 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:

21. Velocity and Pressure ( )

22. Evolutions of semi-major and semi-minor axises ( fixed)

23. Volume Conservation

24. Iterations in BFGS and GMRES

25. (II) Inextensible interface in Stoks Flows • Incompressible Stokes Equations • The interface exerts singular force on the fluid • The motion of the moving interface satisfies

26. 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

27. Finite difference MAC scheme with correction terms (#) (#) Solved by FFT, Multigrid, PCG, etc

28. 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

29. Matrix-vector multiplication (*)

30. Evolving Inextensible Interface: Implementation

31. 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: Streamline pattern at steady state Pressure profile at steady state

32. Shapes of deformed interface at different times

33. Initial (left) and final (right) shape of interface with different initial incidences. Temporal evolution of orientation angle of interfaces with different initial incidences

34. Area conservation and arc length conservation

35. Numerical Results: initially concave interface Initial orientation angle

36. Shapes of deformed interface at different times

37. 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.

38. Thank you !