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 Ø Introduction Ø IIM for simulation of incompressible two-fluid interface • Jump Conditions across the Interface • Numerical Algorithm • Numerical Results Ø IIM for the dynamics of inextensible interfaces • Numerical Algorithm • Numerical Results Ø Conclusions
Introduction q Peskin’s Immersed Boundary Method (IBM) • Fluid dynamics of blood flow in human heart • Biological flows: platelet aggregation, bacterial organisms • Rigid boundaries q Immersed Interface Method (IIM by Le. Veque 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. Ω+ X(s, t) s ΩΓ(t) δΩ • 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 • 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 • The motion of the moving interface satisfies
Coupled Jump Conditions
Decoupled Jump Conditions
Numerical Algorithm: Projection Method A pressure-increment projection algorithm is employed on a MAC staggered grid -mesh point control point No need for pressure boundary conditions dealing with
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](https://present5.com/presentation/7c61770b9607dc8a3d17e873ee87a76c/image-18.jpg "Numerical Results: Rotational Flow • Ω = [-1, 1]×[-1, 1] • Interface: circle r")
Numerical Results: Rotational Flow • Ω = [-1, 1]×[-1, 1] • Interface: circle r = 0. 5, located at (0, 0) • Force strength: • Viscosities:
Numerical Results: Rotational Flow
![Numerical Results: Elastic Membrane • Ω = [-1. 5, 1. 5]×[-1. 5, 1. 5]](https://present5.com/presentation/7c61770b9607dc8a3d17e873ee87a76c/image-20.jpg "Numerical Results: Elastic Membrane • Ω = [-1. 5, 1. 5]×[-1. 5, 1. 5]")
Numerical Results: Elastic Membrane • Ω = [-1. 5, 1. 5]×[-1. 5, 1. 5] • Semi-major axis: 0. 75; semiminor axis: 0. 5 • Unstretched state: 0. 5 • Elastic force:
Velocity and Pressure ( )
Evolutions of semi-major and semi-minor axises ( fixed)
Volume Conservation
Iterations in BFGS and GMRES
(II) Inextensible interface in Stoks Flows • Incompressible Stokes Equations • The interface exerts singular force on the fluid • The motion of the moving interface satisfies
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 2 D 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] •](https://present5.com/presentation/7c61770b9607dc8a3d17e873ee87a76c/image-31.jpg "Numerical Results: initially elliptical interface • Ω = [-3, 3]×[-1. 5, 1. 5] •")
Numerical Results: initially elliptical interface • Ω = [-3, 3]×[-1. 5, 1. 5] • Semi-major axis: 0. 75; semiminor axis: 0. 5 • Initial orientation angle: Streamline pattern at steady state Pressure profile 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
Shapes of deformed interface at different times
Conclusions q A second order accurate IIM for solving viscous incompressible flows with discontinuous viscosity is presented. q An IIM is developed to simulate the dynamics of inextensible interface in a viscous fluid. q Extend our IIM code to 3 D problems.
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