Yongsam Kim

Penalty immersed boundary method for an elastic boundary with mass

The immersed boundary (IB) method has been widely applied to problems involving a moving elastic boundary that is immersed in fluid and interacting with it. Most of the previous applications of the IB method have involved a massless elastic boundary and used efficient Fourier transform methods for the numerical solutions. Extending the method to cover the case of a massive boundary has required spreading the boundary mass out onto the fluid grid and then solving the Navier-Stokes equations with a variable mass density. The variable mass density of this previous approach makes Fourier transform methods inapplicable, and requires a multigrid solver. Here we propose a new and simple way to give mass to the elastic boundary and show that the method can be applied to many problems for which the boundary mass is important. The method does not spread mass to the fluid grid, retains the use of Fourier transform methodology, and is easy to implement in the context of an existing IB method code for the massless case. Two verifications of the method are given. One is a numerical convergence study that shows that our numerical scheme is second-order accurate for a particular test problem. The other is direct comparison with experimental data of vortex-induced vibrations of a massive cylinder, which shows that the results obtained by the present method are quite comparable to the experimental data.

2-D Parachute Simulation by the Immersed Boundary Method

Parachute aerodynamics involves an interaction between the flexible, elastic, porous parachute canopy and the high speed airflow (relative to the parachute) through which the parachute falls. Computer simulation of parachute dynamics typically simplifies the problem in various ways, e.g., by considering the parachute as a rigid bluff body. Here, we avoid such simplification by using the immersed boundary (IB) method to study the full fluid-structure interaction. The IB method is generalized to handle porous immersed boundaries, and the generalized method is used to study the influence of porosity on parachute stability.

Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method

In this paper, we develop an immersed boundary (IB) method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid. In order to take into account the inextensibility constraint of the vesicle, the penalty immersed boundary (pIB) method is used to virtually decouple the fluid and vesicle dynamics. As numerical tests of our current pIB method, the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored, and compared with the previous literature. The method is also validated by a series of convergence study, which confirms its consistent first-order accuracy on the velocity field, the vesicle configuration, the vesicle area and the perimeter errors. In addition, the method is also applied to study a binary-component vesicle problem.

Numerical simulations of two-dimensional foam by the immersed boundary method

In this paper, we present an immersed boundary (IB) method to simulate a dry foam, i.e., a foam in which most of the volume is attributed to its gas phase. Dry foam dynamics involves the interaction between a gas and a collection of thin liquid-film internal boundaries that partition the gas into discrete cells or bubbles. The liquid-film boundaries are flexible, contract under the influence of surface tension, and are permeable to the gas, which moves across them by diffusion at a rate proportional to the local pressure difference across the boundary. Such problems are conventionally studied by assuming that the pressure is uniform within each bubble. Here, we introduce instead an IB method that takes into account the non-equilibrium fluid mechanics of the gas. To model gas diffusion across the internal liquid-film boundaries, we allow normal slip between the boundary and the gas at a velocity proportional to the (normal) force generated by the boundary surface tension. We implement this method in the two-dimensional case, and test it by verifying the von Neumann relation, which governs the coarsening of a two-dimensional dry foam. The method is further validated by a convergence study, which confirms its first-order accuracy.

An immersed boundary method for simulating the dynamics of three-dimensional axisymmetric vesicles in Navier-Stokes flows

In this paper, we develop a simple immersed boundary method to simulate the dynamics of three-dimensional axisymmetric inextensible vesicles in Navier-Stokes flows. Instead of introducing a Lagrange?s multiplier to enforce the vesicle inextensibility constraint, we modify the model by adopting a spring-like tension to make the vesicle boundary nearly inextensible so that solving for the unknown tension can be avoided. We also derive a new elastic force from the modified vesicle energy and obtain exactly the same form as the originally unmodified one. In order to represent the vesicle boundary, we use Fourier spectral approximation so we can compute the geometrical quantities on the interface more accurately. A series of numerical tests on the present scheme have been conducted to illustrate the applicability and reliability of the method. We first perform the accuracy check of the geometrical quantities of the interface, and the convergence check for different stiffness numbers as well as fluid variables. Then we study the vesicle dynamics in quiescent flow and in gravity. Finally, the shapes of vesicles in Poiseuille flow are investigated in detail to study the effects of the reduced volume, the confinement, and the mean flow velocity. The numerical results are shown to be in good agreement with those obtained in literature.

On various techniques for computer simulation of boundaries with mass

Publisher Summary This chapter presents three distinct numerical treatments for immersed boundaries with different mass density than the surrounding fluid. With some background information on the immersed boundary method and its application in the modeling of biological fluids, the chapter demonstrates the selected numerical example of the potential modeling complex biological fluids involving charged particles under both viscous fluid and electro-kinetic forces. The chapter demonstrates the feasibility of three numerical treatments for immersed boundaries with mass. It is found that the first method—namely, a direct method, D’Alembert force approach, is most straightforward, yet it requires small time step for large mass; whereas the second method—namely, a penalty approach, yields virtually the same result as the first method, yet it is not feasible in handling relatively small mass and there also exist high frequency oscillations because of the attached stiff spring-mass system. Finally, the third method—namely, a distribution of boundary mass to the fluid along with a multigrid fluid solver, provides similar results, yet the required solver is not as efficient as the Fast Fourier Transform solver employed in the first two methods.

Numerical study of incompressible fluid dynamics with nonuniform density by the immersed boundary method

We apply the immersed boundary method to the dynamics of an incompressible fluid with a nonuniform density. In order to take into account both the inertial and gravitational effects of the variable density, the penalty immersed boundary (pIB) method is used [Y. Kim and C. S. Peskin, Phys. Fluids 19, 053103 (2007)]. Incompressible fluid motion with a nonuniform density has been extensively explored both experimentally and computationally. We show that the pIB method is a robust and efficient numerical tool for the simulation of fluids with variable density by conducting computations of some example problems: The falling of a heavier fluid surrounded by a lighter fluid and the Rayleigh–Taylor instabilities in two dimensions and three dimensions and the dynamic stabilization of the Rayleigh–Taylor instability.

Blood Flow in a Compliant Vessel by the Immersed Boundary Method

In this paper we develop a computational approach to analyze hemodynamics in the aorta; this may serve as a useful tool in the development of noninvasive methods to detect early onset of diseases such as aneurysms and stenosis in major blood vessels. We introduce a mathematical model which describes the interaction of blood flow with the aortic wall; this model is based on the immersed boundary method. A two-dimensional vessel model is constructed, the velocity at the inlet is prescribed based on the information from the Magnetic Resonance Imaging data measured in the aorta of a healthy subject, and the velocity at the outlet is prescribed by driving the pressure level reproduced from the literature. The mathematical model is validated by comparing with well-known solutions of the viscous incompressible Navier–Stokes equations, i.e., Womersley flow. The hysteresis behavior in the pressure–diameter relation is observed when the viscoelastic material property of the arterial wall is taken into consideration. Five different shapes of aortic wall are considered for comparison of the flow patterns inside the aorta: one for the normal aorta, two for the dilated aorta, and two for the constrictive aorta.

A penalty immersed boundary method for a rigid body in fluid

We extend the penalty immersed boundary (pIB) method to the interaction between a rigid body and a surrounding fluid. The pIB method is based on the idea of splitting an immersed boundary, which here is a rigid body, notionally into two Lagrangian components: one is a massive component carrying all mass of the rigid body and the other is massless. These two components are connected by a system of stiff springs with 0 rest length. The massless component interacts with the surrounding fluid: it moves at the local fluid velocity and exerts force locally on the fluid. The massive component has no direct interaction with the surrounding fluid and behaves as though in a vacuum, following the dynamics of a rigid body, in which the acting forces and torques are generated from the system of stiff springs that connects the two Lagrangian components. We verify the pIB method by computing the drag coefficients of a cylinder and ball descending though a fluid under the influence of gravity and also by studying the int...

An Immersed Boundary Heart Model Coupled with a Multicompartment Lumped Model of the Circulatory System

A new computational model of the circulatory system is developed to investigate the intracardiac blood flow patterns and the motion of the mitral valve. In this work, we couple an existing two-dimensional model for the left heart with a multicompartment lumped model for the whole circulation to analyze the hemodynamics of the normal circulation in humans. The two-dimensional left-heart model is based on the immersed boundary method, and the lumped circulation model is governed by a system of ordinary differential equations. We investigate the intraventricular velocity field and the velocity curves over the mitral ring, between the valve leaflets, and across the aortic outflow tract. The flow and pressure curves are also measured in the left and right hearts and the systemic and pulmonary arteries. Our simulation results are in reasonably good agreement with ones in the literature and comparable to the existing magnetic resonance data, despite the inherent deficiency of a two-dimensional modeling of the left heart.