| dc.description.abstract | This thesis is devoted to developing efficient immersed boundary methods for simulating the dynamics
of fluid-structure interaction problems. It is mainly divided into the following two parts:
In the first part, we propose a novel penalty immersed boundary method for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow to weaken the incompressibility condition, relating the surface divergence to the elastic tension $sigma$ to relax the interface′s inextensibility, and connecting the particle surface-velocity with the particle surface force ${mbf F}$ to regularize the particle′s rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns $p_h$, $sigma_h$ and ${mbf F}_h$ directly, so that we just need to solve a smaller linear system of the velocity approximation ${mbf u}_h$. This advantage is preserved and even enhanced when such an approach is applied to the transient Stokes flow with multiple compound vesicles. Moreover, this smaller linear system is symmetric and negative-definite, which enables us to use efficient linear solvers. Another important feature of the proposed method is that the discretization scheme is unconditionally stable in the sense that an appropriately defined energy functional associated with the discrete system is decreasing and hence bounded in time. We numerically test the accuracy and stability of the immersed boundary discretization scheme. The tank-treading and tumbling motions of inextensible interface with a suspended solid particle in the simple shear flow will be studied extensively. The simulation of the motion of multiple compound vesicles will be performed as well. Numerical results illustrate the superior performance of the proposed penalty immersed boundary method.
In the second part, we propose a simple prediction-correction direct-forcing immersed boundary method, which is combined with the Choi-Moin projection method, for simulating the dynamics of fluid-solid interaction problems. The immersed solid object can be stationary or moving in the fluid with a prescribed velocity. As usual, an Eulerian description is used for the fluid dynamics, while the Lagrangian representation is employed for the immersed solid boundary. The proposed approach can be categorized as a discrete forcing method with a prediction-correction strategy, in which a virtual force of discrete type distributed on the whole solid body is introduced and added to the fluid momentum equations to accommodate the no-slip boundary condition at the immersed solid boundary. More specifically, based on the rate of moment changes of the solid body, the virtual force at the grid points can be first predicted by using the difference between the prescribed solid velocities and the computed velocities which are obtained by the Choi-Moin projection method for the incompressible Navier-Stokes equations without adding any virtual forcing term. Such predicted virtual force is then put into the momentum equations for the correction step to update the velocity field, pressure and virtual force. This prediction-correction procedure can be iterated to generate a more general method, if necessary. Numerical experiments of several benchmark problems are performed to illustrate the simplicity and high performance of the proposed prediction-correction approach. We find that our numerical results are in very good agreement with the previous works in the literature and in most cases, one correction step is good enough. | en_US |