浸入边界法的改进及其应用

The immersed boundary method is a new version of numerical method which is suit for treating complex geometric boundary 、moving boundary、and fluid-solid interation problem. In the immersed boundary method, the structure immersed in fluid is considered as a kind of momentum forcing in the Navier-Stokes equations rather than a body. In addition, the Cartesian grid is used to avoid the difficulty of generating a body conformal grid. Campared with traditional numerical method of fluid dynamics, this method have a series of advantages, such as a simple algorithm 、 easy to generate grid、natural parallel、efficient to handle the complex boundary. In this project， we will focus on the immersed boundary method for the unsteady incompressible Navier-Stokes equations, including (1)a new algorithm is presented by combing the characteristic method and the immersd method;(2)the simulation of the flow around a sphere is performed by the virtue of implicit force method.

浸入边界法是适用于处理形状复杂的几何边界、可移动边界以及流固耦合问题等的一种数值方法。它通过在流体运动方程中加力源项来实现物体边界与流体的相互作用，并在计算过程中采用简单的笛卡尔网格。与传统的数值方法相比， 浸入边界法具有很多优势， 如算法简单， 网格生成简单，并行计算方便，边界容易处理。本项目对于求解非定常Navier-Stokes方程的浸入边界法进行研究。主要内容包括（1）将特征线方法与浸入边界法相结合，提出求解非定常不可压缩Navier-Stokes方程的特征线算法，这种算法不但网格生成简单，且可以使用大的时间步长。(2)利用隐式方法处理固壁边界对于流体的作用，并数值模拟三维球体绕流。

浸入边界法是适用于处理形状复杂的几何边界、可移动边界以及流固耦合问题等的一种数值方法。在计算过程中采用简单的笛卡尔网格。在计算过程中通常采用算子分裂法来简化计算。为此.(1)基于粘性分裂的思想,提出一种求解非定常Navier-Stokes算子分裂方法，并给出了理论分析与数值模拟。.(2)针对拥有柯氏力的非定常不可压缩Navier-Stokes提出一种算子分裂格式，并给出了理论分析和数值模拟。.(3)对于稳态的自然对流问题，将次网格稳定化方法与亏量校正法相结合进行了研究，给出了格式的稳定性和误差分析，并通过数值算例对其进行了验证。