We develop an energy-stable scheme for simulating fluid-particle interaction problems governed by a coupled system consisting of the incompressible Navier-Stokes (NS) equations defined in a time-dependent fluid domain and Newton's second law for particle motion. A modified temporary arbitrary Lagrangian-Eulerian (tALE) method is designed based on a bijective mapping between the fluid regions at different time steps. In the proposed numerical scheme, the tALE mesh velocity, the incompressible NS equations, and Newton's second law are solved simultaneously. We prove that under certain conditions, the new time discretization scheme satisfies an energy law. For the space discretization, the extended finite element method (XFEM) is used to solve the problem on a fixed Cartesian mesh. The developed method is first-order accurate in time and space without being momentum conservative. To verify the accuracy and stability of our numerical scheme, we present numerical experiments including the fitting of the Jeffery orbit by rotating of an ellipse and the free-falling of an elliptic particle in water.

我们开发了一种能量稳定方案，用于模拟由耦合系统控制的流体-粒子相互作用问题，该耦合系统由在随时间变化的流体域中定义的不可压缩的Navier-Stokes（NS）方程和粒子运动的牛顿第二定律组成。基于不同时间步长的流体区域之间的双射映射，设计了一种改进的临时任意拉格朗日欧拉（tALE）方法。在提出的数值方案中，同时求解了tALE网格速度，不可压缩的NS方程和牛顿第二定律。我们证明在一定条件下，新的时间离散方案满足能量定律。对于空间离散化，使用扩展有限元方法（XFEM）来解决固定笛卡尔网格上的问题。所开发的方法在时间和空间上都是一阶精确的，而没有动量保守。为了验证我们的数值方案的准确性和稳定性，我们提供了数值实验，包括通过旋转椭圆和使椭圆粒子在水中自由下落来拟合Jeffery轨道。