基于Navier滑移边界条件的拟域方法模拟流固耦合问题
The Fictitious Domain Method Based on Navier Slip Boundary Condition for Simulation of Flow-Particle Interaction
摘要：
本文提出了一种最小二乘/虚拟区域法，用于直接模拟流体-颗粒界面上具有Navier滑移边界条件的流体颗粒运动。设$Ω$和$B$是$\mathbb{R}^{d}$的两个有界整环，使得$\overline{B} \subset Ω$ .固体颗粒$B$的运动由牛顿方程控制。我们在这里的目标是开发一个虚构的域的方法，其中一个解决了一个变种的原始问题的完整的$Ω$，其次是一个精心选择的校正超过$B$和修正有关的平移速度和角速度的粒子。这种方法是虚拟控制类型的，依赖于最小二乘公式，使问题可以通过在精心选择的控制空间中操作的共轭梯度算法来解决。由于完全显式的计划更新的粒子运动使用牛顿方程是不稳定的，我们提出并实现了一个显式-隐格式，其中，在每一个时间步，显式更新的粒子的位置，Navier-斯托克斯方程和粒子速度的解决方案是由最小二乘/虚拟域方法隐式求解。数值结果验证了我们的数值方法。
英文摘要：
In this article, we develop a least--squares/fictitious domain method for direct simulation of fluid particle motion with Navier slip boundary condition at the fluid--particle interface. Let $Ω$ and $B$ be two bounded domains of $\mathbb{R}^{d}$ such that $\overline{B} \subset Ω$. The motion of solid particle $B$ is governed by Newton's equations. Our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full $Ω$, followed by a well--chosen correction over $B$ and corrections related to translation velocity and angular velocity of the particle. This method is of the virtual control type and relies on a least--squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Since the fully explicit scheme to update the particle motion using Newton's equation is unstable, we propose and implement an explicit--implicit scheme in which, at each time step, the position of the particle is updated explicitly, and the solution of Navier-Stokes equations and particle velocities are solved by the the least--squares/fictitious domain method implicitly. Numerical results are given to verify our numerical method.
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