Yu Huidan, Chen Xi, Wang Zhiqiang, Deep Debanjan, Lima Everton, Zhao Ye, Teague Shawn D
Department of Mechanical Engineering, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana 46202, USA.
Department of Computer Science, Kent State University, Kent, Ohio 44240, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jun;89(6):063304. doi: 10.1103/PhysRevE.89.063304. Epub 2014 Jun 11.
In this paper, we develop a mass-conserved volumetric lattice Boltzmann method (MCVLBM) for numerically solving fluid dynamics with willfully moving arbitrary boundaries. In MCVLBM, fluid particles are uniformly distributed in lattice cells and the lattice Boltzmann equations deal with the time evolution of the particle distribution function. By introducing a volumetric parameter P(x,y,z,t) defined as the occupation of solid volume in the cell, we distinguish three types of lattice cells in the simulation domain: solid cell (pure solid occupation, P=1), fluid cell (pure fluid occupation, P=0), and boundary cell (partial solid and partial fluid, 0<P<1). The formulation of volumetric lattice Boltzmann equations are self-regularized through P and consist of three parts: (1) collision taking into account the momentum exchange between the willfully moving boundary and the flow; (2) streaming accompanying a volumetric bounce-back procedure in boundary cells; and (3) boundary-induced volumetric fluid migration moving the residual fluid particles into the flow domain when the boundary swipes over a boundary cell toward a solid cell. The MCVLBM strictly satisfies mass conservation and can handle irregular boundary orientation and motion with respect to the mesh. Validation studies are carried out in four cases. The first is to simulate fluid dynamics in syringes focusing on how MCVLBM captures the underlying physics of flow driven by a willfully moving piston. The second and third cases are two-dimensional (2D) peristaltic flow and three-dimensional (3D) pipe flow, respectively. In each case, we compare the MCVLBM simulation result with the analytical solution and achieve quantitatively good agreements. The fourth case is to simulate blood flow in human aortic arteries with a very complicated irregular boundary. We study steady flow in two dimensions and unsteady flow via the pulsation of the cardiac cycle in three dimensions. In the 2D case, both vector (velocity) and scalar (pressure) fields are compared to computation results from a well-established Navier-Stokes solver and reasonably good agreements are obtained. In the 3D case, the unsteady flow pattern and wall shear stress are well captured at the representative time instants during the pulsation. The validations demonstrate that the MCVLBM is a relatively simple but reliable computation scheme to deal with static or moving irregular boundaries.
在本文中，我们开发了一种质量守恒的体积晶格玻尔兹曼方法（MCVLBM），用于数值求解具有任意移动边界的流体动力学问题。在MCVLBM中，流体粒子均匀分布在晶格单元中，晶格玻尔兹曼方程处理粒子分布函数的时间演化。通过引入一个定义为单元中固体体积占有率的体积参数P(x,y,z,t)，我们在模拟域中区分出三种类型的晶格单元：固体单元（纯固体占据，P = 1）、流体单元（纯流体占据，P = 0）和边界单元（部分固体和部分流体，0 < P < 1）。体积晶格玻尔兹曼方程的公式通过P进行自正则化，由三部分组成：(1) 碰撞，考虑任意移动边界与流体之间的动量交换；(2) 对流，伴随边界单元中的体积反弹过程；(3) 边界诱导的体积流体迁移，当边界朝着固体单元扫过边界单元时，将残留的流体粒子移入流动域。MCVLBM严格满足质量守恒，并且能够处理相对于网格的不规则边界方向和运动。在四种情况下进行了验证研究。第一种情况是模拟注射器中的流体动力学，重点关注MCVLBM如何捕捉由任意移动活塞驱动的流动的基本物理过程。第二种和第三种情况分别是二维（2D）蠕动流和三维（3D）管道流。在每种情况下，我们将MCVLBM模拟结果与解析解进行比较，并在定量上取得了良好的一致性。第四种情况是模拟具有非常复杂不规则边界的人体主动脉中的血流。我们研究二维稳态流以及通过三维心动周期脉动的非稳态流。在二维情况下，将矢量（速度）和标量（压力）场与成熟的纳维 - 斯托克斯求解器的计算结果进行比较，并获得了合理的良好一致性。在三维情况下，在脉动期间的代表性时刻很好地捕捉到了非稳态流动模式和壁面剪应力。验证表明，MCVLBM是一种相对简单但可靠的计算方案，可用于处理静态或移动的不规则边界。
