In this paper we present a numerical approach to solve the Navier-Stokes equations on moving domains with second-order accuracy. The space discretization is based on the ghost-point method, which falls under the category of unfitted boundary methods, since the mesh does not adapt to the moving boundary. The equations are advanced in time by using Crank-Nicholson. The momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach. To avoid the checkerboard instability for the pressure, a staggered grid is adopted, where velocities are defined at the sides of the cell and the pressure is defined at the centre. The lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown, representing the average divergence of the velocity, and an additional equation to set the average pressure to zero. Several tests are performed to simulate the motion of an incompressible fluid around a moving object, as well as the lid-driven cavity tests around steady objects. The object is implicitly defined by a level-set approach, that allows a natural computation of geometrical properties such as distance from the boundary, normal directions and curvature. Different shapes are tested: circle, ellipse and flower. Numerical results show the second order accuracy for the velocity and the divergence (that decays to zero with second order) and the efficiency of the multigrid, that is comparable with the tests available in literature for rectangular domains without objects, showing that the presence of a complex-shaped object does not degrade the performance.
翻译:在本文中,我们展示了用二阶精确度移动域上的 Navier- Stokes 等式解算数字方法。 空间离散基于幽点方法, 属于不合适的边界方法类别, 因为网状不适应移动边界。 方程式通过使用 Clank- Nicholson 来及时推进。 动速和压力的动和连续性方程式同时通过采用适当的多格方法来解决。 为了避免压力的检查板不稳定, 采用了错开的格子格子, 在单元格两侧定义了天速, 在中心定义了压力。 增加一个未知的标度, 表示速度的平均偏差, 并增加一个将平均压力设定为零的方程式。 为了模拟移动对象周围的压压压性液体运动, 以及稳定对象周围的利定的轨迹测试。 该对象被一个直径直线的天体定义为单元格两侧的天体形, 压力的不具有可比较性平的直径直径直径, 使正常的轨道的平面能够进行自然测序, 。