两种浸入式边界方法的比较

COMPARISONS OF TWO TYPES OF IMMERSED BOUNDARY METHODS IN NUMERICAL SIMULATIONS OF A CYLINDER IN UNIFORM INCOMPRESSIBLE FLOWS

介绍两类不同的浸入式边界方法及其对它的改进.然后采用均匀矩形交错网格和压力校正投影法,对不可压流场中的二维圆柱绕流进行了数值求解并对比了两类方法的精度.计算分析表明,连续显力法具有构造简单,适用性强的优点.但离散隐力法在物面边界精度上要优于前者.改进后,在二阶精度的离散格式下物面边界精度较低的显示力源法的精度有一定提高,同时发现,加密网格以提高数值精度的方法对于连续显力法并不总是有效.而同样格式下,离散隐力法具有更高精度,其中预测-校正离散隐力法可以在此基础上获得更小的计算误差和更快的收敛速度.数值解与文献已有的数值和实验结果吻合得很好,表明边界算法及其程序是可靠和有效的.

The Immersed Boundary （IB） Methods have been proven to be suitable to handle complex geometrical boundaries, moving boundaries and fluid-structure interactions, etc, since only the fixed Cartesian grids are required. C. S. Peskin first introduced this method in the 1970s with a flat development period. In resent years, numerous modifications and refinements have been proposed and variants of this approach now exist. In this study, two different types of IB methods are introduced, the Continuous Explicit Forcing Method （CEFM） and the Discrete Implicit Forcing Method （DIFM）. In CEFM, the forcing terms are introduced into the controlling equations to replace the boundary conditions on the surfaces and the controlling equations can be explicitly solved. In DIFM, the forcing effect is considered in the boundary conditions on the so-called ghost cells inside the surfaces directly. To promote the computational accuracy near the solid surface, this study firstly modifies the CEFM and then proposes a new treatment called a predicted DIFM. They have been validated by numerical simulations of the uniform incompressible flow past a cylinder. The results suggest that the present modification for CEFM leads to a higher accuracy near solid surfaces than the original one, which uses forcing terms in a staggered time-level comparing the time-level of velocities. Moreover, only 1-order or even less of accuracy near surfaces can be obtained since the controlling equations are discretized by a 2-order scheme. Therefore, it may be not useful for this type of methods to refine the meshes near the surfaces to obtain a higher accuracy. Meanwhile, the DIFM can obtain much more accurate results especially and the assumption of pseudo-coincident points is proven to be effective to promote the accuracy from 1-order to 2-order with fine grids, even if the linear assumption of velocity distribution near solid surfaces remains. Hence, the predicted DIFM is proven to lead less numerical errors and a faster convergence, even though it can＇t promote a higher order of accuracy. Moreover, the overall accuracies are not affected obviously by the different methods and good results are obtained in simulations of the cylinder case. Further researches are performed in the cases with moving boundaries, more complex geometrical boundaries and fluid-structure interactions.