A highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions
A highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions
复制标题
使用级数展开处理柱坐标极轴处的流动方程的高精度技术
DOI：
10.1006/jcph.2002.7187
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发表时间：
2002
影响因子：
4.1
通讯作者：
S. Lele
中科院分区：
文献类型：
--
作者：
G. Constantinescu;S. Lele
在柱坐标或球坐标系中求解流动方程的数值方法应该能够捕捉到控制方程的特定形式奇异的区域附近的精确解的行为。在这项工作中，我们专注于这些数值奇异性的有限差分方法的治疗，通过重新解释的正则性条件下开发的伪谱方法。当用高精度有限差分格式（例如，有限差分法）在柱坐标系中求解非轴对称流时，处理极轴处奇点的一种普遍适用的数值方法。Pade格式）在非交错网格上进行了数值模拟。在极轴的流动的控制方程推导出使用一系列的扩展r = 0附近。计算这些方程中的系数所需的唯一信息是流量变量的值及其在前一次迭代（或时间）水平上的径向导数。这些导数在极轴处是多值的，使用从（0，R）*（0，2 π）到（-R，R）*（0，π）的流域映射来计算，而不降低数值方法的精度，其中R是计算域的半径。这允许使用高阶差分方案（例如，紧凑方案）在位于极轴上的点处。通过与线性增长区圆形可压缩强迫射流理论解的比较，验证了该方法的准确性。所提出的技术说明了层流强迫射流和湍流可压缩射流使用大涡模拟（LES）方法的模拟结果。在数值方法的一般鲁棒性和接近极轴的解的光滑性方面，本结果与类似的计算相比非常有利，在类似的计算中，方程在极轴处在笛卡尔坐标中求解，或者通过在径向方向上采用交错网格而没有在r = 0处的网格点来去除奇异性，根据Mohseni和Colonius最近提出的方法[1]。这里所描述的方法的扩展不可压缩流或任何其他一组方程，解决了在圆柱或球坐标系的非交错网格与有限差分格式的各种精度水平是即时的。
Numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular. In this work we focus on the treatment of these numerical singularities for finite-difference methods by reinterpreting the regularity conditions developed in the context of pseudospectral methods. A generally applicable numerical method for treating the singularities present at the polar axis, when nonaxisymmetric flows are solved in cylindrical coordinates using highly accurate finite-difference schemes (e.g.. Pade schemes) on nonstaggered grids, is presented. Governing equations for the flow at the polar axis are derived using series expansions near r = 0. The only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration (or time) level. These derivatives, which are multivalued at the polar axis, are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from (0, R) * (0,2π) to (-R, R) * (0,π), where R is the radius of the computational domain. This allows the radial derivatives to be evaluated using high-order differencing schemes (e.g., compact schemes) at points located on the polar axis. The accuracy of the method is checked by comparison with the theoretical solution corresponding to a circular compressible forced jet in the regime of linear growth. The proposed technique is illustrated by results from simulations of laminar-forced jets and turbulent compressible jets using large eddy simulation (LES) methods. In terms of the general robustness of the numerical method and smoothness of the solution close to the polar axis, the present results compare very favorably to similar calculations in which the equations are solved in Cartesian coordinates at the polar axis, or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r = 0, following the method proposed recently by Mohseni and Colonius [1]. Extension of the method described here for incompressible flows or for any other set of equations that is solved on a nonstaggered mesh in cylindrical or spherical coordinates with finite-difference schemes of various levels of accuracy is immediate.