绕流直接数值模拟误差分析

ERROR IN NUMERICAL SIMULATION OF EXTERNAL FLOW AROUND AN IMMSERD BODY

对于无边界绕流问题的计算流体力学模拟通常是将物体置于“足够大”的槽道中,而通过不断改变槽道尺寸以及离散网格密度,后验对比方式来检查模拟误差。本文结合多种经典流场理论,提出一种简单的先验误差估计方法来确定槽道尺寸以及相应的网格分布。在此方法中,对于槽道尺寸的确定基于线性叠加原理(即在极小雷诺数下采用Stokes理论解叠加,而在其他雷诺数条件下采用势流理论解叠加),来估计槽道尺寸对绕流结果的影响。而对网格尺寸与分布,则是使用多项式逼近中的基本误差分析工具,应用到速度边界层,远场势流,以及Rankine涡等简单流动,从而确定整个绕流问题中的离散误差。为了验证前面的理论分析结果,本文模拟了相当大雷诺数范围内的二维翼型以及三维圆球绕流,所得数值结果非常好地验证了理论分析。结果表明,对于Stokes流动问题,槽道尺寸需要大约100倍于物体特征尺寸来保证其结果与无边界绕流相差不超过1%;而在雷诺数超过大约100时,槽道尺寸只需10倍(二维绕流)或者5倍(三维绕流)于物体特征尺寸来达到同等精度。在此先验误差估计方法可应用于一般化的绕流问题。

A posteriori error estimation is usually adopted in numerical simulations of external flow around an immersed body, where the body is located in a channel of “sufficient size”. During the error estimation, both the channel size and grid resolution are varied in large ranges so as to achieve the converged solution. In this work, a simple technique of a priori error estimation is developed for such simulations. The effect of the channel size is investigated by the superposition of image solutions for Stokes flow or potential flow depending on the actual Reynolds number. The effect of grid resolution is investigated through applying the well-known error estimation in polynomial approximation to specific simple flows, i.e., boundary layer, far-field potential flow, and the Rankine vortex. The global error estimation for external flow simulations is obtained by assembling the results obtained under these three simple characteristic flow models. In order to verify the theoretical findings,numerical simulations of flow around an air foil(2D) and around a sphere(3D) are carried out in a large range of Reynolds numbers. The numerical simulations agree with the theoretical predictions extremely well. Both theoretical and numerical results demonstrate that the channel size needs to be of around 100 folds of the characteristic size of the immersed body in Stokes flows so as to exhibit a discrepancy around 1% from the external flow problem of infinite domain. However, for flows of Reynolds number larger than 100, the channel size only needs to be 10(for 2D) or 5(for 3D) in unit of the body size so as to achieve the same precision. The technique of a priori error estimation can be used for simulations of external flow around an arbitrary body other than the air foil or sphere investigated here.