绕圆柱非定常周期性涡旋脱落的数值模拟

Numerical simulation of unsteady periodic vortex shedding from a circular cylinder

利用非定常流函数涡量方程数值模拟圆柱突然起动尾流涡旋的形成及周期性脱落过程。对求解的流函数的一阶导数即速度项采用四阶精度的 Hermitian公式 ,而方程的对流项则采用四阶精度的差分格式 ,并利用 ADI方法迭代求解差分方程组。当雷诺数 Re不大于 4 0时 ,圆柱尾流为附体的两个对称涡 ,为定常解。当 Re大于 4 0后流动为非定常及非对称的 ,圆柱尾流呈现周期性涡旋交替脱落而形成著名的 Karman涡街。选择 Re =10 0 为例 ,在初始条件未加任何扰动情况下 ,成功地模拟了圆柱非定常涡旋形成与脱落的完整过程(无量纲时间算到 t=2 50及以上 )。所计算的阻力系数与实验结果及其它数值方法的计算结果一致。约在 t=2 0 0 形成严格的 Karman涡街。对涡量方程 ADI求解方法的稳定性进行了分析。对流项采用四阶精度差分格式 ,若应用于定常问题 ,将极大提高数值求解的精度 ,若应用于非定常问题的求解 ,将对求解精度有所改善 ,其中时间空间两阶混合偏导数的处理是关键 。

The unsteady equations of stream and vorticity functions were used for numerical simulation on the process of the vortex formation and periodic shedding from an impulsively started circular cylinder. The Hermitian formulas with the fourth order accuracy were adopted for the first partial derivatives of stream function (i.e. velocities). The fourth order finite difference scheme put forward by the author was applied to the convective terms of the vorticity equation. The finite difference equations were solved with the ADI method. The calculation results show that when the Reynolds number is not greater than 40, there are two symmetric vortices attched to the wake of the circular cylinder and the flow is steady; when the Reynolds number is greater than 40, the rear flow around a circular cylinder is unsteady and asymmetric and the periodic vortices alternately shed from the surfaces of the circular cylinder, which is the famous Karman vortex street. In this paper, the Reynolds number was assumed to be 100 for an example. The unsteady process of the vortex formation and shedding from the circular cylinder was simulated successfully up to the dimensionless time t larger than 250 under the condition that no fluctuations were added to trigger at the beginning of the calculations. The calculation results of drag coefficients are almost the same as the result of the experiment and the results obtained with other numerical methods. At about t=200, the perfect Karman vortex street is formed. In the present paper, the stability of the ADI method for the vorticity equation was analysised. It shows that if the method with the fourth order finite difference scheme for the convective terms of the vorticity equation is used for the steady problems, the numerical accuracy could be raised greatly; if the method is used for the unsteady problems, there will be some improvement in the numerical accuracy. In the unsteady problems the key point lies in the treatment of the second order mixed time and space partial derivatives and it will be further researched.