一种基于边界积分方程的隐式翼型反设计算法

An Implicit Algorithm for Airfoil Inverse Design Based upon Integral Boundary Equation

本文由边界元方法出发,将适用于单连通空间Laplace问题的边界积分方程推广到带环量的多连通空间中,并对离散边界积分方程中的矩阵元积分式解析化,以避免在翼型尾缘处尖点附近直接利用数值积分计算矩阵元导致的数值振荡,对于以翼型表面压力分布为收敛目标的反设计问题,利用Newton-Raphson迭代求解满足该目标压力的非线性方程组,从而构造关于二维不可压缩流动翼型反设计问题的一个简便有效的隐式迭代算法。我们采用不同的数值算例对该算法予以检验,数值算例显示本文提出的隐式反设计算法收敛范围和来流攻角无关,且明显大于依赖于来流攻角的显式反设计算法的收敛范围,另外该算法精度高且具有很快的收敛速度。该翼型反设计算法有望在可变形翼型气动性能的研究中得到应用。

The boundary integral formulation for Laplace＇s equation in simply connected space was generalized into multiply connected space,where the circulation around airfoil was introduced.It is proposed that the influence coefficients calculation for the boundary integral equations by analytical integration instead of by numerical integration can eliminate numerical oscillations in the calculations around the airfoil trailing edge.For the inverse design problem from the given pressure distribution on the airfoil,the Newton-Raphson iterative method was utilized to solve the nonlinear equations.With the above procedure,an implicit algorithm of the airfoil inverse design for two-dimensional incompressible flow was performed.The different numerical examples were utilized to test the proposed algorithm.The numerical results show that the convergence range of the implicit algorithm is much wide than that of the previous explicit algorithm,being independent of the angle of attack,and the convergence range of the implicit algorithm,however,decreases as decreasing the angle of attack.It also shows that the implicit algorithm has a good accuracy with a rapid convergent speed.Hopefully this implicit algorithm of the airfoil inverse design can apply into the study for the aerodynamics properties of the morphing airfoils.