基于控制理论的Euler方程翼型减阻优化设计

Control theory based airfoil design using Euler equations

在Jameson的控制论气动优化设计思想下 ,应用Euler方程研究了翼型的反设计和减阻问题。设计过程中的梯度是通过求解一个伴随偏微分方程而得到的 ,每个设计迭代只需求解一次Euler方程和一次伴随方程 ,计算量与设计变量数无关。在具体数值执行中 :①利用分部积分将目标函数变分的计算转化成了类似于计算Euler方程残值的形式 ,节省了机器时间 ;②根据控制理论的要求 ,将减阻问题转化成了相应地修改伴随方程的物面边界条件和目标函数的变分表达 ,使问题得到了简化 ;③利用特征不变量分析法 ,处理了伴随方程的物面和远场边界条件。设计算例证明了本文方法可靠性好、收敛快。

This paper describes the implementation of optimization techniques based on control theory for airfoil with Euler equations.The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation,and the boundary shape is then modified in a direction of descent.The advantage is that the cost function variation is independent on the flow field variation,with the result that the gradient of cost function with respect to arbitrary number of design variables can be determined without the need for additional flow field evaluations.So each design cycle requires the numerical solution of both the flow and the adjoint equations leading to a computational cost roughly equal to the cost of two flow solutions.Results are presented for both the inverse problem and drag minimization.