一个新的边界层方程及其在形状控制中的应用

本文给出固壁边界上(即一个二维流形上) 的流体速度梯度和压力的二阶偏微分方程, 从而也给出边界上法向应力, 以及流体中运动物体所受的阻力和升力的计算公式. 本方法的创新在于边界上法向速度梯度不是通过在边界层内速度梯度的数值微分达到, 而是通过它与其他变量一起作为一组偏微分方程的解而得到, 证明边界层方程组的适定性问题, 并且给出解关于边界形状的Gâteaux 导数所满足的偏微分方程. 本文将本方法应用于飞机外形的形状最优控制, 给出阻力泛函关于形状第一变分的可计算形式. 数值例子表明, 用本方法得到的阻力精度比通用程序得到要高.

A new boundary layer equation and its applications to the shape optimal control

In this paper, the second order partial differential equations of the velocity gradient and pressure for the flow on a solid boundary （two-dimensional manifold） are proposed. Thus, the normal stress on the boundary, the drag and lift of the moving body in the flow are obtained. The key innovation of our method is that the velocity gradient on the boundary which must be computed in the shape optimization is obtained not from numerical difference of velocity at the boundary layer, but from solving new boundary layer equations which are proposed by us. Moreover, we prove the posedness of boundary layer equations and give the equations satisfied by the GEteaux derivative of the solutions to boundary layer equations with respond to the shape. At last, we give an application, which applies our method to the shape optimal control of the airplane, and the calculated form for the first variation of the drag functional to the shape. Numerical example shows the accuracy of drag obtained by using our method higher than by standard software.