Navier-斯托克斯方程边界形状控制及叶轮叶片型面几何设计方法
Boundary Shape Control of Navier-Stokes Equations and Geometrical Design Method for Blade's Surface in the Impeller
摘要：
本文提出了一种叶轮叶片型面的几何设计方法$\Im$。$\Im$是由著名的Navier-斯托克斯方程和四阶椭圆边值问题组成的耦合系统的解。该耦合系统描述了Navier-斯托克斯方程的解与流体所占区域几何形状之间的关系，为上述区域边界的几何优化设计提供了新的理论和方法。该耦合系统是描述叶轮叶片型面几何设计新原理的最优控制问题的欧拉-拉格朗日方程组。控制变量为叶片表面，状态方程为两叶片间通道内具有混合边界条件的Navier-斯托克斯方程。该目标函数根据叶片表面的几何形状来描述流动的耗散能和叶轮的功率。首先证明了最优控制问题解的存在性。然后利用Navier-斯托克斯方程的一个特殊坐标系导出了与曲面$Θ$显式相关的目标泛函。我们还显示了弱连续的Navier-斯托克斯方程的解相对于叶片的表面的几何形状。
英文摘要：
In this paper A Geometrical Design Method for Blade's surface $\Im$ in the impeller is provided here $\Im$ is a solution to a coupling system consisting of the well-known Navier-Stokes equations and a four order elliptic boundary value problem . The coupling system is used to describe the relations between solutions of Navier-Stokes equations and the geometry of the domain occupied by fluids, and also provides new theory and methods for optimal geometric design of the boundary of domain mentioned above. This coupling system is the Eular-Lagrange equations of the optimal control problem which is describing a new principle of the geometric design for the blade's surface of an impeller. The control variable is the surface of the blade and the state equations are Navier-Stokes equations with mixed boundary conditions in the channel between two blades. The objective functional depending on the geometry shape of blade's surface describes the dissipation energy of the flow and the power of the impeller. First we prove the existence of a solution of the optimal control problem. Then we use a special coordinate system of the Navier-Stokes equations to derive the objective functional which depends on the surface $Θ$ explicitly. We also show the weakly continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface.
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