Geometrical design of blade's surface and boundary control of Navier-Stokes equations

摘要

In this article a new principle of geometric design for blade's surface of an impeller is provided. This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade. We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum. The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived. In addition, under a new curvilinear coordinate system, the flow domain between the two blades becomes a fixed hexahedron, and the surface as a mapping from a bounded domain in R~2 into R~3 , is explicitly appearing in the objective functional. The Navier-Stokes equations, which include the mapping in their coefficients, can be computed by using operator splitting algorithm. Furthermore, derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too. Hence, a conjugate gradient method can be used to solve the optimal control problem.