Optimal shape design for blade's surface of an impeller via the Navier-Stokes equations

In this paper, a new principle of geometric design for blade's surface of an impeller is established. This is a thorough mathematical investigation of a shape control problem for the optimal design of an impeller's blade while 3D rotating Navier-Stokes equations are as the state equations in the control problem. The objective is to design the blade such that the viscous drag is minimized. Establishing a new curvilinear coordinate system, we depart from a standard domain transformation commonly used in the engineering literature and utilizing it we rephrase the design problem in one on a fixed domain. Certain advantages of our approach as no need for mesh regeneration and node redistribution and 3D mesh quality improvement are evident. Furthermore, in new coordinate system, it is easy to obtain gradient of optimal functional with respect to the surface of blade and to derive Euler-Lagrange equations for optimal solution which is an elliptic boundary value problem of fourth order. Moreover, we give the conjugate gradient algorithm for the control problem.