Boundary Shape Control of Navier-Stokes Equations and Geometrical Design Method for Blade's Surface in the Impeller

TL;DR

This paper introduces a new geometrical design method for impeller blades by coupling Navier-Stokes equations with a boundary value problem, enabling optimal blade shape design to improve flow efficiency.

Contribution

It develops a novel optimal control framework coupling Navier-Stokes equations with boundary shape optimization for impeller blade design.

Findings

Existence of solutions for the optimal control problem is proven.

A coordinate system is used to explicitly relate the objective functional to blade shape.

Weak continuity of Navier-Stokes solutions with respect to blade shape is established.

Abstract

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 $\Theta$ 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.