Boundary shape control of the Navier-Stokes equations and applications

Abstract

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gâteaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.

A new boundary layer equation and its application to the flow in turbo‐machinery

“…[1]. Furthermore, the covariant components and contrivariant components (g ij , g ij ) of metric tensor of three Euclidean space 3 in the semigeodesic coordinate system are, respectively, given in the same.…”

“…where f λ k , h λ k can be found in Ref. [4]. γ in = l i ×[0, δ], γ out = l o ×[0, δ], l i and l 0 are blade leading edge and blade trailing edge, respectively.…”

“…For verifying the reliability of BLE quantitatively, we introduce the computation of impeller power and the general formulation is as follows [1],…”

“…Moreover, the basis vectors of the semigeodesic coordinate system are given by Ref. [1], (e θ ) 1 = e θ · e 1 = e θ · (x 2 θ 1 e θ + k) = x 2 θ 1 , (e θ ) 2 = e θ · e 2 = e θ · (x 2 θ 2 e θ + e r ) = x 2 θ 2 , On the other hand, the stress tensor in the semigeodesic coordinate system is, σ 3i = 2νg 33 g ik e 3k (u) − pg 3i , σ 33 = e 33 (u) − p = 2ν ∂u 3 ∂ξ − p 0 = 2νU 3 − p 0 , σ 3α (u, p) = a αβ e 3β (u) = a αβ γ 3β (u) = νa αβ a βλ ∂u λ ∂ξ + * ∇β u 3 = νa αβ (a βλ U λ ) = νU α , Furthermore, the results of BLE are considered; therefore, the formulation of power could be simplified as follows,…”

“…Many researchers propose the dimensional splitting method to compute 3D viscous flow (see Refs. [1,2,5,6,[10][11][12][13]). Here we present the boundary layer equations, which are a part of dimensional splitting method.…”