Optimal control of unsteady compressible viscous flows

Abstract

SUMMARYThe control of complex, unsteady ows is a pacing technology for advances in uid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical ow control systems. However, most of the prior work in this area has focused on incompressible ow which precludes many of the important physical ow phenomena that must be controlled in practice including the coupling of uid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two-dimensional compressible Navier-Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter-rotating viscous vortices above an inÿnite wall which, due to the self-induced velocity ÿeld, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall-normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the in uence of control regularization for each case.

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

“…Control and stability of fluid flow have been a significant topic of study and have numerous useful applications. Many researchers have been interested in the subject of the controllability of fluid flows, more so for incompressible flow (see [1][2][3][4][5][6]) than for compressible flow (see [7,8]). The stability analysis of the linearized compressible Navier-Stokes system is of interest to us in this research.…”

“…8) with respect to t, we have.  1 (t) = ∫ L 0 e −𝜆x 𝜎𝜎 t dx = − ∫ L −𝜆x 𝜎𝜎 x dx − b ∫ L −𝜆x 𝜎u x dx − ∫ L 0 a(x)e −𝜆x 𝜎 2 dx − ∫ 𝜔 e −𝜆x c(x)𝜎(t − 𝜏, x)𝜎(t, x) dx −𝜆x 𝜎 2 dx − e −𝜆L 2 𝜎 2 (t, L) + 1 (t, 0) − b ∫ L −𝜆x 𝜎u x dx − ∫ L 0 a(x)e −𝜆x 𝜎 2 dx − ∫ 𝜔 e −𝜆x c(x)𝜎(t − 𝜏, x)𝜎(t, x) dx −𝜆x 𝜎 2 dx − b ∫ L −𝜆x 𝜎u x dx − ∫ L 0 a(x)e −𝜆x 𝜎…”

Optimization of zeta potential distributions for minimal dispersion in an electroosmotic microchannel

“…Therefore, the researchers [24][25][26] investigated the optimization problems with the governing equations coupled by the flow field and the magnetic field. Also, Collis et al [27] studied on the optimal flow control problem coupled by the flow and the temperature.…”

Adaptive finite element methods for the solution of inverse problems in optical tomography

“…The goal of this paper is therefore to review the numerical techniques necessary to solve nonlinear inverse problems on adaptively refined meshes, using optical tomography as a realistic testcase. The general approach to solving the problem is similar as used in work by other researchers [1,17,18] and related to methods described and analyzed in [26,38,39,79]. However, adaptivity is a rather intrusive component of PDE solvers, and we will consequently have to re-consider all aspects of the numerical solution of this inverse problem and modify them for the requirements of adaptive schemes.…”