Stabilization of two-dimensional Rayleigh–Bénard convection by means of optimal feedback control

doi:10.1016/j.physd.2003.07.001

Physica D: Nonlinear Phenomena

Volume 186, Issues 3–4, 15 December 2003, Pages 185-204

https://doi.org/10.1016/j.physd.2003.07.001 Get rights and content

Abstract

We consider the problem of suppressing the Rayleigh–Bénard convection in a finite domain by adjusting the heat flux profile at the bottom of the system while keeping the heat input the same. The appropriate profile of heat flux at the bottom is determined by the optimal feedback control. When most of the convection modes are taken into consideration in the construction of the feedback controller, the suppressed state is found to be stable for all range of Rayleigh number investigated. With the feedback controller constructed by employing only the dominant convection modes, however, there exists a threshold Rayleigh number beyond which a new convective state emerges due to the hydrodynamic instability. The threshold Rayleigh number is found to increase with the number of modes taken in the feedback controller.

Introduction

Thermal instability may arise when a stagnant fluid layer is heated from below such that its lower side is hotter than its upper. The resulting convective flow is called the Rayleigh–Bénard convection. The dimensionless number corresponding to the heat input or the temperature gradient in the system is the Rayleigh number, and convection sets in only when the Rayleigh number is above the critical one. The determination of the critical Rayleigh number and how intensity of the convection increases with the Rayleigh number are well documented in Chandrasekhar [1] and Drazin and Reid [2]. In many industrial processes, it may be desirable to suppress the intensity of convection without changing the Rayleigh number. In the Czochralski method of processing semiconductor materials, convective flow gives rise to a fluctuating rate of crystal growth which, in turn, produces a microscopically nonuniform distribution of dopant in the crystal. Therefore, it is necessary to suppress natural convection by some means [3].

In the present investigation, we consider a method of suppressing the intensity of the Rayleigh–Bénard convection by adjusting heat flux distribution at the bottom, while keeping the total heat input or the Rayleigh number the same. Specifically, we consider the determination of the time-dependent spatial distribution of heat flux at the bottom boundary to suppress the intensity of convection at the minimum cost of control. The appropriate profile of heat flux can be determined by means of the optimal feedback control [4]. The optimal feedback control determines the heat flux at the bottom proportional to the velocity and temperature fields in the domain by minimizing an objective function. It shall be shown that when most of the convection modes are taken into consideration in the construction of the feedback controller, the suppressed state is stable for all range of Rayleigh number investigated. With the feedback controller constructed by employing only dominant convection modes, however, there exists a threshold Rayleigh number beyond which a new convective state emerges due to the hydrodynamic instability. The threshold Rayleigh number can be determined by means of the linear stability analysis. Recently, Tang and Bau [5], [6] have considered a similar problem of stabilizing the no-motion state in the Rayleigh–Bénard convection. They employed a proportional feedback controller to suppress the convection and investigated how the gain of the proportional controller affects the linear stability of the Rayleigh–Bénard convection. Other related problems are control of shear-driven channel or boundary layer flows [7], [8], [9]. In this work, they simplified the governing equations to facilitate the implementation of the optimal control schemes.

Section snippets

System and governing equations

We consider a two-dimensional rectangular cavity filled with a Boussinesq fluid. Because of the heat flux imposed at the bottom boundary natural convection is induced in the domain. Our aim is to suppress the convective flow by adjusting the heat flux distribution at the bottom under the constraint that the total heat input at each moment being the same. The governing equations for the system may be written in dimensionless variables as follows: $∇· v =0,$ $∂ v ∂t + v ·∇ v =−∇P+Pr ∇^{2} v +R Pr T j,$ $∂T ∂t + v ·∇T=∇^{2} T,$ with

Construction of the feedback controller

The following control problem is considered in this investigation: given a fixed Rayleigh number where the convective flow is significant if the heat flux F(x,t) is spatially constant (=1), we want to suppress the convection by adjusting the profile of the heat flux at a minimum control action. As the initial field, we take the steady convection field with F(x,t)=1. This statement may be rewritten as the following optimization problem: $minimize J= 12 ∫ 0 t_{f} ∫ Ω v^{2} d Ω d t+ ϵ 2 ∫ 0 t_{f} ∫ x=−1 1 [F(x,t)−1]^{2} d x d t, subject$

Suppression of convection by means of feedback control

, , , , where F^f(x,t) is determined by Eq. (47), are solved to assess the performance of the optimal feedback controller constructed in the previous section. The state vector $x =(x_{1},x_{2},…,x_{NM} |x_{NM+1},x_{NM+2},…,x_{NM+NT})^{T}$ to be used in Eq. (47) is obtained from v and Θ as $x_{k} = ∫ Ω v ·φ_{(k)} d Ω M_{k}, 1≤k≤ NM,$ $x_{k} = ∫ Ω Θϕ_{(k)} d Ω N_{k}, NM +1≤k≤ NM + NT.$ Fig. 2 shows how the convection intensity v² is suppressed with time by the optimal feedback controller which is constructed using the reduced order model with NM=45 and NT=45, where

Linear stability analysis for the suppressed state

The threshold Rayleigh number for the controlled Rayleigh–Bénard convection can be determined by means of the linear stability analysis. At the suppressed state, we may safely neglect nonlinear terms in the Boussinesq set since the flow is negligible. Therefore, the linear stability analysis is expected to yield accurate results at the suppressed state. The governing equations for the linear stability analysis can be derived by deleting nonlinear terms, v·∇v and v·∇Θ, in , , , , and employing

Conclusions

The problem of suppressing the intensity of the Rayleigh–Bénard convection in a finite domain is considered, where the heat flux profile at the bottom is adjusted continually while keeping the heat input the same. The optimal state feedback controller is constructed analytically by reducing the Boussinesq equation to a small set of ordinary differential equations with the help of eigenfunctions of the linear stability analysis. Hydrodynamic stability analysis in the finite domain is performed

References (12)

R.W. Series et al.
The use of magnetic fields in semiconductor crystal growth
J. Cryst. Growth
(1991)
T. Bewley et al.
A general framework for robust control in fluid mechanics
Physica D
(2000)
H.C. Ku et al.
Pseudospectral methods for solution of the incompressible Navier–Stokes equation
Comput. Fluids
(1987)
H.M. Park et al.
A solution method of nonlinear convection stability problems in finite domains
J. Comput. Phys.
(2001)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford,...
P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge,...

There are more references available in the full text version of this article.

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Stabilization of two-dimensional Rayleigh–Bénard convection by means of optimal feedback control

Abstract

Introduction

Section snippets

System and governing equations

Construction of the feedback controller

Suppression of convection by means of feedback control

Linear stability analysis for the suppressed state

Conclusions

J. Cryst. Growth

Physica D

Comput. Fluids

J. Comput. Phys.