We consider problems of controlling the intensity of the Rayleigh-Be´nard convection by adjusting the heat flux distribution at the boundary while keeping the heat input the same. The Karhunen-Loe`ve Galerkin procedure is used to reduce the Boussinesq equation to a low dimensional dynamic model, which in turn is employed in a projected gradient method to yield the optimal heat flux distribution. The performance of the Karhunen-Loe`ve Galerkin procedure is assessed in comparison with the traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.
1.
Chandrasekhar, S., 1969, Hydrodynamic and Hydromagnetic Stability, Oxford University Press.
2.
Drazin, P. G., and Reid, W. H., 1981, Hydrodynamic Stability, Cambridge University Press.
3.
Series
, R. W.
, and Hurle
, D. T. J.
, 1991
, “The Use of Magnetic Fields in Semiconductor Crystal Growth
,” J. Cryst. Growth
, 113
, pp. 305
–328
.4.
Sterl
, A.
, 1990
, “Numerical Simulation of Liquid Metal MHD Flows in Rectangular Ducts
,” J. Fluid Mech.
, 216
, pp. 161
–191
.5.
Gunzburger
, M. D.
, Hou
, L.
, and Svobodny
, T. P.
, 1991
, “Boundary Velocity Control of Incompressible Flow with an Application to Viscous Drag Reduction
,” SIAM J. Control Optim.
, 30
, pp. 167
–181
.6.
Park
, H. M.
, and Lee
, M. W.
, 1998
, “An Efficient Method of Solving the Navier-Stokes Equation for the Flow Control
,” Int. J. Numer. Methods Eng.
, 41
, pp. 1131
–1151
.7.
Park
, H. M.
, and Cho
, D. H.
, 1996
, “Low Dimensional Modeling of Flow Reactors
,” Int. J. Heat Mass Transf.
, 39
, pp. 3311
–3323
.8.
Park
, H. M.
, and Kim
, O. Y.
, 2000
, “A Reduction Method for the Boundary Control of the Heat Conduction Equation
,” ASME J. Dyn. Syst., Meas., Control
122
, pp. 435
–444
.9.
Courant, R., and Hilbert, D., 1962, Methods of Mathematical Physics 1, Wiley-Interscience, New York.
10.
Luenberger, D. G., 1984, Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA.
11.
Berggren
, M.
, 1998
, “Numerical Solution of a Flow-Control Problem: Vorticity Reduction by Dynamic Boundary Action
,” SIAM J. Sci. Comput. (USA)
, 19
, pp. 829
–860
.12.
Glowinski
, R.
, 1993
, “Ensuring Well-Posedness through Analogy: Stokes Problem and Boundary Control of the Wave Equation
,” J. Comput. Phys.
, 104
, pp. 189
–221
.13.
Park, H. M., and Lee, W. J., 2002, “Numerical Methods for the Boundary Optimal Control Problems of the Heat Equation,” Int. J. Numer. Methods Eng., (in press).
14.
Park
, H. M.
, and Chung
, O. Y.
, 1999
, “Inverse Natural Convection Problem of Estimating Wall Heat Flux Using a Moving Sensor
,” ASME J. Heat Transfer
, 121
, pp. 828
–836
.15.
Park
, H. M.
, and Jung
, W. S.
, 2001
, “The Karhunen-Loe`ve Galerkin Method for the Inverse Natural Convection Problems
,” Int. J. Heat Mass Transf.
, 44
, pp. 155
–167
.Copyright © 2002
by ASME
You do not currently have access to this content.
