Variational bounds on energy dissipation in incompressible flows .3. Convection

被引:205
作者
Doering, CR [1 ]
Constantin, P [1 ]
机构
[1] UNIV CHICAGO,DEPT MATH,CHICAGO,IL 60637
来源
PHYSICAL REVIEW E | 1996年 / 53卷 / 06期
关键词
D O I
10.1103/PhysRevE.53.5957
中图分类号
O35 [流体力学]; O53 [等离子体物理学];
学科分类号
070204 ; 080103 ; 080704 ;
摘要
Building on a method of analysis for the Navier-Stokes equations introduced by Hopf [Math. Ann. 117, 764 (1941)], a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the Boussinesq equations of motion. When supplied with appropriate test background fields satisfying a spectral constraint, reminiscent of an energy stability condition, the variational formulation produces rigorous upper bounds on the Nusselt number (Nu) as a function of the Rayleigh number (Ra). For the case of vertical heat convection between parallel plates in the absence of sidewalls, a simplified (but rigorous) formulation of the optimization problem yields the large Rayleigh number bound Nu less than or equal to 0.167 Ra-1/2-1. Nonlinear Euler-Lagrange equations for the optimal background fields are also derived, which allow us to make contact with the upper bound theory of Howard [J. Fluid Mech. 17, 405 (1963)] for statistically stationary flows. The structure of solutions of the Euler-Lagrange equations are elucidated from the geometry of the variational constraints, which sheds light on Busse's [J. Fluid Mech. 37, 457 (1969)] asymptotic analysis of general solutions to Howard's Euler-Lagrange equations. The results of our analysis are discussed in the context of theory, recent experiments, and direct numerical simulations.
引用
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页码:5957 / 5981
页数:25
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