### Abstract

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≤0.167 Ra^{1/2}. Non-linear 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.

Original language | English (US) |
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Pages (from-to) | 5957-5981 |

Number of pages | 25 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 53 |

Issue number | 6 SUPPL. A |

DOIs | |

State | Published - Jan 1 1996 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics