Bounds on heat transport for convection driven by internal heating

Abstract

The mean vertical heat transport $⟨wT⟩$ in convection between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number $R$ by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of quadratic auxiliary functions. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound $⟨wT⟩\le 1/2$ can only be improved up to finite $R$. Indeed, we demonstrate analytically that $⟨wT⟩\le 2^{-21/5}{R}^{1/5}$ and therefore prove that $⟨wT⟩<1/2$ for $R<65536$. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than $1/2$ until at least $R=3.4\times 10^5$. While the computational results suggest that the best bound on $⟨wT⟩$ approaches $1/2$ asymptotically from below as $R\to \infty$, we prove that typical analytical constructions cannot be used to prove this conjecture.

arXiv comment33 pages, 12 figures

References in corpus (14)

• Scaling in thermal convection: A unifying theory
• "Ultimate state" of two-dimensional Rayleigh-Benard convection between free-slip fixed temperature boundaries
• Transition to the ultimate regime in a radiatively driven convection experiment
• Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization
• Penetrative internally heated convection in two and three dimensions
• Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems
• Convection driven by internal heating
• Bounds for convection between rough boundaries
• Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming
• Internal heating driven convection at infinite Prandtl number
• Internally heated convection beneath a poor conductor
• Bounds on heat transfer for Bénard-Marangoni convection at infinite Prandtl number
• Optimal bounds with semidefinite programming: an application to stress driven shear flows
• Optimization with affine homogeneous quadratic integral inequality constraints

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• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Internal heating profiles for which downward conduction is impossible
• Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries
• Internally heated convection with rotation: bounds on heat transport
• Ultimate regimes in horizontal and internally heated convection
• Prandtl number dependence of rotating internally heated convection