arXiv 0811.30482008-11-19physics.flu-dyn

具有一般热边界条件的Rayleigh-Benard对流的边界部分1.固定Biot数边界

Bounds on Rayleigh-Benard convection with general thermal boundary conditions. Part 1. Fixed Biot number boundaries

Ralf W. Wittenberg

备注：: Submitted

摘要：

本文研究了湍流Rayleigh-Benard对流边界的热性质对对流热输运的影响。本文利用Doering-Constantin背景流方法，系统地推导了常Biot数η的一般混合热边界条件的Nusselt-Rayleigh数关系的边界原理，该边界原理在已有研究的固定温度（$η= 0$）和固定通量（$η= \infty$）情况之间连续插值，并得到了显式的渐近和严格的边界.引入一个控制参数R作为衡量的驱动，这是在一般情况下不同于通常的瑞利数Ra，我们发现，对于每个$η> 0$，作为R增加的约束上的努塞尔数Nu的方法，为固定通量的问题。具体地说，对于$0 < η\leq \infty$和足够大的R（对于小η为$R > R_s = O(η^{-2})$），努塞尔数被限定为$Nu \leq c(η) R^{1/3} \leq C Ra^{1/2}$，其中C是与η无关的常数。在$R \to \infty$极限中，通常的固定温度假设因此是这个一般边界问题的奇异极限。

英文摘要：

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Benard convection on analytical bounds on convective heat transport. Using the Doering-Constantin background flow method, we systematically formulate a bounding principle on the Nusselt-Rayleigh number relationship for general mixed thermal boundary conditions of constant Biot number ηwhich continuously interpolates between the previously studied fixed temperature ($η= 0$) and fixed flux ($η= \infty$) cases, and derive explicit asymptotic and rigorous bounds. Introducing a control parameter R as a measure of the driving which is in general different from the usual Rayleigh number Ra, we find that for each $η> 0$, as R increases the bound on the Nusselt number Nu approaches that for the fixed flux problem. Specifically, for $0 < η\leq \infty$ and for sufficiently large R ($R > R_s = O(η^{-2})$ for small η) the Nusselt number is bounded as $Nu \leq c(η) R^{1/3} \leq C Ra^{1/2}$, where C is an η-independent constant. In the $R \to \infty$ limit, the usual fixed temperature assumption is thus a singular limit of this general bounding problem.

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