Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection.
Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection.
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解决上限问题的时间步进方法：在二维瑞利-贝纳德对流中的应用。
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
C. Doering
中科院分区：
文献类型：
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作者：
Baole Wen;G. Chini;R. Kerswell;C. Doering
分析了一类由强迫耗散无限维非线性动力系统的严格上界分析引起的变分问题，包括Navier-Stokes方程和Oberbeck-Boussinesq方程，并将其应用于Rayleigh-Bénard对流。对于三种典型流型，证明了该数值算法唯一能收敛到的定常状态是相应变分问题的全局最优解。与其他大多数用于计算“背景场”变分框架内传输量(例如，热或动量)的最优界的数值格式不同，这些数值格式使用牛顿方法的变体，因此需要非常精确的初始迭代，新的计算方法很容易实现，关键是不需要数值连续。该算法被用来确定作为瑞利数(Ra)、普朗特数(Pr)和区域长宽比L的函数的二维无应力等温界面间瑞利-Bénard对流(瑞利最初的1916年对流模型)中关于热传输强化因子的最佳背景方法界限，即努塞尔数(Nu)。计算结果是有意义的，因为分析、实验室实验和数值模拟表明，在假定的NuαPr(β)Ra()标度关系中，指数范围是α和β。计算表明，对于Ra≤10(10)，在固定的L=2√[2]下，Nu≤为0.106Pr(0)Ra(5/12)，这表明在“极限”高Ra区，分子输运一般不能忽略。
An alternative computational procedure for numerically solving a class of variational problems arising from rigorous upper-bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number (Nu), as a function of the Rayleigh number (Ra), Prandtl number (Pr), and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu∼Pr(α)Ra(β) scaling relation. The computations clearly show that for Ra≤10(10) at fixed L=2√[2],Nu≤0.106Pr(0)Ra(5/12), which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.