Wen Baole, Chini Gregory P, Kerswell Rich R, Doering Charles R
Program in Integrated Applied Mathematics, University of New Hampshire, Durham, New Hampshire 03824, USA.
Center for Fluid Physics, University of New Hampshire, Durham, New Hampshire 03824, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):043012. doi: 10.1103/PhysRevE.92.043012. Epub 2015 Oct 16.
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.
分析了一种用于数值求解一类变分问题的替代计算方法，这类变分问题源于对强迫耗散无限维非线性动力系统（包括纳维 - 斯托克斯方程和奥伯贝克 - 布西涅斯克方程）进行严格上界分析时出现的问题，并将其应用于瑞利 - 贝纳德对流。针对三种典型流动构型，给出了证明：该数值算法能够收敛到的唯一稳态是相关变分问题所需的全局最优解。与“背景场”变分框架内用于计算传输量（如热量或动量）最优界的大多数其他数值方案不同，那些方案采用牛顿法的变体，因此需要非常精确的初始迭代值，而这种新的计算方法易于实现，并且至关重要的是，不需要数值延拓。该算法用于确定二维瑞利 - 贝纳德对流在无应力等温边界之间（瑞利 1916 年的原始对流模型）热传输增强因子（即努塞尔数（Nu））的最优背景方法界，作为瑞利数（Ra）、普朗特数（Pr）和域纵横比 L 的函数。计算结果具有重要意义，因为分析、实验室实验和数值模拟在假定的 Nu∼Pr(α)Ra(β)标度关系中暗示了一系列指数α和β。计算结果清楚地表明，对于固定的 L = 2√[2]且 Ra≤10(10)，Nu≤0.106Pr(0)Ra(5/12)，这表明在“最终”的高 Ra 区域，分子传输通常不能被忽略。
