论文标题
通过多项式优化的雷利 - 贝纳德对流模型的截短模型的热传输边界
Heat transport bounds for a truncated model of Rayleigh-Bénard convection via polynomial optimization
论文作者
论文摘要
对于雷利（Rayleigh）1916年的自然热对流模型的八模式盖金截断，获得了时间平均热传输的上限。 ODE模型的边界---通过构造满足足够条件的辅助函数来得出的洛伦兹三联系统的扩展，其中某些多项式表达式必须是无负的。通过要求多项式表达式允许平方表示表示，从而实现了此类条件，从而可以使用半决赛编程来最大程度地减少所得边界。对于模型参数的大量值，计算平均热传输上的尖锐或几乎锐利的边界：瑞利和prandtl数字以及域纵横比。在所有情况下，雷利数足够小，以至于ode模型可以定量地接近PDE模型，则平均热传输是通过稳态最大化的。在某些情况下，在较大的瑞利数字下，时间周期状态最大化截短模型中的热传输。分析参数依赖性界限是使用二次辅助函数得出的，并且它们对于足够小的瑞利数字很清晰。
Upper bounds on time-averaged heat transport are obtained for an eight-mode Galerkin truncation of Rayleigh's 1916 model of natural thermal convection. Bounds for the ODE model---an extension of Lorenz's three-ODE system---are derived by constructing auxiliary functions that satisfy sufficient conditions wherein certain polynomial expressions must be nonnegative. Such conditions are enforced by requiring the polynomial expressions to admit sum-of-squares representations, allowing the resulting bounds to be minimized using semidefinite programming. Sharp or nearly sharp bounds on mean heat transport are computed numerically for numerous values of the model parameters: the Rayleigh and Prandtl numbers and the domain aspect ratio. In all cases where the Rayleigh number is small enough for the ODE model to be quantitatively close to the PDE model, mean heat transport is maximized by steady states. In some cases at larger Rayleigh number, time-periodic states maximize heat transport in the truncated model. Analytical parameter-dependent bounds are derived using quadratic auxiliary functions, and they are sharp for sufficiently small Rayleigh numbers.
