Lin Wenxian, Armfield S W
Solar Energy Research Institute, Yunnan Normal University, Kunming, Yunnan 650092, People's Republic of China and School of Engineering, James Cook University, Townsville, QLD 4811, Australia.
School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Dec;88(6):063013. doi: 10.1103/PhysRevE.88.063013. Epub 2013 Dec 18.
It is of fundamental significance, especially with regard to application, to fully understand the flow behavior of unsteady natural convection boundary layers on a vertical plate heated by a time-dependent heat flux. Such an understanding is currently scarce. In this paper, the scaling analysis by Lin et al. [Phys. Rev. E 79, 066313 (2009)] using a simple three-region structure for the unsteady natural convection boundary layer of a homogeneous Newtonian fluid with Pr>1 under isothermal heating was substantially extended for the case when the heating is due to a time-varying sinusoidal heat flux. A series of scalings was developed for the thermal boundary thickness, the plate temperature, the viscous boundary thicknesses, and the maximum vertical velocity within the boundary layer, which are the major parameters representing the flow behavior, in terms of the governing parameters of the flow, i.e., the Rayleigh number Ra, the Prandtl number Pr, and the dimensionless natural frequency f(n) of the time-varying sinusoidal heat flux, at the start-up stage, at the transition time scale which represents the ending of the start-up stage and the beginning of the transitional stage of the boundary-layer development, and at the quasi-steady stage. These scalings were validated by comparison to 10 full numerical solutions of the governing equations with Ra, Pr, and f(n) in the ranges 10(6)≤Ra≤10(9), 3≤Pr≤100, and 0.01≤f_{n}≤0.1 and were shown in general to provide an accurate description of the flow at different development stages, except for high-Pr runs in which a further, although weak, Pr dependence is present, which cannot be accurately predicted by the current scaling analysis using the simple three-region structure, attributed to the non-boundary-layer nature of the velocity field with high-Pr fluids. Some scalings at the transition time scale and at the quasi-steady stage also produce noticeable deviations from the numerical results when f(n) is reduced, indicating that there may be a further f(n) dependence of the scalings which also cannot be accurately predicted by the current scaling analysis.
充分理解由随时间变化的热流加热的垂直平板上非稳态自然对流边界层的流动特性具有根本重要性，尤其是在应用方面。目前对此类理解还很匮乏。在本文中，Lin等人[《物理评论E》79, 066313 (2009)]针对等温加热条件下Pr>1的均匀牛顿流体非稳态自然对流边界层采用简单三区结构进行的尺度分析，在加热由随时间变化的正弦热流引起的情况下得到了大幅扩展。针对热边界厚度、平板温度、粘性边界厚度以及边界层内最大垂直速度等一系列代表流动特性的主要参数，根据流动的控制参数，即瑞利数Ra、普朗特数Pr以及随时间变化的正弦热流的无量纲固有频率f(n)，分别在启动阶段、代表边界层发展启动阶段结束和过渡阶段开始的过渡时间尺度以及准稳态阶段建立了尺度关系。通过与10个控制方程的完整数值解进行比较，验证了这些尺度关系，其中Ra、Pr和f(n)的取值范围分别为10(6)≤Ra≤10(9)、3≤Pr≤100和0.01≤f_{n}≤0.1，结果表明，一般情况下这些尺度关系能准确描述不同发展阶段的流动，但对于高Pr运行情况，存在进一步的、尽管较弱的Pr依赖性，当前使用简单三区结构的尺度分析无法准确预测，这归因于高Pr流体速度场的非边界层性质。当f(n)减小时，过渡时间尺度和准稳态阶段的一些尺度关系也会与数值结果产生明显偏差，这表明尺度关系可能还存在进一步的f(n)依赖性，当前的尺度分析也无法准确预测。
