Scaling of Reynolds number based on maximum velocity and characteristic Reynolds number in two-dimensional thermal turbulence convection
Authors: He Jian-Chao , Fang Ming-Wei , Bao Yun ,   School of Aeronautics Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China,   School of Aeronautics and Astronautics, Sun Yat-Sen University, Shenzhen 518107, China
Published: June 2022

Rayleigh number (<i>Ra</i>) dependence in Rayleigh-Bénard (RB) convection has been studied by many investigators, but the reported power-law scaling expressions are different in these researches. Previous studies have found that when <i>Ra</i> reaches a critical value, the flow patterns change and a transition appears in the scaling of <i>Nu</i>(<i>Ra</i>) (where <i>Nu</i> represents Nusselt number) and <i>Re</i>(<i>Ra</i>) (where <i>Re</i> denotes Reynold number). The Grossmann-Lohse(GL) model divides the <i>Ra-Pr(</i>where <i>Pr</i> refers to Prandtl number) phase into several regions to predict the scaling expressions of <i>Nu</i>(<i>Ra,Pr</i>) and <i>Re</i>(<i>Ra,Pr</i>), indicating that the thermal dissipation behavior and kinetic dissipation behaviors are diverse in the different regions. Moreover, some physical quantities also show a transition and some structures in the flow fields, such as large scale circulation and boundary layer, change when <i>Ra</i> increases. In this work, we conduct a series of numerical simulations in two-dimensional RB convection with <i>Ra</i> ranging from 10<sup>7</sup> to 10<sup>12</sup> and Pr ranging from 0.25 to 100, which is unprecedentedly wide. The relationship between the maximum velocity and <i>Ra</i> is investigated, and an unexpected drop happens when <i>Ra</i> reaches a critical value <i>Ra</i><sub>c</sub>, and <i>Ra</i><sub>c</sub> increases with Pr increasing. The <i>Re</i> number, which is defined as a maximum velocity, also shows a plateau at <i>Ra</i><sub>c</sub>. Before and after <i>Ra</i><sub>c</sub>, the <i>Ra</i> scaling exponent of <i>Re</i> remains 0.55, which gets smaller at very high <i>Ra</i>. Specially, under different <i>Pr</i> values, the plateau appears at <i>Re</i><sub>c</sub> ≈ 1.4 × 10<sup>4</sup>. In addition, a scaling <i>Ra</i><sub>c</sub>~<i>Pr</i><sup>1.5</sup> is found and the <i>Ra</i> is compensated for by <i>Pr</i><sup>–1.5</sup> to disscuss the relationship between <i>Re</i> and <i>RaPr</i><sup>–1.5</sup>. It is interesting that the <i>Re</i>(<i>RaPr</i><sup>–1.5</sup>) expressons at different <i>Pr</i> values well coincide, indicating a self-similarity of <i>Re</i>(<i>RaPr</i><sup>–1.5</sup>). The plateau appears at <i>RaPr</i><sup>–1.5</sup> = 1 × 10<sup>9</sup>, meaning that <i>Re</i><sub>c</sub> would reach 1.4 × 10<sup>4</sup> at any <i>Pr</i> value when <i>RaPr</i><sup>–1.5</sup> = 1 × 10<sup>9</sup>. To further investigate the plateau of <i>Re</i>, the flow patterns are compared with time-averaged velocity fields and we find that the large scale circulation (LSC) changes from ellipse to circle at <i>Ra</i><sub>c</sub>. In other words, the flow pattern will change into circular LSC at <i>Re</i><sub>c</sub> at different <i>Pr</i> values, and <i>Re</i><sub>c</sub> is a constant as mentioned above. This finding can help us to distinguish the two flow patterns with given <i>Ra</i> and <i>Pr</i>, and to predict the <i>Re</i> scaling in an appropriate range of <i>Ra</i> with different <i>Pr</i> values.