Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 52071272, 1210021247), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2020JC-18), the Fundamental Research Funds for the Central Universities, China (Grant Nos. 3102020HHZY030014, 3102021HHZY030002), and the Open Fund of Henan Key Laboratory of Underwater Intelligent Equipment, China (Grant No. KL01B2101)
Received Date:08 March 2021
Accepted Date:07 May 2021
Available Online:07 June 2021
Published Online:20 September 2021
Abstract:Studying transitions from laminar to turbulence of non-Newtonian fluids can provide a theoretical basis to further mediate their dynamic properties. Compared with Newtonian fluids, transitions of non-Newtonian fluids turning are less focused, thus being lack of good predictions of the critical Reynolds number (Re) corresponding to the first Hopf bifurcation. In this study, we employ the lattice Boltzmann method as the core solver to simulate two-dimensional lid-driven flows of a typical non-Newtonian fluid modeled by the power rheology law. Results show that the critical Re of shear-thinning (5496) and shear-thickening fluids (11546) are distinct from that of Newtonian fluids (7835). Moreover, when Re is slightly larger than the critical one, temporal variations of velocity components at the monitor point all show a periodic trend. Before transition of the flow filed, the velocity components show a horizontal straight line, and after transition , the velocity components fluctuate greatly and irregularly. Through fast Fourier transform for the velocity components, it is noted that the velocity has a dominant frequency and a harmonic frequency when Re is marginally larger than the critical one. Besides, the velocity is steady before transition of flow filed, so it appears as a point on the frequency spectrum. As the flow filed turns to be turbulent, the frequency spectrum of the velocity component appears multispectral. Different from a single point in the velocity phase diagram before transition, the velocity phase diagram after transition forms a smooth and closed curve, whose area is also increasing as Re increases. The center point of the curve moves along a certain direction, while movement directions of different center points are different. Proper orthogonal decompositions for the velocity and vorticity field reveal that the first two modes, in all types of fluids, are the dominant modes when Re is close to the critical one, with energy, occupying more than 95% the whole energy. In addition, for one type of fluid, the dominant modes at different Re values have similar structures. Results of the first and second modes of velocity field show that the modal peak is mainly distributed in vicinity of the cavity wall. Keywords:power-law fluid/ transition/ lattice Boltzmann method/ lid-driven flow
进一步使用POD对牛顿流体、剪切变稀和剪切增稠流体的流场速度、涡量和黏性进行处理, 结果如图7和图8所示. 图7(a)—(c)分别为牛顿流体、剪切变稀流体和剪切增稠流体的水平速度POD能量占比图. 从图7可以发现, 不同类型的流体在第一临界转捩雷诺数附近且发生转捩后, 均呈现前两阶模态能量占主导地位的特点, 能量占比超95%, 且前两阶模态能量占比接近. 在图7(a)Re = 10500时, 出现了多模态的结果, 这里的流场可能是进入Hopf第二分岔第二模式中[5]. 在图7(c)Re = 11500, 12500时, 一阶模态能量占比大幅度高于二阶模态的能量占比, 但结合图6(c)可以发现, 此时流场速度波动量较小几乎为定常流场, 一阶模态与二阶模态相对于平均流场几乎可以忽略. 图 7 速度u的各阶模态的能量占比 (a) 牛顿流体; (b) 剪切变稀流体; (c) 剪切增稠流体 Figure7. Energy share of each order of mode for velocity u: (a) Newtonian fluid; (b) shear thinning-fluid; (c) shear-thickening fluid.
图 8 涡量的各阶模态的能量占比 (a) 牛顿流体; (b) 剪切变稀流体; (c) 剪切增稠流体 Figure8. Vortex energy share of each order of mode: (a) Newtonian fluid; (b) shear-thinning fluid; (c) shear-thickening fluid.
图8(a)—(c)分别为牛顿流体、剪切变稀流体和剪切增稠流体的涡量POD能量占比图. 涡量POD能量占比与水平速度POD能量占比结果相比, 同种流体在同一雷诺数下, 涡量POD一阶模态的占比略高于水平速度POD能量占比, 涡量POD中前两阶模仍占主导地位. 图9为牛顿流体在不同雷诺数下各阶模态的流场模态的速度场云图, 图9(a)和图9(d)分别为Re = 8500与Re = 9000时平均水平速度场云图, 图9(b)和图9(e) 分别为Re = 8500与Re = 9000时水平速度一阶模态云图, 图9(c)和图9(f) 分别为Re = 8500与Re = 9000时水平速度二阶模态云图. 从速度平均场来看, Re = 8500与Re = 9000的结果相近; 从一阶模态的云图来看, 峰值区域均靠近壁面, 且峰值区域的形状相似; 从二阶模态的云图来看, 二阶模态的结果相近, 且峰值主要分布在壁面处. 图 9 牛顿流体的水平速度的各阶模态图 (a), (b), (c) Re = 8500时, 平均场、一阶模态与二阶模态; (d), (e), (f) Re = 9000时, 平均场、一阶模态与二阶模态 Figure9. Modal diagrams of horizontal velocity of Newtonian fluid: (a), (b) and (c) The mean field, the first and second modes when Re = 8500; (d), (e), (f) mean field, first-order mode and second-order mode when Re = 9000
图10为剪切变稀流体在不同Re下各阶模态的流场模态的速度场云图, 图10(a)和图10(d)分别为Re = 5700与Re = 6000时平均速度场云图, 图10(b)和图10(e)分别为Re = 5700与Re = 6000时速度一阶模态云图, 图10(c)和图10(f)分别为Re = 5700与Re = 6000时速度二阶模态云图. 从平均场的结果来看, Re = 5700与Re = 6000结果相近; 从一阶和二阶模态的结果来看, 模态峰值分布区域集中在壁面附近, 峰值分布区域的形状相似, 但峰值分布区的模态值相反. 图 10 剪切变稀流体的水平速度的各阶模态图 (a), (b), (c) Re = 5700时, 平均场、一阶模态与二阶模态; (d), (e), (f) Re = 6000时, 平均场、一阶模态与二阶模态 Figure10. Modal diagrams of horizontal velocity of shear-thinning fluid: (a), (b), (c) The mean field, the first and second modes when Re = 5700; (d), (e), (f) mean field, first-order mode and second-order mode when Re = 6000.
图11为剪切增稠流体在不同Re下各阶模态的流场模态的速度场云图, 图11(a)和图11(d)分别为Re = 13000与Re = 13500时平均速度场云图, 图11(b)和图11(e)分别为Re = 13000与Re = 13500时速度一阶模态云图, 图11(c)和图11(f)分别为Re = 13000与Re = 13500时速度二阶模态云图. 从平均场的结果来看, Re = 13000与Re = 13500结果相近; 从一阶和二阶模态的结果来看, 峰值分布区域集中在壁面附近, 且峰值分布区域的形状相似, 这与牛顿流体和剪切变稀流体的结果是类似的. 剪切增稠流体一阶和二阶模态峰值分布区的模态值相反. 图 11 剪切增稠流体的水平速度的各阶模态图 (a), (b), (c) Re = 13000时, 平均场、一阶模态与二阶模态; (d), (e), (f) Re = 13500时, 平均场、一阶模态与二阶模态 Figure11. Modal diagrams of horizontal velocity of shear-thickening fluid: (a), (b), (c) The mean field, the first and second modes when Re = 13000; (d), (e), (f) mean field, first-order mode and second-order mode when Re = 13500.