Fund Project:Project supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 300102128301, 300102128302), the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2019JQ-507), and the National for the Innovative University Student Foundation of China (Grant No. 201910710174)
Received Date:18 April 2020
Accepted Date:17 June 2020
Available Online:02 December 2020
Published Online:20 December 2020
Abstract:When a container filled with water is subjected to vertical vibration, bubbles in the water may sink. This phenomenon exists widely in the field of engineering, and has a non-negligible influence on aerospace engineering and ship engineering. Therefore, it is of great significance to study the movement of bubble sinking in order to reduce the adverse effect caused by bubble sinking in the project. In previous papers, the effect of Basset force on bubble motion was usually ignored. In this paper, the bubble motion model based on the ideal gas equation is built for spherical bubbles, and the influence of the Basset force on the bubble motion is considered in the model. In the process of solving Basset force, the motion is directly separated and the convergence factor is introduced in theoretical solution. The equal step composite trapezoid formula is applied to the numerical solution. The results of numerical calculation show that the added mass force is important for bubble sinking. We find that the Basset force has no effect on the stable oscillation position of bubble, but it can accelerate the later trajectory of bubble motion. Importantly, we demonstrate that the bubble is hindered by the following component forces: buoyancy, viscous resistance, and flow thrust (which are ordered from large to small value). The movement of the bubble is observed to be in the form of oscillation, and there exists a depth, i.e. a critical depth: the bubble oscillate steadily at this depth, specifically, the bubble rises above this depth and sinks below this depth. When the vibration pressure changes, the location of the bubble’s stable oscillation will also be affected. The origin can be ascribed to the change of added mass force caused by the change of vibration pressure. Meanwhile, on the basis of digital image processing method, denoising, filtering, local stretching, image binarization and image filling are used to extract the characteristic dimension of bubbles. The theoretical value of the critical depth of bubble sinking matches the experimental result and their relative error is less than 5%. These new findings enrich the understanding of the moving bubbles in liquid materials used in nuclear reactors, rocket propulsion fuels and chemical experiments. Keywords:vertical vibration/ bubble sinking/ stable oscillation/ Basset force
为了进一步分析气泡长时间运动后的规律, 以及比较不同初始位置对数值求解的影响, 将求解(17)式转化为求解一阶常微分方程组初值问题, 采用向前差分格式将微分方程组转化为差分方程组, 并进行迭代求解. 选取参数ρ = 998 kg/m3, x = 0.17 m, A = 0.002 m, P0 = 101325 Pa, ω = 100π rad/s, g = 9.8 m/s2, μ = 1.01 × 10–3 Pa·s, R0 = 0.001 m, 并给定初始速度$ {\dot x}(0)=0$, 得到图1所示的结果. 图 1 气泡运动方程的数值模拟 (a) 气泡初始位置为0.1 m; (b) 气泡初始位置为0.05 m; (c) 气泡初始位置为0.076 m Figure1. Numerical simulation of the bubble motion equation$ : $ (a) The initial position of the bubble is 0.1 m; (b) the initial position of the bubble is 0.05 m; (c) the initial position of the bubble is 0.076 m.
对于图1(a), 气泡的初始位置为0.1 m, 气泡从该位置开始运动一直下沉到底部; 对于图1(b), 气泡的初始位置为0.05 m, 气泡从该位置开始运动一直上升至液面; 对于图1(c), 气泡的初始位置为0.076 m, 气泡在该位置附近振荡了一段时间后开始下沉到底部. 数值结果表明, 存在一个临界深度X0, 使得气泡在临界深度以上时上升, 在临界深度以下时下沉, 在临界深度处稳定振荡. 此外, 从图1(b)可以观察到气泡下沉是一个周期性累积的结果. 将Basset力的数值解(19)式和解析解(22)式进行对比分析, 选取参数R0 = 0.001 m, ρl = 998 kg/m3, μ = 1.01 × 10–3 Pa·s, ω = 100π rad/s, φ = 0, A = 0.002 m, g = 9.8 m/s2, P0 = 101325 Pa, B = 1/3A, x(0) = 0.1, x(0) = 0进行数值模拟, 结果如图2所示. 图 2 Basset力解析解和数值解的比较 Figure2. Comparison between the analytical and numerical solutions of Basset forces.
为了研究Basset力对气泡运动的影响, 首先采用以下参数对气泡的运动方程进行数值模拟, 分析Basset力对气泡运动的影响, 其中μ =1.01 × 10–3 Pa·s, R0 = 0.001 m, g = 9.8 m/s2, P0 = 101325 Pa, ρl = 998 kg/m3, ω = 100π rad/s, A = 0.002 m, x(0) = 0.1, x(0) = 0, 结果如图3所示. 图 3 Basset力对气泡运动方程的影响 Figure3. Effect of Basset force on the equation of motion of bubbles.
对两条运动轨迹进行比较, 在第3 s时两者偏移了6%, 在第4 s时两者偏移了20%. 结果表明, Basset力对气泡下沉前期的运动轨迹无明显影响, 但对后期的运动轨迹会产生较大的偏移. 为了进一步分析气泡轨迹偏移的原因, 将气泡运动前期和后期的附加质量力、Basset力、黏性阻力、浮力和水流推力进行对比, 选取与图3相同的参数进行数值模拟, 结果如图4所示. 图 4 气泡受力分析 (a)气泡下沉前期和后期各个分力的对比; (b) 气泡分力对气泡位移的贡献 Figure4. Force analysis of bubble: (a) Comparison of component forces in the early and late stages of bubble sinking; (b) contribution of bubble component force to bubble displacement.