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Gas physisorption impact on prolate dust in free-molecule flows: A static study 气体物理吸附对自由分子流中棱形尘埃的影响:静态研究
Pub Date : 2024-05-01 DOI: 10.1063/5.0207053
Hsin-Chen Yu, Xiaoping Zhang, Lei Wu, Zhongzhou Ren, Peishan He
Gas–solid coupling systems operating at low pressure or the micro/nanoscale generally exist in nature and industrial manufacture. Although the gas-scattering model has been widely used to study this problem on the dust surface, the consideration of gas physisorption was often neglected in previous applications of gas–surface scattering models. Therefore, this study aims to investigate the distribution of gas physisorption on the dust surface and assess its impact on the static force experienced by nonspherical dust in free-molecule flows. In this study, the prolate dust spinning around its minor axis is considered and the in-house direct simulation Monte Carlo code is used. Results show that gas physisorption on prolate dust is influenced by changes in gas number densities, Mach number, and dust shape. Furthermore, the gas physisorption enhances the gas–dust coupling for dust with a smooth surface at low gas pressure, attributed to the increasing ratio of Maxwell diffuse scattering of gas molecules on the gas-adsorbed part of the surface. Hence, gas physisorption was suggested as a potential factor for gas–dust coupling at low gas pressure.
自然界和工业生产中普遍存在低压或微/纳米尺度的气固耦合系统。尽管气体散射模型已被广泛用于研究粉尘表面的这一问题,但在以往气体表面散射模型的应用中,气体物理吸附的考虑往往被忽视。因此,本研究旨在研究气体物理吸附在粉尘表面的分布,并评估其对非球形粉尘在自由分子流中所受静力的影响。本研究考虑了绕其小轴线旋转的棱形尘埃,并使用了内部直接模拟蒙特卡洛代码。研究结果表明,气体物理吸附作用会受到气体密度、马赫数和尘埃形状变化的影响。此外,气体物理吸附增强了低气压下表面光滑的尘埃的气尘耦合,这归因于气体分子在表面气体吸附部分的麦克斯韦漫散射比的增加。因此,气体物理吸附被认为是低气压下气尘耦合的一个潜在因素。
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引用次数: 0
Identification of full-field wind loads on buildings using a mechanism-inspired recursive convolutional neural network with partial structural responses 利用部分结构响应的机制启发递归卷积神经网络识别建筑物的全场风荷载
Pub Date : 2024-05-01 DOI: 10.1063/5.0206423
Fubo Zhang, Ying Lei, Lijun Liu, Jinshan Huang
Indirect identification approaches through structural responses have proven effective for wind load estimation in real-world engineering. Currently, methods for identifying wind loads mainly rely on theoretical inverse identification, with rare research based on the mapping relationship between structural responses and wind loads through machine learning. In this paper, a scheme for identifying full-field wind loads using a recursive convolutional neural network (CNN) inspired by physical mechanisms is proposed. The recursive form of the network, as well as the inspiration for its inputs and outputs, is inspired by the spatial correlation and the mapping relationship between wind loads and structural responses. Thus, the network inputs comprise a fusion of structural acceleration and inter-story displacement responses, while the network outputs represent the independent wind loads on structures. Notably, mismatch test is employed by the network, wherein the training and testing datasets originate from entirely different sources. Specifically, during training, Gaussian white noises that simulate wind loads are utilized, while real wind load data are used for testing. The generalization of the proposed scheme is demonstrated through the identification of full-field wind loads generated by different stationary or non-stationary wind spectra of the 76-story wind-excited benchmark building. Furthermore, the proposed scheme is validated by identifying the full-field wind loads of a 67-story shear wall structure with wind tunnel test data.
在实际工程中,通过结构响应进行间接识别的方法已被证明对风荷载估算非常有效。目前,识别风荷载的方法主要依赖于理论上的反识别,很少有通过机器学习来研究结构响应与风荷载之间的映射关系。本文受物理机制启发,提出了一种利用递归卷积神经网络(CNN)识别全场风荷载的方案。该网络的递归形式及其输入和输出的灵感来自风荷载和结构响应之间的空间相关性和映射关系。因此,网络输入包括结构加速度和层间位移响应的融合,而网络输出则代表结构上的独立风荷载。值得注意的是,该网络采用了错配测试,其中训练数据集和测试数据集的来源完全不同。具体来说,在训练过程中,使用模拟风荷载的高斯白噪声,而测试则使用真实的风荷载数据。通过识别 76 层风激基准建筑的不同静态或非静态风频谱所产生的全场风荷载,证明了所提方案的通用性。此外,还利用风洞试验数据识别了 67 层剪力墙结构的全场风荷载,从而验证了所提出的方案。
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引用次数: 0
Falling film hydrodynamics and heat transfer under vapor shearing from various orientations 不同方向蒸汽剪切作用下的落膜流体力学和热传导
Pub Date : 2024-05-01 DOI: 10.1063/5.0210075
Chuang-Yao Zhao, Qiong Li, Fang-Fang Zhang, Di Qi, Hasan Yildizhan, Jun-Min Jiang
Vapor shearing is a common issue encountered in the operations of falling film heat exchangers. The vapor stream effect depends on its orientation. This study investigates liquid film hydrodynamics and heat transfer performance under the influence of vapor streams from different orientations. The results indicate that both orientation and velocity of vapor determine the encountering time and position of the films on the tube's two sides. The liquid film thickness uniformity and the liquid column deflection vary significantly depending on the orientation and velocity of the vapor. Zones of accelerated liquid film, climbing liquid film, liquid stagnation, and transition of liquid film flow pattern are observed. The gradient of film thickness along the tube axis and the deflection in time-averaged peripheral film thickness increase as the vapor orientation varies from 0° to 90° and subsequently decrease as the vapor orientation varies from 90° to 180°. Vapor streams have more pronounced effects on time-averaged peripheral film thickness in regions close to the liquid inlet and outlet. Vapor streams result in changes in peripheral heat transfer coefficients toward the downstream side depending on the orientation and velocity of the vapor. The impact of vapor streams on the overall heat transfer coefficient does not directly correlate with the velocity of the vapor when maintaining the same orientation.
蒸汽剪切是降膜式热交换器运行过程中经常遇到的问题。蒸汽流的影响取决于其取向。本研究探讨了不同方向蒸汽流影响下的液膜流体力学和传热性能。结果表明,蒸汽的取向和速度决定了薄膜在管子两侧的相遇时间和位置。液膜厚度的均匀性和液柱的挠度随蒸汽的方向和速度的变化而显著不同。观察到了液膜加速区、液膜爬升区、液膜停滞区和液膜流动模式过渡区。当蒸汽取向从 0° 变化到 90° 时,沿管子轴线的液膜厚度梯度和时间平均外围液膜厚度偏转增加;当蒸汽取向从 90° 变化到 180° 时,液膜厚度梯度和时间平均外围液膜厚度偏转减小。在靠近液体入口和出口的区域,蒸汽流对时间平均外围膜厚的影响更为明显。蒸汽流导致下游侧的外围传热系数发生变化,这取决于蒸汽的取向和速度。在保持方向不变的情况下,蒸汽流对整体传热系数的影响与蒸汽速度没有直接关系。
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引用次数: 0
A wall-modeled immersed boundary/large eddy simulation method and its application to simulating heart valve flows 壁模型沉浸边界/大涡流模拟方法及其在模拟心脏瓣膜流动中的应用
Pub Date : 2024-05-01 DOI: 10.1063/5.0198734
Jingyang Wang, T. Pu, Chunhua Zhou
In this work, a wall-modeled immersed boundary (IB)/large eddy simulation (LES) method is extended to the simulation of moving-boundary flows. The used non-equilibrium algebraic wall model is based on an assumed velocity profile, the coefficients of which are determined from physical constraints provided by the full turbulent-boundary-layer equations. To implement the wall model in an IB method named the local domain-free discretization (DFD) method, a local coordinate system fixed on the moving body is introduced. Thus, wall modeling is transformed into a local two-dimensional problem and the complexity of implementation of the wall model is reduced. In the present LES-DFD method, the tangential velocity at an exterior dependent node is determined via wall shear stress prescribed by the wall model. To reduce computational cost for simulating an internal flow with moving boundaries, the stationary boundaries are handled by the body-fitted-grid method and the moving boundaries by the local DFD method. There is no need of an auxiliary grid for solving the non-equilibrium algebraic wall model. Therefore, the inbuilt advantage of an IB method can be retained when simulating moving-boundary problems, and the economy of equilibrium wall models can also be preserved. The present method is applied to simulating the pulsatile flows through a bileaflet mechanical heart valve implanted in a model aorta. The predicted results show an acceptable agreement with the referenced experimental measurements or numerical results at much higher resolution and the applicability of the non-equilibrium wall model to LES of complex moving-boundary flows is verified.
本研究将沉浸边界(IB)/大涡模拟(LES)方法中的壁模型扩展到移动边界流的模拟。所使用的非平衡代数壁模型基于假定的速度剖面,其系数根据完整湍流边界层方程提供的物理约束确定。为了在一种名为局部无域离散(DFD)方法的 IB 方法中实现壁模型,引入了一个固定在运动体上的局部坐标系。这样,壁面建模就转化为局部二维问题,并降低了壁面模型实施的复杂性。在本 LES-DFD 方法中,外部相关节点处的切向速度是通过壁面模型规定的壁面剪应力确定的。为了降低模拟具有移动边界的内部流动的计算成本,静止边界由体拟合网格法处理,移动边界由局部 DFD 法处理。在求解非平衡代数壁模型时不需要辅助网格。因此,在模拟移动边界问题时,可以保留 IB 方法的内在优势,也可以保留平衡壁模型的经济性。本方法被应用于模拟通过植入模型主动脉的双叶机械心脏瓣膜的搏动流。预测结果显示与参考的实验测量结果或分辨率更高的数值结果具有可接受的一致性,并验证了非平衡壁模型对复杂运动边界流的 LES 的适用性。
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引用次数: 0
Effects of compressibility on Richtmyer–Meshkov instability of heavy/light interface 压缩性对重质/轻质界面里氏不稳定性的影响
Pub Date : 2024-05-01 DOI: 10.1063/5.0207779
Jiaxuan Li, Chenren Chen, Z. Zhai, Xisheng Luo
Experimental and numerical studies on the evolution of shock-accelerated SF6/air interface with small initial amplitude are conducted. The effect of compressibility on the early development of perturbation is highlighted by varying shock intensity and fluid properties. The startup process is analyzed when rarefaction waves are reflected and the characteristic time of the startup process is provided. The relationship between the phase inversion process and the startup process under different incident shock strengths is clarified. According to the startup time, a new start point for normalization is given, which can better normalize the amplitude growth at the early stage. In addition, the effects of incident shock strength and physical properties of fluids on the linear growth rate are highlighted through numerical simulations. The incompressible linear model loses validity when the incident shock is strong, and the existing rotational model is verified to provide excellent predictions under any shock strengths. The decrease in adiabatic exponent of the heavy fluid or the increase in adiabatic exponent of the light fluid can reduce the linear growth rate. As the absolute value of Atwood number increases, the adiabatic exponent of the heavy fluid has a more significant effect on the linear growth than that of the light fluid.
对初始振幅较小的冲击加速 SF6/空气界面的演变进行了实验和数值研究。通过改变冲击强度和流体性质,强调了可压缩性对扰动早期发展的影响。分析了稀释波反射时的启动过程,并给出了启动过程的特征时间。阐明了不同入射冲击强度下相位反转过程与启动过程之间的关系。根据启动时间,给出了归一化的新起点,从而更好地归一化早期阶段的振幅增长。此外,还通过数值模拟强调了入射冲击强度和流体物理性质对线性增长率的影响。当入射冲击强度较大时,不可压缩线性模型失去了有效性,而现有的旋转模型在任何冲击强度下都能提供出色的预测,这一点得到了验证。重流体绝热指数的减小或轻流体绝热指数的增大都会降低线性增长率。随着阿特伍德数绝对值的增加,重流体的绝热指数比轻流体的绝热指数对线性增长的影响更大。
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引用次数: 0
Data-driven learning algorithm to predict full-field aerodynamics of large structures subject to crosswinds 预测受横风影响的大型结构全场空气动力学的数据驱动学习算法
Pub Date : 2024-05-01 DOI: 10.1063/5.0197178
Xianjia Chen, Bo Yin, Zheng Yuan, Guowei Yang, Qiang Li, Shouguang Sun, Yujie Wei
Quick and high-fidelity updates about aerodynamic loads of large-scale structures, from trains, planes, and automobiles to many civil infrastructures, serving under the influence of a broad range of crosswinds are of practical significance for their design and in-use safety assessment. Herein, we demonstrate that data-driven machine learning (ML) modeling, in combination with conventional computational methods, can fulfill the goal of fast yet faithful aerodynamic prediction for moving objects subject to crosswinds. Taking a full-scale high-speed train, we illustrate that our data-driven model, trained with a small amount of data from simulations, can readily predict with high fidelity pressure and viscous stress distributions on the train surface in a wide span of operating speed and crosswind velocity. By exploring the dependence of aerodynamic coefficients on yaw angles from ML-based predictions, a rapid update of aerodynamic forces is realized, which can be effectively generalized to trains operating at higher speed levels and subject to harsher crosswinds. The method introduced here paves the way for high-fidelity yet efficient predictions to capture the aerodynamics of engineering structures and facilitates their safety assessment with enormous economic and social significance.
从火车、飞机、汽车到许多民用基础设施,大型结构在各种横风影响下的气动载荷的快速、高保真更新对其设计和使用中的安全评估具有实际意义。在此,我们证明了数据驱动的机器学习(ML)建模与传统计算方法相结合,可以实现对受横风影响的运动物体进行快速而准确的空气动力学预测的目标。我们以一列全尺寸高速列车为例,说明我们的数据驱动模型只需通过少量模拟数据进行训练,就能高保真地预测列车表面在各种运行速度和横风速度下的压力和粘性应力分布。通过从基于 ML 的预测中探索空气动力系数对偏航角的依赖性,实现了空气动力的快速更新,这可以有效地推广到列车在更高的速度水平和更恶劣的横风条件下运行。本文介绍的方法为高保真且高效地预测工程结构的空气动力学性能铺平了道路,并有助于对其进行安全评估,具有巨大的经济和社会意义。
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引用次数: 0
Influence of surface slip on hydrodynamics and flow field around a two-dimensional hydrofoil at a moderate Reynolds number 表面滑移对中等雷诺数下二维水翼周围流体力学和流场的影响
Pub Date : 2024-05-01 DOI: 10.1063/5.0203389
Manfu Zhu, Weixi Huang, Liran Ma, Jianbin Luo
In the present study, the effects of surface slip on the hydrodynamics and flow around a two-dimensional National Advisory Committee for Aeronautics 0012 hydrofoil are systematically investigated by numerical methods. The objective is to fully understand the effects of surface slip on the streamlined body. Three slip positions (both surfaces, the upper surface, the lower surface) and eight slip lengths (in a wide range from 1 to 500 μm) under 0°–10° angles of attack are fully investigated at a moderate Reynolds number of 1.0 × 106. Surface slip has been found to increase lift and reduce drag by postponing the flow transition, laminar separation bubble, and flow separation on the hydrofoil surface under both surfaces and the upper surface slip conditions. Slip has also been found to induce upshift of the mean velocity profile, decrease the displacement thickness, and mitigate the turbulent kinetic energy in the flow field. However, counterintuitive phenomenon occurs under the lower surface slip condition, where the total drag of the hydrofoil is increased compared to that under the no slip condition. Total drag increase is found mainly due to the increase in the pressure drag under small slip lengths and relatively large angles of attack. Flow maps demonstrating the complex interaction between different surface slip conditions and the flow field are further presented. The results suggest that surface slip can not only reduce drag, but also increase the drag of the streamlined body, which shall provide valuable insights for practical applications of slippery materials.
在本研究中,采用数值方法系统地研究了表面滑移对二维国家航空咨询委员会 0012 水翼周围流体力学和流动的影响。目的是全面了解表面滑移对流线体的影响。在中等雷诺数 1.0 × 106 条件下,对 0°-10° 攻角条件下的三个滑移位置(两个表面、上表面、下表面)和八个滑移长度(1 至 500 μm)进行了全面研究。研究发现,在表面和上表面滑移条件下,表面滑移可通过推迟水翼表面的流动过渡、层流分离气泡和流动分离来增加升力和减少阻力。研究还发现,滑移可导致平均速度剖面上移、减小位移厚度并减轻流场中的湍流动能。然而,在下表面滑移条件下出现了与直觉相反的现象,与无滑移条件下相比,水翼的总阻力增加了。总阻力的增加主要是由于小滑移长度和相对较大的攻角下压力阻力的增加。研究还进一步展示了不同表面滑移条件与流场之间复杂相互作用的流场图。结果表明,表面滑移不仅能减少阻力,还能增加流线型机体的阻力,这将为滑移材料的实际应用提供有价值的见解。
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引用次数: 0
Electrohydrodynamic deformation of a compound droplet in an alternating current and direct current superposed electric field 复合液滴在交流和直流叠加电场中的电流体动力变形
Pub Date : 2024-05-01 DOI: 10.1063/5.0209008
Bikash Mohanty, Aditya Bandopadhyay
In this study of a compound droplet subjected to alternating current (AC) and direct current (DC) superposed (AC/DC) electric fields, both core and shell deformations oscillate, albeit with reduced amplitude compared to solely alternating current electric fields. As surface tension relaxes, periodic cyclic deformation ensues, with mean deformation amplifying alongside electric field amplitude. Concurrently, normal and tangential Maxwell stresses escalate with amplitude, thus augmenting interfacial surface velocities. Manipulating the offset ratio of alternating and direct current superposed electric field modulates mean deformations. Across low frequencies, stable deformation remains constant, yet a delayed onset characterizes higher frequencies. The presence of a core affects the electrohydrodynamics of the compound droplet and shell deformation, thereby mitigating phase differences between cyclic deformations. Contrasting alternating current (AC)—only fields, alternating current and direct current superposed (AC/DC) electric field scenarios exhibit heightened surface charge densities and prompter stable deformation onset. Furthermore, the direct current component magnifies mean deformations while harmonizing phase disparities between core and shell deformations. This study illuminates the intricate interplay between alternating current and direct current fields on compound droplet behavior, offering profound insight with broad implications for applications necessitating precise deformations under electric fields.
本研究对受到交流(AC)和直流(DC)叠加(AC/DC)电场作用的复合液滴进行了研究,结果发现,液滴的内核和外壳都会发生振荡变形,尽管与单纯的交流电场相比振幅有所减小。随着表面张力的松弛,周期性的循环变形随之产生,平均变形随着电场振幅的增大而增大。同时,法向和切向麦斯韦尔应力也随振幅增加,从而提高了界面表面速度。操纵交流和直流叠加电场的偏移比可调节平均变形。在低频情况下,稳定变形保持不变,但在高频情况下,变形会延迟发生。核心的存在会影响复合液滴和外壳变形的电流体力学,从而减轻周期变形之间的相位差。与纯交流电场不同,交流电和直流电叠加(AC/DC)电场情况下的表面电荷密度更高,变形开始得更快更稳定。此外,直流电成分还能放大平均变形,同时协调核心和外壳变形之间的相位差。这项研究揭示了交流电场和直流电场对复合液滴行为的复杂相互作用,为需要在电场下精确变形的应用提供了深刻的见解和广泛的影响。
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引用次数: 0
Free surface water waves generated by instability of an exponential shear flow in arbitrary depth 任意深度指数剪切流不稳定性产生的自由水面波浪
Pub Date : 2024-05-01 DOI: 10.1063/5.0208081
M. Abid, C. Kharif
The stability of an exponential current in water to infinitesimal perturbations in the presence of gravity and capillarity is revisited and reformulated using the Weber and Froude numbers. Some new results on the generation of gravity-capillary waves are presented, which supplement the previous works of Morland et al. [“Waves generated by shear layer instabilities,” Proc. Math. Phys. Sci. 433, 441–450 (1991)] and Young and Wolfe [“Generation of surface waves by shear-flow instability,” J. Fluid Mech. 739, 276–307 (2014)] on finite depth. To consider perturbations at much larger scales, special attention is given to the stability of exponential currents only in the presence of gravity. More precisely, the present investigation reveals significant insights into the stability of exponential shear currents under different environmental conditions. Notably, we have identified that the dimensionless growth rate increases with the Froude number, providing a deeper understanding of the interplay between shear layer thickness and surface velocity. Furthermore, our analysis elucidates the dimensional wavelength of the most unstable mode, emphasizing its relevance to the characteristic shear layer thickness. Additionally, within the realm of gravity-capillary instabilities, we have established a sufficient condition for the stability of exponential currents based on the Weber number. Our findings are supported by stability diagrams at finite depth, showing how the size of stable domains correlates with the characteristic thickness of the shear layer. Moreover, we have explored the stability of a thin film of liquid in an exponential shearing flow, further enriching our understanding of the complex dynamics involved in such systems.
利用韦伯数和弗劳德数重新研究和阐述了水中指数流在重力和毛细作用下对无穷小扰动的稳定性。本文提出了一些关于重力-毛细管波产生的新结果,补充了莫兰等人之前的研究成果["Waves generated by shear layer instabilities," Proc.Math.433, 441-450 (1991)] 以及 Young 和 Wolfe ["Generation of surface waves by shear-flow instability," J. Fluid Mech.739, 276-307 (2014)]的有限深度。为了考虑更大尺度的扰动,我们特别关注了指数流在重力作用下的稳定性。更确切地说,本研究揭示了不同环境条件下指数剪切流稳定性的重要见解。值得注意的是,我们发现无量纲增长率随着弗劳德数的增加而增加,从而加深了对剪切层厚度和表面速度之间相互作用的理解。此外,我们的分析还阐明了最不稳定模式的尺寸波长,强调了其与特征剪切层厚度的相关性。此外,在重力-毛细管不稳定性领域,我们根据韦伯数建立了指数流稳定性的充分条件。我们的发现得到了有限深度稳定性图的支持,该图显示了稳定域的大小与剪切层特征厚度的相关性。此外,我们还探索了指数剪切流中液体薄膜的稳定性,进一步丰富了我们对此类系统中复杂动力学的理解。
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引用次数: 0
Pressure characteristics of a nonspherical underwater explosion bubble in a compressible fluid 非球形水下爆炸气泡在可压缩流体中的压力特性
Pub Date : 2024-05-01 DOI: 10.1063/5.0206482
Junliang Liu, Wei Xiao, Xiongliang Yao
The pressures produced by underwater explosions present serious threats to ships, submarines, and other marine structures. A significant part of underwater explosion pressure comes from the explosion bubble. Most computational studies on bubble pressure have considered the case of an incompressible fluid or have confined calculations to the time before the formation of a toroidal bubble, because of the complexity and strong nonlinearity of a compressible flow field with a doubly connected geometry. The few compressible models that are capable of calculating the pressure after jet impact suffer from computational difficulties. In this paper, we calculate the bubble pressure by constructing a new form for solving an auxiliary function based on a boundary integral method that takes account of the compressibility of the flow field. We verify out numerical algorithm by comparison with a classical theoretical model and a set of experimental results. We then compare the bubble pressure in a compressible flow field in both the first and second oscillation periods with that in an incompressible flow field. The results of this comparison confirm that it is necessary to consider the compressibility of the fluid and the multiperiod pulsations of a bubble in pressure calculations. We present a comprehensive discussion of the pressure characteristics in the central axial and circumferential directions induced by a nonspherical bubble in a free field. Finally, we obtain a critical bubble–wall distance rw > 1.8 for which the bubble is prevented from splitting after the first jet impact, and we investigate the pressure characteristics of a bubble near a rigid wall for both cases of rw < 1.8 and rw > 1.8.
水下爆炸产生的压力对船只、潜艇和其他海洋结构构成严重威胁。水下爆炸压力的很大一部分来自爆炸气泡。由于具有双连接几何形状的可压缩流场的复杂性和强非线性,大多数有关气泡压力的计算研究都考虑了不可压缩流体的情况,或将计算局限于环形气泡形成之前的时间。能够计算喷流冲击后压力的可压缩模型为数不多,但都存在计算上的困难。在本文中,我们基于考虑流场可压缩性的边界积分法,通过构建一种新的辅助函数求解形式来计算气泡压力。通过与经典理论模型和一组实验结果的比较,我们验证了这一数值算法。然后,我们将第一和第二振荡周期内可压缩流场中的气泡压力与不可压缩流场中的气泡压力进行了比较。比较结果证实,在压力计算中必须考虑流体的可压缩性和气泡的多周期脉动。我们全面讨论了非球形气泡在自由场中引起的中心轴向和圆周方向的压力特性。最后,我们得到了一个临界泡壁距离 rw > 1.8,在这个距离上,气泡在第一次射流冲击后不会分裂,我们还研究了 rw < 1.8 和 rw > 1.8 两种情况下气泡在刚性壁附近的压力特性。
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引用次数: 0
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Physics of Fluids
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