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Physics-informed machine-learning solution to log-layer mismatch in wall-modeled large-eddy simulation 壁面建模大涡流模拟中对数层错配的物理信息机器学习解决方案
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-26 DOI: 10.1103/physrevfluids.9.084609
Soju Maejima, Kazuki Tanino, Soshi Kawai
This study proposes a physics-informed machine learning to enable using the erroneous flow data at near-wall grid points as the input to the wall model in a wall-modeled large-eddy simulation (LES). The proposed neural network predicts the amount of numerical error in the near-wall grid-point data and inputs the physically correct flow variables into the wall model by correcting the near-wall error. The input and output features of the neural networks are selected based on the physical relations of the turbulent boundary layer for robustness against various Reynolds and Mach number conditions. The proposed neural networks allow the wall model to accurately predict the wall shear stress from the erroneous near-wall information and yields accurate predictions of the turbulence statistics. Additionally, the proposed physics-informed machine-learning approach reproduces the asymmetry in the probability density functions of the predicted wall shear stress observed in direct numerical simulations, while the conventional wall model with input away from the wall does not. The results suggest that using the near-wall information for wall modeling may increase the fidelity of the wall-modeled LES.
本研究提出了一种物理信息机器学习方法,可将近壁网格点的错误流量数据作为壁模型大涡流模拟(LES)中壁模型的输入。所提出的神经网络可预测近壁网格点数据的数值误差量,并通过修正近壁误差将物理上正确的流动变量输入壁模型。神经网络的输入和输出特性是根据湍流边界层的物理关系选择的,以确保在各种雷诺数和马赫数条件下的鲁棒性。所提出的神经网络允许壁面模型从错误的近壁信息中准确预测壁面切应力,并产生准确的湍流统计预测。此外,所提出的物理信息机器学习方法再现了直接数值模拟中观察到的壁面剪应力预测概率密度函数的不对称性,而使用远离壁面输入的传统壁面模型则没有这种不对称性。结果表明,使用近壁信息进行壁面建模可提高壁面建模 LES 的保真度。
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引用次数: 0
Cascades of turbulent kinetic energy and multicomponent scalars in a momentum-scalar coupling turbulence driven by multiple mechanisms under homogeneous and isotropic hypotheses 均质和各向同性假设下多种机制驱动的动量-标量耦合湍流中的湍流动能和多分量标量级联
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-26 DOI: 10.1103/physrevfluids.9.084610
Wei Zhao
Momentum-scalar coupling turbulence, a phenomenon observed in both natural and engineering contexts, involves the intricate interaction between multicomponent scalars and multiscale forces (i.e., multiple coupling mechanisms), resulting in a wide array of manifestations. Despite its importance, limited research has been conducted to comprehend the influence of these multicomponent and multiple coupling mechanisms on turbulence cascades. Hence, this study aims to provide a preliminary and theoretical exploration into how these multiple coupling mechanisms govern the cascades of turbulent kinetic energy and multicomponent scalars. To simplify the mathematical analysis, homogeneous and isotropic hypotheses of flow field have been applied. The key findings of this study can be summarized as follows. The first is validation of quad-cascade processes. The second is an examination of various cases involving single scalar components but multiple coupling mechanisms. Of particular interest is the coexistence of buoyancy-driven turbulence and electrokinetic turbulence, which introduces a new variable flux (VF) subrange resulting from their nonlinear interaction. Another extension considers an exponential modulation function, equivalent to the coexistence of multiple coupling mechanisms acting on a single scalar. The study identifies two new VF subranges. Third, binary scalar components and coupling mechanisms are investigated, indicating coupling mechanisms with significantly different strengths that can also induce complex interactions and new VF subranges. Fourth is the complexity when three or more different scalar components and coupling mechanisms coexist simultaneously: with the exception of certain special cases, closure of the problem becomes unattainable. This highlights the challenges inherent in addressing the simultaneous presence of multiple scalar components and coupling mechanisms. This research endeavor illuminates the theoretical understanding of the diverse scaling properties observed in momentum-scalar coupling turbulence across different scenarios.
动量-标量耦合湍流是一种在自然和工程环境中都能观察到的现象,涉及多分量标量和多尺度力(即多重耦合机制)之间错综复杂的相互作用,从而导致一系列广泛的表现形式。尽管湍流级联非常重要,但在理解这些多组分和多重耦合机制对湍流级联的影响方面的研究还很有限。因此,本研究旨在从理论上初步探讨这些多重耦合机制如何影响湍流动能和多组分标量的级联。为了简化数学分析,本研究采用了流场的均质和各向同性假设。本研究的主要结论可归纳如下。首先是对四级联过程的验证。其次是对涉及单一标量成分但有多种耦合机制的各种情况进行了研究。特别值得关注的是浮力驱动湍流和电动湍流的共存,这引入了一个由它们的非线性相互作用产生的新的可变通量(VF)子范围。另一个扩展考虑了指数调制函数,相当于作用于单个标量的多种耦合机制并存。研究确定了两个新的 VF 子范围。第三,研究了二元标量成分和耦合机制,表明耦合机制的强度存在显著差异,也会诱发复杂的相互作用和新的 VF 子区间。第四,当三个或更多不同的标量成分和耦合机制同时存在时,问题就变得复杂了:除了某些特殊情况,问题的封闭性变得无法实现。这凸显了解决同时存在多个标量成分和耦合机制问题所固有的挑战。这项研究工作阐明了对动量-标量耦合湍流在不同情况下观察到的不同缩放特性的理论理解。
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引用次数: 0
Edge-wave phase shifts versus normal-mode phase tilts in an Eady problem with a sloping boundary 具有倾斜边界的埃迪问题中的边波相位偏移与正常模式相位倾斜的关系
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-26 DOI: 10.1103/physrevfluids.9.083905
J. Mak, N. Harnik, E. Heifetz, G. Kumar, E. Q. Y. Ong
One mechanistic interpretation of baroclinic instability is that of mutual constructive interference of Rossby edge waves. The suppression of baroclinic instability over slopes has been widely established, where previous research argues that a sloping boundary modifies the properties of these Rossby edge waves, but does not provide a mechanistic explanation for the suppression that is valid over all parameter space. In the context of an Eady problem modified by the presence of a sloping boundary, we provide a mechanistic rationalization for baroclinic instability in the presence of slopes that is valid over all parameter space, via an equivalent formulation explicitly in terms of Rossby edge waves. We also highlight the differences between edge-wave phase shifts and normal-mode phase tilts, showing that the edge-wave phase shifts should be the ones that are mechanistically relevant, and normal-mode phase tilt is a potentially misleading quantity to use. Further, we present evidence that the edge-wave phase shifts but not normal-mode phase tilts are well correlated with geometric quantities diagnosed from an analysis framework based on eddy variance ellipses. The result is noteworthy in that the geometric framework makes no explicit reference to the edge-wave structures in its construction, and the correlation suggests the geometric framework can be used in problems where edge-wave structures are not so well defined or readily available. Some implications for parametrization of baroclinic instability and relevant eddy-mean feedbacks are discussed. For completeness, we also provide an explicit demonstration that the linear instability problem of the present modified Eady problem is parity-time symmetric, and speculate about some suggestive links between parity-time symmetry, shear instability, and the edge-wave interaction mechanism.
对巴氏不稳定性的一种机理解释是罗斯比边缘波的相互建构干扰。斜坡上的巴氏不稳定性受到抑制的现象已被广泛证实,以往的研究认为斜坡边界改变了这些罗斯比边缘波的特性,但并没有提供在所有参数空间都有效的抑制机理解释。在因存在倾斜边界而有所改变的埃迪问题背景下,我们通过明确以罗斯比边缘波为基础的等效表述,为斜坡存在时的巴氏不稳定性提供了在所有参数空间内都有效的机理解释。我们还强调了边缘波相位移动和常模相位倾斜之间的差异,表明边缘波相位移动应该是与机理相关的相位移动,而常模相位倾斜则可能会误导我们使用。此外,我们还提出证据表明,边波相移而非正常模式相位倾斜与基于涡度方差椭圆的分析框架诊断出的几何量有很好的相关性。值得注意的是,几何框架在构建过程中没有明确提及边波结构,而这种相关性表明,几何框架可用于边波结构定义不明确或不容易获得的问题。我们讨论了气压不稳定性参数化和相关涡均反馈的一些影响。为完整起见,我们还明确证明了目前修正的埃迪问题的线性不稳定性问题是奇偶时对称的,并推测了奇偶时对称性、剪切不稳定性和边波相互作用机制之间的一些暗示性联系。
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引用次数: 0
Viscous influences on impulsively generated focused jets 粘性对脉冲产生的聚焦射流的影响
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-23 DOI: 10.1103/physrevfluids.9.l082001
Xianggang Cheng, Xiao-Peng Chen, Hang Ding, Chun-Yu Zhang, Haibao Hu, Laibing Jia
Impulsively generated focused jets play a significant role in various applications, including inkjet printing, needle-free drug delivery, and microfluidic devices. As the demand for generating jets and droplets from medium to highly viscous liquids increases, understanding the role of viscosity in jetting dynamics becomes crucial. While previous studies have examined the viscous effects on walls, the impact on free surfaces has not been thoroughly understood. This study aims to bridge this gap by integrating experiments with numerical simulations to investigate the viscous effects on focused jet formation. We demonstrate that mass and momentum transfer along the tangential direction of the free surface contribute to focused jet formation, and viscosity plays a key role in this transfer process. The viscosity-induced diffusion of the shear flow and vorticity near the free surface reduces the jet speed. Based on experimental observations and simulation results, we propose an equation to predict the viscous jet velocity. These findings offer new perspectives on viscous interface dynamics in advanced manufacturing and biomedical applications.
脉冲产生的聚焦射流在各种应用中发挥着重要作用,包括喷墨打印、无针给药和微流体设备。随着从中等粘度到高粘度液体中产生射流和液滴的需求增加,了解粘度在射流动力学中的作用变得至关重要。虽然之前的研究已经研究了粘度对壁面的影响,但对自由表面的影响还没有深入了解。本研究旨在弥合这一差距,将实验与数值模拟相结合,研究粘性对聚焦射流形成的影响。我们证明,沿自由表面切线方向的质量和动量传递有助于聚焦射流的形成,而粘度在这一传递过程中起着关键作用。自由表面附近由粘度引起的剪切流和涡度扩散降低了射流速度。根据实验观察和模拟结果,我们提出了一个预测粘性射流速度的方程。这些发现为先进制造和生物医学应用中的粘性界面动力学提供了新的视角。
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引用次数: 0
Surfactant-laden bubble bursting: Dynamics of capillary waves and Worthington jet at large Bond number 含有表面活性剂的气泡破裂:大邦德数下的毛细管波和沃辛顿射流动力学
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-22 DOI: 10.1103/physrevfluids.9.083606
P. Pico, L. Kahouadji, S. Shin, J. Chergui, D. Juric, O. K. Matar
We present a numerical study of the main substages preceding aerosol formation via bursting bubbles: capillary wave propagation along the bubble, convergence at the bubble's apex, and the ascent of a Worthington jet and its breakup to release liquid drops. We focus on two crucial yet overlooked aspects of the system: the presence of surface-active agents and dynamics driven by non-negligible gravitational effects, quantified by the Bond number. Our results propose a mechanism explaining capillary wave retardation in the presence of surfactants, involving the transition from bi- to unidirectional Marangoni stresses, which pull the interface upwards, countering the motion of the waves. We also quantitatively elucidate the variable nature of the waves' velocity with various surfactant parameters, including surfactant solubility and elasticity, a departure from the constant behavior well documented in clean interfaces.
我们对气泡破裂形成气溶胶之前的主要阶段进行了数值研究:毛细管波沿气泡传播、在气泡顶点汇聚、沃辛顿射流上升及其破裂释放液滴。我们重点研究了系统中两个关键但却被忽视的方面:表面活性物质的存在和由不可忽略的引力效应(以邦德数量化)驱动的动力学。我们的研究结果提出了表面活性剂存在时毛细管波延迟的解释机制,其中涉及从双向马兰戈尼应力到单向马兰戈尼应力的转变。我们还定量阐释了毛细管波速度随各种表面活性剂参数(包括表面活性剂溶解度和弹性)变化的性质,这与清洁界面中记录的恒定行为不同。
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引用次数: 0
Robust experimental data assimilation for the Spalart-Allmaras turbulence model 斯帕拉特-奥尔马拉斯湍流模型的稳健实验数据同化
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-22 DOI: 10.1103/physrevfluids.9.084608
Deepinder Jot Singh Aulakh, Xiang Yang, Romit Maulik
This study presents a methodology focusing on the use of computational model and experimental data fusion to improve the Spalart-Allmaras (SA) closure model for Reynolds-averaged Navier-Stokes solutions. In particular, our goal is to develop a technique that not only assimilates sparse experimental data to improve turbulence model performance, but also preserves generalization for unseen cases by recovering classical SA behavior. We achieve our goals using data assimilation, namely the ensemble Kalman filtering approach, to calibrate the coefficients of the SA model for separated flows. A holistic calibration strategy is implemented via the parametrization of the production, diffusion, and destruction terms. This calibration relies on the assimilation of experimental data collected in the form of velocity profiles, skin friction, and pressure coefficients. Despite using observational data from a single flow condition around a backward-facing step (BFS), the recalibrated SA model demonstrates generalization to other separated flows, including cases such as the two-dimensional (2D) NASA wall mounted hump and the modified BFS. Significant improvement is observed in the quantities of interest, i.e., the skin friction coefficient (Cf) and the pressure coefficient (Cp), for each flow tested. Finally, it is also demonstrated that the newly proposed model recovers SA proficiency for flows, such as a NACA-0012 airfoil and axisymmetric jet, and that the individually calibrated terms in the SA model target specific flow-physics wherein the calibrated production term improves the recirculation zone while destruction improves the recovery zone.
本研究提出了一种方法,重点是利用计算模型和实验数据融合来改进雷诺平均纳维-斯托克斯解的斯帕拉特-阿勒马拉斯(SA)闭合模型。特别是,我们的目标是开发一种技术,它不仅能同化稀疏的实验数据以提高湍流模型的性能,还能通过恢复经典的 SA 行为来保留未见案例的通用性。我们利用数据同化(即集合卡尔曼滤波方法)来校准分离流的 SA 模型系数,从而实现我们的目标。通过对生产、扩散和破坏项进行参数化,实施了整体校准策略。这种校准依赖于以速度剖面、表皮摩擦和压力系数形式收集的实验数据的同化。尽管使用的是后向阶梯(BFS)周围单一流动条件的观测数据,但重新校准的 SA 模型显示出对其他分离流动的普适性,包括二维(2D)NASA 壁装驼峰和改进的 BFS 等情况。在所测试的每种流体中,都能观察到相关量(即表皮摩擦系数 (Cf) 和压力系数 (Cp))的显著改善。最后,还证明了新提出的模型能够熟练地恢复 NACA-0012 机翼和轴对称射流等流动的 SA,而且 SA 模型中的单独校准项针对的是特定的流动物理,其中校准的生产项改善了再循环区,而破坏项改善了恢复区。
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引用次数: 0
Self-scaling generalized Townsend-Perry constants for high-order moments in turbulent boundary layers 湍流边界层高阶矩的自标度广义汤森-佩里常数
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-22 DOI: 10.1103/physrevfluids.9.l082602
Xibo He, Hongyou Liu, Xiaojing Zheng
Inspired by the thought-provoking paper of Meneveau and Marusic [J. Fluid Mech. 719, R1 (2013)], the universal expression of the self-scaling generalized Townsend-Perry constants for the high-order statistical moments is investigated. The measured results deviate from the previous attached-eddy-model–based Gaussian prediction because the wall-non-attached eddies with sub-Gaussian statistics mask the Gaussian behavior of the wall-attached eddies. Leveraging the generalized Gaussian distribution function and the logarithmic law for turbulence intensity, the universal expression of the self-scaling generalized Townsend-Perry constants, regardless of the eddy type, is derived. Moreover, asymptotic expression of the shape parameter in self-scaling generalized Townsend-Perry constants with Reynolds number is further characterized by data in boundary layers and atmospheric surface layers with Reynolds number Reτ spanning over O(103) to O(106).
受 Meneveau 和 Marusic 发人深省的论文[J. Fluid Mech. 719, R1 (2013)]的启发,研究了高阶统计矩的自缩放广义汤森-佩里常数的通用表达式。测量结果偏离了之前基于附壁涡模型的高斯预测,因为具有亚高斯统计量的附壁非附壁涡掩盖了附壁涡的高斯行为。利用广义高斯分布函数和湍流强度的对数定律,推导出了自缩放广义汤森-佩里常数的通用表达式,与涡的类型无关。此外,通过雷诺数 Reτ 跨度从 O(103) 到 O(106) 的边界层和大气表层的数据,进一步描述了自缩放广义汤森-佩里常数的形状参数随雷诺数的渐近表达。
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引用次数: 0
Translation of a sphere towards a rigid plane in an Oldroyd-B fluid 球体在奥尔德罗伊德-B 流体中向刚性平面的平移
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-21 DOI: 10.1103/physrevfluids.9.083303
Tachin Ruangkriengsin, Rodolfo Brandão, Bimalendu Mahapatra, Evgeniy Boyko, Howard A. Stone
We analyze the low-Reynolds-number translation of a sphere towards or away from a rigid plane in an Oldroyd-B fluid under two scenarios: prescribing the sphere's translational velocity, and prescribing the force on the sphere. Leveraging the lubrication approximation and a perturbation expansion in powers of the Deborah number, we develop a comprehensive theoretical analysis that yields analytical approximations for velocity fields, pressures, and forces acting on the sphere. Our framework aids in understanding temporal microstructural changes as the particle-wall gap evolves over time. In particular, we show that alterations in the polymer conformation tensor in response to geometric changes induce additional forces on the sphere. For cases with prescribed velocity, we present a theoretical approach for calculating resistive forces at any order in the Deborah number and utilize a reciprocal theorem to obtain higher-order corrections based on velocity fields in the previous orders. When the sphere translates with a constant velocity, the fluid viscoelasticity decreases the resistive force at the first order. However, at the second-order correction, the direction of the sphere's movement determines whether viscoelasticity increases or decreases the resistive force. For cases with prescribed force, we show that understanding the influence of viscoelasticity on the sphere's translational velocity necessitates a more intricate analysis even at low Deborah numbers. Specifically, we introduce an ansatz for constant force scenarios, and we derive solution forms for general prescribed forces using the method of multiple scales. We find that when a sphere undergoes sedimentation due to its own weight, the fluid viscoelasticity results in a slower settling process, reducing the leading-order sedimentation rate.
我们分析了在奥尔德罗伊德-B 流体中球体朝向或远离刚性平面的低雷诺数平移的两种情况:规定球体的平移速度和规定球体上的力。利用润滑近似和德博拉数幂的扰动扩展,我们进行了全面的理论分析,得出了球体上的速度场、压力和作用力的分析近似值。我们的框架有助于理解颗粒-壁间隙随时间演变时的微观结构变化。我们特别指出,聚合物构象张量随几何变化而发生的改变会对球体产生额外的作用力。对于具有规定速度的情况,我们提出了一种理论方法,用于计算德博拉数任意阶的阻力,并利用倒易定理根据前阶速度场获得高阶修正。当球体以恒定速度平移时,流体粘弹性会降低一阶阻力。然而,在二阶修正时,球体的运动方向决定了粘弹性是增加还是减少阻力。对于有规定力的情况,我们表明,要理解粘弹性对球体平移速度的影响,就必须进行更复杂的分析,即使在低德博拉数的情况下也是如此。具体来说,我们引入了恒定力情况的解析,并利用多尺度方法推导出了一般规定力的求解形式。我们发现,当球体由于自身重量而发生沉降时,流体粘弹性会导致沉降过程变慢,从而降低前阶沉降速度。
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引用次数: 0
Tones and upstream-traveling waves in ideally expanded round impinging jets 理想膨胀圆形撞击射流中的音调和逆流波
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-21 DOI: 10.1103/physrevfluids.9.083904
Igor A. Maia, Maxime Fiore, Romain Gojon
We study the generation of tones by ideally expanded round jets impinging on a flat plate. Data from large-eddy simulations performed for different nozzle-to-plate distances are explored, and we consider closure of the aeroacoustic feedback loop responsible for the tones by guided jet modes. Allowable frequency ranges for resonance, underpinned by the existence of modes with upstream-directed group velocities, are computed using two different models: a cylindrical vortex-sheet model, and a locally parallel stability model which considers a finite-thickness velocity profile. It is shown that inclusion of a finite-thickness velocity profile consistent with the mean flow in the vicinity of the plate improves the agreement between observed tones and model predictions. The frequencies of the largest tones found in the data are found to fall within, or very close to, the frequency limits of the finite-thickness model, correcting discrepancies observed with the vortex-sheet model. The same trend is observed in comparisons with experimental and numerical data gathered from the literature. Pressure eigenfunctions of the stability model are in good agreement with upstream-traveling disturbances educed from the data at the tone frequencies. This provides further evidence for the involvement of guided jet modes in the resonance mechanism.
我们研究了理想膨胀圆形射流撞击平板产生的音调。我们对不同喷嘴到平板距离的大涡流模拟数据进行了探讨,并考虑了通过导引射流模式产生音调的气声反馈回路的闭合问题。我们使用两种不同的模型计算了共振的允许频率范围,其基础是具有上游定向群速度的模式的存在:一种是圆柱形涡流片模型,另一种是考虑了有限厚度速度剖面的局部平行稳定性模型。结果表明,加入与板附近平均流一致的有限厚度速度剖面,可以改善观测到的声调与模型预测之间的一致性。数据中发现的最大音调的频率位于或非常接近有限厚度模型的频率极限,从而纠正了涡片模型中观察到的差异。在与文献中收集的实验数据和数值数据进行比较时,也发现了同样的趋势。稳定模型的压力特征函数与从音调频率数据中得到的上游游动扰动非常吻合。这进一步证明了共振机制中存在导向喷射模式。
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引用次数: 0
Single theoretical model for breakup of viscous thread with and without a fiber 有纤维和无纤维粘性线缆断裂的单一理论模型
IF 2.7 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS Pub Date : 2024-08-20 DOI: 10.1103/physrevfluids.9.084005
Hyejoon Jun, Hyoungsoo Kim
In this study, we introduce a comprehensive theoretical model for viscous liquid systems exhibiting Rayleigh-Plateau instability, accommodating cases both with and without a solid fiber. Employing the lubrication approach and implementing the hydrodynamic interaction at the solid-liquid interface, we formulate one-dimensional evolution equations for the breakup of viscous liquid threads and films on a fiber. Through several validations, we showed that our model exhibits a good agreement with experimental results in comparison to numerical simulations. Finally, our model, which incorporates the flow effect from the inner boundary condition by reconsidering the ansatz of a conventional long-wave approximation, provides a necessary condition for satellite droplet formation and determines the most unstable mode proportional to k*2, where k* is the most unstable wavenumber. In addition, we observed that the volume of the satellite droplets exponentially decays depending on the wavenumber. Moreover, our single model integrates the findings of Goren's liquid film on a fiber and Rayleigh's viscous liquid thread, demonstrating its versatility and relevance to a wide range of systems.
在本研究中,我们为表现出瑞利-普法不稳定性的粘性液体系统引入了一个全面的理论模型,该模型既适用于有固体纤维的情况,也适用于无固体纤维的情况。我们采用润滑方法和固液界面的流体动力学相互作用,建立了纤维上粘性液丝和液膜破裂的一维演化方程。通过多次验证,我们发现与数值模拟相比,我们的模型与实验结果具有良好的一致性。最后,我们的模型通过重新考虑传统长波近似的剖分,纳入了内边界条件的流动效应,为卫星液滴的形成提供了必要条件,并确定了与 k*2 成比例的最不稳定模式,其中 k* 是最不稳定的波数。此外,我们还观察到卫星液滴的体积随波长呈指数衰减。此外,我们的单一模型综合了戈伦的纤维上液体薄膜和瑞利的粘性液体线的发现,证明了它的多功能性和对广泛系统的相关性。
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引用次数: 0
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