A fully meshless method for ‘gas – evaporating droplet’ flow modelling

A meshless method for modelling two-phase flows with phase transition is described. The method is based on consideration of three systems: viscous-vortex blobs, thermal-blobs and droplets; and can be applied for numerical simulation of 2D non-isothermal flows of ‘gas-evaporating droplets’ in the framework of the one-way coupled two-fluid approach. The carrier phase is viscous incompressible gas. The dispersed phase is presented by a cloud of identical spherical droplets, and, due to evaporation, the radius and mass of droplets are time dependent. The carrier phase parameters are calculated using the viscous-vortex and thermal-blob method; the dispersed phase parameters are calculated using the Lagrangian approach. Two applications have been considered: (i) a standard benchmark – Lamb vortex; (ii) a cold spray injected into a hot quiescent gas. In the latter problem three cases corresponding to three droplet sizes were investigated. The smallest droplets (of the three cases considered) are more readily entrained by the carrier phase and form ring-like structures; the flow shows better mixing. Larger droplets evaporate less intensively. The medium sized droplets collect into two narrow bands stretched along the jet axis. The largest droplets form a two-phase jet, which remains close to the jet axis. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

相场方法模拟液滴的动态润湿行为

液滴的动态湿润现象广泛存在于自然界和工业生产中,该现象数值研究的建模需要解决接触线附近的奇异性并引入合理的接触角描述.基于相场方法,结合Yokoi动态接触角模型,建立了考虑动态润湿效应的两相流数值模型,并在OpenFOAM开源平台上实现相应程序.针对液滴撞击壁面的动态湿润过程,数值模拟和对比研究了不同的接触角模型.结果表明:接触角模型的选择对液滴动态润湿过程的模拟结果具有较大的影响,其中基于改进动态接触角模型的结果与文献中的实验结果具有很好的吻合度,反映了提出的数值模型在液滴的动态润湿行为模拟的有效性.     

三维PTT黏弹性液滴撞击固壁面问题的改进SPH模拟

基于光滑粒子动力学(smoothed particle hydrodynamics, SPH)方法,对三维Phan-Thien Tanner(PTT)黏弹性液滴撞击固壁面问题进行了数值模拟.为了有效地防止粒子穿透固壁,且缩减三维数值模拟所消耗的计算时间,提出了一种适合三维数值模拟的改进固壁边界处理方法.为了消除张力不稳定性问题,采用一种简化的人工应力技术.应用改进SPH方法对三维PTT黏弹性液滴撞击固壁面问题进行了数值模拟,精细地捕捉了液滴在不同时刻的自由面,讨论了PTT黏弹性液滴不同于Newton(牛顿)液滴的流动特征,分析了PTT拉伸参数对液滴宽度、高度和弹性收缩比等的影响.模拟结果表明,改进SPH方法能够有效而准确地描述三维PTT黏弹性液滴撞击固壁面问题的复杂流变特性和自由面变化特征.     

二元海水液滴对心碰撞过程数值模拟

为研究海水循环冷却系统中液滴碰撞的基本规律及碰撞结果预测模型,采用流体体积函数(volume of fluid,VOF)方法捕捉两相交界面,利用动态网格自适应技术提高求解精度,对二元海水液滴的对心碰撞过程进行直接数值分析与模拟.首先对氮气中正十四烷液滴的碰撞实验进行数值模拟,验证了数值模型的可靠性.开展了常温常压下等尺寸二元海水液滴对心碰撞数值研究,分析了液滴碰撞过程流场结构及流动机理,研究了不同液滴直径和不同海水浓度对碰撞过程的影响规律,得到了聚合和自反分离两种碰撞结果类型以及二者的临界Weber数.总结出不同Ohnesorge数下海水液滴碰撞结果诺模图.     

A probabilistic model of thermal explosion in polydisperse fuel spray

This work is concerned with an analysis of polydisperse spray droplets distribution on the thermal explosion processes. In many engineering applications it is usual to relate to the practical polydisperse spray as a monodisperse spray. The Sauter Mean Diameter (SMD) and its variations are frequently used for this purpose [13]. The SMD and its modifications depend only on “integral” characterization of polydisperse sprays and can be the same for very different types of polydisperse spray distributions.The current work presents a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of different radii (polydisperse). The polydispersity is modeled using a probability density function (PDF) that corresponds to the initial distribution of fuel droplets size. This approximation of polydisperse spray is more accurate than the traditional ‘parcel’ approximation and permits an analytical treatment of the simplified model. Since the system of the governing equations represents a multi-scale problem, the method of invariant (integral) manifolds is applied.An explicit expression of the critical condition for thermal explosion limit is derived analytically. Numerical simulations demonstrate an essential dependence of these thermal explosion conditions on the PDF type and represent a natural generalization of the thermal explosion conditions of the classical Semenov theory.     

Behavior of droplets for a class of Glauber dynamics at very low temperature

Summary We consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non-increasing fashion.We extend to this class results previously shown for the particular case of Metropolis dynamics [NS]. We show that for fixed volume and external field 0<h<1, at very low temperature small rectangular droplets of sping +1 in a sea of spins –1 tend to shrink, while large droplets tend to grow and cover the whole system. The threshold between the two behaviors is sharply defined, the critical length being 2/h.An example is given which shows that without the assumption of monotonicity of the rates this result may be false.We use the result on critical droplets to show that starting from the configuration with all spins down, the systems in the class that we consider evolve in a metastable fashion until the configuration with all spins up is reached.For similar systems in higher dimensions we show that under analogous conditions on the rates, small enough droplets are likely to shrink, while large enough droplets are likely to grow.Partially supported by CNPq.     

Formation of uniformly sized metal droplets from a capillary jet by electromagnetic force

A new method for producing uniformly sized metal droplets is proposed. In this method, an intermittent electromagnetic pinch force is applied to a capillary jet of liquid metal to generate fluctuations of equal interval on the surface of the jet. As the fluctuations grow, the liquid metal jet breaks into small droplets whose size depends on the frequency of the intermittent electromagnetic pinch force. The breakup of the capillary jet is numerically simulated by performing multiphase fluid flow analysis with surface tracking (volume of fluid method) and electromagnetic force analysis. The simulation results agree well with the results of model experiments. The jet breaks up into uniformly sized droplets when the frequency of the intermittent force equals the frequency that corresponds to the natural disturbance wavelength of the capillary jet.     

Capillary hydrodynamics of the deposition of thin films and droplets on heterogeneous,porous substrates

The work presented here deals with recent research on capillary hydrodynamics of thin viscous liquid films and droplets. The process engineering background comes from the industrial process to deposit thin films or droplets on substrates such as polymer foils or paper [1, 2, 3]. During the production of paper, for example, layers of coating color have to be applied onto the raw paper in order to ensure its usability. Ink–jet paper consists of several different functional layers to optimize the paper with regard to printing quality or paper handling. For the printing process by the ink–jet technique, a controlled application of droplets onto the paper and subsequent penetration of the fluid into the paper are crucial for the final quality of the print. The work summarized here considers the application of films or droplets on substrates as well as the flow of liquids inside porous media. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modélisation multi-fluide eulérienne pour la simulation de brouillards denses polydispersés

In this Note, we derive from the kinetic level a Eulerian multi-fluid model for the simulation of polydisperse dense sprays of vaporizing liquid droplets. This yields an extension of the sectional method developed by Tambour et al. and justified by Laurent et al., to dense sprays where droplets of various sizes, and various velocities due to varying inertia, can coalesce. The velocity partial collisional integrals are first evaluated on Gaussian distribution of velocity and covering the whole range of droplet sizes. The final model is obtained by letting the velocity dispersion go to zero, allowing the collisional integrals to be pre-calculated.     

Singularly perturbed homotopy analysis method

In this paper we combined the homotopy analysis method (HAM) and the method of integral manifold (MIM) to investigate the problem of thermal explosion in two-phases polydisperse combustible mixtures of gas with fuel droplets. The size distribution of the fuel droplets is assumed to be continuous in the form of an exponential distribution and is found from the solution of the kinetic equation for the probability density function. The system of the polydisperse fuel spray takes into account the effects of the thermal radiation and convection. By applying the HAM and the MIM, we derived an analytical solution of the system and we compared our results with the numerical solutions.     

Ice Accretion on Cables of Various Cross-sections

Poots & Rodgers (1976) gave analytical and numerical resultson the icing of a cable having circular cross-section. The windcarrying freezing fog or supercooled cloud droplets impingeson the cable and a spike of rime ice grows outward into thewind. A linear Stokes law of resistance was assumed in calculatingthe drag force on individual droplets. Consequently the theoreticalpredictions on ice accretion are shown to be restricted to smallcable radii and high wind speeds (i.e. for large Stokes number). The purpose of this paper is to give detailed numerical solutionsfor ice accretion on cables of various cross-sections. Empiricaldrag formulae for droplets in motion, with local Reynolds numbersup to 10², are now employed in the numerical solution of droplettrajectories and impaction velocities on the ice accretion surface.Cables having circular, elliptical, square, and equilateraltriangular cross-section are investigated; also included areresults for a cylinder of elliptic cross-section the major axisof which is aligned at an angle to the upstream wind direction.For fixed wind speed and cable size and cross-section the massof ice accreted is given as a function of the time. These resultsare relevant to the evolution of ice accretion on structures.     

液滴碰撞和聚合模型研究

建立了描述液滴碰撞和聚合过程的数学模型,设计了一种高效的计算液滴碰撞对搜索算法,并在已有研究成果的基础上对液滴碰撞的结果做了进一步的发展．借助平滑粒子流体动力学方法两种实现方式各自的优点,将液滴间的相互作用局限在其周围一定数目的液滴之间,并采用积分核函数定义了液滴间碰撞的概率,通过数值模拟探讨了模型的特性．结果表明,所建模型对所采用的计算网格没有明显的依赖性,具有较高的计算精度和计算效率,不但能很好地维持系统动量的守恒性,而且对液滴初始尺寸分布没有明显的依赖性．     

喷射雾化最小等效直径的能量判断准则

通过能量分析的方法考虑喷射雾化问题,认为破碎的液滴存在着表面张力势能,这种势能是由于气流对液体的冲击作用提供的,并由此得到了破碎液滴的最小等效直径的能量判断准则.该判断准则在趋势上与经验公式的符合程度是令人满意的.     

Containerless rapid solidification of undercooled Cu-Co peritectic alloys

Droplets of Co-16%Cu and Co-71.6%Cu peritectic alloys were solidified during containerless processing in a 3_m drop tube. The microstructures of Co-16%Cu alloy droplets were characterized by dendritic or equiaxed α-Co phase with a small amount of Cu-rich solid solution distributed on α-Co phase boundaries. Two thresholds of droplet diameter were observed for Co-16%Cu alloy at which "equiaxed-dendritic-equiaxed" morphological transitions occur to primary α-Co phase. This conspicuous refinement of primary α-Co grains results from the fragmentation of α-Co dendrites caused by recalescence effect. For Co-71.6%Cu alloy, the primary α-Co phase forms as coarse columnar dendrites in large droplets and equiaxed dendrites in small droplets. Theoretical calculations indicate that Marangoni migration contributes more to the growth of disperse Co-rich spheres by stimulating collision and coalescence than Stokes motion caused by the residual gravity in the falling Co-71.6%Cu alloy droplets.     

Dynamics of droplets in an agitated dispersion with multiple breakage. Part I: formulation of the model and physical consistency

In (Proc. Symp. Lectures Appl. Math. 2000; 123–141) a new model for the evolution of a system of droplets dispersed in an agitated liquid was presented, with the inclusion of the so‐called volume scattering effect (a combination of coalescence and breakage). In that paper droplets breakage was considered to be binary, in order to simplify exposition. Here we remove that limitation, investigating the effect of each breakage mode and of scattering with multiple exits. We also allow the breakage kernel, at each mode, to become singular when droplets approach their finite maximum admissible size. Copyright © 2004 John Wiley & Sons, Ltd.     

基于格子Boltzmann方法的液滴撞击具有不同润湿性孔板的研究

基于格子Boltzmann方法,对液滴撞击不同湿润性节流孔板表面进行了数值模拟.主要研究了在液滴撞击过程中,Weber数(We)、孔板表面湿润性和孔板尺寸对液滴通过孔板时不同状态的影响.数值模拟结果表明:孔板为亲水特性时,在较低We下,液滴不会与孔板表面脱离,而是附着在孔板下表面,并且在毛细作用下液滴会在孔道中上升一段...     

On solving the potential equation

We study radial solutions to the generalized Swift-Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as r → ∞. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift-Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo-and heteroclinic orbits, and numerical simulation.     

Internal Flow Analysis for Slow Moving Small Droplets in Contact with Hydrophobic Surfaces

Internal fluid flow behavior for slow moving small droplets in contact with hydrophobic surfaces is analyzed. The shape of the droplet is first computed using the Young-Laplace equation. For this purpose a Finite Element (FE) model [1], in which contact constraints are enforced through Penalty and Augmented Lagrange Multiplier methods, is used. The flow field within the droplet is then analyzed using the Stokes flow model, considering a de-coupled approach. Similar to the membrane deformation model, the formulation for the flow analysis is also expressed in the framework of FE analysis. Both, stabilized (Pressure Stabilizing/Petrov-Galerkin PSPG) and Galerkin FE formulations are considered. The motion of the fluid inside the droplet is governed by the slip condition enforced on the membrane of the droplet. Numerical examples for droplets rolling steadily are presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of the gelation process of hydrogel droplets in 3D bioprinting

The aim of this work is to numerically simulate the gelation of a crosslinking polymer, which is a very complex process involving chemical reactions and phase transitions from a viscous fluid to a viscoelastic solid. A phenomenological model is proposed to simulate the gelation process of agarose droplets, considering the thermal boundary conditions. The numerical model is implemented using the finite element package FlexPDE. The temperature- and time-dependent degree of gelation and the deformation of the droplets during the gelation process are investigated. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Modeling of Liquid Droplets Moving on Rough Surfaces

We present a de-coupled approach for computational modeling of liquid droplets moving on rough substrate surfaces. The computational model comprises solving the membrane deformation problem and the fluid flow problem in a segregated manner. The droplet shape is first computed by solving the Young-Laplace equation where contact constraints, due to the droplet-substrate contact, are applied through the penalty method [1]. The resulting configuration constitutes the domain for the fluid flow problem, where the bulk fluid behavior is modeled by the unsteady Stokes' flow model expressed in Arbitrary Lagrangian-Eulerian (ALE) framework. The entire analysis is performed in the framework of Finite Element Method (FEM). Application of the approach to the case of a droplet moving on a rough surface is presented as an example. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)