Accurately preserving the volume of the dispersed droplets remains a significant challenge in phase-field simulations of droplet-laden turbulence, especially under conditions that feature strong interfacial deformation and breakup. While modified phase-field equations have been developed to mitigate volume loss, their effectiveness has not been systematically assessed in the context of fully developed turbulent flows. In this work, we first evaluate the performance of several representative volume-corrected phase-field models in direct numerical simulations of droplet-laden homogeneous isotropic turbulence. Our results reveal that, at sufficiently high Weber numbers, none of the existing models provides satisfactory droplet-volume preservation. To address this limitation, we then propose a simple yet effective modification of the conservataive Allen-Cahn equation by incorporating a curvature-dependent counter-diffusion correction. Direct numerical simulations in turbulent regimes demonstrate that the proposed model achieves conservation of droplet volume in a statistical sense, while avoiding common adverse effects, such as numerical instability, violation of global mass conservation, increased computational cost, artificial coarsening, or enhanced spurious velocities.
This review focuses on Direct numerical simulations (DNS) of turbulent flows laden with droplets or bubbles. DNS of these flows are more challenging than those of flows laden with solid particles due to the surface deformation in the former. The classification of the discussed numerical methods is based on whether the initial diameter of the bubble/droplet is smaller or larger than the Kolmogorov length scale, and whether the instantaneous surface deformation is fully resolved or obtained via a phenomenological model. Numerical methods that account for the breakup of single droplet/bubble as w…
The motion of a viscous droplet in unbounded Poiseuille flow under the combined influence of bulk-insoluble surfactant and linearly varying temperature field aligned in the direction of imposed flow is studied analytically. Neglecting fluid inertia, thermal convection and shape deformation, asymptotic analysis is performed to obtain the velocity of a force-free surfactant-laden droplet. The present study is focused on two limiting situations of surfactant transport: (i) small surface Peclet number, and (ii) high surface Peclet number. Thermocapillary-induced Marangoni stress, strength of which…
The motion of a surfactant-laden viscous droplet in the presence of background non-isothermal Poiseuille flow is studied analytically and numerically. Specifically, the effect of interfacial Marangoni stress due to non-uniform distribution of surfactants and temperature at the droplet interface on the velocity and direction of motion of the droplet along the centerline of imposed Poiseuille flow is investigated in the presence of linearly varying temperature field. In the absence of thermal convection, fluid inertia and shape deformation, the interfacial transport of bulk-insoluble surfactants…
The present study deals with the effect of interfacial slip on the deformation and emulsion rheology of a dilute suspension of droplets in a linear flow. The droplets are laden with surfactants that are bulk-insoluble and get transported only along the interface. An asymptotic approach is adopted for the present analysis in order to tackle the nonlinearity present due to deformation of droplets. The analysis is carried out under two different limiting scenarios namely: surface diffusion-dominated-surfactant transport and surface convection-dominated surfactant transport. For either of the limi…
In the present study we analytically investigate the deformation and bulk rheology of a dilute emulsion of surfactant-laden droplets suspended in a linear flow. We use an asymptotic approach to predict the effect of surfactant distribution on the deformation of a single droplet as well as the effective shear and extensional viscosity for the dilute emulsion. The non-uniform distribution of surfactants due to the bulk flow results in the generation of a Marangoni stress which affects both the deformation as well as the bulk rheology of the suspension. The present analysis is done for the limiti…
Understanding and modulating the cross-stream motion of a surfactant-coated droplet in pressure driven flow has great implications in many practical applications. A combination of interfacial viscosity and Marangoni stress acting over a surfactant-coated droplet in pressure driven flow offers greater flexibility to modulate the cross-stream motion of it. Despite the intense theoretical and numerical research towards manipulating the surfactant-laden Newtonian droplets in Poiseuille flow, the experimental investigations are seldom explored. Herein, we report our study on understanding the influ…
The motion of a viscous deformable droplet suspended in an unbounded Poiseuille flow in the presence of bulk-insoluble surfactants is studied analytically. Assuming the convective transport of fluid and heat to be negligible, we perform a small-deformation perturbation analysis to obtain the droplet migration velocity. The droplet dynamics strongly depends on the distribution of surfactants along the droplet interface, which is governed by the relative strength of convective transport of surfactants as compared with the diffusive transport of surfactants. The present study is focused on the fo…
In the present study, we investigate the cross-stream migration of a deformable droplet suspended in a non-isothermal Poiseuille flow in the presence of bulk-insoluble surfactants. Owing to the non-linearity present in the system of governing equations, an asymptotic approach is adopted, in an effort to capture the intricate and non-trivial coupling between the various influencing parameters. With the assumption of negligible inertia in fluid flow and convective transport of thermal energy, we obtain the droplet migration velocity through small-deformation perturbation analysis for two differe…
A conservation-consistent boundary condition for nonlinear models of soluble-surfactant-laden falling films
Centre for Applied Sciences (CAS), International Institute of Information Technology（国际信息技术学院应用科学中心）；Universite Claude Bernard Lyon 1, LMFA, UMR5509, CNRS, Ecole Centrale de Lyon, INSA Lyon（里昂第一大学，LMFA，UMR5509，CNRS，里昂中央理工学院，里昂国立应用科学学院）；Department of Mathematics, Vivekananda Mahavidyalaya（维韦卡南达学院数学系）
Atmospheric aerosols can consist of inorganic and organic substances, including surfactants at a significant concentration. Importantly, the latter can reduce the surface tension at the liquid-vapor surfaces, where they preferentially adsorb due to their amphiphilic structure. As a result, processes such as droplet coalescence, development of precipitation and ultimately cloud lifetime, may depend on the presence of surfactants in the aerosols. Here, we present a numerical model for cloud droplet formation, which is based on the Lagrangian particle-based microphysics-scheme super-droplet metho…
The fragmentation of liquid sheets produces a collection of droplets. The size distribution of the droplets has a considerable impact on the coating efficiency of sprays and the transport of contaminants. Although many processes commonly used particulate suspensions, the influence of the particles on the spreading dynamics of the sheet and its subsequent fragmentation has so far been considered negligible. In this paper, we consider experimentally a transient suspension sheet that expands radially. We characterize the influence of the particles on the dynamics of the liquid sheet and the fragm…
Experimental investigation into Lagrangian statistics of droplets in homogeneous isotropic turbulence
New Cornerstone Science Laboratory, Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of MoE, Department of Energy and Power Engineering, Tsinghua University（清华大学能源与动力工程系，燃烧能中心，新Cornerstone科学实验室）；Department of Applied Physics and Science Education, Eindhoven University of Technology（埃因霍温理工大学应用物理与科学教育学系）；Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University（清华大学航天航空学院工程力学系）
The COVID-19 pandemic has inspired several studies on the fluid dynamics of respiratory events. Here, we propose a computational approach in which respiratory droplets are coarse-grained into an Eulerian liquid field advected by the fluid streamlines. A direct numerical simulation is carried out for a moist cough using a closure model for space-time dependence of the evaporation time scale. Estimates of the Stokes number are provided, for the initial droplet size of $10 μ$m, which are found to be <<1 thereby justifying the neglect of droplet inertia. Several of the important features of the mo…
Evaporation-Driven Nanowire Self-Assembly in an Elongated Droplet
Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IET-2), Forschungszentrum Jülich（于利希研究中心埃尔朗根-纽伦堡亥姆霍兹可再生能源研究所）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
Droplets moving in a microfluidic loop device exhibit both periodic and chaotic behaviors based on the inlet droplet spacing. We propose that the periodic behavior is an outcome of a dispersed phase conservation principle. This conservation principle translates into a droplet spacing conservation equation. Additionally, we define a simple technique to identify periodicity in experimental systems with input scatter. Aperiodic behavior is observed in the transition regions between different periodic behaviors. We propose that the cause for aperiodicity is the synchronization of timing between th…
A numerical study is carried out on the droplet-laden two-phase rotating detonation wave (RDW) of kerosene/oxygen-enriched air at normal temperature. Two types of combustors without and with the inlet mixing section (IMS) are constructed to illustrate the effect of IMS on the combustion characteristics of two-phase RDW. The important role of the preheating zone in the IMS after the back-propagation shock on the droplet evaporation is analyzed. The parameter sensitivity of RDW propagation stability to the average droplet diameter d0 is further discussed. Results show that the droplets mainly ev…
A thermodynamically consistent and conservative phase-field model for surfactant contact line dynamics with solid adsorption
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）；The State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology（华中科技大学智能装备与技术国家重点实验室）
The features of turbulence modulation produced by a heavy loaded suspension of small solid particles or liquid droplets are discussed by using a physically-based regularisation of particle-fluid interactions. The approach allows a robust description of the small scale properties of the system exploiting the convergence of the statistics with respect to the regularisation parameter. It is shown that sub-Kolmogorov particles/droplets modify the energy spectrum leading to a scaling law, $E(k)\propto k^{-4}$, that emerges at small scales where the particle forcing balances the viscous dissipation.…
Aiming for the simulation of colloidal droplets in microfluidic devices, we present here a numerical method for two-fluid systems subject to surface tension and depletion forces among the suspended droplets. The algorithm is based on an efficient solver for the incompressible two-phase Navier-Stokes equations, and uses a mass-conserving level set method to capture the fluid interface. The four novel ingredients proposed here are, firstly, an interface-correction level set (ICLS) method; global mass conservation is achieved by performing an additional advection near the interface, with a correc…
This research introduces a new drop fluidics, which uses a deformable and stretchable elastomeric film as the platform, instead of the commonly used rigid supports. Such a soft film impregnated with magnetic particles can be modulated with an external electromagnetic field that produces a vast array of topographical landscapes with varying surface curvature, which, in conjunction with capillarity, can direct and control motion of water droplets efficiently and accurately. When a thin layer of oil is present on this film that is deformed locally, a centrosymmetric wedge is formed. A water dropl…
The conservation of mass is common issue with multiphase fluid simulations. In this work a novel projection method is presented which conserves mass both locally and globally. The fluid pressure is augmented with a time-varying component which accounts for any global mass change. The resulting system of equations is solved using an efficient Schur-complement method. Using the proposed method four numerical examples are performed: the evolution of a static bubble, the rise of a bubble, the breakup of a thin fluid thread, and the extension of a droplet in shear flow. The method is capable of con…
In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and co…
We propose a non-local model for surface tension obtained in the form of an integral of a molecular-force-like function with support $3.5\varepsilon$ added to the Navier-Stokes momentum conservation equation. We demonstrate analytically and numerically that with the non-local model interfaces with a radius of curvature larger than the support length behave macroscopically and microscopically, otherwise. For static droplets, the pressure difference $P_{\varepsilon, in} - P_{\varepsilon, out}$ satisfies the Young-Laplace law for droplet radius greater than $3.5\varepsilon$ and otherwise deviates…
On the size and shape of condensing steam jets and plumes
Laboratoire de Physique de l’Ecole Normale Supérieure, UMR 8023 PSL-SU-UPCité-CNRS（巴黎高等师范学院物理实验室）；Physique et Mécanique des Milieux Hétérogènes, UMR 7636 ESPCI-PSL-SU-UPCité-CNRS（异质介质物理与力学）
Evaporation kinetics of pendant droplets is an area of immense importance in several applications in addition to possessing rich fluid and thermal transport physics. The present article experimentally and analytically sheds insight into the augmented evaporation dynamics of paramagnetic pendent droplets in the presence of a magnetic field stimulus. Literature provides information that solutal advection and solutal Marangoni effect lead to enhancement of evaporation in droplets with ionic inclusions. The major crux of the present article remains to modulate the thermosolutal advection with the …
Modeling of a dense spray regime using an Euler-Lagrange approach is challenging because of local high volume loading. A cluster of droplets, that are assumed subgrid, can lead to locally low void fractions for the fluid phase. Under these conditions, spatio-temporal changes in the fluid volume fractions should be considered in an Euler-Lagrange, two-way coupling model. This leads to zero-Mach number, variable density governing equations. Using pressure-based solvers, this gives rise to a source term in the pressure Poisson equation and a non-divergence free velocity field. To test the validit…
