A Level-Set Method for Analysis of Particle Motion in an Evaporating Microdroplet

A sharp-interface level-set (LS) method is presented for computing particle motion in an evaporating microdroplet. The LS formulation for incompressible two-phase flow is extended to include the effects of evaporation, mass transfer, heat transfer, and dynamic contact angles. A numerical technique for the conservation of particle concentration is incorporated into the LS method, and calculation procedures are also developed and tested for reducing the numerical errors caused in the computation of interface curvature and liquid–gas velocity jump. The improved LS method is applied to the simulation of particle distribution in microdroplet evaporation on a solid surface.     

A Level-Set Method for the Direct Numerical Simulation of Particle Motion in Droplet Evaporation

A sharp-interface level-set (LS) method is presented for direct numerical simulation (DNS) of particle motion in droplet evaporation. The LS formulation for liquid–gas flows is extended to liquid–gas–solid flows by treating the moving solid region as a high-viscosity fluid phase. The evaporation effect is accurately implemented by imposing the coupled temperature and vapor fraction conditions at the interface. The LS method is tested through computations of particle sedimentation in single-phase and two-phase fluids. The DNS of particle motion in droplet evaporation demonstrates the pinning phenomena of the liquid–gas–solid contact line.     

Numerical simulation of droplet impact and evaporation on a porous surface

Numerical simulation is performed for the evaporation of a droplet impacted on a porous surface. A level-set formulation for tracking the droplet deformation is extended to include the effects of evaporation coupled to heat and mass transfer, porosity and porous drag and capillary forces. The local volume averaged conservation equations of mass, momentum, energy and vapor fraction for the porous region are simultaneously solved with the conservation equations for the external fluid region. The computations demonstrate not only the evolution of the liquid-gas interface during the whole period of droplet penetration and evaporation in a porous medium, but also the associated flow, temperature and vapor fraction fields. The effects of impact velocity, porosity and particle size on the droplet deformation and evaporation are quantified.     

Flow and thermal field in sessile droplet evaporation at various environmental conditions

Sessile droplets' evaporation is a complex process that involves fluid flow coupled with heat and mass transfer. In this study, mathematical modelling of sessile droplet evaporation on hydrophobic substrates is developed and simulations are carried out on COMSOL. The model results are validated with the data available in the literature. Postvalidation, the simulation of droplet evaporation is carried out on the various substrate hydrophobicities and various environmental conditions. For these conditions, contours are plotted for temperature, velocity, and mass concentration for the droplet and moist air domain. The result shows that Marangoni convection plays a very important role in droplet evaporation. A high rate of evaporation is observed at the droplet interface at low relative humidity and a large degree of subheating. The effect of air velocity on the evaporation rate is studied, however, its effect is very marginal as compared to relative humidity and degree of subheating. The heat flux at the three-phase contact line is large for a smaller Prandtl number fluid. Overall, the evaporation rate increases with increasing the Prandtl number because it has a large value of Marangoni convection.     

Interfacial heat transfer during microdroplet evaporation on a laser heated surface

A comprehensive experimental and numerical investigation on water microdroplet impingement and evaporation is presented from the standpoint of phase-change cooling technologies. The study investigates microdroplet impact and evaporation on a laser heated surface, outlining the experimental and numerical conditions necessary to quantify the interfacial thermal conductance (G) of liquid-metal interfaces during two-phase flow. To do this, continuum-level numerical simulations are conducted in parallel with experimental measurements facilitating high-speed photography and in-situ time-domain thermoreflectance (TDTR). During microdroplet evaporation on laser heated Al thin-films at room temperature, an effective interfacial thermal conductance of G_eff = 6.4 ± 0.4 MW/m² is measured with TDTR. This effective interfacial thermal conductance (G_eff) is interpreted as the high-frequency (ac) interfacial heat transfer coefficient measured at the microdroplet/Al interface. Also on a laser heated surface, fractal-like condensation patterns form on the Al surface surrounding the evaporating microdroplet. This is due to the temperature gradient in the Al surface layer and cyclic vapor/air convection patterns outside the contact line. Laser heating, however, does not significantly increase the evaporation rate beyond that expected for microdroplet evaporation on isothermal Al thin-film surfaces.     

A numerical model for transport in flat heat pipes considering wick microstructure effects

A transient, three-dimensional model for thermal transport in heat pipes and vapor chambers is developed. The Navier–Stokes equations along with the energy equation are solved numerically for the liquid and vapor flows. A porous medium formulation is used for the wick region. Evaporation and condensation at the liquid–vapor interface are modeled using kinetic theory. The influence of the wick microstructure on evaporation and condensation mass fluxes at the liquid–vapor interface is accounted for by integrating a microstructure-level evaporation model (micromodel) with the device-level model (macromodel). Meniscus curvature at every location along the wick is calculated as a result of this coupling. The model accounts for the change in interfacial area in the wick pore, thin-film evaporation, and Marangoni convection effects during phase change at the liquid–vapor interface. The coupled model is used to predict the performance of a heat pipe with a screen-mesh wick, and the implications of the coupling employed are discussed.     

Numerical Analysis of Film Boiling in Liquid Jet Impingement

A numerical approach is presented for computing film boiling in liquid jet impingement on a high-temperature plate. The conservation equations of mass, momentum and energy are numerically solved in the liquid, vapor, and air phases. The sharp-interface level-set formulation is employed to track the liquid-air interface, as well as the liquid-vapor interface with phase change. A simplified analytical model for a thin vapor film, whose thickness is several orders of magnitude smaller than the liquid layer, is incorporated into the level-set formulation. The multiscale approach is tested through the computations of film boiling in a circular water jet.     

Onset of oscillatory thermocapillary convection in acetone liquid bridges: The effect of evaporation

Experiments have been carried out for half-zones of acetone (Pr = 4.3) to investigate the effects of evaporative cooling on the flow structures and temperature fields during transition from steady to oscillatory convection. The unstable flow phenomena have been measured using a variety of diagnostic techniques to determine the effects of evaporative cooling on Marangoni convection in liquid bridges of intermediate Prandtl number. The results show that Marangoni convection in acetone liquid bridges with and without strong evaporation becomes unstable due to the same mechanism but the evaporation has a strong stabilizing effect on the onset of oscillatory Marangoni convection.     

Marangoni convection,evaporation and interface deformation in liquid films on heated substrates with non-uniform thermal conductivity

Marangoni convection plays an important role in hydrodynamics of liquid films on heated or cooled substrates. In this paper a model describing Marangoni convection, interface dynamics and evaporation in liquid films on composite substrates or substrates of functionally graded materials is developed. Non-uniform thermal conductivity of the substrate causes non-uniformity of temperature distribution at the liquid–gas interface which leads to appearance of Marangoni stresses, convective vortices and film deformation. The film dynamics is described in the framework of long-wave theory. The substrate thermal conductivity non-uniformity has a pronounced effect on transport processes in the liquid film.     

Numerical Analysis of Bubble Growth and Departure from a Microcavity

A numerical approach is presented for analysis of bubble growth and departure from a microcavity during nucleate boiling. The level-set formulation for tracking the phase interfaces is modified to include the effect of phase change on the liquid–vapor interface and to treat the no-slip and contact angle conditions on the immersed (or irregularly shaped) solid surface of the microcavity. Also, the formulation is coupled with a simple and efficient model for predicting the evaporative heat flux from the liquid microlayer on an immersed solid surface. The effects of cavity size and geometry on the bubble growth and departure in nucleate boiling are investigated.     

A level-set method for ultrasound-driven bubble motion with a phase change

A level-set (LS) method is presented for computation of ultrasound-driven bubble motion including the effect of liquid and vapor compressibility as well as the effect of liquid–vapor phase change. The semi-implicit pressure correction formulation is implemented into the LS method to avoid the serous time-step restriction in low Mach number (or near incompressible) flows. The numerical results for one-dimensional compressible flows and spherical bubble motion in a periodic acoustic field show good agreement with the analytical solutions. The effects of phase change and ambient temperature on the ultrasound-driven bubble motion are quantified.     

Onset of Marangoni instability of a two-component evaporating droplet

The temperature and solute concentration reductions across a thin boundary layer near the free surface of an evaporating droplet may induce cellular flow motion in the droplet because of Marangoni instability. The present study is aimed at investigating theoretically the onset of Marangoni instability due to the evaporation of a two-component evaporating droplet.

With the quasi-steady approximation which means that the surrounding gas motion is asymptotically steady, the size change of the droplet is negligible, and the temperature and concentration distributions of the droplet are temporarily frozen at each specified instant of interest, the onset condition for Marangoni instability is obtained through the linear stability analysis.

By assuming the surface tension is a monotonically decreasing function of both temperature and concentration of the higher-volatility substance, the thermocapillary and diffuso-capillary effects augment each other. Therefore, the theoretical analysis predicts a linear relation, with a negative slope, between the onset thermal Marangoni number, Ma_T, and the onset solute Marangoni number, Ma_S. Moreover, when liquid Lewis number Le_l>1, the critical wave number, l_c, may possess different values depending on the variation of the thermocapillary effect and diffuso-capillary effect. In addition, Le_l has a stronger effect on the critical solute Marangoni number Ma_S,C, than on the critical thermal Marangoni number Ma_T,C. That is, as Le_l decreases, Ma_T,C decreases mildly while Ma_S,C increases drastically.     

Topological structure evolvement of flow and temperature fields in deformable drop Marangoni migration in microgravity

Using the level-set method and the continuum interface model, the axisymmetric thermocapillary migration of a deformable liquid drop immerged in an immiscible bulk liquid with a temperature gradient is simulated numerically with constant material properties of the two phases. Steady terminal state of the motion can always be reached. The dimensionless terminal migration velocity decreases monotonously with the increase of the Marangoni number. Good agreements with space experimental data and most of previous numerical studies in the literature are evident. The terminal topological structure of flow field, in which a recirculation identical to Hill’s vortex exists inside the drop, does not change with the Marangoni number. Only slight movement of the location of vortex center can be observed. On the contrary, bifurcations of the terminal topological structure of temperature field occur twice with increasing Marangoni number. At first, the uniform and straight layer-type structure of temperature field at infinitesimal Reynolds and Marangoni numbers wraps inside of the drop due to convective transport of heat as the Marangoni number increases, resulting in the emergence of an onion-type local cooler zone around the center of the drop beyond a lower critical Marangoni number. Expanding of this zone, particularly in the transverse direction, with the increasing of the Marangoni number leads to a cap- or even shell-type structure. The coldest point within the liquid drop locates on the axis. There is a middle critical Marangoni number, beyond which the coldest point will jump from the rear stagnation into the drop, though the topological structure of the temperature field does not change. The second bifurcation occurs at an upper critical Marangoni number, where the shell-type cooler zone inside drops ruptures from the central point and then a torus-type one emerges. The coldest point departs from the axis, and the so-called “cold-eye” appears in the meridian. It is also found that the inner and outer thermal boundary layers along the interface may exist both inside and outside the drop if Ma > 70. But the thickness decreases with the increasing Marangoni number more slowly than the prediction of potential flow at large Marangoni and Reynolds numbers. A velocity shear layer outside the drop is also introduced formally, of which modality may be affected by the convective transports of heat and/or momentum.     

A NUMERICAL STUDY OF SOLIDIFICATION IN THE PRESENCE OF A FREE SURFACE UNDER MICROGRAVITY CONDITIONS

A numerical study of the relative importance of Marangoni effects under microgravity conditions is presented. The mathematical formulation adopted is based on the enthalpy porosity method. One of the advantages of the fixed grid method is that a unique set of equations and boundary conditions is used for the whole domain, including both solid and liquid phases. The governing equations written in a vorticity-velocity formulation are discretized using a finite volume technique on a staggered grid. A fully implicit method has been adopted for the mass and momentum equations, while the temperature field is solved separately in order to evaluate the variation in the local liquid mass fraction. The resulting algebraic system of equations is solved using a preconditioned BI-CGStab method. Numerical results modelling the free surface, including the effects on it of Marangoni convection, are presented. The influence of the presence of argon in the gap above the free surface is investigated. During the numerical simulations presented in this paper 161 2 41 and 641 2 161 uniform meshes on the whole computational domain for values of Marangoni number ( Ma ) up to 16,120 and Rayleigh number ( Ra ) of 5 have been used.     

Fringe probing of an evaporating microdroplet on a hot surface

The phenomenon of droplets impacting and evaporating on a hot surface is of interest in many areas of engineering. Quantitative measurement of these processes provides great help to reveal the physics behind. A novel technique was developed to quantitatively measure the volume evolution and contact diameter of an evaporating microdroplet on a hot surface utilizing interference fringe scattering method. In this method, fine fringes produced by the interference of two coherent laser beams was scattered by the droplet and projected onto a screen. The profile and volume of the droplet can be derived from the spatial fringe spacing on the screen. The number of total fringes measurable on the screen was used to determine the instantaneous contact diameter of the microdroplet. Validation experiments demonstrated that the measurement errors are less than ±5% and ±1% for microdroplet volume and contact diameter, respectively. By using this method, the dynamic of droplet impingement, evaporation and boiling using ethanol, pure water and water solution of a surfactant (sodium dodecyl sulfate) with impact velocity of 7.5 m/s and diameters ranged from 0.19 to 0.46 mm were investigated.     

Numerical Simulation of Liquid Film Evaporation in Circular and Square Microcavities

Numerical simulations are performed for liquid film evaporation in circular and square microcavities, which occurs frequently in ink-jet fabrication. The conservation equations of mass, momentum, energy, and mass fraction in the liquid and gas phases are solved using a sharp-interface level-set method, which is modified to include the effects of evaporation and dynamic contact angles. Three-dimensional computations for a square cavity are efficiently carried out in a reduced domain by introducing an effective mass transfer length for the truncated region. The liquid film evaporation pattern is observed to depend strongly on the dynamic contact angles and cavity geometry.     

蒸发液滴内部非稳态流动的数值计算

在气液两相流VOF(volume of fluid,VOF)模型的基础上耦合CSF(continuum surface force,CSF)表面张力模型,建立了高温平板上的铺展液滴与高温空气中悬浮液滴蒸发过程中内部非稳态流动模型,对液滴蒸发过程中内部非稳态流动进行了研究。基于相变理论,采用用户自定义函数将流体相变模型加入非稳态流动模型中进行耦合计算,获得了高温平板上的铺展液滴与高温空气中悬浮液滴蒸发过程中的内部流动及变化过程。液滴蒸发过程中非稳态内部流动由液滴表面的温度梯度引发,Marangoni流动在液滴内部形成的时间非常短,流体从液滴表面高温区域流向低温区域。计算结果表明:高温平板上随着液滴蒸发的进行,液滴内部一直保持两个对称的涡流,Marangoni流动比较稳定;高温空气环境中随着液滴蒸发的进行,液滴内部四个涡流逐渐转变成两个对称的涡流;液滴内部温度分布因Marangoni流动加强传热而变得均匀,同时由于温度分布变得均匀,Marangoni流动被削弱。     

Simulation of mass transfer from an oscillating microdroplet

Mass transfer from micrometer and sub-micrometer airborne microdroplets arises in various chemical process, material science, and atmospheric phenomena, such as impinging flow reactor and unsteady pollutant diffusion, where microdroplet oscillation can substantially increase the mass transfer rate. Previous theories, however, do not adequately predict this enhancement of mass transfer, especially in the case of relatively large-amplitude oscillations. We have analyzed slow evaporation of an oscillating microdroplet having a sufficiently low vapor pressure such that it remains at the surrounding gas temperature and has a negligibly small rate of change of diameter. We solved the governing convective diffusion equation numerically to obtain the Sherwood number as a function of the system parameters. These include the oscillation frequency, the maximum velocity, and the initial microdroplet diameter. The theoretical results are compared with mass transfer data from the literature for a dodecanol microdroplet levitated in an electrodynamic balance (EDB) and oscillated by varying the dc levitation voltage and the ac amplitude and frequency. The predicted Sherwood numbers agree with the experimental results with a mean deviation of 9.2%. The analysis shows a distinct periodic change in the mass transfer rate or Sherwood number with a period that is one-half the period of oscillation of the microdroplet.     

Numerical analysis of the interactions between evaporating droplets in a monodisperse stream

A numerical simulation of evaporation in a monodisperse droplet stream is proposed, taking into account the transient state of the evaporation, and the non-uniform mass and heat transfer coefficients on the droplet surface. These investigations emphasize the strong interaction effects between closely spaced droplets in a dense spray, reducing significantly the transfer coefficients. Moreover, the Marangoni force becomes more significant than the viscous force, driving the internal motion of the droplet and affecting the temperature fields. Otherwise, a better understanding of the evaporation phenomenon around closely spaced droplets will help to refine the existing models used in dense sprays.     

A level-set based adaptive-grid method for premixed combustion

A numerical method to simulate premixed combustion is analyzed. It consists of a Cartesian cut-cell flow solver for compressible viscous flows coupled with a level-set method which solves the G-equation to describe the kinematics of the premixed flame. The coupling of the two solvers is achieved via a dual hierarchical dynamic adaptive-mesh framework. Both solvers operate on different Cartesian hierarchical meshes sharing a common background grid level through which they are connected. For the flow solver, feature- and G-based adaptive mesh refinement is taken advantage of, while a uniform high-resolution grid is used for the level-set solver. The heat release due to combustion is described by a source-term formulation by which the reaction rate profile of the premixed flame can be attached to the flame front, the motion of which is governed by the G-equation. A flame–vortex interaction problem is discussed in detail to validate the proposed methodology and to demonstrate the benefits of solution-adaptive mesh refinement in the context of the level-set approach for premixed combustion. After a forced laminar Bunsen flame is considered as an example for attached flames, the coalescence of two spherical flame kernels is simulated to assess the performance of the method and the potential savings in terms of computational costs for three-dimensional problems. The results of the test problems show the artificial thickening of the flame and numerical errors in the level-set solution on coarser grids to possess a comparatively small impact on the overall accuracy. The best findings in the sense of efficiency and physical quality are achieved by the combined feature-/G-based adaptation method.