A coupled Volume of Fluid and Immersed Boundary Method for simulating 3D multiphase flows with contact line dynamics in complex geometries

A comprehensive numerical and experimental investigation on micrometer-sized water droplet impact dynamics and evaporation on an unheated, flat, dry surface is conducted from the standpoint of spray-cooling technology. The axisymmetric time-dependent governing equations of continuity, momentum, energy, and species are solved. Surface tension, wall adhesion effect, gravitational body force, contact line dynamics, and evaporation are accounted for in the governing equations. The explicit volume of fluid (VOF) model with dynamic meshing and variable-time stepping in serial and parallel processors is used to capture the time-dependent liquid-gas interface motion throughout the computational domain. The numerical model includes temperature- and species-dependent thermodynamic and transport properties. The contact line dynamics and the evaporation rate are predicted using Blake's and Schrage's molecular kinetic models, respectively. An extensive grid independence study was conducted. Droplet impingement and evaporation data are acquired with a standard dispensing/imaging system and high-speed photography. The numerical results are compared with measurements reported in the literature for millimeter-size droplets and with current microdroplet experiments in terms of instantaneous droplet shape and temporal spread (R/D(0) or R/R(E)), flatness ratio (H/D(0)), and height (H/H(E)) profiles, as well as temporal volume (inverted A) profile. The Weber numbers (We) for impinging droplets vary from 1.4 to 35.2 at nearly constant Ohnesorge number (Oh) of approximately 0.025-0.029. Both numerical and experimental results show that there is air bubble entrapment due to impingement. Numerical results indicate that Blake's formulation provides better results than the static (SCA) and dynamic contact angle (DCA) approach in terms of temporal evolution of R/D(0) and H/D(0) (especially at the initial stages of spreading) and equilibrium flatness ratio (H(E)/D(0)). Blake's contact line dynamics is dependent on the wetting parameter (K(W)). Both numerical and experimental results suggest that at 4.5 < We < 11.0 the short-time dynamics of microdroplet impingement corresponds to a transition regime between two different spreading regimes (i.e., for We < or = 4.5, impingement is followed by spreading, then contact line pinning and then inertial oscillations, and for We > or = 11.0, impingement is followed by spreading, then recoiling, then contact line pinning and then inertial oscillations). Droplet evaporation can be satisfactorily modeled using the Schrage model, since it predicts both well-defined transient and quasi-steady evaporation stages. The model compares well with measurements in terms of flatness ratio (H/H(E)) before depinning occurs. Toroidal vortices are formed on the droplet surface in the gaseous phase due to buoyancy-induced Rayleigh-Taylor instability that enhances convection.

Evaporation of fluid from a pinned drop placed on solid surface proceeds via constant contact radius (CCR) mode, with a continuous reduction in the contact angle. The reduction of contact angle leads to an imbalance of interfacial tensions at the three-phase contact line. When the unbalanced force is sufficiently strong, the drop slips from the pinned contact line and slides inward. Depinning of the drop alters the mode of evaporation to constant contact angle (CCA) mode till it repins onto the surface. The change in evaporation mode from CCR to CCA is usually achieved by tuning the pinning energy barrier by controlling the surface properties of the substrate. Here, we demonstrate that the evaporation mode can be controlled by solely tailoring the surface tension of the drop, which is achieved in microgel particle-laden sessile drops that show spontaneous adsorption of microgels to the air/water interface, leading to a decrease in the interfacial tension. We show that droplets containing a sufficient number of microgels evaporate predominantly in CCR mode even on a hydrophobic surface, and the contact line remains pinned throughout the evaporation of the drop. Interestingly, the contact line dynamics can be controlled by tuning the softness of the microgels and the particle concentration in the drops.

When boundary conditions arising from the usual hydrodynamic assumptions are applied, analyses of dynamic wetting processes lead to a well-known nonintegrable stress singularity at the dynamic contact line, necessitating new ways to model this problem. In this paper, numerical simulations for a set of representative problems are used to explore the possibility of providing material boundary conditions for predictive models of inertialess moving contact line processes. The calculations reveal that up to Capillary number Ca=0.15, the velocity along an arc of radius 10Li (Li is an inner, microscopic length scale) from the dynamic contact line is independent of the macroscopic length scale a for a&amp;gt;103Li, and compares well to the leading order analytical “modulated-wedge” flow field [R. G. Cox, J. Fluid Mech. 168, 169 (1986)] for Capillary number Ca&amp;lt;0.1. Systematic deviations between the numerical and analytical velocity field occur for 0.1&amp;lt;Ca&amp;lt;0.15, caused by the inadequacy of the leading order analytical solution over this range of Ca. Meniscus shapes produced from calculations in a truncated domain, where the modulated-wedge velocity field [R. G. Cox, J. Fluid Mech. 168, 169 (1986)] is used as a boundary condition along an arc of radius R=10−2a from the dynamic contact line, agree well with those using two inner slip models for Ca&amp;lt;0.1, with a breakdown at higher Ca. Computations in a cylindrical geometry reveal the role of azimuthal curvature effects on velocity profiles in the vicinity of dynamic contact lines. These calculations show that over an appropriate range of Ca, the velocity field and the meniscus slope in a geometry-independent region can potentially serve as material boundary conditions for models of processes containing dynamic contact lines.

Dynamic wetting characteristics during droplet formation in a microfluidic T-junction

Simulation of moving contact lines in two-phase polymeric fluids

Deposition of colloidal material in evaporating droplets is important in many applications including DNA sequencing and medical diagnostic testing. When colloidal droplets evaporate, the majority of material is often deposited at the periphery of the resultant deposition in a coffee-ring pattern. Formation of this pattern is the result of contact line pinning and the interplay between evaporative and surface tension effects in the droplet. When the contact line is pinned and the evaporative flux in the droplet is highest at the periphery, a radially outward flow is generated to conserve mass that deposits particles in the fluid at the contact line. Evaporation at the contact line can also create a temperature gradient across the droplet. This gradient gives rise to a surface tension driven flow that can resuspend particles in the droplet. When the evaporative flow dominates, particles are deposited at the contact line in a coffee-ring pattern. The presence of the coffee-ring pattern is undesirable in many printing and medical diagnostic processes. Suppression of the coffee-ring effect has been achieved by addition of surfactant, enhancement of surface tension flow, surface modification, alteration of particle shape, and application of an electric field. Manipulation of the coffee-ring effect has been achieved through the application of both AC and DC electric fields. One result of the presence of this field is the electrowetting force at the contact line which acts to reduce the contact angle and increase contact area. Since this force acts at the contact line, it may disrupt typical contact line dynamics, including evaporative dynamics, which are responsible for the formation of the coffee-ring effect. This work will experimentally examine contact line dynamics of evaporating droplets in the presence of DC electric fields. Droplets of water will be desiccated on a device with a photolithographically patterned electrode covered with a thin layer of SU8-3005. Experimental cases with applied DC fields will be compared with unactuated control cases to examine changes in transient interface shape and contact diameter.

When a solid surface makes contact with a fluid-fluid interface, a three-phase contact line is formed. This contact line may move along the surface, driven by both the external fluid flow and the thermodynamically-determined contact angle that the fluid-fluid interface makes with the solid surface. Although the steady motion of contact lines has been studied extensively, the formation and unsteady motion of contact lines has not. The process of contact-line formation is of obvious interest in any physical situation where contact lines exist. In order to study the formation and motion of contact lines we have analysed a particular model problem: the interaction of an initially flat interface between two inviscid fluids with a thin, semi-infinite, flat plate. We extend the results obtained by Billingham and King (J. Fluid Mech 296 1995), where this problem was studied for the case of normal incidence, to incidence at an arbitrary initial angle.

본 연구에서는 규칙파동장에 있어서 인장계류된 원형부유체의 파랑응답해석에 2차원 Navier-Stokes solver에 기초한 새로운 수치파동수조모델을 제안하였다. 본 수치파동수조모델에서는 이동구조물과 유체와의 상호작용을 해석하기 위하여 직각좌표계에서 임의형상의 불투과경계를 갖는 구조물과 유체와의 연성해석이 가능한 IBM(Immersed Boundary Method)과 자유수면 추적을 위한 VOF(Volume of Fluid)법을 결합하였다. 부유체운동에 대한 수치결과를 기존의 FAVOR(Fractional Area Volume Obstacle Representation)법에 의한 계산결과 및 수리실험과 비교하였다. 게다가, 수치모델의 보다 자세한 검증을 위하여 원형부유체의 동요 및 자유수면변동에 관한 수리모형을 추가로 실시하였고, 제안된 수치모델의 범용성과 타당성을 검증하기 위하여 직사각형부유체에 대한 수치 및 수리실험도 병행하였다. 이로부터 추정된 수치계산결과는 실험결과를 잘 재현하고 있는 것으로 판단되었다. In the present study, we proposed a new numerical wave tank model to analyze the vertical tension-leg circular floating bodies, using a 2-D Navier-Stokes solver. An IBM(Immersed Boundary Method) capable of handling interactions between waves and moving structures with complex geometry on a standard regular Cartesian grid system is coupled to the VOF(Volume of Fluid) method for tracking the free surface. Present numerical results for the motions of the floating body were compared with existing experimental data as well as numerical results based on FAVOR(Fractional Area Volume Obstacle Representation) algorithm. For detailed examinations of the present model, the additional hydraulic experiments for floating motions and free surface transformations were conducted. Further, the versatility of the proposed numerical model was verified via the numerical and physical experiments for the general rectangular floating bodies. Numerical results were compared with experiments and good agreement was archived.

Dynamic contact angles in CFD simulations

This work investigates the onset of wetting failure for displacement of Newtonian fluids in parallel channels. A hydrodynamic model is developed for planar geometries where an advancing fluid displaces a receding fluid along a moving substrate. The model is evaluated with three distinct approaches: (i) the low-speed asymptotic theory of Cox [J. Fluid Mech. 168, 169–194 (1986)], (ii) a one-dimensional (1D) lubrication approach, and (iii) a two-dimensional (2D) flow model solved with the Galerkin finite element method (FEM). Approaches (ii) and (iii) predict the onset of wetting failure at a critical capillary number Cacrit, which coincides with a turning point in the steady-state solution family for a given set of system parameters. The 1D model fails to accurately describe interface shapes near the three-phase contact line when air is the receding fluid, producing large errors in estimates of Cacrit for these systems. Analysis of the 2D flow solution reveals that strong pressure gradients are needed to pump the receding fluid away from the contact line. A mechanism is proposed in which wetting failure results when capillary forces can no longer support the pressure gradients necessary to steadily displace the receding fluid. The effects of viscosity ratio, substrate wettability, and fluid inertia are then investigated through comparisons of Cacrit values and characteristics of the interface shape. Surprisingly, the low-speed asymptotic theory (i) matches trends computed from (iii) throughout the entire investigated parameter space. Furthermore, predictions of Cacrit from the 2D flow model compare favorably to values measured in experimental air-entrainment studies, supporting the proposed wetting-failure mechanism.

Numerical simulation of a square cavity was conducted to validate an implementation of an Immersed Boundary Method (IBM). The cavity consists of two differentially heated side walls and adiabatic top and bottom walls. A Cartesian grid is used with a finite volume, fractional step pressure correction method. Simulations use Dirichlet boundaries for vertical walls and Neumann boundaries for horizontal walls. The Immersed Boundary Method involves modifying the Navier--Stokes equations to include a forcing function in the momentum and energy equations that creates a virtual boundary. This method is useful because the boundary does not necessarily have to coincide with grid points; however, it is much less computationally expensive than other similar methods such as the cut cell method. The IBM is commonly used in simulations involving complex objects and can also accommodate moving boundaries. A standard numerical simulation with grid aligned with the boundary is first compared with previous results. The same geometry is then simulated by tilting the grid at various angles and using the IBM for each of the walls, and comparing these results with those initially obtained. We detail the implementation method and common problems associated with this. Velocity and temperature profiles are presented and the IBM is shown to maintain second order spatial accuracy. References E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161(1):35--60, 2000. doi:10.1006/jcph.2000.6484 T. Gao, Y. H. Tseng, and X. Y. Lu. An improved hybrid cartesian/immersed boundary method for fluid-solid flows. International Journal for Numerical Methods in Fluids, 55(12):1189--1211, 2007. doi:10.1002/fld.1522 M. P. Kirkpatrick and S. W. Armfield. Experimental and large eddy simulation results for the purging of salt water from a cavity by an overflow of fresh water. International Journal of Heat and Mass Transfer, 48(2):341--359, 2005. doi:10.1016/j.ijheatmasstransfer.2004.08.016 M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent. A representation of curved boundaries for the solution of the navier-stokes equations on a staggered three-dimensional cartesian grid. Journal of Computational Physics, 184(1):1--36, 2003. doi:10.1016/S0021-9991(02)00013-X B. P. Leonard and Simin Mokhtari. Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady-state simulation of convection. International Journal for Numerical Methods in Engineering, 30(4):729--766, 1990. doi:10.1002/nme.1620300412 S. H. Peng and L. Davidson. Large eddy simulation for turbulent buoyant flow in a confined cavity. volume 22, pages 323--331, 2001. doi:10.1016/S0142-727X(01)00095-9 C. S. Peskin. Numerical-analysis of blood-flow in heart. Journal of Computational Physics, 25(3):220--252, 1977. doi:10.1016/0021-9991(77)90100-0 J. Salat, S. Xin, P. Joubert, A. Sergent, F. Penot, and P. Le Quere. Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity. International Journal of Heat and Fluid Flow, 25:824--832, 2004. doi:10.1016/j.ijheatfluidflow.2004.04.003 Y. H. Tseng and J. H. Ferziger. A ghost-cell immersed boundary method for flow in complex geometry. Journal of Computational Physics, 192(2):593--623, 2003. doi:10.1016/j/jcp.2003.07.024 N. Zhang and Z. C. Zheng. An improved direct-forcing immersed-boundary method for finite difference applications. Journal of Computational Physics, 221(1):250--268, 2007. doi:10.1016/j.jcp.2006.06.012 N. Zhang, Z. C. Zheng, and S. Eckels. Study of heat-transfer on the surface of a circular cylinder in flow using an immersed-boundary method. International Journal of Heat and Fluid Flow, 29(6):1558--1566, 2008. doi:10.1016/j.ijheatfluidflow.2008.08.009

Spiral motifs are pervasive in nature, art, and technology due to their functional property of providing compact length. Nature is particularly adept at spiral patterning, and yet, the spirals observed in seashells, hurricanes, rams' horns, flower petals, etc. all evolve via disparate physical mechanisms. Here, we present a mechanism for the self-guided formation of spirals from evaporating saline drops via a coupling of crystallization and contact line dynamics. These patterns are in contrast to commonly observed patterns from evaporation of colloidal drops, which are discrete (rings, concentric rings) or continuous (clumps, uniform deposits) depending on the particle shape, contact line dynamics, and evaporation rate. Unlike the typical process of drop evaporation where the contact line moves radially inward, here, a thin film pinned by a ring of crystals ruptures radially outward. This motion is accompanied by a nonuniform pinning of the contact line due to crystallization, which generates a continuous propagation of pinning and depinning events to form a spiral. By comparing the relevant timescales of evaporation and diffusion, we show that a single dimensionless number can predict the occurrence of these patterns. These insights on self-guided crystallization of spirals could be used to create compact length templates.

Gas and water flooding are known as improved oil recovery method. Implementation of these IOR for volatile oil reservoir is essential to oil recovery and project economic. Different IOR schemes provide different mechanisms of flooding and pressure maintenance for volatile oil reservoirs. Hence, understanding of volatile oil behavior and each IOR performance will lead to a proper scheme selection. Compositional simulation study is performed in order to find the optimum way to recover volatile oil. Simulation of natural depletion, waterflooding, gasflooding, and water alternating gas are investigated and compared in aspect of oil recovery performance and project economic. From simulation results, we found that gravity segregation of solution gas is the most influential drive mechanism to volatile oil recovery of natural depletion. Hence, enhancement of this drive mechanism is the key to production optimization and completion design. For water and gas flooding, starting maintaining the reservoir pressure above the bubble point pressure and before free exists in the reservoir will maximize the recovery. For waterflooding, if free gas exists, displacement efficiency is reduced due to gas blockage effect. For gasflooding, existing of free gas is lower capability in miscibility condition achievement because the intermediate components in oil phase already flash to gas phase and cannot be used to develop miscible flood front. And when compared all studied production schemes, waterflooding can recover oil less than gasflooding, but can yield similar net profit due to lower investment cost required. While water alternating gas yields the highest total oil recovery and net profit because it combines the advantages of both water and gas flooding.

This work focuses on the numerical study of electrohydrodynamic multiphase fluid flow problems. Liquid bodies experience deformations caused by strong external electric fields. These deformations can be used in industrial applications to accurately control the motion of single droplets. In the limiting case, strong electric fields can force atomisation of fine microdroplets from much larger liquid bulks. Due to differences in the electrical properties of the different fluid phases, the flow affects in turn the electric field distribution. The analysis of such coupled flows requires numerical modelling of the interdependent hydrodynamic and electrodynamic problems. Since most liquids exhibit some conductivity due to intrinsic ionic species and dissolved impurities, the electrodynamic problem must be modelled by an electroquasistatic model taking into account capacitive, resistive and convective electrical currents. This electroquasistatic problem is coupled to an incompressible fluid flow problem described by the Navier-Stokes equations. Both problems are solved on the same computational grid, using the finite volume method. The fluid-fluid interface is modelled using the volume of fluid method. The resulting diffuse interface captures the motion of the phase boundaries while efficiently handling topology changes. The motion of contact lines is furthermore modelled using a dynamic contact angle model including hysteresis effects. Pinned contact lines and stick-slip contact line motion in transient problems can therefore be represented. The developed solver is readily applicable to a large range of electrohydrodynamic flow problems. This work investigates electrohydrodynamic flows occurring in three technical applications. First, the dynamics of sessile droplets subjected to an AC electric field on the surface of an insulator are considered. The dynamics of the oscillating droplets are compared with experimental data. Partial discharge inception fields are then estimated for similar configurations. Secondly, the detachment dynamics of two liquids in an on-demand droplet generator, where droplet detachment is enforced by electric fields, are considered and compared to experimental data. The different conductivities of the liquids are shown to lead to substantially different detachment dynamics. Relevant parameters in the detachment dynamics are extracted from the simulations. Finally, transient electrosprays in the cone-jet mode are characterised for a number of liquids with different electromechanical properties. The charge-radius correlations of the first ejections are found to obey the scaling laws reported in the literature. Moreover, additional scaling laws are found for the subsequent ejections.

We present an hybrid VOF/embedded boundary method allowing to model two-phase flows in presence of solids with arbitrary shapes. The method relies on the coupling of existing methods: a geometric Volume of fluid (VOF) method to tackle the two-phase flow and an embedded boundary method to sharply resolve arbitrary solid geometries. Coupling these approaches consistently is not trivial and we present in detail a quad/octree spatial discretization for solving the corresponding partial differential equations. Modelling contact angle dynamics is a complex physical and numerical problem. We present a Navier-slip boundary condition compatible with the present cut cell method, validated through a Taylor–Couette test case. To impose the boundary condition when the fluid–fluid interface intersects a solid surface, a geometrical contact angle approach is developed. Our method is validated for several test cases including the spreading of a droplet on a cylinder, and the equilibrium shape of a droplet on a flat or tilted plane in 2D and 3D. The temporal evolution and convergence of the droplet spreading on a flat plane is also discussed for the moving contact line given the boundary condition (Dirichlet or Navier) used. The ability of our numerical methodology to resolve contact line statics and dynamics for different solid geometries is thus demonstrated.