Front Tracking Method Research Articles

Self-driven transport and heat transfer characteristics of a droplet on a wettability-gradient surface by front-tracking method

Droplet manipulation is a multidisciplinary field with broad applications across various industries. It holds significant potential in areas such as microfluidics, oil–water separation, water harvesting, and heat transfer. However, there is still a lack of comprehensive knowledge regarding droplet migration on restricted surfaces. In this study, we conducted a numerical simulation using the front-tracking method to investigate the heat transfer associated with droplet migration on a cold plate with a wettability gradient. We examined the effects of relative temperature differences, surface wettability, low initial impact velocities (We≤10), and wettability constraints (the width of the wettability stripe capable of driving droplet movement) on various droplet-related heat transfer characteristics and the resulting temperature field distribution. Our key findings indicate that as the temperature difference between the droplet and the surface increases, the heat flux experienced by the droplet after deposition also increases. Additionally, the decline in the heat flux curve during the descending phase becomes more significant. The surface contact angle plays a crucial role in the heat transfer dynamics during droplet migration. Droplets reach thermal equilibrium more quickly on hydrophilic surfaces with smaller contact angles. Higher initial impact velocities initially cause droplets to rebound on the surface, leading to more pronounced fluctuations in transient heat flux during the impact phase. However, as droplets transition from the rebound phase to the migration phase, the impact velocity's influence diminishes. Additionally, the restricted wettability (W*) affects the droplet-surface heat transfer through variations in the wetting area. We observed a fourfold difference in the relative wetting area between W*=0.4 and W*=2.5 in the final stage.

A coupled local front reconstruction and immersed boundary method for simulating 3D multiphase flows with contact line dynamics in complex geometries

Multi-phase flows in the presence of complex solid geometries are encountered widely throughout industrial processes to intensify mass and heat transfer processes. The efficiency of these processes relies largely on the predominant fluid dynamics regime inside the reaction vessel (e.g. trickle bed, bubble column reactor). The prevailing flow structure is influenced by many factors including the nature of the fluid-solid interactions (i.e. wetting or non-wetting). Such flows are often studied using front-capturing techniques to capture the fluid-fluid interface in combination with an auxiliary model to account for fluid-solid interactions. However, these front-capturing methods have difficulties in accurately representing the interface. Therefore, we adopted a front-tracking method: the Local Front Reconstruction Method (LFRM). In this study, LFRM is coupled with the Immersed Boundary method to incorporate the no-slip boundary condition at the solid surface. The model performance is assessed using droplet spreading simulations, where the equilibrium droplet shape (i.e. radius, height, and outline), the interfacial pressure difference, and the surface tension force are compared to analytical solutions and literature data. The results show an excellent match with the analytical solutions, and additionally superior performance to state-of-the-art volume tracking models.

Directional self-migration of droplets on an inclined surface driven by wettability gradient

In the current study, the anti-gravity directional self-migration of droplets on an inclined surface driven by wettability gradient (ω) was investigated using a front-tracking method. A unified mechanical model of droplet motion on an inclined wettability gradient wall was derived, considering the driving force generated by ω (Fd), gravity (G), and flow resistance (Ff). The model demonstrates that ω, G, and inclination angle (α) are key parameters affecting droplet motion. By varying ω, Bond number (Bo), and α, the droplet dynamic characteristics were analyzed, and a real-time Capillary number (Ca) was introduced to measure the droplet migration speed. The results indicate that a larger ω generates a greater Fd, leading to faster migration and more pronounced spreading. When the ratio of the channel width to the droplet diameter is 0.7, the droplet can cross three regions, obtaining double Fd, and Ca curve exhibits a bimodal structure. When the ratio of the channel width to the droplet diameter is 1.2, the droplet slides and spreads in the middle region without ω, resulting in a trimodal Ca curve. A larger Bo implies a stronger gravity effect, reducing the net driving force for upward migration and slowing the migration speed. At α=30° and ω=0.54, Bo reaches its critical value at 0.5, where G exceeds Fd, causing the droplet to slide downward along the wall. α affects droplet motion by controlling the gravitational component along the wall (Gx). A larger α results in a smaller net driving force for upward migration, reducing the migration speed.

Modeling and self-supporting printing simulation of fuse filament fabrication

This study presented a comprehensive computational fluid dynamics-based model for fused filament fabrication (FFF) three-dimensional (3D) printing multiphase and multiphysics coupling. A model based on the framework of computational fluid dynamics was built, utilizing the front-tracking method for high precision of multiphase material interfaces, a fully resolved simulation at the mesoscale explores the underlying physical mechanism of the self-supported horizontal printing. The study investigated the influence of printing temperature and velocity on the FFF process, exhibiting a certain self-supporting forming ability over a specific range. The results indicated that during the printing of large-span horizontal extension structures, the bridge deck material transitions from initial straight extension to sagging deformation, ultimately adopting a curved shape. The straight extension distance is inversely proportional to the depth of the sagging deformation. Additionally, the study revealed that printing temperature primarily affected the curing time of the molten material, while printing velocity fundamentally affected the relaxation time of both thermal and dynamic characteristics of the material.

Does dispersed phase inertia affect the shape of sheared emulsion droplets?

Inertial effects on sheared emulsion droplets are a topic of scientific and industrial interest for several applications from processing to microfluidics. Most of the literature have addressed so far the role of inertia of the continuous phase, which is known to affect shear-induced droplet deformation and migration at values of the Reynolds number of the external fluid Rec > 1. However, less attention has been paid to the case of inertial effects inside the droplets, corresponding to values of the Reynolds number of the droplet fluid Red > 1. Such a case is especially relevant when the viscosity ratio λ between the droplet and the external fluid is ≪ 1, which is typical of water-in-oil emulsions where the low values of droplet viscosity can result in Red > 1, while Rec < 1 due to the larger oil viscosity. Here, we focus on the effect of droplet inertia under shear flow at λ ≪ 1 by high-speed video microscopy experiments in a microcapillary and by numerical simulations based on a front-tracking finite-difference method. The results unveil the droplet's three-dimensional shape under shear flow at low viscosity ratios and show that droplet inertia tends to increase droplet deformation and orientation along the flow direction and to form two vortices inside the droplets even at small Rec. The latter findings are at variance with the case of external fluid inertia, where droplets become more aligned with the velocity gradient direction.

Simulation of group of droplets evaporation

Droplet evaporation occurs in many natural phenomena and industrial processes; hence, several studies have been conducted on droplet evaporation. In many applications, a group of droplets evaporate and the interaction between them affects the evaporation process. In this paper, the front-tracking method is used to simulate the droplets groups evaporation. Since the front-tracking method uses a Lagrangian grid for each droplet, this method offers good accuracy in predicting the shape, the displacement and the evaporation rate of droplets. The numerical method has been developed to simulate the evaporation of binary droplets. The fluid surrounding the droplets is modeled as a gas mixture, so the numerical method can be used to simulate multiphase-multicomponent problems. The front-tracking method requires very fine grid resolution to simulate flows at high-density ratios; therefore, the method is rarely used at high-density ratios. In this paper, a two-step method is used to move the front at high-density ratios without requiring a very fine grid resolution. First, a static droplet evaporation is simulated, and the results are compared with the analytical solutions; evaporation of a Decane droplet is then simulated, and the results are compared with the experimental data. Subsequently, the evaporation of a binary droplet is modeled. The evaporation of a group of static droplets is also simulated, and the effect of droplets interaction is investigated. Next, the evaporation of three injected droplets is simulated, and the effect of some parameters on droplets interaction is probed. The evaporation rate and displacement of each droplet are calculated and compared with the single droplet. Finally, the evaporation of the groups of droplets is simulated, and the effect of different arrangements of droplets on the evaporation rate is studied. Understanding the droplets interactions is helpful in predicting droplet spray behavior and developing numerical methods. Thus, the presented results are useful to achieve a better understanding of the droplets interaction phenomenon, its outcomes, and the parameters affecting evaporation rate.

An Edge-based Interface Tracking (EBIT) method for multiphase flows with phase change

In this paper, the Edge-Based Interface Tracking (EBIT) method is extended to simulate multiphase flows with phase change. As a novel Front-Tracking method, the EBIT method binds interfacial markers to the Eulerian grid to achieve automatic parallelization. To include phase change effects, the energy equation for each phase is solved, with the temperature boundary condition at the interface sharply imposed. When using collocated grids, the cell-centered velocity is approximately projected. This will lead to unphysical oscillations in the presence of phase change, as the velocity will be discontinuous across the interface. It is demonstrated that this issue can be addressed by using the ghost fluid method, in which the ghost velocity is set according to the jump condition, thereby removing the discontinuity. A series of benchmark tests are performed to validate the present method. It is shown that the numerical results agree very well with the theoretical solutions and the experimental results.

Dynamic characteristics of droplets impacting an inclined wall based on a front-tracking method

In the current study, the dynamic characteristics of droplets impacting a hydrophilic/hydrophobic inclined wall are investigated using the front-tracking method. The morphological and kinematic features of the droplets are examined across various contact angles (θ), low Weber numbers (We≤10), and inclination angles (α). A sequence of events, including deposition, spreading, contraction, rebound, and subsequent re-spreading, is observed when a droplet impacts a hydrophobic wall, according to the results. The droplet exhibits both a small dimensionless spreading area (A*) and significant fluctuation, along with a noticeable slip on the wall. As We increases, the spread and rebound effects become more pronounced, and the slipping velocity also increases. As α increases, the slipping velocity becomes greater, and both A* and the dimensionless tangential spreading diameter (β) decrease. When a droplet impacts a hydrophilic wall, continuous deposition, spreading, and a subtle slip with inconspicuous rebound are observed. As We increases, A* expands at an accelerating rate. Furthermore, the wetting effect of hydrophilic surfaces on droplets cannot be overlooked. Consequently, it also shows that as α increases, there is a larger diffusion range along the wall with a longer diffusion film and a larger β. Such behaviors stand in contrast to a droplet impacting a hydrophobic wall.

Direct Numerical Simulation of single bubble dynamics in nucleate pool boiling with micro-region modeling and thermal coupling to a solid wall

In this study, we conduct two-dimensional, axisymmetric simulations to investigate the growth and departure of a single bubble, with a particular focus on the conjugate heat transfer between the fluid and the heating wall. A multiscale modeling approach is employed to account for nanoscale effects near the liquid-vapor-solid triple contact line (CL). At the macro (bubble size) scale, the interface dynamics is captured with the front-tracking method by using the open-source code TRUST/TrioCFD. The macro scale algorithm is coupled with a sub-grid micro-region model. It is driven by the wall superheating at the CL as input, and predicts the macroscopic apparent contact angle and heat fluxes. To validate our modeling approach, we first conduct quantitative comparisons using the data on the bubble growth experiment RUBI and its simulations. Next, an investigation of the case of water under atmospheric pressure and gravity is performed. Notably, we observe a significant spatial variation in temperature near the nucleation site during the expansion and contraction of the bubble base. This variation greatly affects the thermal boundary layers in the fluid and solid domains and the bubble growth, demonstrating the need to resolve the conjugate heat transfer in the considered cases.

An inverse front tracking method for modeling solidification process in a smelting furnace

An inverse process based on the conjugate gradient and adjoint problem is developed for predicting the time-varying bank thickness during the solidification/melting process in a smelting furnace. The direct problem rest on a single-phase moving boundary model to simulate the solid bank and refractory brick, simultaneously. The time-varying heat flux at solid/liquid interface, initial bank thickness and time-varying heat transfer coefficient at outer surface of the furnace are considered as the unknowns. Unlike the methods developed so far based on the enthalpy model, the current method does not require any prior information regarding the thermophysical properties of the liquid-phase and complex physical phenomena that occur in it. This improves the uniqueness of the solution that is inherent in the ill-posed inverse methods. To avoid the so-called “inverse crime”, a two-phase enthalpy method is used to generate the simulated measurement. These measurements are used in the present single-phase method for reconstruction of bank history. The model has shown good performance for non-intrusive temperature obtained from the outer surface of the furnace. The inverse model has been validated through an experiment conducted in a metallurgical reactor. A reasonable agreement between the estimated and measured molten salt bank indicates the existence and convergence of the inverse solution.

An edge-based interface tracking (EBIT) method for multiphase-flow simulation with surface tension

We present a novel Front-Tracking method, the Edge-Based Interface Tracking (EBIT) method for multiphase flow simulations. In the EBIT method, the markers are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers. This feature makes the process of marker addition or removal easier than in the traditional Front-Tracking method. The EBIT method also allows almost automatic parallelization due to the lack of explicit connectivity.In a previous journal article we have presented the kinematic part of the EBIT method, that includes the algorithms for piecewise linear reconstruction and advection of the interface. Here, we complete the presentation of the EBIT method and combine the kinematic algorithm with a Navier–Stokes solver. A circle fit is now implemented to improve the accuracy of mass conservation in the reconstruction phase. Furthermore, to identify the reference phase and to distinguish ambiguous topological configurations, we introduce a new feature: the Color Vertex. For the coupling with the Navier–Stokes equations, we first calculate volume fractions from the position of the markers and the Color Vertex, then viscosity and density fields from the computed volume fractions and finally surface tension stresses with the Height-Function method. In addition, an automatic topology change algorithm is implemented into the EBIT method, making it possible the simulation of more complex flows. The two-dimensional version of the EBIT method has been implemented in the free Basilisk platform, and validated with seven standard test cases: stagnation flow, translation with uniform velocity, single vortex, Zalesak's disk, capillary wave, Rayleigh-Taylor instability and rising bubble. The results are compared with those obtained with the Volume-of-Fluid (VOF) method already implemented in Basilisk.

A random free-boundary diffusive logistic differential model: Numerical analysis, computing and simulation

A free boundary diffusive logistic model finds application in many different fields from biological invasion to wildfire propagation. However, many of these processes show a random nature and contain uncertainties in the parameters. In this paper we extend the diffusive logistic model with unknown moving front to the random scenario by assuming that the involved parameters have a finite degree of randomness. The resulting mathematical model becomes a random free boundary partial differential problem and it is addressed numerically combining the finite difference method with two approaches for the treatment of the moving front. Firstly, we propose a front-fixing transformation, reshaping the original random free boundary domain into a fixed deterministic one. A second approach is using the front-tracking method to capture the evolution of the moving front adapted to the random framework. Statistical moments of the approximating solution stochastic process and the stochastic moving boundary solution are calculated by the Monte Carlo technique. Qualitative numerical analysis establishes the stability and positivity conditions. Numerical examples are provided to compare both approaches, study the spreading-vanishing dichotomy, prove qualitative properties of the schemes and show the numerical convergence.

Numerical Investigation of Gas Bubble Interaction in a Circular Cross-Section Channel in Shear Flow

This work’s goal is to numerically investigate the interactions between two gas bubbles in a fluid flow in a circular cross-section channel, both in the presence and in the absence of gravitational forces, with several Reynolds and Weber numbers. The first bubble is placed at the center of the channel, while the second is near the wall. Their positions are set in such a way that a dynamic interaction is expected to occur due to their velocity differences. A finite element numerical tool is utilized to solve the incompressible Navier–Stokes equations and simulate two-phase flow using an unfitted mesh to represent the fluid interface, akin to the front-tracking method. The results show that the velocity gradient influences bubble shapes near the wall. Moreover, lower viscosity and surface tension force account for more significant interactions, both in the bubble shape and in the trajectory. In this scenario, it can be observed that one bubble is trapped in the other’s wake, with the proximity possibly allowing the onset of coalescence. The results obtained contribute to a deeper understanding of two-phase inner flows.

A Coupled Machine Learning and Lattice Boltzmann Method Approach for Immiscible Two-Phase Flows

An innovative coupling numerical algorithm is proposed in the current paper, the front-tracking method–lattice Boltzmann method–machine learning (FTM-LBM-ML) method, to precisely capture fluid flow phase interfaces at the mesoscale and accurately simulate dynamic processes. This method combines the distinctive abilities of the FTM to accurately capture phase interfaces and the advantages of the LBM for easy handling of mesoscopic multi-component flow fields. Taking a single vacuole rising as an example, the input and output sets of the machine learning model are constructed using the FTM’s flow field, such as the velocity and position data from phase interface markers. Such datasets are used to train the Bayesian-Regularized Back Propagation Neural Network (BRBPNN) machine learning model to establish the corresponding relationship between the phase interface velocity and the position. Finally, the trained BRBPNN neural network is utilized within the multi-relaxation LBM pseudo potential model flow field to predict the phase interface position, which is compared with the FTM simulation. It was observed that the BRBPNN-predicted interface within the LBM exhibits a high degree of consistency with the FTM-predicted interface position, showing that the BRBPNN model is feasible and satisfies the accuracy requirements of the FT-LB coupling model.

Mesoscale Modeling of Extrusion and Solidification During Material Extrusion Additive Manufacturing

In this work, we apply a multiphysics approach to fused deposition modeling to simulate extrusion and solidification. Restricting the work to a single line scan, we focus on the application of polylactic acid. In addition to heat, momentum and mass transfer, the solid/liquid/vapor interface is simulated using a front-tracking, level-set method. The results focus on the evolving temperature, viscosity, and volume fraction and are cast within a set of parametric studies, to include the printing and extrusion speed, as well as the extrusion temperature. Among other findings, it was observed that fused deposition modeling can be effectively modeled using a front-tracking method (i.e. the level set method) in concert with a temperature dependent porosity function. The use of the level-set method for discriminating the phase change interface in this context is relatively new and offers considerable advantages over existing methods.

Acoustic black hole designs for circular beams under axial loading

Acoustic Black Hole (ABH), as a non-reflecting wave effect, has been realised in beams and plates by locally changing their thickness. Geometrical ABH designs stem from their transverse-load bearing properties, assuming isotropic material properties associated with engineering materials (e.g., steels). The underlying physics is to manipulate stiffness locally for a transversely-loaded structure, leading to a change in the elastic wave speed. However, the same ABH would have limitations for axial loading, i.e., slender beams for which longitudinal waves dominate. Here, we present a wave propagation approach using wavefront tracking to identify a potential ABH for axially-loaded circular beams. We study wave propagation in three exponential designs of finite length, monitoring the wave-travel time. Indicative of an effective ABH in finite length sections, the wave-travel time increases compared to the case of a beam without ABH. By employing the wave front tracking method for the design of an ABH with axial loading, it is possible to verify the effectiveness of ABHs. Also, various material models, e.g., orthotropic materials such as wood, and different loading conditions can be considered, which opens a new avenue in applications of ABH phenomena beyond conventional vibro-acoustic control problems.

Ferrofluid drops-based actuator in a narrow gap

Directional transport of a liquid is an important issue in microfluidic systems and application purpose. Here, through combining the ideas of pressure-driven gas bubble-induced acoustic streaming flow and magnetic field-deformed ferrofluid drop, we study the ambient flow induced by an oscillating ferrofluid drop as an in situ actuator in a millimeter-sized gap environment. A drop squeezed by two parallel glass sheets, under the influence of a magnetic field, is discovered to undergo multimodal oscillations. The particle image velocimetry technique helps us to reveal the vortex-typed flow structure surrounding the oscillating drop. The shape changes of drop are found including the circular, elliptical, triangular, inverse-triangular, and circular shapes. We employ the numerical front-tracking method and analytical mixed-mode model to elucidate a drop-driven flow. We find that the pulsating, translational, and quadrupole mode oscillation of the drop is capable to describe most features of the flow distribution. Furthermore, we demonstrate an in situ pump by applying a spatially non-uniform pulsating magnetic field onto the arranged ferrofluid drops. The ferrofluid drop-based in situ pump shows the ability to produce a flow rate of 108 μl/min, which should be a great help in microfluidic pumping.

A front-tracking method for simulating interfacial flows with particles and soluble surfactants

In this research, a three-dimensional model is proposed to simulate the behavior of particle-covered droplets that contain soluble surfactant. The model includes a full two-way coupling between surfactant transport and particle motion, with surfactant concentrations at the interface and in the bulk linked to the flow and motion of the interface by a convection-diffusion equation. A front-tracking method is used to track the interfacial flow, while a semi-resolved particle method is used to model the particles, accounting for various forces acting on them, such as the excluded volume force, the direct contact force and the hydrodynamic lubrication force. The model also includes a surfactant-dependent surface tension that can affect the particle-interface interactions. The model is validated against previous numerical data and a parametric study is performed to examine the delicate interplay between flow fields and interfaces associated with surfactant concentration and particle motion. The study then investigates the role of surfactants in the dynamics of a particle-covered droplet, focusing on their coupled influence on droplet deformation. In addition, it is observed that particles bridge colliding droplet interfaces and suppress fluid drainage from the gap, preventing droplet coalescence along with the interparticle repulsion force.

An unstructured finite-volume level set / front tracking method for two-phase flows with large density-ratios

We extend the unstructured LEvel set / froNT tracking (LENT) method [1,2] for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4–8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed ρLENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ρ−/ρ+∈[1,104]. Numerical stability is demonstrated for in terms of the relative L∞ velocity error norm, for density-ratios in the range of [1,104], dynamic viscosity-ratios in the range of [1,104] and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed ρLENT method1 is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm.

Efficient reduction of vertex clustering using front tracking with surface normal propagation restriction

A significant computational expense and source of numerical errors in front tracking is the remeshing of the triangulated front, required due to distortion and compaction of the front following the Lagrangian advection of its vertices. Additionally, in classic front tracking, the remeshing of the front mesh is required not only due to the deformation of the front shape, but also because the vertices of the front are translated in the direction tangential to the front, induced by the front advection. We present the normal-only advection (NOA) front-tracking method with the aim of preventing the tangential motion of the front vertices and the associated vertex clustering, in order to reduce the number of remeshing operations required to retain a high-quality triangulated interface. To this end, we reformulate the velocity used to advect the front at each discrete front-vertex position. The proposed method is validated and tested against the classic front-tracking method, comparing volume conservation, shape preservation, computational costs, and the overall need for front remeshing, as well as experimental results for canonical interfacial flows. The presented results demonstrate that the NOA front-tracking method leads to a typical reduction of remeshing operations by 80% or more compared to the classic front-tracking method for well-resolved cases, and results in a smoother front mesh, which is essential for an accurate representation of the geometrical properties of the front. The volume conservation error is reduced by approximately one order of magnitude with the proposed method compared to the classic front-tracking method, at a similar computational cost.