Front Tracking Method Research Articles

Coupling characteristics of bubbles with a free surface initially disturbed by water waves

The interactions between bubbles and water waves have important applications in ocean engineering, and their coupling characteristics are strongly associated with the wave phase angle, wavelength, and wave amplitude. Based on the assumption that the liquid is inviscid and incompressible, the coupling characteristics between bubbles and water waves are solved by the Euler equations with the finite volume method, and the bubble surface and water wave surface are tracked by the front tracking method. The accuracy of the numerical method is verified by comparison with a spark-generated bubble experiment. Compared with the bubble near the initially plane free surface, the rising height of the water spike is reduced by water waves in the crest state, where a concave shape forms on the falling water wave during bubble contraction when the wavelength λ≤ 4.00 and the wave amplitude h≥ 0.364. The rising height of the water spike is significantly strengthened by water waves in the trough state with smaller wavelengths and larger wave amplitudes, which produce a thinner and higher water spike. The bubble cycle is shortened by water waves in the crest state with smaller wavelengths and prolonged by water waves in the trough state with smaller wavelengths and larger wave amplitudes. The results presented in this paper provide guidance for the study of underwater explosions in complex water wave environments.

Evolution of mesh-like liquid films in multi-port lifted Hele Shaw cells

Multi-port Lifted Hele-Shaw Cells (MLHSCs) can be used to efficiently generate liquid films with regular mesh-like patterns, which in turn may be used as templates for micro-fabrication of mesh-like structures. This work systematically characterizes the spatio-temporal evolution of Newtonian liquid films in MLHSCs using experiments and numerical simulations. The interior holes of the mesh tend to assume circular shapes for low Capillary number, while they approach a square-like shape at high Capillary number. Fingering instability is observed at high Capillary numbers and higher inter-hole spacing. Capillary number can also significantly affect the shape of peripheral and corner holes in the mesh. Numerical simulations, based on front tracking method, are used to characterize the stability of interfaces in a periodic hole array. Data from numerical simulations is used to construct and validate a computationally efficient low-order model for predicting the evolution of mesh-like patterns formed in the film.

Thermocapillary droplet migration in a vertical temperature gradient controlled by thermal radiations

Thermocapillary migration of a droplet in a vertical temperature gradient controlled by uniform and non-uniform thermal radiations is theoretically analyzed and numerically investigated. A non-dimensionlized thermal radiation number is proposed to quantitatively depict the intensity ratio of the thermal radiation flux to the uniform temperature gradient. From the momentum and energy equations at zero limits of Reynolds and Marangoni numbers, analytical results for the uniform and non-uniform thermal radiations are determined. The steady migration velocity raises with the increasing of the thermal radiation number. By using the front-tracking method, it is observed that thermocapillary droplet migration under the uniform thermal radiation at moderate Marangoni and moderate thermal radiation numbers reaches a steady process. The steady migration velocity decreases with the increasing of Marangoni number and increases with the increasing of thermal radiation number. Moreover, the intensity of thermal energy transferred from the interface to both fluids depends on the volume heat capacity ratio. For the larger/smaller volume heat capacity ratio, more heat is transferred into the continuous phase fluid/the droplet. Furthermore, when the uniform thermal radiation is replaced by the non-uniform ones, the time evolutions, the structures of temperature fields, and parameter dependencies of thermocapillary droplet migration at moderate Marangoni and moderate thermal radiation numbers remain qualitatively unchanged. This study provides a profound understanding of thermocapillary droplet migration in a vertical temperature gradient controlled by thermal radiations, which is of great significance for practical applications in microgravity and microfluidic fields.

A front-tracking method for two-phase flow simulation with no spurious currents

Using current cell-centered numerical methods to calculate two-phase flows result in spurious currents developing at the interface between both phases when the capillary number is low enough. Treatments of the interface such as: computing surface tension more accurately, smoothing of the indicator function, and balancing the pressure and surface tension, are often used to reduce spurious currents. These treatments can affect the physics of the interface, and these approaches struggle when the density ratios are high or when the meshes are distorted. This paper presents a methodology that uses a moving unstructured staggered mesh that can handle extremely high-density ratios of at least 1,000,000,000 and extremely high surface tensions of at least 500 N/m without generating spurious currents. Instead of spurious currents, only physically correct low-amplitude and low energy capillary waves at the interface result from the errors in the surface discretization. The method is demonstrated on 2D static and dynamic droplet cases.

Mathematical Modeling and Associated Numerical Simulation of Fusion/Solidification Front Evolution in the Context of Severe Accident of Nuclear Power Engineering

Considering transient processes where liquid/solid phase change occurs, this paper focuses on the associated modeling and numerical treatment in the frame of “Computational Fluid Dynamics” simulations. While being of importance in many industrial applications involving solidification and melting of mixed materials, including power and manufacturing engineering, the first application of this work pertains to the analysis of severe accidents in a nuclear reactor. Indeed, in this context, the molten core materials (a.k.a. corium) can form a high-temperature multiphase liquid pool at the boundary of which fusion and solidification phenomena are of prime importance. In this context, even if materials at play are treated as pure components, it is mandatory to distinguish two different phase change temperatures with a solid fusion temperature and a liquid solidification temperature. Accordingly, in the frame of a sharp interface representation, the paper introduces non-classical heterogeneous conditions at the liquid/solid boundary in such a way that both moving interface (through Stefan conditions associated with fusion or solidification) and static interface (imposing heat flux continuity) are supported at the same time on different spatial locations along this boundary. Within a monolithic resolution of Navier–Stokes and heat conduction equations, this interface is explicitly tracked with combined Front-Tracking and VOF methods. In order to ensure zero velocity in the solid phase, an Immersed Boundary Method and a direct forcing penalization are also introduced. The main relevant features of this combination of numerical methods are discussed along with their implementation in the TrioCFD code taking advantage of the pre-existing code capabilities. Numerical simulations including both verification tests and a case of interest for our industrial application are reported and demonstrate the applicability of the proposed triptych model+methods+code to treat such problems. The numerical tools and the simulation code developed in this work could be used not only in the several accident context but also to simulate melting, solidification and fusion processes occurring in aerodynamics, hypersonic reentry vehicles and laser applications to cite but a few.

A machine learning strategy for computing interface curvature in Front-Tracking methods

In this work we have described the application of a machine learning strategy to compute the interface curvature in the context of a Front-Tracking framework. Based on angular information of normal and tangential vectors between marker points, the interface curvature is predicted using a neural network. The Front-Tracking-Machine-Learning method is validated using a sine wave and then applied in combination with a Marker-And-Cell method for solving a complex free surface flow. Our results indicate that it is feasible to employ machine learning concepts as an alternative approach for computing curvatures in Front-Tracking schemes.

Subgrid-scale fire front reconstruction for ensemble coupled atmosphere-fire simulations of the FireFlux I experiment

This paper introduces the Blaze fire model based on a Eulerian level-set front-tracking method and solved using high-order numerical schemes. Blaze includes an original and efficient subgrid-scale fire front reconstruction to substantially reduce computational cost and better localize surface heat fluxes compared to a weighted-averaged method. In this study, Blaze is coupled to the MesoNH atmospheric model to evaluate its performance against the FireFlux I experimental data set. Results show good agreement between simulations and measurements for both 25-m and 10-m atmospheric resolutions combined with a 5-m fire resolution. The fire-induced atmospheric flow below 10 m is correctly captured in the two-way coupled mode and leads to a realistic spread rate trend between the two instrumented towers compared to one-way forcing modes (forced and fire replay modes). A more realistic air temperature near the ground is obtained by considering heat fluxes in the already burnt area and not only at the flaming front. Also, the significant impact of inflow turbulence on both fire spread and fire-induced flow is highlighted. This study motivates the use of a statistical ensemble technique to account for near-surface turbulence and more generally, environmental variability at the scale of an experimental fire such as FireFlux I.

Multiphysics simulation of single pulse laser powder bed fusion: comparison of front capturing and front tracking methods

PurposeDuring thermal laser processes, heat transfer and fluid flow in the melt pool are primary driven by complex physical phenomena that take place at liquid/vapor interface. Hence, the choice and setting of front description methods must be done carefully. Therefore, the purpose of this paper is to investigate to what extent front description methods may bias physical representativeness of numerical models of laser powder bed fusion (LPBF) process at melt pool scale.Design/methodology/approachTwo multiphysical LPBF models are confronted: a Level-Set (LS) front capturing model based on a C++ code and a front tracking model, developed with COMSOL Multiphysics® and based on Arbitrary Lagrangian–Eulerian (ALE) method. To do so, two minimal test cases of increasing complexity are defined. They are simplified to the largest degree, but they integrate multiphysics phenomena that are still relevant to LPBF process.FindingsLS and ALE methods provide very similar descriptions of thermo-hydrodynamic phenomena that occur during LPBF, providing LS interface thickness is correctly calibrated and laser heat source is implemented with a modified continuum surface force formulation. With these calibrations, thermal predictions are identical. However, the velocity field in the LS model is systematically underestimated compared to the ALE approach, but the consequences on the predicted melt pool dimensions are minor.Originality/valueThis study fulfils the need for comprehensive methodology bases for modeling and calibrating multiphysical models of LPBF at melt pool scale. This paper also provides with reference data that may be used by any researcher willing to verify their own numerical method.

Three-phase solidification of a liquid compound droplet on a curved surface

A compound droplet solidifying on a cold curved surface in a gaseous environment is numerically studied by an axisymmetric three-phase front tracking method. The compound droplet containing an inner core of gas is initially assumed to be part of a sphere, and the curved surface is maintained at temperature below the solidification point of the shell liquid. The effects of the radius of the cold surface, volume change (in terms of the solid-to-liquid density ratio ρsl), supercooling degree (in terms of the Stefan number St), growth angle ϕgr and shell liquid thickness on the solidified droplet shape, the solidification rate and the solidification times are under consideration. Volume expansion (ρsl < 1.0) induces an apex, and volume shrinkage (ρsl > 1.0) results in a cavity on top of the outer surface while volume change has a minor effect on the inner surface of the solidified droplet. A more convex surface causes the solidification process to finish earlier, whereas a more concave surface prolongs the entire solidification time. However, the mean solidification rate, the tip angle at the droplet top and the additional height are not affected by the radius of the curved surface. The tip angle is also slightly affected by the outer shape of the initial droplet, the shell thickness and St while these parameters strongly affect the rate of solidification. It is found that the completion of solidification around the inner core is not affected by the growth angle, while it takes longer when increasing ρsl, the size of the inner core or decreasing the Stefan number.

Invasive behaviour under competition via a free boundary model: a numerical approach.

What will happen when two invasive species are competing and invading the environment at the same time? In this paper, we try to find all the possible scenarios in such a situation based on the diffusive Lotka-Volterra competition system with free boundaries. In a recent work, Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) considered a weak-strong competition case of this model (with spherical symmetry) and theoretically proved the existence of a "chase-and-run coexistence" phenomenon, for certain parameter ranges when the initial functions are chosen properly. Here we use a numerical approach to extend the theoretical research of Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) in several directions. Firstly, we examine how the longtime dynamics of the model changes as the initial functions are varied, and the simulation results suggest that there are four possible longtime profiles of the dynamics, with the chase-and-run coexistence the only possible profile when both species invade successfully. Secondly, we show through numerical experiments that the basic features of the model appear to be retained when the environment is perturbed by periodic variation in time. Thirdly, our numerical analysis suggests that in two space dimensions the population range and the spatial population distribution of the successful invader tend to become more and more circular as time increases no matter what geometrical shape the initial population range possesses. Our numerical simulations cover the one space dimension case, and two space dimension case with or without spherical symmetry. The numerical methods here are based on that of Liu et al. (Mathematics, 6(5):72, 2018, Int J Comput Math, 97(5): 959-979, 2020). In the two space dimension case without radial symmetry, the level set method is used, while the front tracking method is used for the remaining cases. We hope the numerical observations in this paper can provide further insights to the biological invasion problem, and also to future theoretical investigations. More importantly, we hope the numerical analysis may reach more biologically oriented experts and inspire applications of some refined versions of the model tailored to specific real world biological invasion problems.

The effect of droplet coalescence on drag in turbulent channel flows

We study the effect of droplet coalescence on turbulent wall-bounded flows by means of direct numerical simulations. In particular, the volume-of-fluid and front-tracking methods are used to simulate turbulent channel flows containing coalescing and non-coalescing droplets, respectively. We find that coalescing droplets have a negligible effect on the drag, whereas the non-coalescing ones steadily increase drag as the volume fraction of the dispersed phase increases: indeed, at 10% volume fraction, the non-coalescing droplets show a 30% increase in drag, whereas the coalescing droplets show less than 4% increase. We explain this by looking at the wall-normal location of droplets in the channel and show that non-coalescing droplets enter the viscous sublayer, generating an interfacial shear stress, which reduces the budget for viscous stress in the channel. On the other hand, coalescing droplets migrate toward the bulk of the channel forming large aggregates, which hardly affect the viscous shear stress while damping the Reynolds shear stress. We prove this by relating the mean viscous shear stress integrated in the wall-normal direction to the centerline velocity.

Coupling characteristics between bubble and free surface in a shallow water environment

The bubble motion in the shallow water environment has significant applications in underwater explosion and seabed exploration, where the bubble characteristics are mainly associated with the two types of stand-off distance parameters γf and γs (the parameters are defined in Section 2.3). Based on the assumption that the liquid is inviscid and incompressible, the influences of γf and γs on the coupling characteristics between the bubble and the free surface in the shallow water environment are solved by the Euler equations with the finite volume method, and the bubble surface and the free surface are tracked by the front tracking method. For the case γs = 0.8, (i) the water spike always keeps rising during the bubble expansion and contraction when γf<1.25, (ii) the water spike velocity increases with the decrease of γf, and the maximum bubble jet velocity is generated when γf = 0.8, (iii) the bubble top elongated is no longer obvious when γf>1.5. For the case γf = 0.6, (i) the toroidal bubble is split into two sub-toroidal bubbles when γs≤ 0.6 and γs≥1.1, (ii) the water spike velocity and the step pressure decrease with the increase of γs.

Radial integral boundary element method for simulating phase change problem with mushy zone

A radial integral boundary element method (BEM) is used to simulate the phase change problem with a mushy zone in this paper. Three phases, including the solid phase, the liquid phase, and the mushy zone, are considered in the phase change problem. First, according to the continuity conditions of temperature and its gradient on the liquid-mushy interface, the mushy zone and the liquid phase in the simulation can be considered as a whole part, namely, the non-solid phase, and the change of latent heat is approximated by heat source which is dependent on temperature. Then, the precise integration BEM is used to obtain the differential equations in the solid phase zone and the non-solid phase zone, respectively. Moreover, an iterative predictor-corrector precise integration method (PIM) is needed to solve the differential equations and obtain the temperature field and the heat flux on the boundary. According to an energy balance equation and the velocity of the interface between the solid phase and the mushy zone, the front-tracking method is used to track the move of the interface. The interface between the liquid phase and the mushy zone is obtained by interpolation of the temperature field. Finally, four numerical examples are provided to assess the performance of the proposed numerical method.

CFD Investigations of Bath Dynamics in a Pilot-Scale TSL Furnace

The hydrodynamics of a Top Submerged Lance (TSL) slag bath are investigated here by means of Computational Fluid Dynamics (CFD) simulation. The object of the study is the pilot-scale furnace located at TU Bergakademie Freiberg, where air is injected beneath the slag bath with a top lance. The fluid dynamics system is evaluated at operating conditions, with experimentally measured slag physical properties and real flow rates. The numerical approach is based on the Volume Of Fluid (VOF) model, a front-tracking method that allows the interface to be geometrically reconstructed. Using a fine computational grid, the multiphase interactions are calculated with a high level of detail, revealing the mechanisms of bubble formation and bath dynamics. Two lance configurations are compared, with and without a swirler, and the effect on the hydrodynamics is discussed with regards to key features of the process, such as bubble dynamics, slag splashing, the interface area, rotational sloshing, and bath mixing. The model predicts bubble frequencies in the range of 2.5 to 3 Hz and captures rotational sloshing waves with half the frequencies of the bubble detachment. These results agree with real furnace data from the literature, proving the reliability of the computing model and adding value to the empirical understanding of the process, thanks to the direct observation of the resolved multiphase flow features. The comparative study indicates that the air swirler has an overall positive effect in addition to the proposed enhancement of lance cooling, with an increase in the bath mixing and a reduction in the splashing.

A numerical study on the thermocapillary migration of a droplet under microgravity with periodic thermal boundaries using the front-tracking method

This paper mainly studied the thermocapillary migration of deformable droplets induced by periodic temperature boundary under microgravity conditions. The finite-difference/front-tracking (FD/FT) method was used to solve the Navier–Stokes equation coupled with the energy equation, and the continuum surface force (CSF) model was used to simplify the surface tension of the phase interface. The results showed that the maximum droplet migration velocity increased with the increase of temperature amplitude, and the droplet cycle period became shorter with the increase of temperature angular frequency. In the 1/4 cycle, the initial movement time of droplet decreased with the increase of temperature phase. If the phase was reversed, the initial movement direction of the droplet changed. With the increase of Reynolds number (Re), the droplet tended to maintain its motion inertia.

Dynamics of a contracting fluid compound filament with a variable density ratio

Introduction: Compound fluid filaments appear in many applications, e.g., drug delivery and processing or microfluidic systems. This paper focuses on the numerical simulation of an incompressible, immiscible, and Newtonian fluid for the contraction process of a fluid compound filament by solving the Navier-Stokes equations. The front-tracking method is used to solve this problem, which uses connected segments (Lagrangian grid) that move on a fixed grid (Eulerian grid) to represent the interface between the liquids.&#x0D; Methods: The interface points are advected by the velocity interpolated from those of the fixed grid using the area weighting function. The coordinates of the interface points are used to construct the indicators specifying the different fluids and compute the interfacial tension force.&#x0D; Results: The simulation results show that under the effects of the interfacial tension, the capsuleshaped filament can transform into a spherical compound droplet (i.e., non-breakup) or can break up into smaller spherical compound and simple droplets (i.e., breakup). When the density ratio of the outer to middle fluids increases, the filament changes from non-breakup to breakup upon contraction.&#x0D; Conclusion: Increasing the density ratio enhances the breakup of the compound filament during contraction. The breakup is also promoted by increasing the initial length of the filament.

Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method

Abstract We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in $u$ and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method—a numerical method that is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with discontinuous flux. We also present numerical experiments verifying the convergence rate results and comparing numerical solutions computed with the front tracking method to finite volume approximations.

Modelling cell guidance and curvature control in evolving biological tissues

Tissue geometry is an important influence on the evolution of many biological tissues. The local curvature of an evolving tissue induces tissue crowding or spreading, which leads to differential tissue growth rates, and to changes in cellular tension, which can influence cell behaviour. Here, we investigate how directed cell motion interacts with curvature control in evolving biological tissues. Directed cell motion is involved in the generation of angled tissue growth and anisotropic tissue material properties, such as tissue fibre orientation. We develop a new cell-based mathematical model of tissue growth that includes both curvature control and cell guidance mechanisms to investigate their interplay. The model is based on conservation principles applied to the density of tissue synthesising cells at or near the tissue’s moving boundary. The resulting mathematical model is a partial differential equation for cell density on a moving boundary, which is solved numerically using a hybrid front-tracking method called the cell-based particle method. The inclusion of directed cell motion allows us to model new types of biological growth, where tangential cell motion is important for the evolution of the interface, or for the generation of anisotropic tissue properties. We illustrate such situations by applying the model to simulate both the resorption and infilling components of the bone remodelling process, and to simulate root hair growth. We also provide user-friendly MATLAB code to implement the algorithms.

Evaluation of ANN and ANFIS Methods in Study of the Motion of a Bubble in A Combined Couette-Poiseuille Flow

The equilibrium position of a deformable bubble in a combined Couette-Poiseuille flow is investigated numerically by solving the full Navier-Stokes equations using a finite-difference/front-tracking method. The present approach is examined to predict the migration of a bubble in a combined Couette-Poiseuille flow at finite Reynolds numbers of 5, 10, and 15. The related unsteady incompressible full Navier-Stokes equations are solved using a conventional finite-difference method with a structured staggered grid. The purpose of this study is to evaluate ANN and ANFIS methods in study of the lateral migration of the bubble. Evaluation criteria of accuracy in test set derived from ANFIS demonstrates that estimated values of correlation coefficient (r), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE) are 0.97, 0.001, and 0.0014, respectively. The ANN model with RMSE of 0.0007, MAE of 0.0004 and r of 0.99, is better than ANFIS model. It is also demonstrated that the bubble position estimated by the ANN and ANFIS models closely follows the one achieved from front tracking method.

A numerical study of liquid compound filament contraction

Droplets resulting from liquid filament contraction have been widely used in industrial processes. However, detailed investigations of liquid compound filament contraction processes are lacking in the literature. Therefore, this study provides a numerical investigation of the contraction of a two-layered compound filament. The simulations are based on an axisymmetric front-tracking method. It is found that because of the interfacial tension force, the initially long cylindrical filament contracts to a compound droplet without any breakup or breaks up into smaller droplets during contraction. Unlike simple filaments, the presence of the inner filament inside the compound filament results in a more complicated compound filament breakup process with various droplet types, e.g., simple droplets, single-core compound droplets, and multi-core compound droplets. We find that the inner filament breaks up into droplets, while the outer does not induce breakup. Such a breakup mode produces a multi-core compound droplet after contraction. In some cases, while the inner filament only contracts to a single droplet, its enclosing filament breaks up to produce simple droplets at each end. We also find a breakup mode that combines these two modes, where both the inner and outer filaments perform breakup. In addition, the breakup of the compound filament occurs via one of two mechanisms: end-pinching and necking. These breakup modes and mechanisms are affected by various parameters such as the inner and outer aspect ratios, the Ohnesorge number, the interfacial tension ratio, and the viscosity ratios. Based on these parameters, various regime diagrams of breakup and non-breakup are proposed.