Conservative Level Set Method Research Articles

The log-conformation formulation for single- and multi-phase axisymmetric viscoelastic flows

A computational model for predicting axisymmetric single- and multi-phase viscoelastic flows based on the log-conformation formulation of the constitutive equation is presented. The governing equations are discretised using the finite element method and advanced in time using a projection scheme. The single-phase benchmark problem of viscoelastic flow past a sphere in a cylinder is considered and excellent agreement is found with the literature for the drag coefficient for an extensive range of Weissenberg numbers. The cause of the breakdown in stability due to a velocity inflection near the sphere is investigated. For multi-phase flows, enhanced stability of the conservative level-set method is obtained by introducing a diffused interface approach for normal calculations. This novel method is used to investigate the jump discontinuity phenomenon exhibited for a bubble rising in a viscoelastic fluid. Qualitative and quantitative agreement is found with contemporary literature in terms of the critical bubble volume as well as other phenomena such as the negative wake and trailing cusp.

Flow and Heat Transfer in Two-Phase Flow Immiscible Droplets in Microchannels

ABSTRACT The enhancement of heat transfer in microchannels without phase change is a significant area of study, primarily driven by the internal fluid recirculation in two-phase flows. This investigation focuses on a circular microchannel, 100 μm in diameter, where mineral oil droplets are introduced into a water flow. The study utilizes the conservative level set method for precise interface tracking and liquid film thickness measurement. This research introduces a modified Nusselt number, specifically tailored to describe the heat transfer characteristics of multiphase flows. The study delves into the effects of varying droplet sizes, from small spheres to a slug. The findings indicate that the most significant heat transfer enhancement occurs with droplets whose volume closely matches that of a sphere fitting within the channel. Moreover, the investigation explores the impact of parameters like inlet velocity, primary-phase slug length, and contact angle. Notably, higher inlet velocities lead to improved heat transfer, resulting in a substantial increase in the Nusselt number compared to single-phase flows. The study underscores the delicate balance between recirculation intensity and droplet heat capacity concerning slug length, as excessive variations can harm thermal performance. It also highlights the pivotal role of surface wettability, showing improved thermal performance on hydrophobic surfaces.

A Level Set-Based Solver for Two-Phase Incompressible Flows: Extension to Magnetic Fluids

Development of a two-phase incompressible solver for magnetic flows in the magnetostatic case is presented. The proposed numerical toolkit couples the Navier–Stokes equations of hydrodynamics with Maxwell's equations of electromagnetism to model the behaviour of magnetic flows in the presence of a magnetic field. To this end, a rigorous implementation of a second-order two-phase solver for incompressible nonmagnetic flows is introduced first. This solver is implemented in the finite-difference framework, where a fifth-order conservative level set method is employed to capture the evolution of the interface, along with an incompressible solver based on the projection scheme to model the fluids. The solver demonstrates excellent performance even with high density ratios across the interface (Atwood number ≈ 1 ), while effectively preserving the mass conservation property. Subsequently, the numerical discretisation of Maxwell's equations under the magnetostatic assumption is described in detail, utilising the vector potential formulation. The primary second-order solver for two-phase flows is extended to the case of magnetic flows, by incorporating the Lorentz force into the momentum equation, accounting for high magnetic permeability ratios across the interface. The implemented solver is then utilised for examining the deformation of ferrofluid droplets in both quiescent and shear flow regimes across various susceptibility values of the droplets. The results suggest that increasing the susceptibility value of the ferrofluid droplet can affect its deformation and rotation in low capillary regimes. In higher capillary flows, increasing the magnetic permeability jump across the interface can further lead to droplet breakup as well. The effect of this property is also investigated for the Rayleigh–Taylor instability growth in magnetic fluids.

Computational fluid dynamics simulations of silicon μ-structure formation during excimer laser modification of silicon-nanoparticle layers

The laser modification of silicon-nanoparticle layers with a nanosecond pulsed excimer laser leads to the self-organized formation of crystalline, μ-cone-shaped silicon structures with good electronic properties, which have allowed the demonstration of their potential for printed flexible electronics. With the current nanosecond laser process, silicon exhibits only short melting times, resulting in a method-defined substrate contact angle, instead of this parameter being defined by the substrate surface energy as expected for equilibrium conditions. This substrate material-independent non-equilibrium contact angle of the silicon melt was experimentally determined in this study to be Θ = 68 ± 10°. To gain deeper insight into the process of the sequential melting and the formation of the silicon μ-cone structures during laser modification, a two-dimensional computational fluid dynamics simulation was performed in COMSOL Multiphysics® solving the Navier–Stokes equation for incompressible fluids. The simulation uses an effective medium approach by applying the conservative level set method to describe the porous silicon-nanoparticle layer. Its sequential melting during the pulsed laser modification is modeled using a newly developed simulation methodology, which uses a time- and depth-dependent dynamic viscosity of the molten silicon. The simulation was carried out for different laser energy densities and verified using scanning electron microscopy images of corresponding laser-modified samples. The simulation results agree well with the experiment and were subsequently used to optimize the laser process.

CFD simulation of a small bubble motion in 3D flow domain: effect of liquid density, viscosity and surface tension

The motion of spherical and slightly deformed bubbles rising rectilinearly in a stagnant liquid was numerically simulated in a fully three-dimensional domain using the CFD solver COMSOL Multiphysics. The interface was tracked by the built-in conservative Level set method on a fixed numerical grid. The purpose of this work was to study the single bubble motion in three industrially used liquids that differ significantly in surface tension, density, and viscosity. The motion of bubbles with diameters up to 1.6 mm was also studied experimentally using a high-speed camera. The data obtained together with the results of theoretical models for bubble motion were used for the validation of the simulation data. Using a 3D domain, very good agreement was obtained in both bubble shape deformations and bubble terminal velocity. The best results were achieved for propanol with low surface tension and low viscosity. In the case of high surface tension and low viscosity liquid (water), both the bubble deformation and the bubble velocity were slightly underestimated. In the case of glycerol (high surface tension and viscosity), the negligible bubble deformation is correctly calculated, but the velocity is again slightly underestimated.Graphical abstract

A stabilised finite element framework for viscoelastic multiphase flows using a conservative level-set method

The development of stable numerical schemes for viscoelastic multiphase flows has implications for many areas of engineering applications. The principal original contribution of this paper is the implementation of a conservative level-set method to define implicitly the interface between fluid phases, fully integrated into the mathematical framework of viscoelastic flow. The governing equations are discretized using the finite element method and stabilisation of the constitutive equation is achieved using either the discontinuous Galerkin (DG) or streamline upwinding (SU) method. The discrete elastic viscous stress splitting gradient (DEVSS-G) formulation is also employed in the Navier-Stokes equations to balance the hyperbolic characteristics of the polymeric stress tensor. The numerical scheme is validated with reference to several benchmark problems and excellent quantitative agreement with published data is found for Newtonian and viscoelastic fluids, for both single and multiphase flows. The motion of a gas bubble rising in a viscoelastic fluid is studied in detail. The influence of polymer concentration, surface tension, fluid elasticity and shear-thinning behaviour, on flow features such as the development of filaments and cusps and the generation of negative wakes is explored.

Effects of surface wettability and flow rates on the interface evolution and droplet pinch-off mechanism in the cross-flow microfluidic systems

In this study, the effect of surface wettability on the two-phase immiscible fluids flow and dynamics of droplet pinch-off in a T-junction microchannel has been numerically investigated using the finite element method. A conservative level set method (CLSM) has been adopted to capture the interface topology in the squeezing regime (Cac<10-2) for wide flow rate ratio (1/10⩽Qr⩽10) and contact angle (120°⩽θ⩽180°). This study has revealed that the wettability is a dominant factor in determining the hydrodynamic features of the droplet. Based on the instantaneous phase flow profiles, the droplet formation stages are classified as: initial, filling, squeezing, pinch-off and stable droplet. Wettability effects are insignificant in the filling stage. However, the hydrophobic effects are more vital in the squeezing and pinch-off stages. In general, it is shown that engineering parameters have complex dependence on the dimensionless parameters (Cac,Qr,θ). Capturing the instantaneous interface evolution has revealed that the droplet shape is sensitive to the contact angle. Interface shape profiles transform from convex into concave immediately for hydrophobic conditions (120°⩽θ⩽150°) whereas slowly for the super hydrophobic conditions (150°<θ⩽180°). In contrast to the literature, the pressure in the dispersed phase is not constant, but it is an anti-phase with the pressure in the continuous phase. Maximum pressure in the continuous phase, and neck width of the interface are complex function of the governing conditions (Cac,Qr,θ). Comparison of the filling and pinch-off time based on the pressure and phase profiles has brought new insights that the droplet pinch-off mechanism can be elucidated by installing the pressure sensors even without the flow visualization and phase profiles. The interface curvature adopts a flattened to a more concave shape when the Laplace pressure varies from a smaller to higher value. The interface neck width (2r) shows an increasing trend up to a threshold value and then decreases linearly with the contact angle.

A stable interface-preserving reinitialization equation for conservative level set method

Existing artificial compression based reinitialization approach for conservative level set method has a few drawbacks, like distortion of fluid–fluid interface and formation of unphysical fluid patches away from the interface. In this paper, a novel reinitialization approach has been presented which circumvents these limitations by reformulating the existing reinitialization equation. The new reinitialization equation by design ensures that the interface contour is kept invariant under the action of level set reinitialization. The unphysical patch formation away from the interface is also resolved here by avoiding the use of ill-conditioned contour normal vectors. Owing to the use of much simpler terms, a significant reduction in the numerical computations is achieved with the new reinitialization equation. The new reinitialization equation also enables one to choose a larger time step during the reinitialization iteration, leading to an overall reduction in computational efforts. In order to evaluate the performance of the present formulation, a set of test problems involving reinitialization of stationary level set functions is carried out first. Then, the efficacy of the proposed reinitialization approach is demonstrated using a set of standard two-dimensional scalar advection based test problems and incompressible two-phase flow problems. Finally, the ability to deal with complex mesh types is demonstrated by solving a test problem on an unstructured mesh consisting of finite volume cells having triangular and quadrilateral shapes.

CLS 방법을 이용한 Pore-scale 다상유동 해석에 대한 연구

The fluid invasion patterns in a porous medium, which affect carbon sequestration, enhanced oil recovery and groundwater remediation, are numerically investigated by employing a conservative level-set method combined with a distance level-set method. An immersed boundary method is also implemented for pore-scale simulation of the fluid invasion process. The porous medium is assumed to consist of two-dimensional microscale solid cylinders which are arranged in a rectangle. The conservation equations are solved with no-slip and contact-angle conditions on the immersed solid cylinders. The fluid displacement patterns are investigated by varying the contact angle and the properties of the injected fluid. The effect of preferential path created in the porous medium on invasion patterns is also studied.

Study on the Formation and Separation Process of Droplets in the Medical Piezoelectric Atomization Device Induced by Intra-hole Fluctuation

Traditional atomization devices always exhibit many drawbacks, such as non-uniform atomization particle sizes, instability of transient atomization quantity and uncontrollability of precise energy, which seriously restrict further practical application of atomization inhalation therapy. The formation and separation process of droplets belongs to a microphenomenon of atomization. The investigation of the droplet formation and separation process will be favorable for understanding the atomization mechanism. In present work, the Conservative Level Set Method (CLSM) is successfully applied on the simulation of the formation and separation of droplets in a medical piezoelectric atomization device induced by intra-hole fluctuation. The intra-hole fluctuation mechanism is systematically explored and analyzed, and also the expression of the volume change in the micro cone hole is built and evaluated. Both the control equation and simulation model of droplet formation and separation process have been well established by meshing the simulation model, and thereby the process of droplet formation and separation is simulated. The corresponding results demonstrate that the breaking time of droplets is decreased with the inlet velocity and liquid temperature rising, while enhanced with the liquid concentration increasing. Meanwhile, the volume of droplet is decreased with the inlet velocity and liquid concentration increasing, but increased with the liquid temperature rising. The velocity of droplet is enhanced with the inlet velocity and liquid temperature rising, and reduced with the increase of liquid concentration. When the large side diameter of micro-cone hole is set as 79 μm, the breaking time of the droplet reaches a minimum value of 38.7 μs, whereas the volume and the velocity of droplet reach a maximum value of 79.8 pL and 4.46 m/s, respectively. This study provides theoretical guidance for the design of medical piezoelectric atomization devices and contributes to the promotion of inhalation therapy in practical use.

Analytical and numerical study of capillary rise in sinusoidal wavy channel: Unveiling the role of interfacial wobbling

Capillary rise is ubiquitous in engineering applications and natural phenomena. In straight channels, the dynamics of capillary rise have been thoroughly investigated and are well understood. However, for nonuniform channels of varying radius, the dynamics remain largely unclear. In this study, the capillary rise in a sinusoidal wavy channel is investigated both analytically and numerically. Specifically, the capillary rate-of-rise of water in sinusoidal channels with different contraction frequencies and amplitudes is derived based on the principle of energy conservation. The change in capillary velocity and height over time is further validated by two-phase flow simulations based on the conservative level-set method. The results reveal a strong viscous dissipation in the interfacial region resulting from the wave-like wobbling motion of the liquid–air interface, constituting more than 50% of the total viscous dissipation when the channel profile changes rapidly. Failing to account for this interfacial effect will result in significant overestimations of the capillary velocity and erroneous predictions of the capillary rise curve, typically more than 4 times difference in the capillary velocity and more than 2.5 times difference in the time taken to arrive at the maximum height.

Simulation of Two Phase Flow Using a Modified Accurate Conservative Level Set Method

Simulation of Two Phase Flow Using a Modified Accurate Conservative Level Set Method

A conservative level set method for liquid-gas flows with application in liquid jet atomisation

In this paper, a methodology for modelling two-phase flows based on a conservative level set method in the framework of finite volume method is presented. The novelty of the interface capturing method used here lies on the advection of level set which is solved with a WENO scheme and corrected with a novel re-initialisation method for retaining its signed distance function character. The coupling with the volume of fluid method is done with a simple algebraic approach, and with the new algorithm the accumulated mass conservation errors remain reasonably low. The paper presents a unique coupling between the level set method and the Eulerian-Lagrangian Spray Atomisation approach for modelling spray dispersion in liquid atomisation systems. The method is shown to have good accuracy providing similar results to other numerical codes for the classical tests presented. Preliminary results are also shown for three-dimensional simulations of the primary break-up of a turbulent liquid jet obtaining results comparable to direct numerical simulations. Consequently, the coupled method can be used for simulating various two-phase flow applications offering an accurate representation of the interface dynamics.

Experiments and Simulations of Free-Surface Flow behind a Finite Height Rigid Vertical Cylinder

We present the results of a combined experimental and numerical study of the free-surface flow behind a finite height rigid vertical cylinder. The experiments measure the drag and the wake angle on cylinders of different diameters for a range of velocities corresponding to 30,000 &lt;Re&lt; 200,000 and 0.2&lt;Fr&lt;2 where the Reynolds and Froude numbers are based on the diameter. The three-dimensional large eddy simulations use a conservative level-set method for the air-water interface, thus predicting the pressure, the vorticity, the free-surface elevation and the onset of air entrainment. The deep flow looks like single phase turbulent flow past a cylinder, but close to the free-surface, the interaction between the wall, the free-surface and the flow is taking place, leading to a reduced cylinder drag and the appearance of V-shaped surface wave patterns. For large velocities, vortex shedding is suppressed in a layer region behind the cylinder below the free surface. The wave patterns mostly follow the capillary-gravity theory, which predicts the crest lines cusps. Interestingly, it also indicates the regions of strong elevation fluctuations and the location of air entrainment observed in the experiments. Overall, these new simulation results, drag, wake angle and onset of air entrainment, compare quantitatively with experiments.

Modeling of Droplet Generation in a Microfluidic Flow-Focusing Junction for Droplet Size Control.

In this paper, we study the parameters that affect the generation of droplets in a microfluidic flow-focusing junction. Droplets are evaluated based on the size and frequency of generation. Droplet size control is essential for microfluidic lab-on-a-chip applications in biology, chemistry, and medicine. We developed a three-dimensional numerical model that can emulate the performance of the physical system. A numerical model can help design droplet-generation chips with new junction geometries, different dispersed and continuous phase types, and different flow rates. Our model uses a conservative level-set method (LSM) to track the interface between two immiscible fluids using a fixed mesh. Water was used for the dispersed phase and mineral oil for the continuous phase. The effects of the continuous-to-dispersed flow rate ratio () and the surfactant concentration on the droplet generation were studied both using the numerical model and experimentally. The numerical model was found to render results that are in good agreement with the experimental ones, which validates the LSM model. The validated numerical model was used to study the time effect of changing on the generated droplet size. Properly timing when the flow rates are changed enables control over the size of the next generated droplet, which is useful for single-droplet size modulation applications.

Stability of the foundation trench of the immersed tunnel subjected to combined wave and current loading

The determination of stability for the submarine artificial slope is important in the construction of offshore structures such as immersed tunnels and embedded pipelines. Aimed at this issue, a two-dimensional (2D) integrated model is proposed to investigate the slope stability of the foundation trench for the immersed tunnel under the combined wave and current loading, as in the real marine environment. The Reynolds-Averaged Navier-Stokes (RANS) equations are solved to simulate the wave field, meanwhile the current is realized by setting boundary inlet and outlet velocity at both sides. The interface between water and air is tracked by conservative Level Set Method (LSM). The wave-current induced pore pressure in seabed is calculated by Darcy's law, and the shear strength reduction method is adopted to judge the damage of the elastic-plastic seabed slope which is described through Mohr-Coulomb constitutive model. The wave-current model is validated through comparing with experimental data and analytical solution. The consolidation status of foundation trench after excavation is obtained and set as the initial condition for analysis. The factor of stability (FOS) for the slope is demonstrated to depend on the relative distance between wave crest and slope. The minimum of FOS corresponding to the most dangerous situation is regarded as the stability index for the slope with specific slope ratio in the whole process of dynamic wave-current loading. Through the parametric analysis based on this new evaluation method, the effects of soil strength parameters, slope ratio and current direction on the slope stability are discussed in detail.

Gas bubble dynamics in airlift photo-bioreactors for microalgae cultivation by level set methods

Coupled level set and CFD (computational fluid dynamics) methods are adopted in this work to track the moving gas–liquid interfaces in the riser of an external loop airlift photobioreactor (ALR) in which microalgae are used to produce biofuels and capture CO2 from flue-gas. Modeling the behavior of gas bubbles is a crucial aspect for the fine-tuning of the operation of the reactor when inserted into a closed-loop biorefinery at the pilot-scale. The experimental data used for simulation were completely acquired or calculated from hydrodynamic experimental campaigns carried out on the ALRs. The rise, coalescence, and shape dynamics of the bubbles of the flue-gas are simulated in a rectangular domain representing the vertical section of the ALR riser. Different correction approaches, such as the conservative level set method (CLSM), are proposed to face the volume loss characteristic of LSM. Computational results evidenced strong agreement with the experimental data (bubble shapes and trajectories). The physically-based CLSM model was then effectively used for the fine-tuning of the multiphase flow regime inside the ALRs, suggesting operating conditions for the outdoor cultivation with Reynolds number = 10000 – 11000, Sherwood number between 1400 and 1800, and spherical-caps bubbles in the upper half of the riser, mildly churning the microalgae while avoiding damages to their cells.

Computational Study in Bottom Gas Injection Using the Conservative Level Set Method

This paper presents a computational study on bottom gas injection in a cylindrical tank. The bubble formation at submerged orifices, bubble rising, and interactions between bubbles and bubbles with the free surface were studied using the conservative level set method (CLSM). Since the gas injection is an important technique in various fields and this process is quite complicated, the scenario was chosen to quantify the efficacy of the CLSM to describe the gas-liquid complex interactions with fast changes in the surface tension force and buoyancy force. The simulation accuracy is verified with the grid convergence index (GCI) approach and Richardson Extrapolation (RE) and is validated by comparing the numerical results with experimental observations, theoretical equations, and published data. The results show that the CLSM accurately reproduces the bubble formation frequency, and that it can handle complicated bubble shapes. Moreover, it captures the challenging phenomena of interaction between bubbles and free surface, the jet of liquid produced when bubbles break through the free surface, and the rupture of the film of liquid. Therefore, the CLSM is a robust numerical technique to describe gas-liquid complex interactions, and it is suited to simulate the gas injection operation.

Numerical investigation of falling ferrofluid droplets under magnetic fields

A methodical analysis on the dynamic interaction behavior between a pair of uneven sized ferrofluid droplets freely falling under gravity and uniform magnetic fields is presented in this article. Here, a conservative level set method (LSM) is adopted to precisely calculate the free interface curvature of the droplet, which again couples both magnetic and flow fields. The results indicate that at a unity viscosity ratio (i.e., λ = 1) and a fixed Galilei number (i.e., Ga = 1600), in the absence of any external forces except gravity, a critical initial vertical separation distance between the droplets prevails (ΔYcr⋆ ≥ 6), which prohibits them from undergoing coalescence phenomenon before hitting the bottom wall of the computational domain. However, enacting a magnetic field along α = 0∘ hinders coalescence, while it is expedited by the implementation of the magnetic field along α = 90∘. Contrarily, at α = 45∘, the droplets exhibit downward lateral migration along the secondary diagonal of the domain, leading to a larger separation between them at higher magnetic Bond numbers Bom (i.e., Bom = 8). Additionally, the investigation on the effects of surface tension suggests an increase in the vertical separation between droplets at higher Eötvös numbers (i.e., Eo = 9.48). Moreover, a magnetic field along α = 0∘ results in the formation of round-bottom hull shaped droplets, whereas they transform into teardrop shaped droplets before coalescence at α = 90∘. Furthermore, the magnetic field along α = 45∘ greatly impedes the coalescence phenomenon, which eventually leads to migration of droplets along divergent lateral directions. Finally, if the droplets are dispersed in air, they do not exhibit any coalescence event before the impact under gravity or gravity and magnetic fields.

Analysis of Capillary Flow in a Parallel Microchannel-Based Wick Structure with Circular and Noncircular Geometries.

Capillary flow in porous media is of great significance to many different applications including microfluidics, chromatography, and passive thermal management. For example, heat pipe has been widely used in the thermal management of electronic system due to its high flexibility and low thermal resistance. However, the critical heat flux of heat pipe is often limited by the maximum capillary-driven liquid transport rate through the wicking material. A significant number of novel porous material with complex structures have been proposed in past studies to provide enhanced capillary-driven flow without substantial reduction in pore size and porosity. However, the increasing level of structural complexity often leads to a more tortuous flow path, which deprives the merits of enhanced capillarity. In this study, we examined the capillary performance of a porous material with simple geometric structures both analytically and numerically. Specifically, the capillary rate of rise of water in parallel hollow microchannels with different cross-sectional shapes is derived by solving the momentum transport equation. The relationships between the capillary flow rate and wicking height are further validated by two-phase flow simulation based on the conservative level-set method. The results demonstrate that parallel microchannel configuration, despite its geometric simplicity, provides superior capillary performance than most existing porous media in terms of both capillary flow rate and ultimate wicking height. In addition, design of noncircular cross section reduces the viscous drag and increases the packing density of the microchannels in the bulk solid without affecting the capillary pumping pressure. These features contribute to a further enhancement in the capillary performance by up to 32%. These results provide important guidance to the rational design of porous material with enhanced fluid transport property in a variety of microfluidic systems.