Piecewise Linear Interface Construction Research Articles

Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm.

The motion of air bubbles within a liquid plays a crucial role in various aspects including heat transfer and material quality. In the context of non-Newtonian fluids, such as elastoviscoplastic fluids, the presence of air bubbles significantly influences the viscosity of the liquid. This study presents the development of an interface-capturing method for multiphase viscoelastic fluid flow simulations. The proposed algorithm utilizes a geometric volume of fluid (isoAdvector) approach and incorporates a reconstructed distance function (RDF) to determine interface curvature instead of relying on volume fraction gradients. Additionally, a piecewise linear interface construction (PLIC) scheme is employed in conjunction with the RDF-based interface reconstruction for improved accuracy and robustness. The validation of the multiphase viscoelastic PLIC-RDF isoAdvector (MVP-RIA) algorithm involved simulations of the buoyancy-driven rise of a bubble in fluids with varying degrees of rheological complexity. First, the newly developed algorithm was applied to investigate the buoyancy-driven rise of a bubble in a Newtonian fluid on an unbounded domain. The results show excellent agreement with experimental and theoretical findings, capturing the bubble shape and velocity accurately. Next, the algorithm was extended to simulate the buoyancy-driven rise of a bubble in a viscoelastic shear-thinning fluid described by the Giesekus constitutive model. As the influence of normal stress surpasses surface tension, the bubble shape undergoes a transition to a prolate or teardrop shape, often exhibiting a cusp at the bubble tail. This is in contrast to the spherical, ellipsoidal, or spherical-cap shapes observed in the first case study with a bubble in a Newtonian fluid. Lastly, the algorithm was employed to study the buoyancy-driven rise of a bubble in an unbounded elastoviscoplastic medium, modeled using the Saramito-Herschel-Bulkley constitutive equation. It was observed that in very small air bubbles within the elastoviscoplastic fluid, the dominance of elasticity and capillary forces restricts the degree of bubble deformation. As the bubble volume increases, lateral stretching becomes prominent, resulting in the emergence of two tails. Ultimately, a highly elongated bubble shape with sharper tails is observed. The results show that by applying the newly developed MVP-RIA algorithm, with a tangible coarser grid compared to the algebraic VOF method, an accurate solution is achieved. This will open doors to plenty of applications such as bubble columns in reactors, oil and gas mixtures, 3D printing, polymer processing, etc.

The Effect of the Aeration Condition on the Liquid–Solid Material Mixing in a Stirred Tank with a Single-Layer Impeller

In order to increase industrial production quality and efficiency, it is essential to understand how the aeration and no-aeration condition affects liquid and solid material mixing in the stirred tank. Due to complicated shear flows, the related mass-transfer mechanism confronts numerous difficulties. This paper put forward an improved computational fluid dynamics and discrete element method (CFD–DEM) modeling approach to explore the effect mechanism of aeration conditions on liquid–solid material mixing. Firstly, a mass-transfer dynamic model is set up with a volume of fluid and piecewise linear interface construction (VOF–PLIC) coupling strategy to explore flow modes and vorticity evolution trends under aeration control. Then, a self-developed interphase coupling interface is utilized to modify the coupling force and porosity of the porous media model in the DEM module, and random dispersion properties of the particle phase under non-aeration and aeration are obtained. Results show that the aeration and flow-blocking components transform fluid tangential speeds into axial and radial speeds, which can improve the material mixing quality and efficiency. The mixed flow field can reach a greater turbulent process under the impeller rotation, making the particles have an intensive disorder and complex flow patterns. The enhanced motion efficiency of the vortex clusters encourages their nesting courses and improves cross-scale mixed transport. It can serve as some reference for the three-phase flow mixing mechanism, vorticity distribution law, and particle motion solution and has a general significance for battery homogeneous mixing, biopharmaceutical processes, and chemical process extraction.

Dynamic Evolution Characteristics of the Gear Meshing Lubrication for Vehicle Transmission System

The transmission in automobiles is the core component to ensure operational stability. Heat accumulation in the meshing process will reduce the transmission efficiency and affect the service life. Here, the essential physical process to improve transmission heat dissipation is the dynamic evolution process and the thermal transfer characteristics of lubricating oil fields during gear meshing. This paper presents a modeling and solving method for gear meshing lubrication and thermal transfer features based on the volume of the fluid model and piecewise linear interface construction (VOF-PLIC). The dynamic mesh technique combines spring smoothing and reconstruction to optimize the numerical solution process. The dynamic evolution law of gear meshing lubrication and thermal transfer is obtained by analyzing the lubrication evolution process under different speed/steering conditions. The results show that the proposed modeling and solving method could well reveal the lubrication and thermal transfer laws of the gear meshing. The temperature of the gear meshing regions was higher than that in the other regions, and the lubrication temperature showed an increasing linear trend with the stirring process. As the gear speed increased, the meshing resistance moment increased, the transmission power loss increased, and the lubrication oil temperature was larger than that of the gearbox. The power loss under the clockwise rotation of the driving gear G4 was higher than that under the counterclockwise rotation of the driving gear. The relevant results can provide theoretical references for the dynamic analysis of automotive transmission lubrication and technical support for gear profile design and lubrication optimization.

A comparative study of split advection algorithms on the moment-of-fluid (MOF) method for incompressible flow

The moment-of-fluid method (MOF) is known as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). In this study, several directional splitting advection algorithms are extended from the VOF-PLIC method to the MOF method. These methods, along with some existing directional splitting algorithms, are presented in detail, especially on the geometrical and non-geometrical nature of the advection algorithm. In addition, a simple and efficient analytic form to calculate the volume and centroid of the intersection polyhedron in 3D is proposed. Several numerical tests, including both 2D and 3D tests, are carried out to investigate mass conservation and geometrical error. Numerical results suggest that the mixed EI and LE scheme (EILE2D) (Aulisa et al., 2003) has the overall best performance in 2D, and Weymouth and Yue’s scheme (WY) has the overall best performance in 3D.

A three-dimensional numerical scheme for modeling discontinuous pinning at sharp edges using the Volume-of-Fluid method

We present a novel three-dimensional (3D) numerical scheme for modeling the discontinuous contact line pinning along sharp straight edges. The proposed scheme is devised for multi-phase flow solvers that rely on the Volume-of-Fluid (VOF) method, although its fundamental concepts can be extended and applied to other methods. Following the Piecewise-Linear-Interface-Construction (PLIC) approach in VOF, the discontinuous pinning is modeled by adjusting the orientation of PLIC polygons located near a sharp edge according to the pinning stage. That is achieved by solving a root-finding problem and using a 3D geometrical toolbox, where the advancing contact angle determines critical volume fractions in numerical cells neighboring the sharp edge. Implementing the proposed scheme in our multi-phase flow solver, we assessed its performance using several test cases where contact line pinning effects dominate. To demonstrate the scheme's efficacy, we present quantitative comparisons of our results at various grid resolutions and with a theoretical study. Furthermore, we show quantitatively that without a numerical treatment of contact line pinning, the simulation results will be drastically different. Contact line pinning plays a critical role in several technologies including separation, lithography, lens fabrication, micro-fluidic flow control among numerous others. The proposed scheme will help to accurately capture the pinning effects in computational simulations of such applications.

Effect of Wall Proximity and Shape on Bubble Enhanced Heat Transfer from a Vertical Heated Surface

<div class="section abstract"><div class="htmlview paragraph">Numerical simulation has been performed to study the heat transfer enhancement from the vertical heated wall surfaces with the help of rising bubbles due to the buoyancy force. The effect of wall proximity and bubble shapes are investigated for three wall shapes such as plane wall, wavy wall and triangular wall. Numerical solution is obtained by solving both the thermos-fluid governing equations and the Volume of Fluid (VOF) advection equation along with the Piecewise-Linear Interface Construction (PLIC) algorithm available in ANSYS-Fluent, an FVM based commercial CFD code. The results observed in the three types of wall geometries were showing the heat transfer differently for the 3 mm bubble. For the plane wall from the rise of the bubble to 0.3 seconds the temperature gradient is 10 K whereas for the curved and triangular wavy walls these gradients are 9.6 K and 17.23 K respectively. and after 0.6 seconds, this gradient is almost the same for all the wall shapes. In the latter two cases, the heat transfer by the effect of the wall was changing because of the alternating widening and narrowing channel. The research also discusses the varying distance of the injection of the bubble from the heated wall viz. 6 mm, 12 mm, 8 mm, 14 mm. It is observed that in addition to the well-known wake enhanced heat transfer by rising bubble, the contribution in the heat transfer enhancement by the wall modification from plane surface to wavy/triangular surfaces is substantial. It is also found that, as expected, this enhancement is significantly decreased as the wall proximity is increased.</div></div>

A higher-order accurate VOF interface curvature computation scheme for 3D non-orthogonal structured meshes

This paper proposes a higher-order accurate curvature computation method for volume-of-fluid (VOF) interfaces on 3D non-orthogonal structured meshes. Analogous to the height-function (HF) method, a novel approach is introduced to identify columns of control volumes that straddle the interface. This is followed by a volume conserving piecewise-linear interface construction (PLIC). In the interest of efficiency and generality, the latter employs a novel sweep-plane algorithm with bracketing on convex-decomposed control volumes in each column. The emphasis is on the generalisation of a PLIC-based least squares polynomial surface fitting procedure owing to its invariance with respect to local coordinate rotations in 3D. The PLIC representation of the interface is then smoothed by iteratively refining the PLIC facet normals. The interface curvature is finally computed analytically by fitting either second- or fourth-order polynomial surfaces to local stencils of PLIC facets in a least squares manner. Formal second- and fourth-order accuracy of interface curvature is demonstrated by comparing numerical results with a variety of analytical interface definitions.

Numerical investigation on the water entry of bow-flare section with different bulbous bow shapes

Water entry is one of the important topics in ship and offshore structures. Among the structures, water entry of bow-flare section with bulbous bow could be one of the most difficult problems due to its complex geometry. To investigate the problem, a numerical model which can simulate water entry of complex objects is developed in this paper. In the framework of Cartesian grid, Volume of Fluid method combined with Piecewise Linear Interface Construction (VOF-PLIC) are used to capture free interface. Besides, Volume of Solid based Immersed Boundary (VOS-IB) method are adopted to enforce the body boundary condition. To validate the model, water entry of a wedge and a ship hull section with bulbous bow in calm water are simulated. Time histories of pressure and local force are compared with previous experimental results. Then, a series of simulations are conducted for the water entry of bow-flare sections with different bulbous bows. Three bulbous bow shapes namely ‘Delta’, ‘Oval’ and ‘Nabla’ are considered. The influence of bulbous bow shape on the hydrodynamic forces are analyzed and three peaks are observed. A detailed discussion on the three peaks is performed with special attention paid to the second peak. It is observed that the shape of bulbous bow shape has non-negligible influence on the pressure distribution during the secondary flare impact. Based on the impact characteristics, recommendations for which shape of bulbous bow to use are given.

Analytic Solution to the Piecewise Linear Interface Construction Problem and Its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes

The plane–cube intersection problem has been discussed in the literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding computing time, so a faster analytic solution to the plane–cube intersection would greatly reduce the computing time for such simulations. We derive an analytic solution for all intersection cases and compare it to the previous solution from Scardovelli and Zaleski (Scardovelli, R.; Zaleski, S. Analytical relations connecting linear interfaces and volume fractions in rectangular grids. J. Comput. Phys.2000, 164, 228–237), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding computing time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton–Raphson is fastest with compiler optimization enabled, while analytic solutions perform better than iterative solutions without. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC—curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.

A PLIC-based method for species mass transfer at free fluid interfaces

A finite volume based method to govern mass transfer of a diluted transition component across moving fluid–fluid interfaces is presented. The concentration jump at the interface is described by a constant distribution coefficient, while Marangoni effects and volume changes due to mass transfer are neglected. The interface position is captured by means of a level set function and subsequently reconstructed using a PLIC (piecewise linear interface construction) algorithm. For the discretisation of the species transport equation, cells containing the interface are split at the reconstructed planes. The concentration of the transition component is represented by two variables, each being valid in one phase only. The cell splitting and the definition of two concentration fields is essential to avoid artificial mass transfer due to discretisation errors. The interfacial boundary conditions are the concentration jump and the continuity of species fluxes. They are implemented in a robust and fast converging algorithm.The method has been validated for two liquid–liquid systems and one gas-liquid system by simulating a single moving droplet/bubble. The results are in good agreement with reference data, even for the gas–liquid system with thin liquid-side concentration boundary layers. In contrast to one-field formulations, this method is suitable for convection-dominated mass transport and significant movement of the interface due to the prevention of non-physical effects.

Gas-liquid-solid swirling flow polishing and bubble collapse impact characteristics

The mechanism of three-phase abrasive flow polishing method is first analyzed and then volume of fluid (VOF) method is introduced into three-phase abrasive flow modeling, combined with piecewise linear interface construction (PLIC) method to realize the tracking and calculation of bubble interface in the constrained flow channel. It shows the jet effect produced by bubble collapse can effectively enhance the velocity and turbulent kinetic energy of local fluid; bubbles collapse under the action of turbulent vortex, which in turn induce fluid to generate turbulence, which is an action-reaction process; the directional randomness of the turbulent vortex can avoid the directional strips induced by particles polishing in the same direction. The particle image velocimetry (PIV) measurement was used to observe the bubble collapse near the wall and the three-phase swirling flow field for validating the model and polishing experiments show that the target surface roughness can be down to 2.59 nm.

A Parallel Coupled Lattice Boltzmann-Volume of Fluid Framework for Modeling Porous Media Evolution.

In this paper, we present a framework for the modeling and simulation of a subset of physical/chemical processes occurring on different spatial and temporal scales in porous materials. In order to improve our understanding of such processes on multiple spatio-temporal scales, small-scale simulations of transport and reaction are of vital importance. Due to the geometric complexity of the pore space and the need to consider a representative elementary volume, such simulations require substantial numerical resolutions, leading to potentially huge computation times. An efficient parallelization of such numerical methods is thus vital to obtain results in acceptable wall-clock time. The goal of this paper was to improve available approaches based on lattice Boltzmann methods (LBMs) to reliably and accurately predict the combined effects of mass transport and reaction in porous media. To this end, we relied on the factorized central moment LBM as a second-order accurate approach for modeling transport. In order to include morphological changes due to the dissolution of the solid phase, the volume of fluid method with the piece-wise linear interface construction algorithm was employed. These developments are being integrated into the LBM research code VirtualFluids. After the validation of the analytic test cases, we present an application of diffusion-controlled dissolution for a pore space obtained from computer tomography (CT) scans.

Simulation of Shear-Thickening Liquid Transfer between U-Shaped Cell and Flat Plate

Based on the actual measurement of the shear-thickening properties of water-based inks, in order to improve the ink transfer rate, the PLIC (Piecewise Linear Interface Construction) interface tracking method and the VOF (Volume of Fluid) method are used to simulate the transfer process of the shear-thickening liquid between the U-shaped cell and the upwardly moving plate. The effects of substrate surface wettability, cell contact angle, and cell depth on liquid transfer were studied. The results showed that all can increase the liquid transfer rate, and the change of the cell contact angle also led to the difference in the breaking time of the liquid filament. In addition, the shallow plate effect was discovered in the study of cell depth. The shallow plate effect is a phenomenon by which the amount of liquid transferred and the liquid transfer rate are greatly improved when the depth of the cell decreases to a certain limit value. In addition, for the U-shaped cell, the optimization method combining the shallow printing plate effect and fillet can greatly improve the liquid transfer rate and solve the undesirable problems such as plate blocking. After optimization, a liquid transfer rate of about 85% can be achieved.

NPLIC: A machine learning approach to piecewise linear interface construction

Volume of fluid (VOF) methods are extensively used to track fluid interfaces in numerical simulations, and many VOF algorithms require that the interface be reconstructed geometrically. For this purpose, the Piecewise Linear Interface Construction (PLIC) technique is most frequently used, which for reasons of geometric complexity can be slow and difficult to implement. Here, we propose an alternative neural network based method called NPLIC to perform PLIC calculations. The model is trained on a large synthetic dataset of PLIC solutions for square, cubic, triangular, and tetrahedral meshes. We show that this data-driven approach results in accurate calculations at a fraction of the usual computational cost, and a single neural network system can be used for interface reconstruction of different mesh types.

Numerical study for the improvement of bead spreading architecture with modified nozzle geometries in additive manufacturing of polymers

Purpose This paper aims to develop a numerical model of bead spreading architecture of a viscous polymer in fused filament fabrication (FFF) process with different nozzle geometry. This paper also focuses on the manufacturing feasibility of the nozzles and 3D printing of the molten beads using the developed nozzles. Design/methodology/approach The flow of a highly viscous polymer from a nozzle, the melt expansion in free space and the deposition of the melt on a moving platform are captured using the FLUENT volume of fluid (VOF) method based computational fluid dynamics code. The free surface motion of the material is captured in VOF, which is governed by the hydrodynamics of the two-phase flow. The phases involved in the numerical model are liquid polymer and air. A laminar, non-Newtonian and non-isothermal flow is assumed. Under such assumptions, the spreading characteristic of the polymer is simulated with different nozzle-exit geometries. The governing equations are solved on a regular stationary grid following a transient algorithm, where the boundary between the polymer and the air is tracked by piecewise linear interface construction (PLIC) to reconstruct the free surface. The prototype nozzles were also manufactured, and the deposition of the molten beads on a flatbed was performed using a commercial 3D printer. The deposited bead cross-sections were examined through optical microscopic examination, and the cross-sectional profiles were compared with those obtained in the numerical simulations. Findings The numerical model successfully predicted the spreading characteristics and the cross-sectional shape of the extruded bead. The cross-sectional shape of the bead varied from elliptical (with circular nozzle) to trapezoidal (with square and star nozzles) where the top and bottom surfaces are significantly flattened (which is desirable to reduce the void spaces in the cross-section). The numerical model yielded a good approximation of the bead cross-section, capturing most of the geometric features of the bead with a reasonable qualitative agreement compared to the experiment. The quantitative comparison of the cross-sectional profiles against experimental observation also indicated a favorable agreement. The significant improvement observed in the bead cross-section with the square and star nozzles is the flattening of the surfaces. Originality/value The developed numerical algorithm attempts to address the fundamental challenge of voids and bonding in the FFF process. It presents a new approach to increase the inter-bead bonding and reduce the inter-bead voids in 3D printing of polymers by modifying the bead cross-sectional shape through the modification of nozzle exit-geometry. The change in bead cross-sectional shape from elliptical (circular) to trapezoidal (square and star) cross-section is supposed to increase the contact surface area and inter-bead bonding while in contact with adjacent beads.

Flow around Ship Spoiler.(Dept.M)

To control the roll, pitch motion and speed of the ship, a proposed system of spoilers is mounted at the bottom of the ship. The bow spoilers consist of an even number of sections arranged port and starboard forward of the center of mass of me ship. The stem spoilers consist of an even number of sections arranged port and starboard attached to the ship's transom or transom plate. Injecting exhaust gas stabilizes cavities behind spoilers. The forces and moments acting on the ship's bottom depend on its inclination, spoiler extensions and positions of the spoilers relative to the ship's center of mass. To improve the understanding of the parameters affecting the flow field around such spoiler, we computed the flow field with different spoiler position, and inclination angle with air flow injection. We used a three-dimensional Navier-Stokes code (CFDRC, 2000) to model the two-phase (low field around a ship spoiler with the Free Surface simulation In Piecewise Linear Interface Construction (PUC) method. The governing equations are discretized on a structured grid using an upwind difference scheme. For different conditions, we detemline the bubbles shape, the two-dimensional flowfield around the spoiler body and the pressure variation on the wake of the spoiler body. Higher spoiler inclination angle gives higher pressure difference between front and back of the spoiler. Spoiler with injection at a distance L from ship spoiler gives higher pressure difference between front and back of the spoiler than injection just beside the spoiler In spoiler with 45o inclination angle the cavity has high stability with long bubble foot shape. The bubble shape depends on the spoiler position and inclination. These parameters affect the roll, pitch and speed of the ship.

Numerical and Experimental Analysis of Transient 2D Flow around Ship Spoiler.(Dept.M)

An experimental analysis and numerical computation were conducted for flow around ship spoiler mounted at the bottom of the ship and subjected to air injection The air was injected beside the spoiler and at a distance equals the spoiler length to stabilize cavities behind spoilers. This system was proposed to control the roll, pitch motion and speed of the ship, with injecting exhaust gas behind spoilers. The forces and moments acting on the ship's bottom depend on spoilers' inclination, extensions and positions relative to the ship's center of mass. To improve the understanding of the parameters affecting the flow field aroundsuch spoiler, experimental images of bubble formation are triggered and compared with the computed flow field at different spoiler inclination angle, and air injection position. Two dimensional Navier-Stokes code is used to model the two-phase flow field around a ship spoiler with the free surface simulation in Piecewise Linear Interface Construction (PLIC) method. The governing equations are discretized on a structured grid using an upwind difference scheme. For different conditions, the bubbles shape, the two-dimensional flow field around the spoiler body and the pressure variation on the wake of the spoiler body are computed, Flow field and bubble formation around the spoiler are recorded experimentally at different condition and time sequence with scientific video camera The Froude number was about 1.6 based on spoiler length. The computational and experimental results show that there is a good matching of bubble formation in the spoiler with 90', but in the condition of 45° the experimental images show that bubble filled the space between the bubble and the body of the ship. This is may be due to the two-dimension computational model consideration. So, a water layer was isolated between the bubble and the ship body. In the case of spoiler with 45° inclination angle, the flow is circulated inside the bubble. The bubble shape is longer which gives more negative pressure on larger area. The moment around the 450 spoiler fixationpoint is seven times of that around the 90° spoiler. In double injection condition beside and at a distance equal the horizontal spoiler length downstream the spoiler, the bubbles are more stable and the moment is higher by 7.9 and 17.8 times after 1.2 and 2 seconds, respectively, than that around the 90° spoiler. Thus, changing inclination spoiler angle and position relative to injection angle will give different bubble shapes which give different forces to control the roll, pitch motion and speed of the ship.

Monolithic Solvers for Incompressible Two-Phase Flows at Large Density and Viscosity Ratios

In this paper, we investigate the accuracy and robustness of three classes of methods for solving two-phase incompressible flows on a staggered grid. Here, the unsteady two-phase flow equations are simulated by finite volumes and penalty methods using implicit and monolithic approaches (such as the augmented Lagrangian and the fully coupled methods), where all velocity components and pressure variables are solved simultaneously (as opposed to segregated methods). The interface tracking is performed with a Volume-of-Fluid (VOF) method, using the Piecewise Linear Interface Construction (PLIC) technique. The home code Fugu is used for implementing the various methods. Our target application is the simulation of two-phase flows at high density and viscosity ratios, which are known to be challenging to simulate. The resulting strategies of monolithic approaches will be proven to be considerably better suited for these two-phase cases, they also allow to use larger time step than segregated methods.

Volume of Fluid Computations of Gas Entrainment and Void Fraction for Plunging Liquid Jets to Aerate Wastewater

Among various mechanisms for enhancing the interfacial area between gases and liquids, a vertical liquid jet striking a still liquid is considered an effective method. This method has vast industrial and environmental applications, where a significant application of this method is to aerate industrial effluents and wastewater treatment. Despite the huge interest and experimental and numerical efforts made by the academic and scientific community in this topic, there is still a need of further study to realize improved theoretical and computational schemes to narrow the gap between the measured and the computed entrained air. The present study is a numerical attempt to highlight the air being entrained by water jet when it intrudes into a still water surface in a tank by the application of a Volume of Fluid (VOF) scheme. The VOF scheme, along with a piecewise linear interface construction (PLIC) algorithm, is useful to follow the interface of the air and water bubbly plume and thus can provide an estimate of the volume fraction for the gas and the liquid. Dimensionless scaling derived from the Fronde number and Reynolds number along with geometric similarities due to the liquid jet’s length and nozzle diameter have been incorporated to validate the experimental data on air entrainment, penetration and void fraction. The VOF simulations for void fraction and air-water mixing and air jet’s penetration into the water were found more comparable to the measured values than those obtained using empirical and Euler-Euler methods. Although, small overestimates of air entrainment rate compared to the experiments have been found, however, VOF was found effective in reducing the gap between measurements and simulations.

A geometric VOF method for interface resolved phase change and conservative thermal energy advection

We present a novel numerical method to solve the incompressible Navier-Stokes equations for two-phase flows with phase change. Separate phases are tracked using a geometric Volume-Of-Fluid (VOF) method with piecewise linear interface construction (PLIC). Thermal energy advection is treated in conservative form and the geometric calculation of VOF fluxes at computational cell boundaries is used consistently to calculate the fluxes of heat capacity. The phase boundary is treated as sharp (infinitely thin), which leads to a discontinuity in the velocity field across the interface in the presence of phase change. The numerical difficulty of this jump is accommodated with the introduction of a novel two-step VOF advection scheme. The method has been implemented in the open source code PARIS and is validated using well-known test cases. These include an evaporating circular droplet in microgravity (2D), the Stefan problem and a 3D bubble in superheated liquid. The accuracy shown in the results were encouraging. The 2D evaporating droplet showed excellent prediction of the droplet volume evolution as well as preservation of its circular shape. A relative error of less than 1% was achieved for the Stefan problem case, using water properties at atmospheric conditions. Two cases of a bubble in superheated liquid was performed, at respective Jacob numbers of 0.5 and 2.15. For the final radius of the bubbles in both cases, a relative error of less than 7% was obtained on the coarsest grid, with less than 1% on the finest.