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We describe the implementation of a second-order accurate Volume-of-Fluid interface tracking algorithm in the open source finite element code ASPECT that is designed to model convection and other processes in the Earth’s mantle. This involves the solution of the incompressible Stokes equations coupled to an advection diffusion equation for the temperature, a Boussinesq approximation that governs the dependence of the density on the temperature, and an advection equation for a marker indicating two initial (constant) density states, that are passively advected in the underlying flow field. The Volume-of-Fluid method in ASPECT is fully parallelized and fully integrated with ASPECT’s adaptive mesh refinement algorithm. We present the results of several interface tracking benchmarks in order to demonstrate the accuracy of the method, as well as the results of several benchmarks commonly used in the computational mantle convection community. Finally, we present the results of computations with and without adaptive mesh refinement of a model problem involving thermochemical convection in a computationally stratified fluid designed to provide insight into how thermal plumes, that eventually reach the Earth’s surface as ocean island basalts, originate at structures near the core-mantle boundary known as Large Low Shear wave Velocity Provinces or “LLSVPs”. LLSVPs are structures in parts of the lowermost portion of the Earth’s mantle, characterized by slow shear wave velocities and higher density than the surrounding mantle, which were discovered by seismic tomography of the deep Earth.

An accurate numerical model, coupled with the modified volume-of-fluid method and the hybrid Cartesian immersed boundary method, has been developed to investigate the evolution of undulation of free surface influenced by moving topographies. The dynamic processes comprise topology changes of moving boundaries, at which the air-liquid interface and the solid-liquid interface are both involved. To accurately predict the topology changes along the air-liquid interface, a modified volume-of-fluid method is proposed. The resulting features of evolving surface wave and its possible amplitude amplification are mainly concerned. An optimal local advection scheme associated with the sub-grid resolution of interface tracking technique is developed in a semi-Lagrangian description. Additionally, the hybrid Cartesian immersed boundary method with bilinear interpolation enables to place solid bodies of arbitrary shape in fluid flows, while referring to the ease of mesh generation and its inherent simplicity to implement moving bodies on Cartesian grids. A second-order spatial accuracy is maintained as the two combined method is conducted. To show the accuracy and capability of the proposed model, a number of examples with increasing complexity of bottom boundary are well validated with the existing experimental data and numerical results.

This article presents a sharp-interface model for solving solid-liquid phase change of pure materials. An interface capturing method combining VOF and level set, VOSET, is employed to track the moving interface during phase change, and the level-set function generated in VOSET is used to consider the effect of the phase boundary when solving the fields of temperature and velocity. In order to deal with the interaction between the solid and liquid phases, an immersed boundary method (IBM) is incorporated into SIMPLER algorithm. The fundamental numerical approaches presented in this article were individually assessed by test problems, all showing good agreements with benchmark solutions. The established sharp-interface model was then applied in simulating melting and solidification without and with fluid flow, where competitive performance in accuracy was demonstrated by the numerical results.

A novel way to handle surface tension gradient driven flows is developed in the volume-of-fluid (VoF) framework. Using an open source Navier-Stokes solver, {\it Basilisk}, and the present formulation, we investigate thermocapillary migration of drops/bubbles in a surrounding medium. Several validation exercises have been performed, which demonstrate that the present solver is a robust one to investigate interfacial flows with variable surface tension. It is well known that it is a challenging task to numerically model the tangential and normal surface forces arising due to interfacial tension. We have shown that the present method does not require the artificial smearing of surface tension about the interface, and thus predicts the theoretical value of the terminal velocity of bubble/drop migrating due to an imposed temperature gradient very well. It is also demonstrated that the present solver provides accurate results for problems exhibiting the gravity and thermocapillary forces simultaneously, and useful for systems with high viscosity and density ratios., 30 pages, 16 figures, submitted to Computers and Fluids

A hybrid multiphase method is proposed to simulate the cavitation inside the nozzle and the primary breakup of a diesel spray. The improved method coupling the cavitation sub-model with the Multi-fluid-quasi-VOF (volume of fluid) model is developed in the OpenFOAM code. In the cavitation sub-model, heterogeneous nucleation of bubble is considered. By reformulating the volume fraction transport equations for liquid and vapor phases in the Multi-fluid-quasi-VOF model, the mass transfer source terms due to cavitation are considered. Moreover, the drag force, virtual mass force and surface tension force of diesel are considered in momentum equation. The computational model is validated by comparing the simulated mass flow rates and the cavitation distributions in an optical nozzle at different inlet pressure against published experimental data. The details of in-nozzle flow and the primary breakup of a diesel spray, such as flow separation, wall shear, cavitation in nozzle and the fragmentation of the jet in spray chamber are captured by applying the proposed method. The area of the evaporation region is smaller than the cavitation area, which leads to the detached tail of the cavitation from the nozzle wall in developing cavitation. The improved method reveals the velocity difference is fairly large at the gas-liquid phase interface and the umbrella shape head bears a great drag force. In addition, plenty of vortexes emerge as the results of the sufficient velocity difference across the interface between air and diesel, which stimulates the Kelvin–Helmholtz instability to promote the initial development of waves on the surface of the diesel column jet and subsequent rapid growth.

As a novel micro-reactor, the microporous tube-in-tube microchannel reactor (MTMCR) has been applied to some processes such as gas absorption, preparation of nanomaterials for the past few years. In this work, the three-dimension simulations in the microporous tube-in-tube microchannel reactor (MTMCR) have been carried out via computational fluid dynamics (CFD) to study the influence of various parameters on gas–liquid flow behaviors, and the coupled level set and volume of fluid method was applied to capture the gas–liquid interface. The liquid distribution in MTMCR by CFD was basically in agreement with the results of visualization experiment and the time-averaged pressure drops were in agreement with literature data within deviation of ±5%. The simulation results showed that the interfacial area and pressure drop increased with the increase of gas or liquid flow rates. The smaller micropores size and annular channel width led to a larger interfacial area and higher pressure drop. Three typical liquid flow patterns were revealed: droplet flow, elongated-droplet flow and churn flow, and the interfacial areas ranged from 976 to 9172 m2/m3 under operating conditions. This work indicated the great potential of the MTMCR for intensifying the gas–liquid mass transfer process, and provided theoretical basis of the MTMCR for practical applications.

Circular hydraulic jump (CHJ) and its deviation from the azimuthal symmetry due to non-orthogonal impingement of liquid jet on a solid surface are studied using finite volume based numerical framework. Asymmetric interfacial jump has been modelled from volume of fluid (VOF) prediction for a different angle of inclination of a jet from normal of the surface. The extent of asymmetry is found to be increasing as the jet becomes more and more horizontal, deviation from orthogonal impingement. To understand the role of wall adhered jet on jump, flow physics has been studied which reveals back flow in the jump region causing circulation. Moreover, equipotential and heterogeneous jump–jump interactions are studied to predict fluidic features like fountain formation and upwash. Numerical simulations reveal that strength of interacting wall jets play a major role in the formation of a pattern as consequence of neighbouring twin jet impingement. A vertical flat liquid protrusion is observed at equidistant stagnation line from the impingement points of equipotential jets. On the other hand, a curvilinear upwash film has been found out as characteristics of the interaction between heterogeneous strengths of interacting jets. At a higher ratio of jet strengths, complete engulfment of a smaller hydraulic jump as an eye of larger one has been obtained from careful numerical simulation. Physical insights of these rich fluidic phenomena are described using well-resolved velocity vectors in the wall jet and upwash.

The hydrodynamic loadings on underwater pipelines are essential parameters for pipeline design. This study integrates a Large Eddy Simulation (LES) model and the Volume of Fluid (VOF) method to examine the interaction between free surface flows and a horizontally-mounted circular cylinder. The simulated water depths and surface pressures on the cylinder were verified by the results of laboratory experiments. The numerical model was then utilized to inspect the effects of the Froude number, gap ratio and depth ratio on the flow field and the force coefficients of the submerged circular cylinder. The simulation results revealed that the effects of the water surface occur when the gap ratio h/D ≤ 1.50. The Strouhal number first increased, and then decreased as the cylinder came close to the water surface. The vortex shedding was completely suppressed when the water depth drop behind the cylinder and the downstream was a super-critical flow. In addition, when the downstream water depth changed significantly, the drag coefficient CD of the cylinder was larger than that of the cylinder in an unbounded flow. The lift coefficient increased from 0 to 0.80 as the gap ratio h/D decreased, due to the asymmetric pressure distribution.

The nonlinear interaction between a bubble and a nearby free surface is an important phenomenon, which becomes more complex when the bubble breaks and re-closes at the free surface. In this paper, an axisymmetric bubble dynamics model based on the Eulerian finite element method with the interface tracked by the volume of fluid (VOF) method is established to numerically investigated the breaking and re-closure of a bubble near a free surface. An experimental validation of the numerical model reveals its good accuracy. The motion, breaking, and re-closure of small and large bubbles at a free surface are simulated. Complex phenomena are observed in the simulation after the break of the bubble. Under the effects of inward gas flow and the pressure difference between its inside and outside, a broken bubble finally recloses. The convergence and impact of a spike wall generate an upward water spray higher than that from an intact bubble. Furthermore, the critical depth at which the bubble period reaches its peak has a negative relationship with the weight of the explosive charge used to generate the bubble.

The Volume-of-Fluid (VoF) model with interface-capturing scheme, available in many commercial and open-source CFD packages, may be widely used for simulations of free-surface hydrodynamics but is certainly not used without difficulties. While practitioners have come to expect conservation of mass and momentum, unconditional stability, second-order of accuracy and good iterative convergence, none of these qualities are evident when a free surface gets involved. In this paper, we investigate how these qualities are affected by the discontinuous nature of the volume fraction, which requires specialized numerical techniques for advection, for the interpolation of material properties, for the hydrostatic balance and for the pressure-weighted interpolation used in finite-volume methods with co-located variables. We find that conservation properties and good iterative convergence can still be attained, but not second-order of accuracy and unconditional stability. Code verification is presented to substantiate these findings, with cases ranging from one to three dimensions, from uniform to locally refined grids and from the stand-alone volume fraction equation to the complete set of mass, momentum and volume fraction equations. For the latter, a novel manufactured solution based on a sinusoidal wave in deep water is introduced. These tests permit not only to quantitatively assess the results for comparison with theoretical predictions but also ensure that our findings are characteristic of the VoF method and not of a particular implementation.

Two-phase flow inside a water retarder is numerically studied to understand the water retarder performance and momentum dissipation mechanism for partially and fully-charged conditions at different rotational speeds, and the effects of external cooling flow into the water retarder. In order to accurately capture the flow interaction between the blades of different rotating speeds, the transient simulations are performed using unsteady turbulent flow assumption using a sliding mesh technique. The Realizable k-e turbulence model and the Volume of Fluid (VOF) method are adopted to simulate two-phase motion and interface of air and water mixture inside the retarder. A momentum index analysis is proposed for investigation of flow momentum and its dissipation. Results of partially-charged cases demonstrate that as a general trend, air is accumulated at the center of the hydraulic system forming a toroidal shape. At a low charged water volume flow, circulation inside the retarder and, hence, the retarding torque are trivial. Furthermore, it is observed that the retarding torque becomes very small for the charged water values lower than a special value. Finally, the simulations of the fully-charged open system with external cooling flow (through an inlet and an outlet) reveal that the retarding torque and also flow circulation between wheels decrease with the increase in flowrate.

In this paper we present a numerical method for the simulation of two-material flows governed by the compressible unsteady Euler equations. A system of six equations is used, with a pressure relaxation step which ensures convergence to the equivalent five-equation model. The numerical method employs a volume-of-fluid interface tracking method in order to maintain a sharp interface indefinitely, along with a robust update procedure for interfacial cells which ensures density and internal energy positivity. Numerical results are presented for a range of test cases, including for liquid–gas shock–bubble interaction and cylindrically convergent Richtmyer–Meshkov instability between metal and air. Results are in agreement with theoretical data, and compare well to other approaches designed to sharpen the material interface in the five-equation model.

Probability density functions (PDF) and the relevant methods have been widely used to describe non-linear phenomena in the realm of turbulence modelling and CFD. In order to solve PDF transport equations, the main trend of previous studies rely on Monte Carlo method with Lagrangian particle tracking. However, as with any Lagrangian based approach, the scalability of the parallelized simulations of such method is less than satisfactory. An Eulerian stochastic field model has been presented recently by Dumond et al. [1] to simulate cavitating flows. Their model uses a fully compressible density based solver. Here we present an adapted version using an iso-thermal cavitation model adopting the homogeneous mixture assumption in a pressure based flow solver which is more relevant to engine simulations. A PDF method is used to represent a distribution of vapour volume fractions, based on which the Eulerian stochastic field (ESF) method is applied to perform a three-dimensional large eddy simulation (LES) of the cavitation phenomena inside an academic fuel injector configuration. The numerical model is based on a volume of fluids approach and coupled with a pressure based solver for the flow field, and is implemented in the framework of the open source C++ toolbox OpenFOAM. The result of the ESF simulation is compared against that from a typical single volume fraction solver for validation. Vortex structures and its correspondence to cavitation are shown, and the behaviour of the PDF at different probe locations at different times are acquired to demonstrate the potential of the ESF model in capturing both transient and stochastically steady cavitation.

The three-dimensional (3D) Rayleigh–Taylor instability (RTI) problem is numerically investigated using the coupled level set and volume of fluid (CLSVOF) method. The CLSVOF method combines the level set and volume of fluid methods to ensure good mass conservation performance, enabling the curvature of the interface captured by numerical simulations to be accurately calculated. A classical single-mode RTI (with only one “mushroom” generated under the initial setting) is adopted for numerical verification, and the shape of the interface and the penetration depth with respect to time are determined. Based on this initial two-dimensional single-mode RTI problem, the influence of factors such as the surface tension, density ratio, Reynolds number, and viscosity ratio on the 3D effects caused by the velocity gradient on both sides of the interface are thoroughly analyzed. This analysis allows us to obtain the basic laws governing the influence of different physical factors on the 3D effects of the RTI.

Numerical simulation of the liquid break-up and prediction of the droplet size distribution of injection devices is a challenging task. For predicting the disintegration process, interface capturing techniques, e.g. the volume-of-fluid method, are mandatory. The extensive computational effort caused by the high resolution requirements in space and time determine the limitation for many applications. Validations of such simulations with experimental data can be only rarely found in the literature. This study presents a comprehensive numerical and experimental investigation of the break-up process at an industrial low-pressure three-hole injector. The numerical simulations for three different injection pressures are performed with an efficient volume-of-fluid approach. At certain time instants the volume fraction field is scanned for coherent liquid structures, and the ligament size distribution is evaluated. The simulated mean droplet diameters and the droplet size distribution functions show a good agreement with the performed measurements. The high computational effort, caused by the fine mesh resolution, limits the axial extension of the simulated domain, here fifty times the nozzle diameter. In order to extend the simulated spray domain, an off-line coupling approach with a Lagrangian particle method is proposed. Therefore, the detached droplets from the volume-of-fluid simulation are used as initial conditions for the subsequent Lagrangian spray simulation on a coarser computational grid. This presented second step determines the link between liquid disintegration on a micro-scale, and spray simulation on a macro-scale. The aim of this study was to develop a predictive simulation tool for determination of the injector characteristics.

Cavitation bubble always exists in the droplets breaking up from the fuel jet when supercavitation phenomenon occurs. The collapse of cavitation bubble plays an important role in secondary breakup of droplets. The collapse process of cavitation bubble within the diesel droplet considering the phase change or not have been simulated numerically based on the volume of fluid (VOF) method using OpenFOAM. The results show that collapse process without phase change consists of multiple collapse and rebound stages, similar to the vibration process of a damping spring oscillator. Considering the phase change, there is only collapse stage and no rebound stage in collapse process, and the variation of bubble volume with time is no longer cyclic. The collapse speed become faster with the increasing of environmental pressure, while the collapse speed become faster with the decreasing of saturation vapor pressure. The effect of surrounding pressure on collapse process is greater than the effect of saturation vapor pressure.

A three-dimensional volume-of-fluid (VOF) interface-sharpening method is developed on the general curvilinear grid for two-phase incompressible flows. In this method, a VOF discretization scheme is formulated for the advection of a two-fluid interface. To maintain interface sharpness, a treatment is applied by solving an interface-sharpening equation after each advection time step, thereby reducing the numerical diffusion error in the solution of the discretization scheme. To demonstrate the accuracy and capability of the advection scheme, several numerical experiments involving three benchmark tests of pure advection were conducted. The results show that the method can realize a sharp interface reliably and efficiently, and reasonable mass conservation is obtained. For the flow field of viscous incompressible flows, the Navier–Stokes equations are solved by adopting the dual-time preconditioning method. A fully implicit method with a highly efficient lower-upper symmetrical Gauss–Seidel (LU-SGS) algorithm based on a dual-time stepping technique as a sub-iteration scheme is employed to advance the solution in time. To validate the proposed method for computing incompressible free surface flows, a dam-break flow over a dry horizontal bed and the water entry of a hemisphere with one degree of freedom are simulated. Comparisons of the predicted results with the available experimental data are presented herein.

A parallel computing Eulerian/Lagrangian multi-scale coupling procedure for diesel spray simulation is presented. Early breakup of the diesel jet is captured by using a compressible Volume of Fluid (VOF) method. In regions where the phase interface can no longer be sufficiently resolved, separated and small scale liquid structures are described by a Lagrangian Parcel Tracking (LPT) approach, in conjunction with secondary breakup modelling and a turbulence stochastic dispersion model. The coupling of these two descriptions utilises a Region Coupling Method and an efficiently parallelised droplet identification and extraction procedure. This approach enables run-time VOF-LPT field coupling and filters small-scale liquid structures that are suitable candidates for Eulerian-liquid-structure/Lagrangian droplet conversion, preserving their position, mass and momentum. The coupling procedure is initially applied to model the atomisation of a simple liquid jet and the results are compared with that of a statistical coupling approach to demonstrate the performance of the developed coupling procedure. Its application is then extended to simulate a real diesel spray from a nozzle with a sharp entrance. Coupling in-nozzle phenomena such as flow separation, flow detachment and turbulence to the primary and secondary spray atomisation, provides a tool for the prediction of complex spray dynamics.

The volume of fluid (VOF) method is widely used to simulate the flow of immiscible fluids. It uses a discrete and sharp volume fractions field to represent the fluid-fluid interface on a Eulerian grid. The most challenging part of the VOF method is the accurate computation of the local interface curvature which is essential for evaluation of the surface tension force at the interface. In this paper, a machine learning approach is used to develop a model which predicts the local interface curvature from neighbouring volume fractions. A novel data generation methodology is devised which generates well-balanced randomized data sets comprising of spherical interface patches of different configurations/orientations. A two-layer feed-forward neural network with different network parameters is trained on these data sets and the developed models are tested for different shapes i.e. ellipsoid, 3D wave and Gaussian. The best model is selected on the basis of specific criteria and subsequently compared with conventional curvature computation methods (convolution and height function) to check the nature and grid convergence of the model. The model is also coupled with a multiphase flow solver to evaluate its performance using standard test cases: i) stationary bubble, ii) oscillating bubble and iii) rising bubble under gravity. Our results demonstrate that machine learning is a feasible approach for fairly accurate curvature computation. It easily outperforms the convolution method and even matches the accuracy of the height function method for some test cases.

The present study proposes a new method for numerical simulations of spray break-up of liquid jets with cavitation in the nozzle. A three-phase system consisting of liquid, vapor and gas is applied for the Volume-of-Fluid simulation of the liquid disintegration in order to resolve the liquid–gas interface. To keep the numerical effort moderate, the liquid–vapor interface is not resolved by the computational grid, mass and momentum transfer are described within the Eulerian–Eulerian framework. The numerical method is validated against experimental data from the literature, where a simplified injector-like geometry is operated at different injection pressures in order to show different cavitation regimes. The good agreement in the comparison of the velocity profiles in the injector, and in the visual comparison of cavitation cloud and spray pattern, demonstrate the feasibility of the method.

In this paper the characteristics of the primary breakup of a liquid jet is analyzed numerically. We applied the Volumes of Fluids (VOF) approach utilizing the Direction Averaged Curvature (DAC) model, to estimate the interface curvature, and the Direction Averaged Normal (DAN) model, to propagate the interface. While being used for the first time to predict liquid atomization, this methodology showed a high accuracy. The influence of varying the fluid properties, namely liquid-gas density and viscosity ratio, and injection conditions is discussed related to the required grid resolution. Resulting droplet sizes are compared to distributions obtained through the One-Dimensional Turbulence (ODT) model. (C) 2016 The Authors. Published by Elsevier Ltd.

A new PLIC (piecewise linear interface calculation)-type VOF (volume of fluid) method, called APPLIC (approximated PLIC) method, is presented. Although the PLIC method is one of the most accurate VOF methods, the three-dimensional algorithm is complex. Accordingly, it is hard to develop and maintain the computational code. The APPLIC method reduces the complexity using simple approximation formulae. Three numerical tests were performed to compare the accuracy of the SVOF (simplified volume of fluid), VOF/WLIC (weighed line interface calculation), THINC/SW (tangent of hyperbola for interface capturing/slope weighting), THINC/WLIC, PLIC, and APPLIC methods. The results of the tests show that the APPLIC results are as accurate as the PLIC results and are more accurate than the SVOF, VOF/WLIC, THINC/SW, and THINC/WLIC results. It was demonstrated that the APPLIC method is more computationally efficient than the PLIC method.

A coupled level-set and volume-of-fluid (CLSVOF) method is proposed for tracking interface of incompressible two-phase flows, where the VOF advection is carried out only, and a zero level-set (LS) formula and LS reinitialization equation are employed to reestablish the signed distanced field near the interface, which can program easily. Meanwhile, a mass-correction scheme based on the interface curvature is used to conserve the liquid mass correctly. Firstly, the performance of the CLSVOF method is evaluated through two- and three-dimensional deformation field tests. Then, the CLSVOF method is adopted to simulate the rising bubble and droplet falling problems. Finally, a shape LS function is used to describe and treat the irregular surface (immersed boundary) of mold cavity. With the aid of the shape function, the coupled immersed boundary (IB) and CLSVOF (IB-CLSVOF) method is applied to the melt filling process in the complex mold cavity with insets. The numerical results show that the proposed IB-CLSVOF method has the capability to cope with complex interface variations accurately. And also, the numerical results demonstrate that IB-CLSVOF method is able to successfully depict the phenomena of race-tracking, air entrapment and weldline development, which are very helpful for determining processing conditions and designing mold cavities.

A three-dimensional two-phase flow model is developed to study wave interaction with structures by using a virtual boundary force (VBF) method. This model solves the Navier–Stokes equations for two-phase flows. The second-order accurate volume-of-fluid (VOF) method is used to track the free surface. In order to satisfy the no-slip condition on the rigid body surface, the virtual boundary forces are applied along the surface of the body in the computation. The model is first validated for wave interaction with a structure with the curved surface. This model is then used to simulate various types of two-dimensional and three-dimensional wave–structure interaction problems, i.e., solitary wave runup and rundown on a sloping beach, nonlinear wave transformation above a submerged breakwater, and wave diffraction around a large circular cylinder. The computational results compare well with available analytical solutions and experimental data. Lastly, as a demonstration, the three-dimensional wave breaking on a floating truncated circular cylinder is simulated and discussed.

Coupling techniques that take advantage of the mass conservation property of the volume-of-fluid (VOF) method and the sharpened interface computation of the Level Set (LS) approach are widely used for computations with high accuracy order demands. In this paper, an enhanced coupling method for interface computations in incompressible two-phase flows is presented. In the proposed method, the solution of the re-initialization LS function is reformulated in a conservative form before it is applied to estimate the interface curvature and interface normal vector. The estimated interface exhibits significant improvements in smoothness and accuracy compared to those obtained with the original VOF method and a previous coupling method. The proposed method is then implemented in an incompressible Navier–Stokes solver (interFoam) in the OpenFOAM platform to solve several benchmark tests. Good agreement between the simulated results and the analytical/benchmark solutions with well-preserved mass conservation is obtained for standard tests, including a reversed single vortex, static droplet, and rising bubble, thus demonstrating the potential of the proposed technique for both academic research and practical applications.

Multiphase, compressible and viscous flows are of crucial importance in a wide range of scientific and engineering problems. Despite the large effort paid in the last decades to develop accurate and efficient numerical techniques to address this kind of problems, current models need to be further improved to address realistic applications. In this context, we propose a numerical approach to the simulation of multiphase, viscous flows where a compressible and an incompressible phase interact in the low-Mach number regime. In this frame, acoustics are neglected but large density variations of the compressible phase can be accounted for as well as heat transfer, convection and diffusion processes. The problem is addressed in a fully Eulerian framework exploiting a low-Mach number asymptotic expansion of the Navier-Stokes equations. A Volume of Fluid approach (VOF) is used to capture the liquid-gas interface, built on top of a massive parallel solver, second order accurate both in time and space. The second-order-pressure term is treated implicitly and the resulting pressure equation is solved with the eigenexpansion method employing a robust and novel formulation. We provide a detailed and complete description of the theoretical approach together with information about the numerical technique and implementation details. Results of benchmarking tests are provided for five different test cases.

International audience; Free surfaces and fluid interfaces are encountered in a wide variety of gas-liquid configurations. Although many numerical approaches exist to solve such flows, there is still a need for improved simulation methods. Recently, a new efficient geometric VoF method for general meshes, called isoAdvector, was implemented in OpenFOAM®. More recently, the isoAdvector method was significantly improved by introducing a variant using a reconstructed distance function (RDF) in the interface reconstruction step. Elementary quantitative benchmarks are essential for validation and comparison of interfacial solvers. We present here three benchmarks results that were used for validation of the original and new variant of the isoAdvector method. Comparisons are made with reference data and OpenFOAM® VoF original solver, interFoam, employing the MULES limiter. The first case is the static bubble under zero gravity. The RDF reconstruction method demonstrates better prediction of the interface curvature, pressure jump between the phases and strong reduction of the spurious currents. The second and third validation cases are single rising bubbles in a quiescent liquid, with a spiraling path for the third case. Either on hexahedral or tetrahedral grids, the RDF reconstruction method demonstrates a better behavior.

In this study, to simulate the pollution treatment as a separated phase a 3D incompressible turbulence model under isothermal conditions was considered. The hybrid RANS-LES method with a sub-grid scale(SGS) technique within the framework of two-phase flow was used to study pollution distribution in urban street canyons. An area source of pollutant with a constant concentration, located in the ground of the street canyon, was used to simulate vehicular emissions. The near-wall regions were simulated using the RANS(k-e) turbulence model and the rest of the domain was simulated with the LES model. To measure the effect of building height on pollutant dispersion, the parameter AR(Aspect Ratio) was calculated. Also to determine the effect of street width, a new parameter SWV(Street Width Value) was introduced. Governing equations were discretized using the well-known finite volume method on a staggered grid. To increase the accuracy, the second-order Van Leer Method(VLM) was applied and the Volume of Fluid(VOF) method was employed for the description of the interfaces. The Iterative line-by-line TDMA solution procedure and the Strongly Implicit Procedure(SIP) was used for the Pressure Correction Equation. A fine comparison of the results revealed that by increasing SWV, the number of vertical vortices in the canyon streets generally decreased and the vortices grew larger. In addition, increasing AR caused a rise in the pollution volume fraction in the first canyon street. The current numerical results indicate that the AR and SWV influence the flow field and volume fraction of phases inside an isolated street canyon, significantly.

This research work deals with the computational simulation of the transitions of dripping to jetting which occurs during the flow of Newtonian Fluid. Herein, Volume of Fluid (VOF) model is mapped for the detention of various regimes of drop formation. The transition of drop formation from dripping to jetting is easily categorised through Weber number. The present work reveals that jetting regimes are observed at high Weber number, whereas for the small values of Weber number, dripping and dripping faucet regimes are seen. Critical conditions under which the dripping to jetting transition takes place are arrived by analysing different nozzle diameters subjected to various flow rates.

We extend a higher-order finite volume cell-centered hydrodynamic (CCH) formulation to include an interface-aware subscale closure model and a multi-material remap for simulating 3D compressible hydrodynamic problems within an arbitrary Lagrangian-Eulerian (ALE) framework. This CCH formulation involves a multidirectional approximate Riemann solution using quadratic polynomial reconstructions of the stress tensor and the velocity. At the subscale level, we determine pair-wise material interactions by solving a distinct approximate Riemann problem at the common interface, using the volume of fluids (VOF) method to find the interface. Material interactions are constrained to ensure smooth pressure equilibration among materials. The accuracy and robustness of the ALE method is demonstrated by simulating a suite of 3D Cartesian multi-material problems covering both gas and solid dynamics, where each test case has two or more materials.

There are different approaches and methods to predict the interface changes in two-phase flows. Among these methods, volume-of-fluid (VOF) and level-set (LS) are some of the most famous ones. Common VOF schemes are mass conservative but cannot predict the surface tension with a good accuracy. In contrast, LS uses a continuous sign function which in turn computes surface tension more accurately than VOF. But, re-distancing the LS function causes mass gain/loss which violates the mass conservation. To have the advantages of both methods, a scheme called VOSET is introduced in the present research which couples the two methods fully geometrically. VOSET is used in the structure of the code developed to study the interface topological changes of gas–liquid two-phase flows. The results show that in addition to being mass-conserved, the method computes the surface tension with a good accuracy. Then the code is used to study the bubble topology rising in a quiescent liquid for different Morton and Eotvos numbers. The simulation results show a good agreement with available experimental data. Finally the effect of bubble initial shape on its terminal shape and velocity is investigated by VOSET. For the simulation cases, the effect of initial bubble topology was not noticeable.

Numerical simulations of rainwater morphology evolution on a cylinder subjected to wind are investigated. Direct numerical simulations (DNS) and large eddy simulations (LES) are combined to simulate this complicated process based on a 2-D multiscale model. A Lagrange particle method is used to simulate rain droplets in the LES zone, far away from the cylinder surface. When droplets fall into the DNS zone, i.e., near the cylinder surface, the particles are converted into liquid droplets. The volume-of-fluid (VOF) method is then adopted to capture the morphology of the interface of the liquid–gas system. Rainwater morphology evolution along the circumference of cylinder can be categorized into four patterns over the analysis of the entire evolution process: collision–splashing, accumulation–slipping, formation–breaking, and dynamic equilibrium. The morphological characteristics of rainwater of each pattern are summarized. The circumferentially vibrating upper rivulet across the cylinder surface is observed. The resultant multiscale model is validated by comparison with the previous results from related studies.

This paper presents a three-dimensional VOSET method, which combines both advantages of the VOF and level set. The basic idea is the same with that in 2D VOSET, but new methods are proposed to deal with new geometric problems in three dimensions. An iterative root-finding method is used for implementing Piecewise Linear Interface Construction (PLIC) in three dimensions. An iterative geometric method is presented to calculate the level-set function. The feasibility and accuracy of this method are analyzed by some classical test problems. Combining the use of projection method that solves the flow field, the 3D VOSET is adopted to simulate a liquid dam break problem and a single-bubble rising problem. Our numerical results are found to be in good accordance with those in previous studies.

A new volume of fluid (VOF) based two-phase flow solver has been developed that has been parallelized for general purpose graphical processor units (GPGPU). Incompressible Navier–Stokes equation is explicitly solved using finite volume method on a 2-D Cartesian grid. The pressure Poisson equation is solved on the GPU in conjunction with the geometric multigrid method. This solver performance is established by validating it against standard benchmark test cases. Furthermore, the solver acceleration is determined by comparing it with the serial version of the code. The solver was then used to study the primary jet breakup of a 2-D planar liquid jet due to a co-flowing gas stream. The effect of liquid to gas phase velocity ratio on jet breakup was investigated.

In this work we develop, test and apply a height-function-based numerical code for the simulation of multiphase magnetic fluids. New hierarchical height function algorithms for interface normal and curvature (Bornia et al., 2011) are adapted and integrated into the Volume of Fluid component of our Navier-Stokes-Maxwell code for magnetic fluids and charge-free quasi-static magnetic fields. Accuracy and convergence results are presented for highly elongated ellipses characteristic of magnetic fluids, for static non-magnetic fluid droplets, and for magnetic fluid droplets elongated by an imposed uniform field. The code is applied to Magnetic Drug Targeting to demonstrate the response of magnetically controlled fluid regions to an imposed quadratic flow profile in a model vein.

International audience; The aim of this study is to discus different numerically models for the simulation of moving contact lines in the context of a Volume of Fluid–Continuum Surface Force (VoF–CSF) method. We focus on the particular situation of spreading drops. We first present the numerical methods used for the simulation of moving contact line i.e. static contact angle versus dynamic contact angle, no slip condition versus slip condition. A grid and time convergence is performed for the different models. We show that the integration of the Continuum Surface Force using the finite volume method results in a grid dependence at the onset of the spreading. The static and dynamic models are compared to experiments. It is shown that the dynamic models based on the Cox’s relation for the dynamic contact angle are able to reproduce experiments while static models overestimate the spreading time and are not able to reproduce the Tanner regime. The difference between static and dynamic models is shown to increase with the Ohnesorge number.

Suspending a mesh cylinder in a tube modulates flow patterns with gas bubbles in the annular region to form thin liquid film on the wall. We investigate the effect of gravity levels on the modulated flow patterns. The volume of fluid (VOF) method simulates the slug bubble train flow in bare tube and modulated flow sections. The bubble population density along the flow direction (β) and averaged liquid film thickness (δα) synthesize a parameter β/δa to characterize the enhancement of phase change heat transfer. It is found that at the normal gravity on earth, counter-flow appears with fast upward flow in the annular region and downward liquid flow in the core region. A lower β due to sparsely populated bubbles and thin liquid film form a larger β/δa to enhance the phase change heat transfer. Convective heat transfer in liquid plugs is enhanced by the fast fluid movement in the annular region and liquid circulations at three length scales. At the miniature gravity, quasi-co-current flow happens with upward flows in both annular region and core region, except that a liquid layer inside mesh cylinder flows downward. Bubbles are more densely populated than those at the normal gravity. Liquid circulation occurs only at the bubble length scale. At the micro gravity, the two-phases are thoroughly separated with co-current flows in both annular region and core region. Gas flows slowly and the residence time of gas is increased to result in β = 1. Τhe liquid film is ultra-thin. These two factors create a significantly large β/δa to enhance the phase change heat transfer. We demonstrate the effectiveness of the phase separation concept at miniature and micro gravity environment.

A generic methodology for the design of interface capturing schemes for immiscible multi-fluid flows using the VOF (Volume-Of-Fluid) approach is outlined. Interface capturing schemes are devised as a blend of high-resolution and compressive schemes and their efficiency and accuracy are dependent on the choice of the constituent schemes and the blending function. On the basis of a set of design principles proposed in this work, we show that interface capturing schemes proposed in literature may be encompassed into a single class of GPL (Generalised Piecewise Linear)- κ schemes allowing for a unified approach for development of such schemes. This methodology is used to de-construct four existing schemes and to propose two new schemes for interface capture in an unstructured finite volume framework. The schemes are tested on challenging advection problems using both structured and unstructured grids to evaluate their performance. The new schemes perform as well as existing schemes and even outperform them on unstructured grids, while also exhibiting near Courant number independence for all test problems and grid topologies. The CUIBS (Cubic Upwind Interpolation based Blending Scheme) proposed in the study shows the best performance among all interface capturing schemes discussed herein and is applied to the study of immiscible binary fluid flows. Numerical simulations of viscous sloshing in a tank and Rayleigh–Taylor instability demonstrate the ability of the CUIBS scheme to resolve fluid–fluid interfaces with minimal diffusion.