Volume Of Fluid Equation Research Articles

VOF method in two-stage fourth order time-stepping framework

The fraction volume of fluid (VOF) method is a successful approach to track the dynamics of free boundaries thanks to its flexibility, efficiency and simplicity. And various efficient numerical methods have been developed to solve the associated VOF equation. This paper applies the two-stage fourth order accurate time discretization in [21] for the evolution of the VOF equation, aiming to provide a compact diffuse-interface algorithm simple for implementation, in order to incorporate with CFD shock-capturing schemes for the simulation of multi-material/phase problems with sharp interfaces. As such, the algorithm is decoupled from those for fluid flows as long as the velocity of the interface is given. Numerical results demonstrate the high resolution and high-fidelity for the interface capturing.

OpenFOAM modeling of beryllium melt motion and splashing from first wall in ITER

Beryllium (Be) is a material which will be used as a plasma facing component in ITER due to its unique properties of high thermal conductivity, low density, and high strength. However, under extreme conditions of high temperature and pressure, Be can melt at the surface of tiles and molten droplets can be ejected into the reactor leading to disruption of fusion plasma. The pressure, mass density, velocity of Be vapor, and variations of temperature at the melt layer interface can influence the splashing of Be melt. The Computational Fluid Dynamics (CFD) model based on the OpenFOAM toolbox, a free open source CFD software package, was developed to treat the coupled flow of liquid Be metal and its vapor. The vapor-melt interface is modeled using the volume of fluid (VOF) approach implemented in the interCondensatingEvaporatingFoam solver that solves the continuity, momentum, heat conduction, and VOF equations. This CFD model is capable to predict the hydrodynamic effects of Be vapor on the melt layer motion, splashing, non-linear growth of melt waves, and ejection of molten droplets. The modeling accounts for the effects of thermal, viscous, gravitational, and surface tension forces at the vapor-melt interface. In this research, we used the interCondensatingEvaporatingFoam solver to simulate the effects of Be phase change and the development of melt motion with the droplets ejected from the surface. The CFD model accounts for inter-phase change between Be liquid and Be vapor. The evaporation model was validated against the Stefan phase-change problems. The influence of heat and mass transfer across the vapor-melt interface on melt layer stability is also investigated. The results provide an understanding of how the rate of phase change affects the development of melt structures and waves at the vapor-melt interface.

A numerical model for simulating the interaction of dam-break waves and floating structures

Using OpenFOAM, a numerical model is proposed to simulate the interaction of dam-break waves and floating structures. A geometric volume-of-fluid (VOF) method (isoAdvector) is combined with overset grid techniques. To avoid undesired air bubbles appearing in bulk water regions, a source term is introduced to the VOF equation. Attempts are made to alleviate mass conservation defects of the overset grid. It is found that: (i) Among the mesh interpolation schemes considered, the kernel scheme is the most preferable because it reaches a better balance between accuracy and efficiency; (ii) Extending donor cell stencils is not recommended because it magnifies the mass conservation defect of the overset grid; (iii) A strong coupling strategy is recommended because it alleviates the “water leak” phenomenon. Then, to validate the numerical model, two cases are simulated. The first case is a three-dimensional floating body case. The result demonstrates that this model performs better in terms of mass conservation than the model proposed by Wang et al. (2021). The second case is a two-dimensional experiment of the interaction of a dam-break wave and a floating box. The result demonstrates that this model has better temporal and spatial convergence than the default overset grid solver in OpenFOAM.

Analysis of Hysteresis in the Regime Transition of Cocurrent Liquid–Gas Flow

The dynamics of cocurrent liquid–gas flow control the flow patterns and phase distribution inside the submerged entry nozzle (SEN) in the continuous casting. This regime transition from bubbly flow to annular flow is usually associated with a hysteresis effect that is not fully understood yet. Herein, the regime transition in an analogous liquid–gas flow is investigated using the volume of fluid (VOF) method. A downward turbulent water flow in a vertical pipe is considered as the computational domain at the top of which the gas is injected as a volumetric source in the VOF equation. By temporal variation of the gas volume rate following a linear ramp‐up and then ramp‐down, a transition from bubbly to annular flow and vice versa is observed. However, the transition occurs at different operation points and the numerical simulation pictures this hysteresis phenomenon. The regime transition is connected to the evolution of interfacial turbulence in each phase represented by the amount of vortical energy, that is, enstrophy. In addition, the temporal variation of different enstrophy generation/destruction mechanisms is evaluated. The hysteresis phenomenon is explained by the differences in the history of these mechanisms and the difference in the enstrophy generation level upon transition.

An improved VOF method with anti-ventilation techniques for the hydrodynamic assessment of planing hulls-Part 1: Theory

The numerical prediction of the hydrodynamic characteristics for planing hulls has always been challenging due to numerical ventilation (NV), which is the unphysical entrainment of air under a hull. The capture of the wetted surface, the pressure distribution, the spray, and wave-making phenomena may suffer from this. In this paper series, the characteristics of the NV were analyzed, and anti-ventilation techniques based on the volume of fluid (VOF) method were developed and applied. The NV was formed by the unphysically transported air phase in the wetted surface. To counteract the unphysical transport, pseudo diffusive fluxes in specific directions were introduced by adding an artificial diffusion term to the VOF equation for the pressure area (PA). The unphysical results caused by other numerical issues could also be dissipated by this term. The boundness of the underlying discretization scheme for the modified VOF equation was proved by stability analysis. Based on the anti-ventilation techniques, a novel systematic procedure for the hydrodynamic assessment of planing hulls was developed to improve the accuracy, stability, and convergence rate. Part 1 of this paper series contains the details of the NV and the theory of the anti-ventilation techniques. Part 2 provides the application and validation cases.

Modeling of interfacial flows based on an explicit volume diffusion concept

A novel volume of fluid (VoF) model called explicit volume diffusion (EVD) is developed for the simulation of interfacial flows, including those with turbulence and primary spray atomization. It is derived by volume averaging the VoF equations over an explicit, physically-defined length scale. Fluctuations at the sub-volume scales, which can arise due to interface dynamics or turbulence, are attenuated by the volume averaging process and their effects appear in unclosed terms representing sub-volume flux, sub-volume stress, and volume averaged surface tension force. Models are proposed for each of these. The sub-volume flux is closed by a gradient diffusion model and involves an explicit volume diffusion coefficient, that is, linked to the explicit length scale. The sub-volume stress closure introduces an explicit volume viscosity that can be augmented by a turbulent viscosity in turbulent flows. The volume averaged surface tension force closure is based on a fractal representation of the wrinkled sub-volume interface. To test the closures individually and the EVD model overall, a series of two-dimensional laminar and three-dimensional turbulent interfacial shear flows and a laboratory-scale airblast spray jet are considered. Numerical convergence is demonstrated by keeping the explicit volume length scale constant while refining the numerical grid such that the numerical diffusion diminishes and becomes overwhelmed by the explicit volume diffusion. Sensitivity analyses are also conducted for variations in the explicit length scale which may be linked to the boundary layer thickness on the light fluid side of the interface.

Numerical convergence of volume of fluid based large eddy simulations of atomizing sprays

This paper presents three volume of fluid (VoF)-based methods for large eddy simulations of atomizing sprays with different treatments of the unresolved interface. The turbulent filtered VoF model uses conventional turbulent viscosity models to close the combined interfacial and turbulent sub-grid fluctuations. The hybrid turbulence filtering and artificial compression model includes an additional artificial compression term that is applied along regions where the liquid–air interface is continuous, while conventional turbulence filtering is activated in regions with discrete liquid objects. The new explicit volume diffusion model (EVD) is based on the concept of averaging the VoF equations over explicitly defined physical volumes that are independent of the numerical grid. Closure models of the sub-volume flux and stress terms introduce explicit volume diffusion and explicit volume viscosity that are physically defined and linked to the volume size. Numerical convergence is achieved by reducing the grid size while keeping the explicit volume size constant. The models are tested for two experimental atomizing spray cases with different Weber numbers. The superior numerical convergence of the EVD model is demonstrated by analysis of the mean and rms of the volume fraction and velocity fields. Two models for the surface tension force are investigated for the EVD simulations. Compared with the simple surface tension model which neglects sub-grid fluctuations, an improved volume-averaged model based on fractal properties of wrinkled sub-volume interfaces gives better predictions of the mean volume fraction relative to the experimental data but requires selection of a model constant.

Effect of heat transfer between melt puddle and cooling wheel on amorphous ribbon formation

Planar Flow melt spinning is a one step manufacturing process to produce amorphous ribbons by rapid solidification technique. Formation of amorphous structure in the ribbon highly depends on the rate of heat transfer between the melt puddle and the cooling wheel. The present study numerically analyzes the heat transfer in the melt puddle at the melt wheel contact and its influence on the amorphous structure in the end product. Space between cooling wheel and nozzle wall is taken as the computational domain initially filled with air. Flow, Energy, momentum equations along with Volume of fluid equation are employed to analyze the heat transfer and ribbon growth in the melt puddle. Geometric re-construction scheme is employed to show the melt and air boundaries in a two fluid flow domain. Numerical analysis shows a temperature gradient across the ribbon thickness. With increase in thickness the gradient is observed to increase. Experimental investigation has shown an increase in crystalline structure in the ribbon with thickness and shows a difference in XRD taken on both sides of the ribbons. The temperature gradient across the thickness of the ribbon is responsible for this phenomenon. As the model could predict this phenomenon, it can be used to predict the type of ribbon obtained at a set of process conditions prior to the experiment, thereby helps to avoid production of undesirable crystalline ribbons.

Numerical Hydrodynamic Performance of the Stepped Planing Craft and Its Step Height Effect

One of the most efficient methods of reducing drag on planing craft is the use of transverse step on the bottom of a hull. Applying steps on the hull reduces the contact area with water and as a result, it reduces drag of the craft. Planing craft are able to have one or two transverse steps. In this paper, numerical hydrodynamic performance of the stepped planing craft and its step height effect is investigated by making use of finite volume method (FVM). The Reynolds-Averaged Navier-Stokes (RANS) equations are coupled with the standard k- e turbulence model and volume of fluid equations are solved to simulate transient turbulent free surface flow surrounding the hull by ANSYS-CFX. In order to predict the motion of the craft, equations of two degrees of freedom for rigid body are coupled with governing equations of fluid flow. In order to validate the numerical model presented in this paper, the obtained numerical results are compared with the available experimental data. The numerical results obtained for drag, dynamic trim, rising of center of mass and the pressure distribution on the body at different speeds and different heights of the steps are presented and discussed.

VOF MODELING AND ANALYSIS OF FILMWISE CONDENSATION BETWEEN VERTICAL PARALLEL PLATES

In this study, a computational model has been developed to predict condensation heat transfer between vertical parallel plates. Transient simulations of filmwise condensation in a small two-dimensional parallel plate passage are performed. The Volume of Fluid (VOF) method is used to track the vapor liquid interface. The Geometric Reconstruction Scheme, which is a Piecewise Linear Interface Calculation (PLIC) method, is employed to keep the interface sharp. The governing equations and the VOF equation with relevant source terms for condensation are solved explicitly. The surface tension is taken into account in the modeling and it is evaluated by the Continuum Surface Force (CSF) approach. Different methods to evaluate the source terms in the VOF and energy equations are summarized based on previous studies. The simulation is performed using the CFD software package, Ansys Fluent, and an in-house developed code. This in-house code is specifically developed to calculate the source terms associated with phase change, which are deduced from the Hertz-Knudsen equation based on the kinetic gas theory. The predicted results show that a laminar regime exists at the top of the wall, where the film is the thinnest. A wavy regime appears as a series of regular ripples/waves of condensate moving downwards under the action of both gravity and shear stress in the interface area. As a further step, the simulations have been run under different surface tension, wall temperature, and inlet velocity conditions. The predictions also indicate that the wave peak height decreases with increasing surface tension and decreases with increasing inlet velocity. This has an effect on the heat transfer characteristics of the condensation process. The condensation heat transfer increases sharply by increasing the temperature difference between the wall and saturation temperature of the inlet steam. (Less)

Transient simulations of cavitating flows using a modified volume-of-fluid (VOF) technique

In this study, transient 2D/axisymmetric simulations of cavitating flows are performed using a modified ‘Volume-of-Fluid’ (VOF) technique. Simulation of the cavitation is based on a homogenous equilibrium flow model. To predict the shape of the cavity, the Navier–Stokes equations in addition to an advection equation for the liquid volume fraction are solved. Mass transfer between the phases is treated as a sink term in the VOF equation. The numerical method is used for different geometries in a wide range of cavitation numbers. Computed shapes of cavities were found to be in good agreement with those of the reported experiments. The simulation results also compared well with those obtained from analytical relations.

Front tracking in recirculating flows: a comparison between the TVD and RCM methods in solving the VOF equation

The modeling and simulation of mold filling must include a method to capture the interface formed between the inlet fluid and the fluid that was initially in the mold. A commonly used front-capturing method in a Eulerian mesh is the volume-of-fluid (VOF) method. The VOF advection equation solution may show numerical diffusion and/or dispersion and high-order numerical schemes, such as the TVD schemes with dimensional splitting, have to be employed to discretize the convective terms. The present contribution explores the use of RCM for solution of the VOF color-function equation during mold filling with recirculating flows. The Navier-Stokes equations are solved by a finite-volume method using the SIMPLER algorithm. Filling simulations using the TVD and RCM methods are compared. RCM was able to generate diffusion-free results, sharply defining the interface, even when topological changes (generation of droplets) occur.

NUMERICAL SIMULATION OF CONDENSATION ON A CAPILLARY GROOVED STRUCTURE

Condensation in a capillary groove is investigated using the volume of fluid (VOF) model. The governing equations are written in a generalized form and are applicable to both liquid and vapor phases. Condensation on the fin top and at the meniscus is modeled by introducing additional source terms in the continuity, VOF, and energy equations. The effects of temperature drop, contact angle, surface tension, and fin thickness on the condensation heat transfer are also investigated.

A free-front tracking algorithm for a control-volume-based Hele-Shaw method

With free surface flow problems the position of the free front at a particular time step needs to be predicted. When simulating the flow of hot molten polymer in injection moulds this position is essential for the prediction of the position of the weld lines on the final product. Weld lines are important in that they indicate positions of diminished strength and spoil the aesthetics of the product. A code using a finite volume approach on the thin gap Hele–Shaw flow equations to simulate the filling stage of the injection moulding process has been developed which requires a suitable algorithm for the free-front tracking. The Volume of Fluid (VOF) method was implemented in this study for the tracking of the free front. The discretization and various formulations of the VOF equation are presented as well as the results obtained. The impact on the existing equations is also investigated as well as the development of a packing routine that is used to eliminate the diffusive nature of the VOF equation. © 1998 John Wiley & Sons, Ltd.

A time-implicit filling algorithm

The volume of fluid (VOF) equation is a well-known method for tracking the liquid/air interface during a filling process. Standard numerical approximations of the VOF equation fail to preserve a sharp interface. Methods that ensure a sharp interface in VOF calculations are investigated in this paper. At first, two previous VOF schemes that limit the smearing of the interface, via appropriate approximation of the convective flux, are outlined. These schemes are explicit in nature and hence subject to time step restrictions. A new scheme for approximating the convective flux is introduced. Implicit solutions based on this modified scheme are smear free and are not subject to any time step restrictions. Solutions to example filling problems are in close agreement with solutions based on traditional VOF approaches. Some potential drawbacks of the new method are investigated. In the context of filling problems it is shown that these drawbacks are not significant.