Height Function Method Research Articles

An edge-based interface tracking (EBIT) method for multiphase-flow simulation with surface tension

We present a novel Front-Tracking method, the Edge-Based Interface Tracking (EBIT) method for multiphase flow simulations. In the EBIT method, the markers are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers. This feature makes the process of marker addition or removal easier than in the traditional Front-Tracking method. The EBIT method also allows almost automatic parallelization due to the lack of explicit connectivity.In a previous journal article we have presented the kinematic part of the EBIT method, that includes the algorithms for piecewise linear reconstruction and advection of the interface. Here, we complete the presentation of the EBIT method and combine the kinematic algorithm with a Navier–Stokes solver. A circle fit is now implemented to improve the accuracy of mass conservation in the reconstruction phase. Furthermore, to identify the reference phase and to distinguish ambiguous topological configurations, we introduce a new feature: the Color Vertex. For the coupling with the Navier–Stokes equations, we first calculate volume fractions from the position of the markers and the Color Vertex, then viscosity and density fields from the computed volume fractions and finally surface tension stresses with the Height-Function method. In addition, an automatic topology change algorithm is implemented into the EBIT method, making it possible the simulation of more complex flows. The two-dimensional version of the EBIT method has been implemented in the free Basilisk platform, and validated with seven standard test cases: stagnation flow, translation with uniform velocity, single vortex, Zalesak's disk, capillary wave, Rayleigh-Taylor instability and rising bubble. The results are compared with those obtained with the Volume-of-Fluid (VOF) method already implemented in Basilisk.

Numerical investigation on unstable oscillation of oxygen jet condensation in cryogenic liquid rocket

Oxygen jet condensation always occurs with intense unstable oscillation in the cryogenic delivery pipe of a space launch vehicle. This phenomenon threatens the safety of the propulsion pipeline. It is essential to explore the transient physical features of oxygen jet condensation precisely, including the spectrum of pressure oscillation. Aiming at a deeper understanding of the thermal hydraulic characteristic of oxygen jet condensation, abundant simulations are carried out using a modified phase change method. In this model, the height function method is introduced to capture the interfacial curvature to reveal the mechanism of periodic fluctuation. The simulation proves that three typical flow patterns are found at variant operating conditions in oxygen jet condensation: chugging, oscillation, and swinging. The condensation regime diagram is summarized according to the flow pattern distribution. The chugging flow, along with the phenomenon of liquid oxygen sucked-back flow, shows the most intense pressure oscillation with a maximum amplitude of 133 kPa. In the oscillation flow, the oxygen vapor plume shrinks periodically with a low frequency of about 10 Hz. The swinging flow is a stable flow pattern with a slight pulsation of oxygen vapor plume wake. Its amplitude of pressure oscillation is approximately 1–3 kPa. The influence of liquid oxygen mass flow rate and subcooling on the unstable characteristic is also obtained. These findings offer ample theoretical direction for the development of the space launch vehicle.

Comparison of methods for curvature estimation from volume fractions

This paper evaluates and compares the accuracy and robustness of curvature estimation methods for three-dimensional interfaces represented implicitly by discrete volume fractions on a Cartesian mesh. The height function (HF) method is compared to three paraboloid fitting methods: fitting to the piecewise linear interface reconstruction centroids (PC), fitting to the piecewise linear interface reconstruction volumetrically (PV), and volumetrically fitting (VF) the paraboloid directly to the volume fraction field. A shape-independent test of the curvature estimation is introduced through the use of randomly oriented and shaped paraboloids as reference interfaces. The coupling of the curvature estimation, interface transport, and Navier–Stokes surface tension force is evaluated through the measurement of spurious velocities in simulations of stationary and translating droplets. These studies find that while the curvature error from the VF method converges with second-order accuracy as with the HF method for static interfaces represented by exact volume fractions, the PV method best balances low curvature errors with low computational cost when errors are introduced in the volume fraction field either artificially or through the interface advection and reconstruction steps.

Piecewise circular interface construction using height functions

AbstractA piecewise circular interface construction (PCIC) method is described, where height functions based curvature estimates are directly utilised for accurate interface reconstruction under the framework of volume of fluid method. The present work is an attempt to develop a robust and accurate higher order interface reconstruction algorithm that is capable of accurate simulation of surface tension dominated flows. The proposed hybrid method (H‐PCIC) is thus able to take advantage of merits of both PCIC and HF methods, achieving at least second order convergence with respect to both interface reconstruction and curvature computation. This is in addition to the significantly superior quality of the reconstructed interface with respect to PLIC methods. This seamless blending of the HF and PCIC quantities is enabled by c0‐correction procedures applied to base PLIC and initial PCIC steps. More recent variants of the height function method with variable stencil size are used for calculation of radius of curvature. The capability of this proposed method towards simulation of flow problems within a well‐balanced two‐phase solver is established with help of multiple complex two‐phase flow problems. This validation exercise also demonstrates the capability of PCIC class of methods towards solutions of two‐phase flows with intricate physics.

WELL-BALANCED ALGORITHM AND HEIGHT FUNCTION METHOD FOR DYNAMIC CONTACT ANGLE IN TWO-PHASE SYSTEMS

The well-balanced algorithm combined with dynamic contact angle was well studied in the literature but was never implemented with the pressure-implicit with splitting of operators (PISO) algorithm in a collocated grid commonly used in an incompressible, transient simulation. This article presents a well-balanced algorithm for PISO schemes coupling with the height function method for curvature estimation. The dynamic contact angle model from Kistler and Cox is also integrated to improve the modelling of the curvature at the wall boundary. In collocated finite volume schemes, the well-balanced PISO algorithm is developed by modifying both the calculation of the gradients in the momentum equation and the Rhie and Chow algorithm. This new gradient calculation method ensures that surface tension force and pressure gradient are identically discretized at the same location. The Rhie and Chow algorithm is also modified by adding the surface tension force to balance the pressure forces. The stationary droplet case in two-dimensions is presented first to validate the proposed methodology. The well-balanced algorithm coupling with the height function method shows its benefits by damping spurious currents by two to three orders of magnitude. The 3D surface-driven flow and water-spreading droplets are then simulated; the results show that the new scheme coupled with dynamic contact angle model outperforms the unbalanced scheme of the smooth void fraction method for theoretical and experimental comparisons.

TwoPhaseFlow: A Framework for Developing Two Phase Flow Solvers in OpenFOAM

We present a new OpenFOAM based open-source framework, TwoPhaseFlow, enabling fast implementation and testing of new phase change and surface tension force models for two-phase flows including interfacial heat and mass transfer. Capitalizing on the runtime-selection mechanism in OpenFOAM, the new models can easily be selected and benchmarked against analytical solutions and existing models. The framework currently includes the following three interface curvature calculation methods for surface tension: 1) the height function method, 2) the parabolic fit method and 3) the reconstructed distance function method. As for phase change, two models are available: 1) Interface heat resistance and 2) direct heat flux. These can be combined in three solvers: 1) InterFlow for isothermal, incompressible two-phase flow, 2) compressibleInterFlow for compressible, non-isothermal two-phase flow and 3) multiRegionPhaseChangeFlow for compressible, non-isothermal two-phase flow with conjugated heat transfer. By design, addition of new models and solvers is straightforward and users are encouraged to contribute their specific models, solvers, and validation cases to the library.

Numerical study on the effect of mass flux on the low frequency pressure oscillation occurred in oxygen jet condensation

AbstractIn the delivery pipeline of cryogenic liquid rockets, low‐frequency pressure oscillation always occurs with the oxygen jet condensation. It is a potential hazard to the normal operation of the propulsion system. It is necessary to study the characteristic of low‐frequency pressure oscillation precisely, especially the influence factors of frequency and amplitude. Aiming at the effect of mass flux on low‐frequency pressure oscillation, an oxygen injection condensation simulation is carried out based on a modified mass transfer model. The height function method is adopted to calculate the interfacial curvature, which is the key factor to capture the characteristic of pressure oscillation. The numerical result indicates that there are two typical oxygen jet condensation patterns under different mass fluxes: suck‐back chugging and slight pulsation. The amplitude of suck‐back chugging is 30–120 kPa, while that of slight pulsation is 1–3 kPa. When it comes to suck‐back chugging, the liquid oxygen suck‐back flow phenomenon occurs. The upper and lower boundaries are found to distinguish the suck‐back chugging and slight pulsation patterns, which are caused by the turbulent effect of liquid oxygen and the subcooling of liquid oxygen, respectively. The highest amplitude of pressure oscillation is about 114 kPa, appearing at the intersection of the upper and lower boundary. It is the most dangerous operation condition, which should be avoided in the design of the propulsion system. These conclusions provide sufficient theoretical guidance for the design of the cryogenic liquid rocket.

Mechanisms of low frequency pressure oscillation occurred in oxygen jet condensation based on height function method

In a propulsion system of cryogenic liquid rockets, low frequency pressure oscillation is an important topic on the oxygen jet condensation. It is a challenge to capture the numerical characteristic of pressure oscillation precisely, especially the frequency and amplitude. Aiming at the mechanism and excitation process of pressure oscillation, an oxygen jet condensation simulation is conducted by a novel mass transfer model based on the height function method. The height function method is used to calculate the interfacial curvature between gas-liquid, which is the key factor to determine the frequency of pressure oscillation. The new model is validated by a steam jet condensation simulation and the corresponding result is in great agreement with the experimental data. The research shows that the low frequency pressure oscillation is caused by the periodic condensation mass transfer rate of the continuous oxygen vapor plume, along with a GOX suck-back flow phenomenon in the gas inlet of condenser pipe. The excitation frequency of pressure oscillation is 10.6 Hz, consistent with the experiment data of engine test run. It is close to the natural frequency of propulsion system. It can be inferred that oxygen jet condensation might be one of the excitation source of self-excited Pogo vibration. These conclusions provide a theoretical basis for overcoming the Pogo vibration eventually.

An Improved Cellular Automata Solidification Model Considering Kinetic Undercooling

A cellular automata (CA) model has been developed for solidification simulation considering the kinetic undercooling at the interface. The state-of-the-art model incorporates a decentered growth algorithm to suppress the grid anisotropy and a generalized height function method to calculate the curvature accurately. To develop a CA model which is independent of the mesh size, a new diffusion term is proposed to handle the diffusion between the interface cells and liquid cells. The developed CA model is employed to simulate the single-dendritic solidification of an Al–3Cu (wt pct) alloy. The simulated tip velocities agree with the prediction of the Kurz–Giovanola–Trivedi (KGT) model. Further studies show that the developed CA model converges to an equilibrium model with increasing kinetic mobility values. Moreover, it is found that the virtual liquid cell assumption which is commonly used in existing CA models may lead to a deviation in the mass balance. The mass balance error has been resolved by redistributing solutes from neighboring liquid cells in each time step. The developed CA model could be potentially used in solidification simulations with a high undercooling, which is common in welding and additive manufacturing.

Numerical investigation of oxygen bubble condensation in subcooled quiescent liquid oxygen based on height function method

Bubble condensation is a fundamental issue in the direct contact condensation (DCC) process, which is extremely hard to predict due to the intense mass transfer and flow instability. Aiming at the cryogenic oxygen bubble evolution and pressure oscillation, a modified energy balance mass transfer model is adopted based on the height function method. The height function method is used to calculate the real-time interfacial curvature, modifying the mass transfer rate that emerged in CFD simulation. The modified model is validated to be convincing and accurate by a steam bubble condensation simulation. The results indicate that the cryogenic bubbles condense faster with a more intense mass transfer process, compared with the near-room-temperature mediums. The corresponding amplitude of pressure oscillation for the cryogenic mediums is approximately 3 ∼ 4 times as larger as that for near-room-temperature mediums. It is found that both the condensation rate and the frequency of pressure oscillation are repressed with the increase of bubble volume. That means that a big bubble condensation might produce a low-frequency pressure oscillation. In addition, both the condensation rate and the intensity of the pressure oscillation increase with the increase of subcooling. The average amplitude of pressure oscillation has a power function law with subcooling. These conclusions provide a theoretical basis for a deep understanding of the cryogenic DCC process.

Development of a modified mass transfer model based on height function method to capture the pressure oscillation occurred in direct contact condensation process

For direct contact condensation (DCC) simulation, it is a challenge to capture the numerical characteristic of pressure oscillation precisely, especially the frequency and amplitude. It is the most crucial key issue to calculate the mass transfer across the vapor-liquid interface. Up to now, the pressure characteristic is generally obtained by experiment and the existing mass transfer models always lead to unacceptable deviation from the experiment result. In order to estimate the accurate pressure characteristic, a modified mass transfer model is originally proposed based on the height function method in this paper. An estimation method for the interfacial area density in this model is derived from the algebraic spherical model with the mean bubble diameter. This method correlates the interfacial area concentration with the measurable interfacial curvature and the interfacial curvature is calculated by the height function method. The proposed model is then employed based on the VOF method for the verification of Stefan's analytical problem and steam bubble condensation problem. The corresponding numerical results are both in great agreement with the analytical and experimental results. In addition, the proposed model achieves its advantage for the steam flow condensation to obtain more accurate results of pressure oscillation than other mass transfer models. The numerical result of pressure oscillation frequency is 8.9 Hz, close to the experimental data of 10 Hz with a deviation of 11%. However, the energy balance model and Lee model predict that the frequencies are 31.5 Hz and 19 Hz relatively with the corresponding deviation of 215% and 90%. It is validated to be convincing with excellent accuracy and shows promise for extensive numerical simulation for direct contact condensation applications.

An approximated volume of fluid method with the modified height function method in the simulation of surface tension driven flows

Surface tension in two-phase flow problems plays a dominant role in many micro-flow phenomena and has an important influence on the development of flow instability phenomena that contain free surfaces. In this study, the multi-moment finite volume method is extended for direct numerical simulation of two-phase flow problems. A constraint interpolation profile–CSL (semi-Lagrangian) scheme is used for discretization of the advection part in the momentum equation. A compact volume of fluid method–approximated piecewise linear calculation method without flux limiter is proposed for capturing the moving interface. For modeling the surface tension accurately, the logic in curvature estimation is redesigned based on the height function (HF) method. The isolated volumetric fractions that may reduce accuracy in HF integration are excluded, and the numerical solution shows that the accuracy in the curvature estimation is improved for a coarse mesh. The present method is implemented with a parallel block-structured adaptive mesh refinement (BAMR) strategy; thus, the computational cost can be reduced significantly. Numerical tests show that the present BAMR solver is capable of reproducing the theoretical predictions of capillary wave instability problems with high accuracy. The simulation of droplet collisions further demonstrates the accuracy of the surface tension model. Finally, we extend it to the liquid jet atomization. The wavy disturbance, film breakup, liquid filament pinch-off, and droplet generation are well reproduced. The droplet size distribution satisfies the experimental measurement and theoretical predictions power-law. BAMR shows a huge advantage in computational efficiency than the traditional Cartesian grid. The findings of this study can help for a better understanding of the micro-mechanism of surface tension driven flows.

PArallel, Robust, Interface Simulator (PARIS)

Paris (PArallel, Robust, Interface Simulator) is a finite volume code for simulations of immiscible multifluid or multiphase flows. It is based on the “one-fluid” formulation of the Navier–Stokes equations where different fluids are treated as one material with variable properties, and surface tension is added as a singular interface force. The fluid equations are solved on a regular structured staggered grid using an explicit projection method with a first-order or second-order time integration scheme. The interface separating the different fluids is tracked by a Front-Tracking (FT) method, where the interface is represented by connected marker points, or by a Volume-of-Fluid (VOF) method, where the marker function is advected directly on the fixed grid. Paris is written in Fortran95/2002 and parallelized using MPI and domain decomposition. It is based on several earlier FT or VOF codes such as Ftc3D, Surfer or Gerris. These codes and similar ones, as well as Paris, have been used to simulate a wide range of multifluid and multiphase flows. Program summaryProgram Title: PArallel Robust Interface Simulator — ParisCPC Library link to program files:https://doi.org/10.17632/5cb2yrfx7r.1Licensing provisions: GPLv3.Programming language: Fortran95/2002. Parallelized using MPI and domain decomposition.Nature of problem:Paris is a free code, or software, for computational fluid dynamics (CFD) of multiphase flows (or computational multiphase fluid dynamics (CMFD)), specialized in the numerical simulation of interfacial fluid flows, involving droplets, bubbles and waves, as described in the book by Tryggvason, Scardovelli and Zaleski [1]. It solves the Euler or Navier–Stokes equations in the one-fluid formulation of two-phase flow, including a surface tension force. It computes complex flows such as fast atomizing jets or droplets, expanding cavitation bubble clusters and multiphase flow through porous media.Solution method: The code mostly implements the methods described in the book by Tryggvason, Scardovelli and Zaleski [1]. Time stepping is performed using a first-order or a second-order in time predictor–corrector method with an explicit projection step for the pressure. Spatial discretization is by finite volumes on a regular cuboid grid. Interface tracking is performed with the Front-Tracking (FT) method or the Volume-of-Fluid (VOF) method. In the VOF version Paris uses either the Lagrangian-Explicit (LE) advection method or the exactly mass-conserving method of Weymouth and Yue [2]. The normal computation is performed using the Mixed-Youngs-Centered (MYC) scheme. A mass–momentum advection method has been also implemented that is consistent with the VOF advection [3]. Curvature is computed with the Height Function (HF) method. This is combined with the balanced Continuous Surface Force (CSF) method to compute surface tension forces.If the dynamics of a phase can be neglected, Paris can also run as a free-surface code by specifying a homogeneous pressure, at most varying with time, in the neglected phase. In the case of atomizing jets, an algorithm has been implemented in Paris that can detect isolated droplets, advects them as Lagrangian point-particles and possibly merge them again with the main streamAdditional comments:Paris is extended from or inspired by the following codes: •Ftc3D: Front Tracking code for 3D simulations by Gretar Tryggvason and Sadegh Dabiri.•Surfer: VOF code for 3D simulations by Stephane Zaleski, Jie Li, Ruben Scardovelli and others.•Gerris: multiphase flow solver with Adaptive Mesh Refinement (AMR) by Stephane Popinet.

Natural oscillations of a sessile drop on flat surfaces with mobile contact lines

Oscillation of sessile drops is important to many applications. In the present study, the natural oscillation of a sessile drop on flat surfaces with free contact lines (FCL) is investigated through numerical and theoretical analysis. The FCL condition represents a limit of contact line mobility, i.e. the contact angle remains constant when the contact line moves. In the numerical simulation, the interfaces are captured by the volume-of-fluid method and the contact angle at the boundary is specified using the height-function method. The oscillation frequencies for sessile drops with FCL are mainly controlled by the contact angle and the Bond number and a parametric study is carried out to characterize their effects on the frequencies for the first and high-order modes. Particular attention is paid to the frequency of the first mode, since it is usually the dominant mode. An inviscid theoretical model for the first mode is developed. The model yields an explicit expression for the first-mode frequency as a function of the contact angle and the Bond number, with all parameters involved fully determined by the equilibrium drop theory and the simulation. The predicted frequencies for a wide range of contact angles agree very well with the simulation results for small Bond numbers. The frequencies for both the first and high-order modes decrease with the contact angle and increase with the Bond number. For the high-order modes, the frequencies for different modes generally scale with the Rayleigh frequencies. The scaling relation performs better for small Bond numbers and large contact angles. A simple model is proposed to predict the frequencies of high-order modes for large contact angles and a good agreement with the simulation results is observed.

A mass-momentum consistent, Volume-of-Fluid method for incompressible flow on staggered grids

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with capillary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d’Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner “consistent” with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.

A higher-order accurate surface tension modelling volume-of-fluid scheme for 2D curvilinear meshes

This paper proposes a higher-order accurate curvature calculation method for volume-of-fluid (VOF) based surface tension modelling on 2D curvilinear meshes. The scheme extends the interface reconstruction component of the height function (HF) method to curvilinear meshes in 2D. Linear reconstructions are computed in each column overlapping the interface, which results in a piecewise-linear (PLIC) representation of the free-surface. This is done in a volume conservative manner by employing a novel sweep-line algorithm. The interface curvature is then computed analytically by fitting either second- or fourth-order polynomials to local stencils of PLIC facets. Formal second- and fourth-order accuracy of interface curvature is demonstrated by comparing numerical results with a variety of analytical interface definitions. The curvature algorithm is implemented into a balanced-force continuum-surface-force surface tension scheme for variable-density flows. Second- and fourth-order accuracy are again demonstrated for the Laplace pressure jump in the simulation of a stationary bubble with a high liquid-gas density ratio. Lastly, second-order accuracy is demonstrated in computing the frequency of an inviscid oscillating droplet in zero gravity.

Computing interface curvature from volume fractions: A machine learning approach

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.

On the Application of OpenFOAM® in Gravity Water Wave Generation and Wave-Structure Interaction

This article presents an investigation on the best practice for modeling of water wave generation and wave-structure interaction using a widely spread two-phase flow solver with a specific interface compression technique in OpenFOAM® (OpenFOAM Foundation Ltd., London, United Kingdom). A series of numerical experiments were conducted to examine the effects of the employed schemes, mesh resolution, time step resolution, and compression coefficient. Both surface elevation and velocity profile were considered as the criteria for assessment of the quality of the generated waves. The numerical experiments showed that by using a blending scheme between the Crank-Nicolson and Euler scheme, relatively high quality waves were produced, where the spurious current at the interface region was effectively reduced. Meanwhile, it was also recommended to apply a compression coefficient Cα = 1, the Courant number limit Co = .1, and a mesh resolution of 18 cells per wave height. This set of parameters was used to validate the numerical model for two sets of cases for wave forces on half-immersed horizontal cylinders. The results in general agreed well with the experimental data, although the inline forces were slightly but consistently overestimated. 1. Introduction In the past few decades, because of the increase in the computing ability and resources, computational fluid dynamic (CFD) approach has been gaining popularity in the naval hydrodynamic communities. Considering that a large portion of naval hydrodynamic problems are concerned with the free-surface flows, e.g., ocean waves and its interaction with structures, a free-surface model is needed to couple with the Navier-Stokes solver. Eventually, as a result of direct application of the first principles, the solver can model both inertia and viscous dominant flows. Even local wave breaking and violent flows can be properly resolved. Different methods have been proposed to track the interface of free-surface flows. This includes height function method and line segment method, which are the simplest techniques to represent and configure a free surface. However, they have deficiencies when two surfaces intersect or when a surface folds over on itself, which are overcome by the marker and cell method proposed in Harlow and Welch (1965). In their method, massless particles are used to track the free surface in a Lagrangian manner. Alternatively, the level-set (LS) method or volume of fluid (VOF) method can be used to implicitly capture the surface in an Eulerian way. The LS method was introduced in Osher (1988) where the interface is defined as the one on which a level-set function is equal to zero. This LS function is continuous across the interface. Therefore, it is not significantly influenced by the numerical diffusion. However, the originally proposed LS method suffers from mass conservation problems, which has been resolved in, e.g., Olsson and Kreiss (2005) and Olsson et al. (2007).

A comparison of methods for the calculation of interface curvature in two-dimensional cellular automata solidification models

Four methods for the calculation of interface curvature in cellular automata models for solidification are evaluated. Three of the methods are based on the level-set interpretation of the solid fraction field and calculate the required spatial derivatives by finite differences or by a Taylor approximation in different neighborhoods. The fourth method is the height function method, an interface reconstruction method based on the row- or columnwise summation of solid fractions.Two benchmark problems are investigated, representative for two types of cellular automata models for solidification where interface cells exist in either Moore or von Neumann neighborhoods of solid cells. For the case of interface cells in the Moore neighborhood of solid cells, the height function method yields the most accurate results, while all level-set based methods exhibit significant numerical scatter. For the case of interface cells existing in the von Neumann neighborhood of solid cells, all methods yield a higher level of numerical scatter. The best curvature calculations for this case are achieved by the height function method and by the level set method with a Taylor approximation in an extended neighborhood.

Quantitative cellular automaton model and simulations of dendritic and anomalous eutectic growth

A quantitative cellular automaton (CA) model for dendritic and anomalous eutectic growth has been developed. Solid/liquid(SL) interface curvature plays an important role on the dendritic and eutectic growth kinetics. In order to improve the accuracy of calculated interface curvature, a height function method was used in this research. Simulation results showed that the precision of the interface curvature has been significantly improved. And the quantitative ability of present CA model has also been improved through the simulations of equilibrium shapes. Using the quantitative CA model, directional solidification of dendritic and eutectic growth has been investigated, as well as the anomalous eutectic growth during laser remelting process. For anomalous eutectic, it is found that the very fine lamellar eutectic growing at much smaller cooling rate is the reason to observe the lamellar to anomalous transition.