Revisit to the THINC scheme: A simple algebraic VOF algorithm

This short note presents an improved multi-dimensional algebraic VOF method to capture moving interfaces. The interface jump in the THINC (tangent of hyperbola for INterface capturing) scheme is adaptively scaled to a proper thickness according to the interface orientation. The numerical accuracy in computing multi-dimensional moving interfaces is significantly improved. Without any geometrical reconstruction, the proposed method is extremely simple and easy to use, and its numerical accuracy is superior to other existing methods of its kind and comparable to the conventional PLIC (piecewise linear interface calculation) type VOF schemes.     

A new volume-of-fluid method with a constructed distance function on general structured grids

A second-order volume-of-fluid method (VOF) is presented for interface tracking and sharp interface treatment on general structured grids. Central to the new method is a second-order distance function construction scheme on a general structured grid based on the reconstructed interface. A novel technique is developed for evaluating the interface normal vector using the distance function. With the normal vector, the interface is reconstructed from the volume fraction function via a piecewise linear interface calculation (PLIC) scheme on the computational domain. Several numerical tests are conducted to demonstrate the accuracy and efficiency of the present method. In general, the new VOF method is more efficient than both the high-order level set and the coupled level set and volume-of-fluid (CLSVOF) methods. The results from the new method are better than those from the benchmark VOF method, particularly in the under-resolved regions, and are comparable to those from the CLSVOF method. Breaking waves over a submerged bump and around a wedge-shaped bow are simulated to demonstrate the application of the new method and sharp interface treatment in a two-phase flow solver on curvilinear grids. The computational results are in good agreement with the available experimental measurements.     

Application of the high-resolution Godunov method to the multi-fluid flow calculations

In this paper, we have numerically solved the multi-fluid problems using an operator-split two-step high-resolution Godunov PPM (parabolic piecewise method) for the flow in complex geometries. By using the front capturing method,the PPM integrator captures the interface in the solution process. The basic multi-fluid integrator is coupled to a Cartesian grid algorithm where a VOF (volume of fluid) representation of the fluid interface is also used. As an application of this method, we test the 2D interracial advection example and simulate an experimental hypervelocity launcher model from Sandia National Laboratories. The computational design of the hypervelocity launcher is also given in the paper.     

On the properties and limitations of the height function method in two-dimensional Cartesian geometry

In this study we define the continuous height function to investigate the approximation of an interface line and its geometrical properties with the height function method. We show that in each mixed cell the piecewise linear interface reconstruction and the approximation of the derivatives and curvature based on three consecutive height function values are second-order accurate. We also discuss the quadratic reconstruction and fourth-order accurate expressions of the normal and curvature. We present a hierarchical algorithm to compute the normal vector and curvature of an interface line with the height function method that switches automatically between second- and fourth-order approximations and that can be applied also when the local radius of curvature is of the order of the grid spacing.     

Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions

We present a generalization of the finite volume evolution Galerkin scheme [M. Luká?ová-Medvid’ová, J. Saibertov’a, G. Warnecke, Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comp. Phys. (2002) 183 533– 562; M. Luká?ová-Medvid’ová, K.W. Morton, G. Warnecke, Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems, SIAM J. Sci. Comput. (2004) 26 1–30] for hyperbolic systems with spatially varying flux functions. Our goal is to develop a genuinely multi-dimensional numerical scheme for wave propagation problems in a heterogeneous media. We illustrate our methodology for acoustic waves in a heterogeneous medium but the results can be generalized to more complex systems. The finite volume evolution Galerkin (FVEG) method is a predictor–corrector method combining the finite volume corrector step with the evolutionary predictor step. In order to evolve fluxes along the cell interfaces we use multi-dimensional approximate evolution operator. The latter is constructed using the theory of bicharacteristics under the assumption of spatially dependent wave speeds. To approximate heterogeneous medium a staggered grid approach is used. Several numerical experiments for wave propagation with continuous as well as discontinuous wave speeds confirm the robustness and reliability of the new FVEG scheme.     

A PLIC–VOF method suited for adaptive moving grids

This paper presents a new variant of the volume-of-fluid (VOF) color function C advection algorithm based on the piecewise linear interface construction (PLIC) method suitable for use on general moving grids. From several existing methods for reconstructing the linear interface we adopted the least squares volume-of-fluid interface reconstruction algorithm (LVIRA) which can be easily implemented on general grids. The distinguishing step in the advection algorithm that takes into account the grid movement is the construction of the donating region containing the fluid passing through corresponding cell-faces in a single time-step. The donating regions are constructed utilizing fluid velocity in cell corners relative to grid (corner) velocities. The method is conservative as it complies with the space conservation law (SCL) and requires a proper definition of the grid velocities and fluxes due to the grid movement. The accuracy of the presented advection algorithm is assessed with standard test cases. It is comparable with other PLIC based algorithms on fixed grids, while the applicability on adaptive moving grids enables a considerable reduction in the number of grid cells.     

A novel coupled level set and volume of fluid method for sharp interface capturing on 3D tetrahedral grids

We present a new three-dimensional hybrid level set (LS) and volume of fluid (VOF) method for free surface flow simulations on tetrahedral grids. At each time step, we evolve both the level set function and the volume fraction. The level set function is evolved by solving the level set advection equation using a second-order characteristic based finite volume method. The volume fraction advection is performed using a bounded compressive normalized variable diagram (NVD) based scheme. The interface is reconstructed based on both the level set and the volume fraction information. The novelty of the method lies in that we use an analytic method for finding the intercepts on tetrahedral grids, which makes interface reconstruction efficient and conserves volume of fluid exactly. Furthermore, the advection of volume fraction makes use of the NVD concept and switches between different high resolution differencing schemes to yield a bounded scalar field, and to preserve both smoothness and sharp definition of the interface. The method is coupled to a well validated finite volume based Navier–Stokes incompressible flow solver. The code validation shows that our method can be employed to resolve complex interface changes efficiently and accurately. In addition, the centroid and intercept data available as a by-product of the proposed interface reconstruction scheme can be used directly in near-interface sub-grid models in large eddy simulation.     

A new interface tracking method: The polygonal area mapping method

We present a new method, the polygonal area mapping (PAM) method, for tracking a non-diffusive, immiscible material interface between two materials in two-dimensional incompressible flows. This method represents material areas explicitly as piecewise polygons, traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon-clippings in a discrete manner. The new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately. In addition to high accuracy, the PAM method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications. The mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons. The results from a set of widely used benchmark tests show that the PAM method is superior to existing volume-of-fluid (VOF) methods.     

A volume of fluid approach for crystal growth simulation

A new approach to simulating the dendritic growth of pure metals, based on a recent volume of fluid (VOF) method with PLIC (piecewise linear interface calculation) reconstruction of the interface, is presented. The energy equation is solved using a diffuse-interface method, which avoids the need to apply the thermal boundary conditions directly at the solid front. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of a distance function to the reconstructed interface. The advection equation of a discretized solid fraction function is solved using the unsplit VOF advection method proposed by López et al. [J. Comput. Phys. 195 (2004) 718–742] (extended to three dimensions by Hernández et al. [Int. J. Numer. Methods Fluids 58 (2008) 897–921]), and the interface curvature is computed using an improved height function technique, which provides second-order accuracy. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.     

基于近似Riemann解的有限体积ALE方法

研究二维平面坐标系和二维轴对称坐标系中四边形网格上可压缩流体力学的有限体积ALE(Arbitrary Lagrangian Eulerian)方法.数值方法采用节点中心有限体积法,数值通量采用适用于任意状态方程的HLLC(Harten-Lax-Van Leer-Collela)通量.空间二阶精度通过用WENO(weighted essentially non-oscillatory)方法对原始变量进行重构获得,时间离散采用两步显式Runge-Kutta格式.数值例子显示,方法具有良好的激波分辨能力和高精度的数值逼近能力.     

Third-order Energy Stable WENO scheme

A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. We also present new weight functions which drastically improve the accuracy of the third-order ESWENO scheme. Based on a truncation error analysis, we show that the ESWENO scheme is design-order accurate for smooth solutions with any number of vanishing derivatives, if its tuning parameters satisfy certain constraints. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO scheme of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities.     

液面冲击引起液滴飞溅问题的CIP方法数值模拟

利用自主研发的CIP-ZJU（Constrained Interpolation Profile Method in Zhejiang University）高精度数学模型研究强非线性自由表面流动问题.模型在直角坐标系统下建立,采用紧致插值曲线CIP方法作为流场的基本求解器,通过多相流的方式实现固-液-气耦合同步求解,采用平衡格式的VOF（VOF/WLIC：Volume of Fluid/Weighted LineInterface Calculation）自由面捕捉方法改进了原模型,利用浸入边界方法处理运动物体.利用改进的CIP模型开展了不同类型的物体冲击液面引起的液滴飞溅现象的数值模拟,重点分析液滴飞溅过程中的自由液面变形、作用荷载和物体位置等.通过数值结果与实验结果的比较验证模型的可靠性.结果表明：平衡格式的VOF自由面捕捉方法能更精确地重构自由面,本文的数学模型可精确预测强非线性自由表面流动问题.     

An improved particle correction procedure for the particle level set method

The particle level set method [D. Enright, R. Fedkiw, J. Ferziger, I. Mitchell, A hybrid particle level set method for improved interface capturing, J. Comput. Phys. 183 (2002) 83–116.] can substantially improve the mass conservation property of the level set method by using Lagrangian marker particles to correct the level set function in the under-resolved regions. In this study, the limitations of the particle level set method due to the errors introduced in the particle correction process are analyzed, and an improved particle correction procedure is developed based on a new interface reconstruction scheme. Moreover, the zero level set is “anchored” as the level set functions are reinitialized; hence the additional particle correction after the level set reinitialization is avoided. With this new scheme, a well-defined zero level set can be obtained and the disturbances to the interface are significantly reduced. Consequently, the particle reseeding operation will barely result in the loss of interface characteristics and can be applied as frequently as necessary. To demonstrate the accuracy and robustness of the proposed method, two extreme particle reseeding strategies, one without reseeding and the other with reseeding every time step, are applied in several benchmark advection tests and the results are compared with each other. Three interfacial flow cases, a 2D surface tension driven oscillating droplet, a 2D gas bubble rising in a quiescent liquid, and a 3D drop impact onto a liquid pool are simulated to illustrate the advantages of the current method over the level set and the original particle level set methods with regard to the smoothness of geometric properties and mass conservation in real physical applications.     

Volume-of-fluid algorithm with different modified dynamic material ordering methods and their comparisons

Volume-of-fluid (VOF) interface reconstruction methods are used to define material interfaces to separate different materials in a mixed cell. These material interfaces are then used to evaluate transport flux at each cell edges in multi-material hydrodynamic calculations. Most of the VOF interface reconstruction methods and volume transport schemes rely on an accurate material order unique to each computational cell. Similarly, to achieve overshoot-free volume fractions, a non-intersecting interface reconstruction procedure has to be performed with the help of a ‘material-order list’ determined prior to interface reconstruction. It is, however, the least explored area of VOF technique especially for ‘onion-skin’ or ‘layered’ model. Also, important technical details how to prevent intersection among different material interfaces are missing in many literature. Here, we present an efficient VOF interface tracking algorithm along with modified ‘material order’ methods and different interface reconstruction methods. The relative accuracy of different methods are evaluated for sample problems. Finally, a convergence study with respect to mesh-size is performed.     

Conservative Volume-of-Fluid method for free-surface simulations on Cartesian-grids

This paper contributes to the state of the art in Cartesian-grid methods through development of new advection and reconstruction Volume-of-Fluid (VOF) algorithms which are applicable to two and three-dimensional flows. A computationally efficient and second-order VOF reconstruction method is presented which uses no inversions to determine the interface normal direction. Next, the lack of conservation of fluid volume in previous VOF advection methods are shown to be due to improper treatment of one-dimensional stretching in the velocity field. This paper uses simple explicit time stepping and a cell-center estimate of the volume fraction in the dilatation term to achieve a completely conservative advection method. The new methods are simple, robust and shown to out perform existing approaches for canonical test problems relevant to breaking wave flows.     

Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids

The present paper deals with an efficient and accurate limiting strategy for the multi-dimensional hyperbolic conservation laws on unstructured grids. The multi-dimensional limiting process (MLP) which has been successfully proposed on structured grids is extended to unstructured grids. The basic idea of the proposed limiting strategy is to control the distribution of both cell-centered and cell-vertex physical properties to mimic multi-dimensional nature of flow physics, which can be formulated to satisfy so called the MLP condition. The MLP condition can guarantee high-order spatial accuracy and improved convergence without yielding spurious oscillations. Starting from the MUSCL-type reconstruction on unstructured grids followed by the efficient implementation of the MLP condition, MLP slope limiters on unstructured meshes are obtained.Thanks to its superior limiting strategy and maximum principle satisfying characteristics, the newly developed MLP on unstructured grids is quite effective in controlling numerical oscillations as well as accurate in capturing multi-dimensional flow features. Numerous test cases are presented to validate the basic features of the proposed approach.     

使用界面追踪对膜态沸腾的数值研究

提出一种计算带相变的自由界面的数值算法.基于分段线性界面重构(PLIC)的VOF(volume-of-fluid)方法用于追踪自由界面,并对汽液交界面上的相变导致的不连续速度场给出处理方法.此方法容易实施且被证明是有效的.流场的求解使用SIMPLE方法,表面张力使用连续表面力模型(CSF)进行计算.在三维直角坐标系下,模拟了水平壁面上的膜态沸腾,在二维适体坐标系下,模拟了竖直圆头柱体表面的自然对流膜态沸腾.计算结果与理论关系式符合较好.     

模拟射流雾化过程的VOF-LPT耦合方法

探讨模拟射流雾化的若干典型VOF(Volume of Fluid)界面重构方法, 并在开源流体力学计算软件OpenFOAM中运用VOF-LPT(Lagrangian Particle Tracking)耦合算法; 该算法兼顾模拟的准确性和计算效率, 可较为准确地模拟从连续液体到离散小液滴的射流雾化过程, 并为更大尺度的模拟提供了一种可行的高效模拟方案。     

Gradient-augmented hybrid interface capturing method for incompressible two-phase flow

Motivated by inconveniences of present hybrid methods,a gradient-augmented hybrid interface capturing method(GAHM) is presented for incompressible two-phase flow.A front tracking method(FTM) is used as the skeleton of the GAHM for low mass loss and resources.Smooth eulerian level set values are calculated from the FTM interface,and are used for a local interface reconstruction.The reconstruction avoids marker particle redistribution and enables an automatic treatment of interfacial topology change.The cubic Hermit interpolation is employed in all steps of the GAHM to capture subgrid structures within a single spacial cell.The performance of the GAHM is carefully evaluated in a benchmark test.Results show significant improvements of mass loss,clear subgrid structures,highly accurate derivatives(normals and curvatures) and low cost.The GAHM is further coupled with an incompressible multiphase flow solver,Super CE/SE,for more complex and practical applications.The updated solver is evaluated through comparison with an early droplet research.     

An investigation of interface-sharpening schemes for multi-phase mixture flows

This work evaluates several approaches for sharp phase interface-capturing in computations of multi-phase mixture flows. Attention is focused on algebraic interface-capturing strategies that fit directly within a finite-volume MUSCL-type framework, in which dimension-by-dimension reconstruction of interface states based on extrapolated fluid properties is the norm. In this scope, linear, sine-wave, and tangent hyperbola volume-fraction reconstructions are examined for a range of problems, including advection of a volume-fraction discontinuity, the Rayleigh–Taylor instability, a dam-break problem, an axisymmetric jet instability, the Rayleigh instability, and flow within an aerated-liquid injector. An implicit dual-time stepping approach, applied directly to a preconditioned form of the governing equations, is used for time-advancement. The results show that the sharpening strategies are successful in providing two-to-three-cell capturing of volume-fraction discontinuities.