A new volume of fluid method in three dimensions—Part II: Piecewise‐planar interface reconstruction with cubic‐Bézier fit

A new interface reconstruction method in 3D is presented. The method involves a conservative level‐contour reconstruction coupled to a cubic‐Bézier interpolation. The use of the proposed piecewise linear interface calculation (PLIC) reconstruction scheme coupled to a multidimensional time integration provides solutions of second‐order spatial and temporal accuracy. The accuracy and efficiency of the proposed reconstruction algorithm are demonstrated through several tests, whose results are compared with those obtained with other recently proposed methods. An overall improvement in accuracy with respect to other recent methods has been achieved, along with a substantial reduction in the central processing unit time required. Copyright © 2008 John Wiley & Sons, Ltd.     

A PLIC volume tracking method for the simulation of two‐fluid flows

Numerical methodologies for computer simulations of two‐fluid flows are presented. These methodologies fall into the category of volume tracking methods with piecewise‐linear interface calculation (PLIC). The scope of this work is limited to laminar flows of immiscible, non‐reacting, incompressible Newtonian fluids, without phase change, in planar two‐dimensional geometries. The following novel or enhanced procedures are proposed: a parallelogram scheme for multidimensional advection of the volume‐fraction field; a circle‐fit technique for the orientation of the interface segments and the calculation of curvature; a novel contact angle treatment; and a staggered formulation for volumetric body forces that can accurately balance pressure forces in the vicinity of the interface. In addition, surface‐tension‐derived and hydrostatic‐derived pressure adjustments are introduced as a means of accurately calculating pressure forces in cells that contain the interface, so as to minimize the non‐physical flows that afflict many available volume tracking methods. The proposed method is validated using four test problems that involve simulations of pure advection, a static drop, an oscillating bubble, and a static meniscus. Copyright © 2006 John Wiley & Sons, Ltd.     

A new volume of fluid advection algorithm: the defined donating region scheme

This paper presents a new volume of fluid (VOF) advection algorithm, termed the defined donating region (DDR) scheme. The algorithm uses a linear piecewise method of free surface reconstruction, coupled to a fully multi‐dimensional method of cell boundary flux integration. The performance of the new scheme has been compared with the performance of a number of alternative schemes using translation, rotation and shear advection tests. The DDR scheme is shown to be generally more accurate than linear constant and flux limited schemes, and comparable with an alternative linear piecewise scheme. The DDR scheme conserves fluid volume rigorously without local redistribution algorithms, and generates no fluid ‘flotsam’ or other debris, making it ideal in applications where stability of the free surface interface is paramount. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

From level set to volume of fluid and back again at second‐order accuracy

We present methods for computing either the level set function or volume fraction field from the other at second‐order accuracy. Both algorithms are optimal in that O(N) computations are needed for N total grid points and both algorithms are easily parallelized. This work includes a novel interface reconstruction algorithm in three dimensions that requires a smaller local block of volume fractions than existing algorithms. A compact local solver leads to better algorithm portability and efficiency: for example, fewer restrictions must be imposed on an adaptive mesh, and fewer grid cells must be communicated between processors in a parallel implementation. We also present a fast sweeping method for computing a unique approximation of the signed distance function to a piecewise linear interface. All of the numerical examples confirm second‐order accuracy on both uniform and tree‐based adaptive grids. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical errors of the volume‐of‐fluid interface tracking algorithm

One of the important limitations of the interface tracking algorithms is that they can be used only as long as the local computational grid density allows surface tracking. In a dispersed flow, where the dimensions of the particular fluid parts are comparable or smaller than the grid spacing, several numerical and reconstruction errors become considerable. In this paper the analysis of the interface tracking errors is performed for the volume‐of‐fluid method with the least squares volume of fluid interface reconstruction algorithm. A few simple two‐fluid benchmarks are proposed for the investigation of the interface tracking grid dependence. The expression based on the gradient of the volume fraction variable is introduced for the estimation of the reconstruction correctness and can be used for the activation of an adaptive mesh refinement algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems

The Multidimensional Optimal Order Detection (MOOD) method for two‐dimensional geometries has been introduced by the authors in two recent papers. We present here the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids, and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set of numerical tests for both the convection equation and the Euler system that the optimal high order of accuracy is reached on smooth solutions, whereas spurious oscillations near singularities are prevented. At last, the intrinsic positivity‐preserving property of the MOOD method is confirmed in 3D, and we provide simple optimizations to reduce the computational cost such that the MOOD method is very competitive compared with existing high‐order Finite Volume methods.Copyright © 2013 John Wiley & Sons, Ltd.     

A pressure correction‐volume of fluid method for simulation of two‐phase flows

A pressure correction method coupled with the volume of fluid (VOF) method is developed to simulate two‐phase flows. A volume fraction function is introduced in the VOF method and is governed by an advection equation. A modified monotone upwind scheme for a conservation law (modified MUSCL) is used to solve the solution of the advection equation. To keep the initial sharpness of an interface, a slope modification scheme is introduced. The continuum surface tension (CST) model is used to calculate the surface tension force. Three schemes, central‐upwind, Parker–Youngs, and mixed schemes, are introduced to compute the interface normal vector and the gradient of the volume fraction function. Moreover, a height function technique is applied to compute the local curvature of the interface. Several basic test problems are performed to check the order of accuracy of the present numerical schemes for computing the interface normal vector and the gradient of the volume fraction function. Three physical problems, two‐dimensional broken dam problem, static drop, and spurious currents, and three‐dimensional rising bubble, are performed to demonstrate the efficiency and accuracy of the pressure correction method. Copyright © 2010 John Wiley & Sons, Ltd.     

A first‐order volume of fluid convection model in three‐dimensional space

To improve the numerical analysis of free surface convections and reconstruction in a three‐dimensional space, a first‐order algorithm is developed based on the volume of fluid (VOF) theory. The methodology applied to the first‐order method (FOM) is to define a first‐order surface as near to the horizontal as possible while satisfying the defined volume fraction of a cell. The developed method is compared against the donor cell method of zeroth‐order through simulation of the transitional and rotational convection of liquid spheres. Although the donor cell method shows relatively good predictions for the sphere of a large diameter, it shows poor performance of large distortions for a sphere of a relatively small diameter. However, the FOM developed in this study always shows quite satisfactory prediction results for free surface convection. Copyright © 2001 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of element order and interface reconstruction in FEM/volume‐of‐fluid incompressible flow simulation

In previous studies, the moment‐of‐fluid interface reconstruction method showed dramatic accuracy improvements in static and pure advection tests over existing methods, but this did not translate into an equivalent improvement in volume‐tracked multimaterial incompressible flow simulation using low‐order finite elements. In this work, the combined effects of the spatial discretization and interface reconstruction in flow simulation are examined. The mixed finite element pairs, Q₁Q₀ (with pressure stabilization) and Q₂P _{? 1} are compared. Material order‐dependent and material order‐independent first and second‐order accurate interface reconstruction methods are used. The Q₂P _{? 1} elements show significant improvements in computed flow solution accuracy for single material flows but show reduced convergence using element‐average piecewise constant density and viscosity in volume‐tracked simulations. In general, a refined Q₁Q₀ grid, with better material interface resolution, provided an accuracy similar to the Q₂P _{? 1} element grid with a comparable number of degrees of freedom. Moment‐of‐fluid shows more benefit from the higher‐order accurate flow simulation than the LVIRA, Youngs', and power diagram interface reconstruction methods, especially on unstructured grids, but does not recover the dramatic accuracy improvements it has shown in advection tests. Published 2012. This article is a US Government work and is in the public domain in the USA.     

A novel fully implicit finite volume method applied to the lid‐driven cavity problem—Part I: High Reynolds number flow calculations

A novel implicit cell‐vertex finite volume method is described for the solution of the Navier–Stokes equations at high Reynolds numbers. The key idea is the elimination of the pressure term from the momentum equation by multiplying the momentum equation with the unit normal vector to a control volume boundary and integrating thereafter around this boundary. The resulting equations are expressed solely in terms of the velocity components. Thus any difficulties with pressure or vorticity boundary conditions are circumvented and the number of primary variables that need to be determined equals the number of space dimensions. The method is applied to both the steady and unsteady two‐dimensional lid‐driven cavity problem at Reynolds numbers up to 10000. Results are compared with those in the literature and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.     

A semi‐implicit finite volume implementation of the CSF method for treating surface tension in interfacial flows

We present an implementation of Hysing's (Int. J. Numer. Meth. Fluids 2006; 51 :659–672) semi‐implicit method for treating surface tension, for finite volume models of interfacial flows. Using this method, the surface tension timestep restriction, which is often very stringent, can be exceeded by at least a factor of 5 without destabilizing the solution. The surface tension force in this method consists of an explicit part, which is the regular continuum surface force, and an implicit part which represents the diffusion of velocities induced by surface tension on fluids interfaces. The surface tension force is applied to the velocity field by solving a system of equations iteratively. Since the equations are solved only near interfaces, the computational time spent on the iterative procedure is insignificant. Copyright © 2008 John Wiley & Sons, Ltd.