The MAC method

In this article recent advances in the Marker and Cell (MAC) method will be reviewed. The MAC technique dates back to the early 1960s at the Los Alamos Laboratories and this article starts with a historical review, and then a brief discussion of related techniques. Improvements since the early days of MAC (and the Simplified MAC - SMAC) include automatic time-stepping, the use of the conjugate gradient method to solve the Poisson equation for the corrected velocity potential, greater efficiency through stripping out the virtual particles (markers) other than those near the free surface, and more accurate approximations of the free surface boundary conditions, the addition of bounded high accuracy upwinding for the convected terms (thereby being able to solve higher Reynolds number flows), and a (dynamics) flow visualization facility. More recently, effective techniques for surface and interfacial flows and, in particular, for accurately tracking the associated surface(s)/interface(s...

International Journal for Numerical Methods in Fluids, 2012

This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front-tracking method. The velocity field is computed using a finite-difference discretization of a modification of the Navier-Stokes equations. These equations together with the continuity equation are solved for the two-dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient.

2005

Multiphase flows associated with interfacial dynamics, steep jump in fluid-properties, and moving boundaries between different phases and materials pose substantial computational challenges. Modeling the interfacial dynamics often involves a compromise between the accuracy and efficiency. Recent progress made in handling the geometry of three-dimensional interfaces, adaptive grid refinement, and multigrid techniques is presented, in which the interface is tracked explicitly using markers on triangulated surface grid, while the field equations are solved using the Cartesian grid. A phase volume preserving technique for marker addition and deletion on three dimensional interfaces has been devised to ensure that the conservation laws are satisfied. Illustrative physical applications are presented to highlight the present technique.

Journal of Computational Physics, 2005

A new method is presented for the simulation of three-dimensional, incompressible, free surface fluid flow problems. The new technique, the Eulerian-Lagrangian marker and micro cell (ELMMC) method, is capable of simulating incompressible fluid flow problems in Cartesian coordinates where the free surface can undergo severe deformations, including impact with solid boundaries and impact between converging fluid fronts. The method is also capable of handling the breakup of a fluid front from the main body of the fluid as well as their eventual coalescence. The basic solution methodology solves the continuity and the Navier-Stokes equations with a projection scheme and is even able to incorporate a basic k-e turbulence modeling capability. New approaches are presented for the advection of the free surface, as well as for the calculation of the tentative velocity, final velocity, and pressure fields. The capabilities of the new method are demonstrated by comparing numerical results with experimental studies while the convergence of the new method is demonstrated by spatial and temporal refinement studies.

Computers & Fluids, 2007

The CFD modeling of two-dimensional multiphase flows is a useful tool in industry, although accurate modeling itself remains a difficult task. One of the difficulties is to track the complicated topological deformations of the interfaces between different phases. This paper describes a marker-particle method designed to track fluid interfaces for fluid flows of at least three phases. The interface-tracking scheme presented in this paper is the first part of a series of papers presenting our complete model based on a one-field Godunov markerparticle projection scheme (GMPPS). In this part, we shall focus on the presentation of the interface-tracking scheme and the kinematic tests we conducted to examine the scheme's ability to accurately track interfacial movements typified by vorticity-induced stretching and tearing of the interface. Our test results show that for a set of carefully designed and commonly used error measures, relative percentage errors never exceed 2% for all of the tests and grid sizes considered, provided a sufficient number of marker particles are used. We shall also demonstrate that the method is of second-order accuracy and the interface transition width remains constant never exceeding three cell widths.

44th AIAA Aerospace Sciences Meeting and Exhibit, 2006

For computational multiphase flow involving detailed interfacial dynamics, the outstanding issues include topological changes, geometric capturing, local resolution, conservative treatment, and spurious characteristics associated with property jumps across the interface. In this work, a marker-based, conservative immersed boundary with triangulated surface grid representation for 3D interfaces and local adaptive grid refinement for resolution enhancement is presented. Markers are dynamically added and deleted from the interface using a conservative restructuring to preserve the phase volume and maintain a satisfactory interface resolution. The topological changes are handled using the level-contour reconstruction technique while preserving the logical connectivity information of the interface, which provides enhanced flexibility as compared to the connectivity-free markerbased tracking methods with minimal added algorithmic-complexity. An adaptive Cartesian grid technique has been employed to refine the field computation. The overall accuracy of the algorithms has been established via studies of spurious velocity currents for static bubbles, rising bubble simulations with various terminal shapes, and head-on and off-axis coalescence of two bubbles.

This work presents the development of a numerical technique for simulating two-dimensional viscoelastic free surface flows of an Oldroyd-B fluid. The governing equations for an Oldroyd-B fluid are considered. The time derivative is approximated by a high order method. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries. The full free surface stress conditions are employed. The governing equations are solved by the finite difference method on a staggered grid. Numerical results demonstrating the capabilities of this numerical technique in solving two-dimensional flows of an Oldroyd-B fluid are given for a number of problems involving unsteady free surface flows. In addition, validation and convergence results are presented.

Applied Ocean Research, 2018

In this study, an extended validation of an improved volume-of-fluid (VOF)-based method for 3D highly nonlinear, complex, and breaking free-surface problems in engineering is presented. The VOF interfacetracking scheme is implemented with a Navier-Stokes solver in a curvilinear coordinate system. The numerical procedure is advanced by implementing the scheme on moving Chimera (overset) grids for application in complex engineering problems. The overset grids are used to facilitate the flow simulations of complex geometries and arbitrary motions of objects and improve computational productivity by taking advantage of their flexibility. To solve the six degrees of freedom (6DOF) motions of a rigid body simultaneously, the nonlinear 6DOF motion equations in a vector form system are strongly combined with a flow solver. A collection of problems is carefully selected such that a large density ratio and complex cases under a wide variety of Froude numbers are presented with the aim of demonstrating the capabilities of the new enhanced method on a complicated moving overset grid system. The method is validated for a wide range of parameters. The results of different free-surface problems are presented. The numerical results are compared with numerical alternatives and experimental measurements, and accurate approximations and good agreement are obtained for complex flows.

ArXiv, 2017

We adapt and extend a formulation for soluble surfactant transport in multiphase flows recently presented by Muradoglu & Tryggvason (JCP 274 (2014) 737-757) to the context of the Level Contour Reconstruction Method (Shin et al. IJNMF 60 (2009) 753-778) which is a hybrid method that combines the advantages of the Front-tracking and Level Set methods. Particularly close attention is paid to the formulation and numerical implementation of the surface gradients of surfactant concentration and surface tension. Various benchmark tests are performed to demonstrate the accuracy of different elements of the algorithm. To verify surfactant mass conservation, values for surfactant diffusion along the interface are compared with the exact solution for the problem of uniform expansion of a sphere. The numerical implementation of the discontinuous boundary condition for the source term in the bulk concentration is compared with the approximate solution. Surface tension forces are tested for Marangoni drop translation. Our numerical results for drop deformation in simple shear are compared with experiments and results from previous simulations. All benchmarking tests compare well with existing data thus providing confidence that our adapted LCRM formulation for surfactant advection and diffusion is accurate and effective in three-dimensional multiphase flows. We also demonstrate that this approach applies easily to massively parallel simulations.

In this article we present a 2D and 3D practical interface tracking algorithm to reconstruct and advect the interfaces of interfacial flows. The improved volume-of-fluid (VOF) method in this work is composed of three major components: namely (1) the modified piecewise linear interface calculation (PLIC) based interfacial reconstruction; (2) the Lagrangian split fluid advection and (3) redistribution of volume fraction. Most advanced VOF methods employed by the PLIC technique have faced some problems to extend from 2D to 3D study, because the increase in complexity of the geometry primitives involved has made implementations excessively difficult and ultimately infeasible. The improved algorithm in this study is used to capture the interface of the immiscible fluids which are applicable both in 2D and 3D spaces. A computationally efficient and second-order accurate interface reconstruction method is applied. The normal estimation of the interface is approximated by combining the centered column scheme and the Youngs' method. Besides, a linear mapping technique is implemented to improve the efficiency of numerical simulations with regard to the approximation of capturing the interface. The sequence of the Lagrangian advection in each direction is considered to restrain the fragmentation caused by the strong interface deformation as the fluids propagate. It is significant to note that the method can be accurately accomplished by a regular structured mesh without any geometrical modifications such as boundary fitted grids. Next, the mass conservation is numerically assessed, thus allowing computations to reach the machine precision. The computational results include widely used benchmarks in 2D and 3D cases, such as the solid-body translations and rotations, the swirled single vortex and the deformation fields of fluid body. Good results are obtained through some numerical tests by using present algorithm as comparing with other numerical solutions. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t