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54
A Hybrid Particle Level Set Method for Improved Interface Capturing
 J. Comput. Phys
, 2002
"... In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are underresolved. This is ofte ..."
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Cited by 215 (25 self)
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In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are underresolved. This is often the case for flows undergoing stretching and tearing. The overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method. Our method compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution. The method is presented in three spatial dimensions.
An Adaptive Level Set Approach For Incompressible TwoPhase Flows
 J. Comput. Phys
, 1998
"... In Sussman, Smereka and Osher (1994), a numerical method using the level set approach was formulated for solving incompressible twophase flow with surface tension. In the level set approach, the interface is represented as the zero level set of a smooth function; this has the effect of replacing th ..."
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Cited by 135 (12 self)
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In Sussman, Smereka and Osher (1994), a numerical method using the level set approach was formulated for solving incompressible twophase flow with surface tension. In the level set approach, the interface is represented as the zero level set of a smooth function; this has the effect of replacing the advection of density, which has steep gradients at the interface, with the advection of the level set function, which is smooth. In addition, the interface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the interface in order to robustly compute flows with high density ratios and stiff surface tension effects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible NavierStokes equations, in order to achieve higher resolution of the interface with a minimum of additional expense. We present twodimensional axisymmetric and fully threedimensional results of air bubble and water dr...
A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier–Stokes Equations
 JOURNAL OF COMPUTATIONAL PHYSICS 142, 1–46 (1998)
, 1998
"... In this paper we present a method for solving the equations governing timedependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection–diffusion equations to predict in ..."
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Cited by 131 (28 self)
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In this paper we present a method for solving the equations governing timedependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection–diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergencefree vector fields. Our treatment of the first step uses a specialized secondorder upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and freestream preservation. Our approach to adaptive refinement uses a nested hierarchy of logicallyrectangular girds with simultaneous refinement of the girds in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here
Reconstructing Volume Tracking
 J. Comput. Phys
, 1997
"... A new algorithm for the volume tracking of interfaces in two dimensions is presented. The algorithm is based upon a welldefined, secondorder geometric solution of a volume evolution equation. The method utilitizes local discrete material volume and velocity data to track interfaces of arbitrari ..."
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Cited by 131 (3 self)
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A new algorithm for the volume tracking of interfaces in two dimensions is presented. The algorithm is based upon a welldefined, secondorder geometric solution of a volume evolution equation. The method utilitizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology. A linearitypreserving, piecewise linear interface geometry approximation ensures that solutions generated retain secondorder spatial accuracy. Secondorder temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme. We detail our geometricallybased solution method, in which material volume fluxes are computed systematically with a set of simple geometric tasks. We then interrogate the method by testing its ability to track interfaces through large (yet controlled) topology changes, whereby an initially simple interface configuration is subjected to vortical flows. Numerical results for these strenuous test problems provide evi...
A coupled level set and volumeoffluid method for computing 3D and axisymmetric incompressible twophase flows
 Journal of Computational Physics
, 1006
"... We present a coupled level set/volumeoffluid (CLSVOF) method for computing 3D and axisymmetric incompressible twophase flows. This method combines some of the advantages of the volumeoffluid method with the level set method to obtain a method which is generally superior to either method alone. ..."
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Cited by 96 (2 self)
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We present a coupled level set/volumeoffluid (CLSVOF) method for computing 3D and axisymmetric incompressible twophase flows. This method combines some of the advantages of the volumeoffluid method with the level set method to obtain a method which is generally superior to either method alone. We present direct comparisons between computations made with the CLSVOF method and computations made with the level set method, the volumeoffluid method, and the boundary integral method. We also compare our computations to the exact solution for an oscillating ellipse due to Lamb and experimental results obtained for a rising gas bubble in liquid obtained by Hnat and Buckmaster. Our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow. c ° 2000 Academic Press Key Words: level set; volumeoffluid; curvature; incompressible.
The immersed interface method for the Navier–Stokes equations with singular forces
 J. Comput. Phys
"... Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, t ..."
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Cited by 83 (5 self)
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Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be firstorder accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a secondorder projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully secondorder accuracy for the velocity and nearly secondorder accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper. c ○ 2001 Academic Press Key Words: Navier–Stokes equations; interface; discontinuous and nonsmooth solution; immersed interface method; immersed boundary method; projection method; level set method. 1.
A FrontTracking Algorithm for Accurate Representation of Surface Tension
, 1999
"... We present a front tracking algorithm for the solution of the 2D incompressible NavierStokes equations with interfaces and surface forces. More particularly, we focus our attention on the accurate description of the surface tension terms and the associated pressure jump. We consider the stationary ..."
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Cited by 73 (9 self)
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We present a front tracking algorithm for the solution of the 2D incompressible NavierStokes equations with interfaces and surface forces. More particularly, we focus our attention on the accurate description of the surface tension terms and the associated pressure jump. We consider the stationary Laplace solution for a bubble with surface tension. A careful treatment of the pressure gradient terms at the interface allows us to reduce the spurious currents to the machine precision. Good results are obtained for the oscillation of a capillary wave compared with the linear viscous theory. A classical test of RayleighTaylor instability is presented.
Modelling Pinchoff and Reconnection in a HeleShaw Cell I: The Models and their Calibration
, 2000
"... This is the first paper in a twopart series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a HeleShaw cell. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [69]) to flow ..."
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Cited by 27 (2 self)
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This is the first paper in a twopart series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a HeleShaw cell. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [69]) to flow in a HeleShaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper (Part II [66]), we analyze the behavior of the models in the fully nonline...
Volume Tracking of Interfaces Having Surface Tension in Two and Three Dimensions
, 1996
"... . Solution algorithms are presented for tracking interfaces with piecewise linear (PLIC) volumeoffluid (VOF) methods on Eulerian grids (structured and unstructured) in two and three dimensions. We review the theory of volume tracking methods, derive appropriate volume evolution equations, identify ..."
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Cited by 23 (3 self)
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. Solution algorithms are presented for tracking interfaces with piecewise linear (PLIC) volumeoffluid (VOF) methods on Eulerian grids (structured and unstructured) in two and three dimensions. We review the theory of volume tracking methods, derive appropriate volume evolution equations, identify and present solutions to the basic geometric functions needed for interface reconstruction and volume fluxing, and provide algorithm templates for modern 2D and 3D PLIC VOF interface tracking methods. We discuss some key issues for PLIC VOF methods, namely the method used for time integration of fluid volumes (operator splitting, unsplit, RungeKutta, etc.) and the estimation of interface normals. We also present our latest developments in the continuum surface force (CSF) model for surface tension, namely extension to 3D and variable surface tension effects. We identify and focus on CSF model issues that become especially critical on fine meshes with high density ratio interfacial flows...
Face offsetting: A unified approach for explicit moving interfaces
 J. Comput. Phys
"... Dynamic moving interfaces are central to many scientific, engineering, and graphics applications. In this paper, we introduce a novel method for moving surface meshes, called the face offsetting method, based on a generalized Huygens’ principle. Our method operates directly on a Lagrangian surface m ..."
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Cited by 16 (4 self)
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Dynamic moving interfaces are central to many scientific, engineering, and graphics applications. In this paper, we introduce a novel method for moving surface meshes, called the face offsetting method, based on a generalized Huygens’ principle. Our method operates directly on a Lagrangian surface mesh, without requiring an Eulerian volume mesh. Unlike traditional Lagrangian methods, which move each vertex directly along an approximate normal or userspecified direction, our method propagates faces and then reconstructs vertices through an eigenvalue analysis locally at each vertex to resolve normal and tangential motion of the interface simultaneously. The method also includes techniques for ensuring the integrity of the surface as it evolves. Face offsetting provides a unified framework for various dynamic interface problems and delivers accurate physical solutions even in the presence of singularities and large curvatures. We present the theoretical foundation of our method, and also demonstrate its accuracy, efficiency, and flexibility for a number of benchmark problems and a realworld application.