A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection

Almgren, Ann S.; Bell, John B.; Szymczak, William G.

doi:10.1137/s1064827593244213

Cited by 176 publications

(226 citation statements)

References 11 publications

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“…Our methods make use of level set methods for tracking the fluid interface boundaries, coupled to projection methods to solving the associated fluid flows. A large number of background references for projection methods and level set methods are given in [28]; here we briefly mention the original paper on projection methods for incompressible flow by Chorin [8], second-order Godunov-type improvements by Bell, Colella, and Glaz [4], the finite-element approximate projection by Almgren et al [2], and the extension of these techniques to quadrilateral grids (see, for example, Bell et al [6]) and to moving quadrilateral grids (see Trebotich and Colella[26]). On the interface tracking side, level set methods, introduced in Osher and Sethian [15], rely in part on the theory of curve and surface evolution given in Sethian [18,19] and on the link between front propagation and hyperbolic conservation laws discussed in Sethian [20]; these techniques recast interface motion as a time-dependent Eulerian initial value partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…For a general introduction and overview, see Sethian [21]. For details about projection methods and their coupling to level set methods, see Almgren et al [2,3], Bell et al [4], Bell and Marcus [5], Chang et al [7], Chorin [8], Puckett et al [16], Sussman and Smereka [24], Sussman et al [23,25], and Zhu and Sethian [30]. On the viscoelastic side, in recent years there has been considerable interest in projection-type schemes for viscoelastic flow, see, for example, Trebotich et.…”

Section: Introductionmentioning

confidence: 99%

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Two-phase viscoelastic jetting

Sakai

Sethian

2007

Journal of Computational Physics

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A coupled finite difference algorithm on rectangular grids is developed for viscoelastic ink ejection simulations. The ink is modeled by the Oldroyd-B viscoelastic fluid model. The coupled algorithm seamlessly incorporates several things: (1) a coupled level set-projection method for incompressible immiscible two-phase fluid flows; (2) a higher-order Godunov type algorithm for the convection terms in the momentum and level set equations; (3) a simple first-order upwind algorithm for the convection term in the viscoelastic stress equations; (4) central difference approximations for viscosity, surface tension, and upper-convected derivative terms; and (5) an equivalent circuit model to calculate the inflow pressure (or flow rate) from dynamic voltage.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two-phase viscoelastic jetting

Sakai

Sethian

2007

Journal of Computational Physics

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“…The method used to update the velocity and pressure is a variable density extension of the approximate projection method described by Almgren et al [3]. A projection method is a fractional step scheme in which a discretization of (2) is first used to approximate the velocity at the new time, then an elliptic equation for pressure, which results from taking the divergence of (2), is used to impose the divergence constraint on the new velocity and to update the pressure.…”

Section: Projection Methodologymentioning

confidence: 99%

“…The resulting approximate projection satisfies the divergence constraint to second-order accuracy and the overall algorithm is stable. The reader is referred to Almgren et al [3] for a detailed discussion of this approximate projection. In three dimensions, instead of using the 27-point analog to the nine-point stencil presented here, we use a seven-point stencil as discussed in [2].…”

Section: Discretization Of the Projectionmentioning

confidence: 99%

An Adaptive Level Set Approach for Incompressible Two-Phase Flows

Sussman

Almgren

Bell

et al. 1999

Journal of Computational Physics

559

198

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In Sussman, Smereka and Osher (1994), a numerical method using the level set approach was formulated for solving incompressible two-phase 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 Navier-Stokes equations, in order to achieve higher resolution of the interface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations.

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“…The Crank-Nicholson scheme is used to compute diffusion terms. A divergence constraint is satisfied using the approximate pressure projection method of Almgren et al (1996). The density is computed as…”

Section: The Numerical Algorithmmentioning

confidence: 99%

Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique

2011

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Fluid particulate flows are common phenomena in nature and industry. Modeling of such flows at micro and macro levels as well establishing relationships between these approaches are needed to understand properties of the particulate matter. We propose a computational technique based on the direct numerical simulation of the particulate flows. The numerical method is based on the distributed Lagrange multiplier technique following the ideas of Glowinski et al. (1999). Each particle is explicitly resolved on an Eulerian grid as a separate domain, using solid volume fractions. The fluid equations are solved through the entire computational domain, however, Lagrange multiplier constrains are applied inside the particle domain such that the fluid within any volume associated with a solid particle moves as an incompressible rigid body. Mutual forces for the fluid-particle interactions are internal to the system. Particles interact with the fluid via fluid dynamic equations, resulting in implicit fluid-rigid-body coupling relations that produce realistic fluid flow around the particles (i.e., no-slip boundary conditions). The particle-particle interactions are implemented using explicit force-displacement interactions for frictional inelastic particles similar to the DEM method of Cundall et al. (1979) with some modifications using a volume of an overlapping region as an input to the contact forces. The method is flexible enough to handle arbitrary particle shapes and size distributions. A parallel implementation of the method is based on the SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) library, which allows handling of large amounts of rigid particles and enables local grid refinement. Accuracy and convergence of the presented method has been tested against known solutions for a falling sphere as well as by examining fluid flows through stationary particle beds (periodic and cubic packing). To evaluate code performance and validate particle contact physics algorithm, we performed simulations of a representative experiment conducted at the University of California at Berkley for pebble flow through a narrow opening.

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A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection

Cited by 176 publications

References 11 publications

Two-phase viscoelastic jetting

Two-phase viscoelastic jetting

An Adaptive Level Set Approach for Incompressible Two-Phase Flows

Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique

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