A Numerical Method for Solving Incompressible Flow Problems with a Surface of Discontinuity

Helenbrook, Brian T.; Martinelli, Luigi; Law, Chung K.

doi:10.1006/jcph.1998.6115

Cited by 40 publications

(39 citation statements)

References 26 publications

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“…Another sharp interface method in which the jump conditions in the coefficients was taken into account in constructing the algorithm was developed in [11] and [12] for the incompressible Navier-Stokes equations.…”

Section: Figmentioning

confidence: 99%

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The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

Lai

2001

Journal of Computational Physics

320

300

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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 first-order 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 second-order projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order 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.

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Section: Figmentioning

confidence: 99%

“…Besides the differences in the methodology, our method distinguishes from [16,29,30] in accuracy, and distinguishes from [11,12] in dealing with different problems. There are some other models and methods for interface problems, notably, the phase field model and the finite volume method, for example, [36].…”

Section: Figmentioning

confidence: 99%

The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

Lai

2001

Journal of Computational Physics

320

300

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show abstract

“…Near the interface, the velocity of the unreacted material contains large O(1) numerical errors where it has been nonphysically forced to be continuous with the velocity of the reacted material. Partial solutions to these problems where proposed in [9] where the authors were able to remove the numerical smearing of the normal velocity obtaining a sharp interface profile. Unfortunately, the interface treatment in [9] was considerably intricate and the calculation had to be terminated if two flame fronts were significantly close to each other, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Partial solutions to these problems where proposed in [9] where the authors were able to remove the numerical smearing of the normal velocity obtaining a sharp interface profile. Unfortunately, the interface treatment in [9] was considerably intricate and the calculation had to be terminated if two flame fronts were significantly close to each other, i.e. this method cannot handle the simple merging of flame discontinuities.…”

Section: Introductionmentioning

confidence: 99%

A Boundary Condition Capturing Method for Incompressible Flame Discontinuities

Nguyen

Fedkiw

Kang

2001

Journal of Computational Physics

149

116

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In this paper, we propose a new numerical method for treating two phase incompressible flow where one phase is being converted into the other, e.g. the vaporization of liquid water. We consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow. This method was designed as an extension of the Ghost Fluid Method [4] and relies heavily on the boundary condition capturing technology developed in [13] for the variable coefficient Poisson equation and in [12] for multiphase incompressible flow. Our new numerical method admits a sharp interface representation similar to the method proposed in [9]. Since the interface boundary conditions are handled in a simple and straightforward fashion, the code is very robust, e.g. no special treatment is required to treat the merging of flame fronts. The method is presented in three spatial dimensions, with numerical examples in one, two and three spatial dimensions.

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“…The previously mentioned body-fitted grid and the moving unstructured grid methods are examples of these attempts. Another, more recent, approach is to retain the stationary structured grid and improve the interface treatment by, for example, introducing special difference formulas that incorporate the jump across the interface [45][46][47][48]. One of these is the Ghost Fluid method, in which the jump conditions that hold at the interface are captured implicitly.…”

Section: Numerical Methods For Two Phase Flowsmentioning

confidence: 99%

Vapor bubbles in confined geometries : a numerical study

Can¹

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A Numerical Method for Solving Incompressible Flow Problems with a Surface of Discontinuity

Cited by 40 publications

References 26 publications

The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

A Boundary Condition Capturing Method for Incompressible Flame Discontinuities

Vapor bubbles in confined geometries : a numerical study

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