A High-Order Godunov Method for Multiple Condensed Phases

Miller, Gregory H.; Puckett, Elbridge Gerry

doi:10.1006/jcph.1996.0200

Cited by 191 publications

(180 citation statements)

References 21 publications

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“…Our approach is based on discretizing a transport equation for a volume fraction density for each phase, the averages of which over cells are the volume fractions . The specific form of the transport equation is motivated by previous work in multifluid representations of interfaces [13,26]; the connection to the mathematics and physics being modeled here is discussed in Section 3.2. Our discretization of the volume fraction transport equation uses local tangent-plane approximations to the interface constructed from the volume fractions ␣ , as described in Section 3.1.…”

Section: Overview Of Methodsmentioning

confidence: 99%

“…Following [13,26] we evolve material phase fluid fractions by solving the fluid fraction advection equation…”

Section: Interface Advectionmentioning

confidence: 99%

“…The one-dimensional two-material Riemann problems are solved approximately, using rarefaction shock approximations [12,26]. Use of this approximation is justified by the observation that the rarefaction wave curves and the corresponding shock Hugoniot are C 2 , and so the error in treating a rarefaction as a shock is third order, thus negligible in a numerical scheme whose overall order of accuracy is 2 or less.…”

Section: Solution Of the Solid-fluid Riemann Problemmentioning

confidence: 99%

“…According to the derivation in [26] this modulus acts to normalize ␣ such that ␣ ␣ = 1, while maintaining equality of pressure at the interface. For fluids, then, M is isentropic bulk modulus K S .…”

Section: Moduli Of Incompressibilitymentioning

confidence: 99%

“…However, the combination has not been applied to problems involving material interfaces. More typically, volume-of-fluid representations have been used in material interface problems in a hybrid mode, i.e., in which the material properties are multiply valued in a cell, but the velocity is single valued [13,26,30]. In the present approach, all of the state variables are represented as having distinct values in each phase, with appropriate jump conditions applied at the phase boundaries.…”

Section: Introductionmentioning

confidence: 99%

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A Conservative Three-Dimensional Eulerian Method for Coupled Solid–Fluid Shock Capturing

Miller

Colella²

2002

Journal of Computational Physics

194

193

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A new method is presented for the explicit Eulerian finite difference computation of shock capturing problems involving multiple resolved material phases in three dimensions. We solve separately for each phase the equations of fluid dynamics or solid mechanics, using as interface boundary conditions artificially extended representations of the individual phases. For fluids we use a new 3D spatially unsplit implementation of the piecewise parabolic (PPM) method of Colella and Woodward. For solids we use the 3D spatially unsplit Eulerian solid mechanics method of Miller and Colella. Vacuum and perfectly incompressible obstacles may also be employed as phases.A separate problem is the time evolution of material interfaces, which are represented by planar segments constructed with a volume-of-fluid method. The volume fractions are advanced in time using a second-order 3D spatially unsplit advection routine with a velocity field determined by solution of interface-normal two-phase Riemann problems. From the Riemann problem solutions we also determine crossinterface momentum and energy fluxes.The volume fractions in mixed cells may be arbitrarily small, which would ordinarily make the Courant-Friedrichs-Lewy time step stability limit arbitrarily small as well. We overcome this limitation using the mass-redistribution formalism to conservatively redistribute generalized mass in the neighborhood of the split cells. 3D EULERIAN SHOCK CAPTURING METHOD 27We present an application of this method to an explosion contained in a metal can: a reactive fluid (approximating PBX 9404) is encased within an elastic-plastic solid (approximating copper) surrounded by vacuum. Our implementation is in parallel, and with adaptive mesh refinement. c 2002 Elsevier Science (USA)

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Section: Overview Of Methodsmentioning

confidence: 99%

“…Following [13,26] we evolve material phase fluid fractions by solving the fluid fraction advection equation…”

Section: Interface Advectionmentioning

confidence: 99%

Section: Solution Of the Solid-fluid Riemann Problemmentioning

confidence: 99%

Section: Moduli Of Incompressibilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Conservative Three-Dimensional Eulerian Method for Coupled Solid–Fluid Shock Capturing

Miller

Colella²

2002

Journal of Computational Physics

194

193

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An arbitrary Lagrangian–Eulerian method based on the adaptive Riemann solvers for general equations of state

Tian

Shen

Jiang

et al. 2008

Numerical Methods in Fluids

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SUMMARYApproximate or exact Riemann solvers play a key role in Godunov-type methods. In this paper, three approximate Riemann solvers, the MFCAV, DKWZ and weak wave approximation method schemes, are investigated through numerical experiments, and their numerical features, such as the resolution for shock and contact waves, are analyzed and compared. Based on the analysis, two new adaptive Riemann solvers for general equations of state are proposed, which can resolve both shock and contact waves well. As a result, an ALE method based on the adaptive Riemann solvers is formulated. A number of numerical experiments show good performance of the adaptive solvers in resolving both shock waves and contact discontinuities.

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A comparative study of various pressure relaxation closure models for one‐dimensional two‐material Lagrangian hydrodynamics

Kamm

Shashkov

Fung

et al. 2011

Numerical Methods in Fluids

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SUMMARYLagrangian hydrodynamics of strength-free materials continues to present open issues, even in one dimension. We focus on the problem of closing a system of equations for a two-material cell under the assumption of a single velocity model. There are several existing models and approaches, each possessing different levels of fidelity to the underlying physics and each exhibiting unique features in the computed solutions. We consider three models that take different approaches to breaking the assumption of instantaneous pressure equilibration in the mixed-material cell. The first of these is the well-known method of Tipton, in which a viscosity-like pressure relaxation term is coupled with an otherwise isentropic pressure update to obtain closed-form expressions for the materials' volume fractions and corresponding sub-cell pressures. The second is the physics-inspired, geometry-based pressure relaxation model of Kamm and Shashkov, which is based on an optimization procedure that uses a local, exact Riemann problem. The third model is the unique aspect of this paper, inspired by the work of Delov and Sadchikov and Goncharov and Yanilkin. This sub-scale dynamics approach is motivated by the linearized Riemann problem to initialize volume fraction changes, which are then modified, via the materials' SIEs, to drive the mixed cell toward pressure equilibrium. Each of these approaches is packaged in the framework of a two-step time integration scheme. We compare these multi-material pressure relaxation models, together with corresponding pure-material calculations, on idealized, two-material problems with either ideal-gas or stiffened-gas equations of state.

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A High-Order Godunov Method for Multiple Condensed Phases

Cited by 191 publications

References 21 publications

A Conservative Three-Dimensional Eulerian Method for Coupled Solid–Fluid Shock Capturing

A Conservative Three-Dimensional Eulerian Method for Coupled Solid–Fluid Shock Capturing

An arbitrary Lagrangian–Eulerian method based on the adaptive Riemann solvers for general equations of state

A comparative study of various pressure relaxation closure models for one‐dimensional two‐material Lagrangian hydrodynamics

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