On Zero Pressure Gas Dynamics

Bouchut, François

doi:10.1142/9789814354165_0006

Cited by 192 publications

(236 citation statements)

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“…Let us remark that the system (7) for the spray contains the pressureless gas dynamics system which has been studied by several authors [2,4]. Indeed, this system can be written:…”

Section: Pressureless Gas Dynamicsmentioning

confidence: 99%

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A high order moment method simulating evaporation and advection of a polydisperse liquid spray

Kah

Laurent

Massot

et al. 2012

Journal of Computational Physics

109

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In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N , N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N -dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.

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“…Let us remark that the system (7) for the spray contains the pressureless gas dynamics system which has been studied by several authors [2,4]. Indeed, this system can be written:…”

Section: Pressureless Gas Dynamicsmentioning

confidence: 99%

“…In the case of spray, the first order scheme also satisfies the maximum principle on the velocity. Other properties, like entropy inequalities or the TVD property on the velocity, can also be proved, see [2].…”

Section: First Order Kinetic Schemementioning

confidence: 99%

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

Kah

Laurent

Massot

et al. 2012

Journal of Computational Physics

109

View full text Add to dashboard Cite

show abstract

“…This definition gives natural generalizations of the classical definition of the weak L ∞ -solutions and specifies the definition of measuresolutions given in [1], [26], [28].…”

Section: Singular Solutions To Hyperbolic Systems Of Conservation Lawsmentioning

confidence: 99%

“…Here as was already mentioned above, the functions F (u, v), G(u, v), H(u, v) are linear with respect to v. As was shown in [1], [6]- [15], [26], [28], in order to solve this problem for some "nonclassical cases", it is necessary to introduce new elementary singularities called δ-shock waves. These are generalized solutions of hyperbolic systems of conservation laws of the form u(x, t) = u 0 + u 1 H(−x + ct), v(x, t) = v 0 + v 1 H(−x + ct) + e(t)δ(−x + ct), (1.5) where e(t) is a smooth function such that e(0) = 0 and δ(ξ) is the Dirac delta function.…”

Section: ) Where F (U V) G(u V) H(u V) Are Smooth Functions LImentioning

confidence: 99%

Untitled

2005

Quart. Appl. Math.

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Abstract. For two classes of hyperbolic systems of conservation laws new definitions of a δ-shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of δ-shocks. The weak asymptotics method developed by the authors is used to describe the propagation of δ-shock waves to the three types of systems of conservation laws and derive the corresponding Rankine-Hugoniot conditions for δ-shocks.

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“…Such a delta-shock theory has been developed rigorously in Bouchut (1994), Keyfitz (1999), Li (2001), Li, Zhang & Yang (1998), Yang (1999, and here we will only be concerned with the practicalities of the theory which can be described using ad hoc expansions for the variables. In particular, we avoid the technicalities of the precise definition of products of distributions.…”

Section: Introductionmentioning

confidence: 99%

Non-classical shallow water flows

Edwards¹,

Howison²,

Ockendon³

et al. 2007

IMA Journal of Applied Mathematics

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This paper deals with violent discontinuities in shallow water flows with large Froude number F .On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory we show that, over a certain timescale, this discontinuity may be described by a delta-shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted.For flows on a sloping base, we show that for flow with an aspect ratio of O(F −2 ) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a deltashock. The physical manifestation of this discontinuity is a small 'tube' of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base.

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On Zero Pressure Gas Dynamics

Cited by 192 publications

References 0 publications

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

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Non-classical shallow water flows

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