Front tracking and two-dimensional Riemann problems

Glimm, James; Klingenberg, Christian; McBryan, Oliver A.; Plohr, Bradley J.; Sharp, David H.; Yaniv, Sara

doi:10.1016/0196-8858(85)90014-4

Cited by 110 publications

(51 citation statements)

References 7 publications

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“…Nothing about the steady-state solutions of the conservation equations appears to exclude the stagnant wedge solution to the shooting problem. Although the exclusion of this solution as a formal consequence of plausible assumptions by Glimm et al (1985) and others is initially justifiable, it is worth noting that same assumptions led Glimm to exclude the solution shown in figure 4(j ). This solution is familiar and occurs in the free expansion of an inviscid supersonic jet.…”

Section: Discussionmentioning

confidence: 99%

“…It comes as no surprise therefore that this formulation appears to unify many of the classical problems of gasdynamics (see figure 4). In the context of Glimm et al (1985), we have obtained a complete set of elementary waves in one angular coordinate. To address the further goal of defining a complete set of wave interactions in one angular coordinate, additional development is required and this is elaborated in the following sections.…”

Section: Synthesis Of Solutionsmentioning

confidence: 99%

“…Given this ambiguity, it has proved helpful to apply concepts from bifurcation theory in order to elucidate the non-uniqueness and non-existence problems that arise in the complex manifolds of possible solutions (see for example the review by Glimm 1988). The most complete description of the Riemann problem for gasdynamics in two spatial dimensions is that of Glimm et al (1985) who unambiguously categorize the possible forms of two-dimensional steady-flow wave interactions based on certain physical assumptions and the fundamental solutions originally deduced by Meyer (1908). As discussed by Glimm et al (1985), the solutions for the interaction problem in two spatial dimensions represent the elementary waves for the open two-dimensional Riemann problem (two spatial dimensions and time).…”

Section: Introductionmentioning

confidence: 99%

“…The most complete description of the Riemann problem for gasdynamics in two spatial dimensions is that of Glimm et al (1985) who unambiguously categorize the possible forms of two-dimensional steady-flow wave interactions based on certain physical assumptions and the fundamental solutions originally deduced by Meyer (1908). As discussed by Glimm et al (1985), the solutions for the interaction problem in two spatial dimensions represent the elementary waves for the open two-dimensional Riemann problem (two spatial dimensions and time). Attempting now to put the current study in context, we formulate the problem in two spatial dimensions and time, but immediately reduce the problem to one angular dimension and time by a scaling argument.…”

Section: Introductionmentioning

confidence: 99%

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Gasdynamic wave interaction in two spatial dimensions

Sanderson

2004

J. Fluid Mech.

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We examine the interaction of shock waves by studying solutions of the twodimensional Euler equations about a point. The problem is reduced to linear form by considering local solutions that are constant along each ray and thereby exhibit no length scale at the intersection point. Closed-form solutions are obtained in a unified manner for standard gasdynamics problems including oblique shock waves, PrandtlMeyer flow and Mach reflection. These canonical gas dynamical problems are shown to reduce to a series of geometrical transformations involving anisotropic coordinate stretching and rotation operations. An entropy condition and a requirement for geometric regularity of the intersection of the incident waves are used to eliminate spurious solutions. Consideration of the downstream boundary conditions leads to a formal determination of the allowable downstream matching criteria. By retaining the time-dependent terms, an approach is suggested for future investigation of the open problem of the stability of shock wave interactions. IntroductionRiemann problems in two spatial dimensions are germane to many problems in experimental, analytical and computational gasdynamics. Familiar examples include the interaction of oblique shock waves in steady supersonic flow, the self-similar problem of transition to Mach reflection from a wedge and the treatment of discontinuous shock fronts in numerical shock fitting algorithms.The well-known approach to the interaction of shock waves has its basis in the work of Rankine (1870) and Hugoniot (1889) regarding the propagation of shock waves. The solution for oblique shock waves was then obtained by Meyer (1908) through a Galilean transformation that imposed a velocity component parallel to the shock front. Also due to Meyer is the development of the Prandtl-Meyer function for expansion through an expansion fan. The development of the theory of shock wave interactions by Courant, Friedrichs, von Neumann and independently by Weise is described by Courant & Friedrichs (1948). By hypothesizing the existence of a vortex sheet at the intersection point, solutions for several interacting shock waves and/or expansion fans may be constructed that are consistent with the experimental observations. A similar approach has been developed for shock-shear layer interactions. The interaction solutions are represented graphically by the intersection in the pressure-flow deflection angle, p-δ, plane of the loci of possible downstream states for the interacting waves. Because of the nonlinear interaction of the waves, problems of non-uniqueness

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Section: Discussionmentioning

confidence: 99%

Section: Synthesis Of Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gasdynamic wave interaction in two spatial dimensions

Sanderson

2004

J. Fluid Mech.

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“…As an alternative, fixed-grid methods have been developed that regularized the interface [4]. These include the volume-of-fluid (VOF) method [14,15], the front-tracking method [16,17] and the level-set method [18][19][20]. All these approaches have the advantage of converting the Lagrangian description of a geometric motion into the Eulerian description.…”

mentioning

confidence: 99%

An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

Feng

Shen

et al.

Modeling of Soft Matter

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Abstract. The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate Navier-Stokes systems for two-phase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less self-contained exposition of the method. We will briefly review the literature, present the governing equations and discuss a numerical scheme based on different numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further.

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The interaction of nonlinear hyperbolic waves

Glimm

1988

Comm Pure Appl Math

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Nonlinearities in wave equations lead to focusing and defocusing of solutions. Focusing causes sharply defined wave fronts. The interaction of such sharply defined wave fronts and more generally of nonlinear hyperbolic waves is of fundamental importance and includes such phenomena as Mach triple point formation, shock wave diffraction patterns and the study of Riemann problems in one and higher dimensions.Recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures. This mathematical theory should be useful in the design of improved computational algorithms and in part was motivated by such considerations.It is also of considerable interest for its own sake as new mathematical phenomena as well as in terms of the direct insight it provides into physical phenomena.Within the subject matter and point of view adopted here, we have attempted to present a broad and, we hope, a representative account of recent progress.

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Front tracking and two-dimensional Riemann problems

Cited by 110 publications

References 7 publications

Gasdynamic wave interaction in two spatial dimensions

Gasdynamic wave interaction in two spatial dimensions

An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

The interaction of nonlinear hyperbolic waves

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