The Riemann Problem for the Euler Equations with Nonconvex and Nonsmooth Equation of State: Construction of Wave Curves

Müller, Siegfried; Voß, Alexander

doi:10.1137/040619909

Cited by 47 publications

(56 citation statements)

References 22 publications

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“…We utilize here a semi-analytical approach, by which we understand an analytical solution that involves the numerical solution of ordinary differential equations and non-linear systems of equations. Semi-analytical methods have been applied to systems of conservation laws in various applications in numerous works including [1,27,28,40,42,43,49]. In [27], a general algorithm for the solution of Riemann problems is proposed and applied to three-phase flow in porous media (see also [28]).…”

Section: Related Workmentioning

confidence: 99%

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On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions

Berres

Bürger²

2007

Z Angew Math Mech

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This paper analyses a continuum model of the sedimentation of polydisperse suspensions, where the volume concentrations of the solids species are the sought unknowns. This leads to systems of conservation laws which are non-genuinely nonlinear ("non-convex") in the sense that they are only piecewise genuinely nonlinear. The solution of a Riemann problem is represented as a concatenation of elementary waves, where the linear degeneracy requires using the Liu entropy condition. After a general description of the composition of elementary waves, an algorithm for the computational construction is given. The solution of the Riemann problem is then utilized as a problem-adapted building block of a front tracking method. For the model of bidisperse suspensions, Riemann problems are classified by the location of the left and right state in the phase space of unknown variables. The front tracking method is applied to solve the initial value problem of a first-order hyperbolic 2 × 2 system of conservation laws describing the settling of a bidisperse suspension. The solution obtained by front tracking is compared with results obtained by an experiment and a finite difference scheme.

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Section: Related Workmentioning

confidence: 99%

“…Another non-genuinely nonlinear system are the Euler equations with non-convex flux function. For these equations, Müller and Voß [43] construct an elementary curve, from which an elementary wave is extracted.…”

Section: Related Workmentioning

confidence: 99%

On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions

Berres

Bürger²

2007

Z Angew Math Mech

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“…The analysis of the wave structure in the evolution of classical hydrodynamics described by Euler equations closed with a non-ideal EOS has been widely studied in the literature 1,13,14,[28][29][30][31][32] . It is well known that the structure and dynamics of waves are deter-mined by the properties of the material through a thermodynamic magnitude, the funda-…”

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confidence: 99%

Anomalous wave structure in magnetized materials described by non-convex equations of state

Serna

Marquina

2014

Physics of Fluids

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We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations closed by a general equation of state (EOS) and propose a complete spectral decomposition of the fluxes that allows us to derive an expression of the nonlinearity factor as the mathematical tool to determine the nature of the wave phenomena. We prove that the possible formation of non-classical wave structure is determined by both the thermodynamic properties of the material and the magnetic field as well as its possible rotation. We demonstrate that phase transitions induced by material properties do not necessarily imply the loss of genuine nonlinearity of the wavefields as is the case in classical hydrodynamics. The analytical expression of the nonlinearity factor allows us to determine the specific amount of magnetic field necessary to prevent formation of complex structure induced by phase transition in the material. We illustrate our analytical approach by considering two non-convex EOS that exhibit phase transitions and anomalous behavior in the evolution. We present numerical experiments validating the analysis performed through a set of one-dimensional Riemann problems. In the examples we show how to determine the appropriate amount of magnetic field in the initial conditions of the Riemann problem to transform a thermodynamic composite wave into a simple nonlinear wave.

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“…Thanks to its simplicity, the van der Waals model with constant c v (also referred to as the polytropic van der Waals model) has frequently been employed in studies on negative and mixed nonlinearities (see, e.g. Cramer & Sen 1986;Cramer 1991;Argrow 1996;Brown & Argrow 1997;Müller & Voß 2006). Indeed, as pointed out by many authors (see, e.g.…”

Section: Problem Formulationmentioning

confidence: 99%

Exact solutions to non-classical steady nozzle flows of Bethe–Zel’dovich–Thompson fluids

Guardone

Vimercati

2016

J. Fluid Mech.

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Steady nozzle flows of Bethe-Zel'dovich-Thompson fluids -substances exhibiting non-classical gasdynamic behaviour in a finite vapour-phase thermodynamic region in close proximity to the liquid-vapour saturation curve -are examined. Non-classical flow features include rarefaction shock waves, shock waves with either upstream or downstream sonic states and split shocks. Exact solutions for a mono-component single-phase fluid expanding from a reservoir into a stationary atmosphere through a conventional converging-diverging nozzle are determined within the quasi-onedimensional inviscid flow approximation. The novel analytical approach makes it possible to elucidate the connection between the adiabatic, possibly non-isentropic flow field and the underlying local isentropic-flow features, including the possible qualitative alterations in passing through shock waves. Contrary to previous predictions based on isentropic-flow inspection, shock disintegration is found to occur also from reservoir states corresponding to a single sonic point. The global layout of the flow configurations produced by a monotonic decrease in the ambient pressure, namely the functioning regime, is examined for reservoir conditions resulting in single-phase flows. Accordingly, a classification of steady nozzle flows into 10 different functioning regimes is proposed. Flow conditions determining the transition between the different classes of flow are investigated and each functioning regime is associated with the corresponding thermodynamic region of reservoir states.

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The Riemann Problem for the Euler Equations with Nonconvex and Nonsmooth Equation of State: Construction of Wave Curves

Cited by 47 publications

References 22 publications

On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions

On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions

Anomalous wave structure in magnetized materials described by non-convex equations of state

Exact solutions to non-classical steady nozzle flows of Bethe–Zel’dovich–Thompson fluids

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