Multi-shock structure of roll waves

Boudlal, Abdelaziz; Liapidevskii, V. Yu.

doi:10.1016/j.crme.2004.03.014

Cited by 3 publications

(3 citation statements)

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“…We shall also consider the persistence of roll-waves with characteristic discontinuities. This is of interest from a physical point of view, since they may appear when the pressure term in shallow water equations is not convex like in stratified flows (see [2] for more details).…”

Section: Discussionmentioning

confidence: 98%

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Persistence of Roll Waves for the Saint Venant Equations

Noble¹

2009

SIAM J. Math. Anal.

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The purpose of the article is to study the linear and nonlinear "stability" of roll-waves that are periodic and discontinuous entropic travelling wave solutions of the Saint Venant equations. More precisely, we prove that the Cauchy problem with initial data close to a roll-wave and satisfying suitable compatibility conditions has a solution on a sufficiently small interval. Introduction.Roll-waves are well-known nonlinear patterns appearing in shallow waters under the effect of gravity and bottom friction. This type of flow is commonly modeled by the "shallow water equations" that can be formally derived from the Navier-Stokes system. This is a hyperbolic system for the fluid height h and the average velocity u that is similar to the isentropic Euler equations. An empiric friction term, so-called Chezy friction term, is added to the momentum equation to model the friction of the bottom; see [7] for more details on the derivation of such models. In [3], Dressler proved the existence of periodic travelling waves that are mathematical solutions of the shallow water equations: those travelling waves are discontinuous, the discontinuity being a Lax shock. Similarly to shocks in hyperbolic systems, several questions arise at this stage: first, the existence of continuous roll-wave solutions of a viscous perturbation of the shallow water equations, the convergence to the inviscid roll-waves in the vanishing viscosity limit and their stability. The existence of continuous travelling waves for viscous shallow water equations follows from a classical Hopf bifurcation argument [9]. More involved is the question of the convergence of those solutions in the vanishing viscosity limit: one can prove, using geometric singular perturbation arguments (Fenichel theorems) that continuous roll-waves are ε close to an inviscid roll-wave with spatial period T and wave speed c 0 for a suitable wave speed c(ε, T ) with ε, the size of the viscosity [10], [11]. Only partial results are known concerning the stability of viscous roll-waves: those patterns are proved to be spectrally stable under large wavelength and small wavelength perturbations and are linearly stable provided that they are strongly spectrally stable; see [12] for more details.In order to obtain more information on stability of viscous roll-waves, at least in the vanishing viscosity limit, we shall consider the inviscid case. In the case of a shock wave, the connection between the viscous and inviscid shocks has been established by Rousset [15] in the one-dimensional (1D) case and Gues and coworkers [6] in the multidimensional case. In the hyperbolic setting, the question of the "stability" of inviscid roll-waves then arises either under 1D or multidimensional perturbations.

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Section: Discussionmentioning

confidence: 98%

“…Integrating (2) in the form ξ = ξ 0 + dh P1(h) , it is proved that (2) has no continuous periodic solution [3]. Thus, we are looking for periodic solutions with discontinuities that satisfy admissibility conditions.…”

Section: Existence Of Roll-wavesmentioning

confidence: 97%

Persistence of Roll Waves for the Saint Venant Equations

Noble¹

2009

SIAM J. Math. Anal.

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“…In the elliptic domain long-wave perturbations of any steady-state flow are growing exponentially and the solution cannot be realized in long channels (the Kelvin-Helmholtz instability). Roll waves for the one-layer flow in an inclined channel with arbitrary cross-section have been investigated in [6]. A steadystate solution of (2.5) from the hyperbolic region is unstable when the velocity of the kinematic wave exceeds the velocity of long waves [17].…”

Section: Governing Equationsmentioning

confidence: 99%

Roll waves and plugs in two-layer flows

Boudlal¹

2008

Eur. J. Appl. Math

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In this paper I consider a system of equations describing two stratified fluids flowing in closed, slightly inclined ducts. In the framework of the shallow water approximation with turbulent friction acting on the wall and at the interface, I investigate a special class of periodic travelling waves containing stable moving jumps, namely, roll waves and periodic slug flows. The modulation equations of roll waves and slugs are derived and a nonlinear stability criterion is obtained. As for plug flow, only a gas-liquid system is considered. The stability criterion is expressed in terms of an integro-differential relation. For self-similar cross-section, this criterion is simplified to a relation for a function of one variable only.

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