Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions

Jens Lang; Pascal Mindt

doi:10.3934/nhm.2018008

Article Contents

2018, Volume 13, Issue 1: 177-190. Doi: 10.3934/nhm.2018008

This issue Previous Article Green's function for elliptic systems: Moment bounds Next Article

Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions

Jens Lang and
Pascal Mindt

Department of Mathematics, Technische Universität Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany

Received: April 2017

Revised: September 2017

Published: March 2018

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Abstract

This paper is concerned with a set of novel coupling conditions for the 3× 3 one-dimensional Euler system with source terms at a junction of pipes with possibly different cross-sectional areas. Beside conservation of mass, we require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. Previously used coupling conditions include equality of pressure or dynamic pressure. They are restricted to the special case of a junction having only one pipe with outgoing flow direction. Recently, Reigstad [SIAM J. Appl. Math., 75:679-702,2015] showed that such pressure-based coupling conditions can produce non-physical solutions for isothermal flows through the production of mechanical energy. Our new coupling conditions ensure energy as well as entropy conservation and also apply to junctions connecting an arbitrary number of pipes with flexible flow directions. We prove the existence and uniqueness of solutions to the generalised Riemann problem at a junction in the neighbourhood of constant stationary states which belong to the subsonic region. This provides the basis for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for initial data with sufficiently small total variation.

Keywords:

Conservation laws,
networks,
Euler equations at junctions,
coupling conditions of compressible fluids.

Mathematics Subject Classification: Primary: 35L60, 35L65, 35Q31, 35R02, 76N10.

Citation:

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Figure 1. Possible wave patterns in the solution of Riemann problems for the Euler equations: shock (S), contact (C) and rarefaction (R).

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Figure 2. Connection of the regions $L$, $L\ast$, $R\ast$, and $R$ with the Lax curve $\mathcal{L}_3$ for incoming pipes (a) and the Lax curves $\mathcal{L}_2\!\circ\!\mathcal{L}_3$ for outgoing pipes (b).

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