ZAMM · Z. Angew. Math. Mech. 93, No. 6 – 7, 465 – 475 (2013) / DOI 10.1002/zamm.201100176 Criteria for nonuniqueness of Riemann solutions to compressible duct flows Ee Han ∗ , Maren Hantke ∗∗ , and Gerald Warnecke ∗∗∗ 1 Institut f¨ ur Analysis und Numerik, Otto-von-Guericke-Universit¨ at Magdeburg, Universitaetsplatz 2, 39106 Magdeburg, Germany Received 9 December 2011, revised 25 October 2012, accepted 16 November 2012 Published online 17 December 2012 Key words Riemann problem, nonuniqueness, duct flows, axisymmetric Euler system, L–M curves. Dedicated to Professor Wolfgang L. Wendland on the occasion of his 75th birthday The Riemann solutions without vacuum states for compressible duct flows have been completely constructed in the paper [11]. However, the nonuniqueness of Riemann solutions due to a bifurcation of wave curves in state space is still an open problem. The purpose of this paper is to single out the physically relevant solution among all the possible Riemann solutions by comparing them with the numerical results of the axisymmetric Euler equations. Andrianov and Warnecke in [2] suggested using the entropy rate admissibility criterion to rule out the unphysical solutions. However, this criterion is not true for some test cases, i.e. the numerical result for axisymmetric three dimensional flows picks up an exact solution which does not satisfy the entropy rate admissibility criterion. Moreover, numerous numerical experiments show that the physically relevant solution is always located on a certain branch of the L–M curves. c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction We study the nonuniqueness of Riemann solutions to the nonconservative Euler equations modeling compressible fluid flows through a duct of discontinuous cross-section, which can be written in the form ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂a ∂t =0, ∂aρ ∂t + ∂aρu ∂x =0, ∂aρv ∂t + ∂a(ρu 2 + p) ∂x = p ∂a ∂x , ∂aρE ∂t + ∂au(ρE + p) ∂x =0, (1) where a(x) is the duct area, and the dependent variables ρ, u, and p denote, respectively, the density, velocity, and pressure of the fluid. The specific total energy is given as E = e + u 2 2 , where e is the internal energy. The polytropic equation of state e = p ρ(γ - 1) (2) is used to close the system, where γ is the ratio of specific heats and satisfies 1 <γ< 5 3 . The Riemann initial data for the system (1) are (a, ρ, u, p)(x, 0) =  (a L ,ρ L ,u L ,p L ) , x< 0, (a R ,ρ R ,u R ,p R ) , x> 0, (3) ∗ Corresponding author E-mail: han.ee@st.ovgu.de ∗∗ E-mail: maren.hantke@ovgu.de ∗∗∗ E-mail: gerald.warnecke@ovgu.de. c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim