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Keywords:
non-uniqueness; inviscid gas flow; compressible Euler equations; quasi-one-dimensional; axisymmetric; finite volume method
Summary:
Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with the Mach number equal to one (sonic states). Analytical quasi-one-dimensional results are supplemented by quasi-one-dimensional and axisymmetric three-dimensional finite volume computations. Good agreement between quasi-one-dimensional and axisymmetric results, including the presence of multiple stationary solutions, is presented for axisymmetric nozzles with reasonably small slopes of the radius.
References:
[1] M.  Feistauer: Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Harlow, 1993. Zbl 0819.76001
[2] J.  Felcman, P.  Šolín: On the construction of the Osher-Solomon scheme for 3D  Euler equations. East-West J.  Numer. Math. 6 (1998), 43–64. MR 1629850
[3] C. Hirsch: Numerical Computation of Internal and External Flows, Vol.  2. J.  Wiley & Sons, Chichester, 1990.
[4] D. D.  Knight: Inviscid Compressible Flow. The Handbook of Fluid Dynamics. CRC, 1998.
[5] L. D.  Landau, E. M.  Lifschitz: Fluid Mechanics. Pergamon Press, London, 1959.
[6] H. Ockendon, A. B.  Tayler: Inviscid Fluid Flow. Springer-Verlag, New York-Heidelberg-Berlin, 1983. MR 0693294
[7] H.  Ockendon, J. R.  Ockendon: The Fanno Model for Turbulent Compressible Flow. Preprint, OCIAM. Mathematical Institute, Oxford University, Oxford, 2001. MR 1875697
[8] S.  Osher, F. Solomon: Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comp. 38 (1982), 339–374. MR 0645656
[9] A.  H.  Shapiro: The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1. The Ronald Press Co., New York, 1953.
[10] J. L.  Steger, R. F.  Warming: Flux vector splitting of the inviscid gasdynamics equations with applications to finite-difference methods. J. Comput. Phys. 40 (1981), 263–293. MR 0617098
[11] A.  Terenzi, N.  Mancini, F.  Podenzani: Transient Compressible Flow in Pipelines: a Godunov-type Solver for Navier-Stokes Equations. Preprint, 2000.
[12] E.  Truckenbrodt: Fluidmechanik, Band  2. Springer-Verlag, Berlin-Heidelberg-New York, 1980. Zbl 0445.76001
[13] G. Vijayasundaram: Transonic flow simulation using upstream centered scheme of Godunov type in finite elements. J.  Comput. Phys. 63 (1986), 416–433. MR 0835825
[14] A. J.  Ward-Smith: Internal Fluid Flow, The Fluid Dynamics of Flow in Pipes and Ducts. Clarendon Press, Oxford, 1980.
[15] P.  Wesseling: Principles of Computational Fluid Dynamics. Springer-Verlag, Berlin, 2000. MR 1796357 | Zbl 0960.76002
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