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Networks and Heterogeneous Media

2014, Volume 9, Issue 1: 65-95. doi: 10.3934/nhm.2014.9.65
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Numerical network models and entropy principles for isothermal junction flow

  • 1.
    Dept. of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim
  • Received: 01 September 2013 Revised: 01 November 2013

  • Primary: 35L65, 76N15.

  • We numerically explore network models which are derived for the isothermal Euler equations. Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal cross-sectional areas for all connected pipe sections. In the present paper we extend this proof to the case of pipe sections of different cross-sectional areas.
        We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical high-resolution Roe scheme. We propose a numerical treatment of the boundary conditions at the pipe-junction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
        Numerical results are provided for two different network configurations and for three different network models. Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness. Analytical predictions for junctions connecting three pipe sections are verified for both network configurations. Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different cross-sectional area is shown.

    Citation: Gunhild A. Reigstad. Numerical network models and entropy principles for isothermal junction flow[J]. Networks and Heterogeneous Media, 2014, 9(1): 65-95. doi: 10.3934/nhm.2014.9.65

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  • We numerically explore network models which are derived for the isothermal Euler equations. Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal cross-sectional areas for all connected pipe sections. In the present paper we extend this proof to the case of pipe sections of different cross-sectional areas.
        We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical high-resolution Roe scheme. We propose a numerical treatment of the boundary conditions at the pipe-junction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
        Numerical results are provided for two different network configurations and for three different network models. Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness. Analytical predictions for junctions connecting three pipe sections are verified for both network configurations. Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different cross-sectional area is shown.


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    [22] G. A. Reigstad, T. Flåtten, N. E. Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, Submitted, (2013). Preprint available from: http://www.math.ntnu.no/conservation/2013/007.html.
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