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New coupling conditions for isentropic flow on networks

  • * Corresponding author: Yannick Holle

    * Corresponding author: Yannick Holle 
This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) projects 320021702/GRK2326 Energy, Entropy, and Dissipative Dynamics (EDDy) and HE5386/18, 19.
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  • We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling conditions can be derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the generalized Riemann problem is globally in state space. Furthermore, non-increasing energy at the junction and a maximum principle are proven. A numerical example is given in which the new conditions are the only known conditions leading to the physically correct wave types. The approach generalizes to full gas dynamics.

    Mathematics Subject Classification: 35L65, 76N15, 82C40.

    Citation:

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  • Figure 3.  The boundary Riemann set $ \mathcal V(\rho_*, 0) $

    Figure 1.  Construction of $ (\rho_\alpha, u_\alpha) $ and $ (\rho_\beta, u_\beta) $

    Figure 2.  The sets $ \mathcal A_* $, $ \mathcal B_* $, $ \mathcal C_* $ and $ \mathcal J_* $

    Figure 4.  Level sets for different coupling conditions

    Table 1.  Initial data

    pipeline $ \hat\rho_{k} $ $ \hat\rho_{k} \hat u_{k} $
    1 $ +1.0000 $ $ -1.0000 $
    2 $ +1.0000 $ $ +0.5000 $
    3 $ +1.0000 $ $ +0.5000 $
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical results

    Equal density Equal momentum flux Equal stagnation enthalpy Equal artificial density
    pipeline $ \bar\rho_{k} $ $ \bar\rho_{k} \bar u_{k} $ $ \bar\rho_{k} $ $ \bar\rho_{k} \bar u_{k} $ $ \bar\rho_{k} $ $ \bar\rho_{k} \bar u_{k} $ $ \bar\rho_{k} $ $ \bar\rho_{k} \bar u_{k} $
    1 $ +1.0000 $ $ -1.0000 $ $ +0.8964 $ $ -1.1981 $ $ +0.8518 $ $ -1.2670 $ $ +1.1776 $ $ -0.5417 $
    2 $ +1.0000 $ $ +0.5000 $ $ +1.0266 $ $ +0.5991 $ $ +1.0356 $ $ +0.6335 $ $ +0.9346 $ $ +0.2708 $
    3 $ +1.0000 $ $ +0.5000 $ $ +1.0266 $ $ +0.5991 $ $ +1.0356 $ $ +0.6335 $ $ +0.9346 $ $ +0.2708 $
    Energy dissipation $ -7.5000\times 10^{-2} $ $ -1.725\times 10^{-2} $ $ \approx 0 $ $ -1.3852\times 10^{-1} $
     | Show Table
    DownLoad: CSV

    Table 3.  Wave types

    pipeline Equal density Equal momentum flux Equal stagnation enthalpy Equal artificial density
    1 no waves rarefaction wave rarefaction wave shock
    2 no waves shock shock rarefaction wave
    3 no waves shock shock rarefaction wave
     | Show Table
    DownLoad: CSV
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