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On the 1D modeling of fluid flowing through a Junction

  • * Corresponding author: Rinaldo M. Colombo

    * Corresponding author: Rinaldo M. Colombo 

The authors are supported by the GNAMPA–2018 project

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  • A compressible fluid flows through a junction between two different pipes. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather costly. This note compares different 1D approaches to this phenomenon.

    Mathematics Subject Classification: Primary: 35L65.

    Citation:

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  • Figure 1.  Left, the densities $ \check \varphi_{{l}} (\bar u) $ and $ \hat \varphi_{{l}} (\bar u) $, along a $ 1 $–Lax curve; right, the densities $ \check \varphi_{{r}} (\bar u) $ and $ \hat \varphi_{{r}} (\bar u) $ along a reversed $ 2 $-Lax curve; see (6)

    Figure 2.  The situations of corollaries 1, 2 and 3

    Figure 3.  Singular limit on the section of the pipe. Left, a pipe with a section satisfying (12). From left to right, the section of the pipe gets steeper and, right, it ends being a step function

    Table 1.  Various definitions of $\Phi_2$

    $\Phi_2(a_l, u_l, a_r, u_r)$ Meaning
    (L) $a_r P(u_r) - a_l P(u_l)$ Conservation of linear momentum, see [7]
    (p) $p(\rho_r) - p(\rho_l)$ Equal pressure, typically motivated by static equilibrium, see [4,5]
    (P) $P(u_r) - P(u_l)$ Equal dynamic pressure, see [6,8]
    (S) $ \begin{array}{l} \!\! a_r P(u_r) - a_l P(u_l)\\ - \int_{a_l}^{a_r} p \left( R(\alpha;\rho_l, q_l) \right) \, d\alpha\!\! \end{array} $ Limit of the condition for smooth variations of the pipes' sections, see [11,20]
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