On the 1D modeling of fluid flowing through a Junction

Rinaldo M. Colombo; Mauro Garavello

doi:10.3934/dcdsb.2019149

Article Contents

2020, Volume 25, Issue 10: 3917-3929. Doi: 10.3934/dcdsb.2019149

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On the 1D modeling of fluid flowing through a Junction

Rinaldo M. Colombo^1, , and
Mauro Garavello^2,

1.
INdAM Unit – University of Brescia, Via Branze, 38 – 25123 Brescia, Italy
2.
Department of Mathematics and its Applications – University of Milano - Bicocca, Via R. Cozzi, 55 – 20125 Milano, Italy

^* Corresponding author: Rinaldo M. Colombo
^* Corresponding author: Rinaldo M. Colombo

Received: October 31, 2018

Revised: December 31, 2018

Early access: July 2019

Published: October 2020

The authors are supported by the GNAMPA–2018 project

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Abstract

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.

Keywords:

$ p $-system at junctions,
fluid flows in pipe networks.

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)

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

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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

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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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