American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2019149

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

Received  November 2018 Revised  January 2019 Published  July 2019

Fund Project: The authors are supported by the GNAMPA–2018 project.

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: Rinaldo M. Colombo, Mauro Garavello. On the 1D modeling of fluid flowing through a Junction. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019149
References:
[1]

A. Agrawal, L. Djenidi and R. A. Antonia, Simulation of gas flow in microchannels with a sudden expansion or contraction, Journal of Fluid Mechanics, 530 (2005), 135-144.  doi: 10.1017/S0022112005003691.  Google Scholar

[2]

J. Alastruey, S. M. Moore, K. H. Parker, T. David, J. Peiró and S. J. Sherwin, Reduced modelling of blood flow in the cerebral circulation: Coupling 1-D, 0-D and cerebral auto-regulation models, Internat. J. Numer. Methods Fluids, 56 (2008), 1061-1067.  doi: 10.1002/fld.1606.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. New York, 2000.  Google Scholar

[4]

M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295–314 (electronic). doi: 10.3934/nhm.2006.1.295.  Google Scholar

[5]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41–56 (electronic). doi: 10.3934/nhm.2006.1.41.  Google Scholar

[6]

R. M. Colombo and M. Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media, 1 (2006), 495–511 (electronic). doi: 10.3934/nhm.2006.1.495.  Google Scholar

[7]

R. M. Colombo and M. Garavello, On the $p$-system at a junction, In Control Methods in PDE-dynamical Systems, volume 426 of Cont. Math., pages 193–217. AMS, Providence, 2007. doi: 10.1090/conm/426/08189.  Google Scholar

[8]

R. M. Colombo and M. Garavello, On the Cauchy problem for the $p$-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.  doi: 10.1137/060665841.  Google Scholar

[9]

R. M. Colombo, G. Guerra, M. Herty and V. Sachers, Modeling and optimal control of networks of pipes and canals, SIAM J. Math. Anal., 48 (2009), 2032-2050.   Google Scholar

[10]

R. M. Colombo, M. Herty and V. Sachers, On $2\times2$ conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.  doi: 10.1137/070690298.  Google Scholar

[11]

R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the $p$-system, J. Math. Anal. Appl., 361 (2010), 440-456.  doi: 10.1016/j.jmaa.2009.07.022.  Google Scholar

[12]

J. C. de Almeida, J. A. Velásquez and R. Barbieri, A methodology for calculating the natural gas compressibility factor for a distribution network, Petroleum Science and Technology, 32 (2014), 2616-2624.  doi: 10.1080/10916466.2012.755194.  Google Scholar

[13]

J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.  doi: 10.1016/S0005-1098(03)00109-2.  Google Scholar

[14]

E. Dekama and J. Calverta, Pressure losses in sudden transitions between square and rectangular ducts of the same cross-sectional area, Int. J. Heat Fluid Flow, 9 (1988), 2-7.  doi: 10.1016/0142-727X(88)90023-9.  Google Scholar

[15]

M. Á. Fernández, V. Milišić and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs, Multiscale Model. Simul., 4 (2005), 215–236 (electronic). doi: 10.1137/030602010.  Google Scholar

[16]

L. Formaggia, D. Lamponi, A. Veneziani and D. Tuveri, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart, Computer Methods in Biomechanics and Biomedical Engineering, 9 (2006), 273-288.  doi: 10.1080/10255840600857767.  Google Scholar

[17]

L. Formaggia, A. Quarteroni and A. Veneziani, editors, Cardiovascular Mathematics, volume 1 of MS & A, Springer-Verlag Italia, Milan, 2009. doi: 10.1007/978-88-470-1152-6.  Google Scholar

[18]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, volume 9 of AIMS Series on Applied Mathematics, AIMS, Springfield, MO, 2016.  Google Scholar

[19]

P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902.  doi: 10.1016/j.anihpc.2004.02.002.  Google Scholar

[20]

G. Guerra, F. Marcellini and V. Schleper, Balance laws with integrable unbounded sources, SIAM J. Math. Anal., 41 (2009), 1164-1189.  doi: 10.1137/080735436.  Google Scholar

[21]

M. Gugat, Nodal control of conservation laws on networks. Sensitivity calculations for the control of systems of conservation laws with source terms on networks, Cagnol, John (ed.) et al., Chapman & Hall/CRC. Lecture Notes in Pure and Appl. Math., 240 (2005), 201–215. doi: 10.1201/9781420027426.ch16.  Google Scholar

[22]

M. Gugat, M. Herty and S. Müller, Coupling conditions for the transition from supersonic to subsonic fluid states, Netw. Heterog. Media, 12 (2017), 371-380.  doi: 10.3934/nhm.2017016.  Google Scholar

[23]

M. Gugat and G. Leugering, Global boundary controllability of the de St. Venant equations between steady states, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 1-11.  doi: 10.1016/S0294-1449(02)00004-5.  Google Scholar

[24]

M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612.  doi: 10.1137/070688535.  Google Scholar

[25]

M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Internat. J. Numer. Methods Fluids, 56 (2008), 485-506.  doi: 10.1002/fld.1531.  Google Scholar

[26]

H. Holden and N. H. Risebro, Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497–515 (electronic). doi: 10.1137/S0036141097327033.  Google Scholar

[27]

G. Leugering and E. J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164–180 (electronic). doi: 10.1137/S0363012900375664.  Google Scholar

[28]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[29]

T.-M. Liou, C.-F. Kao and S.-M. Wu, The flow in a rectangular channel with sudden contraction and expansion, Chinese Institute of Engineers Journal, 10 (1987), 139-146.   Google Scholar

[30]

T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys., 83 (1982), 243-260.  doi: 10.1007/BF01976043.  Google Scholar

[31]

G. Montenegro and A. Onorati, Modeling of silencers for I.C. engine intake and exhaust systems by means of an integrated 1D-multiD approach, volume 1 of SAE Int. J. Engines, pages 466–479. SAE 2008 Int. Congress & Exp., Detroit, Michigan, 2008. doi: 10.4271/2008-01-0677.  Google Scholar

[32]

E. Rathakrishnana and A. K. Sreekanthb, Rarefied flow through sudden enlargements, Fluid Dynamics Research, 16 (1995), 131-145.  doi: 10.1016/0169-5983(95)00006-Y.  Google Scholar

[33]

G. A. Reigstad, T. Flåtten, N. Erland Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbolic Differ. Equ., 12 (2015), 37-59.  doi: 10.1142/S0219891615500022.  Google Scholar

[34]

J. S. Vrentas and J. L. Duda, Flow of a newtonian fluid through a sudden contraction, Flow, Turbulence and Combustion, 28 (1973), 241-260.  doi: 10.1007/BF00413071.  Google Scholar

[35]

G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley-Interscience Publication. doi: 10.1002/9781118032954.  Google Scholar

[36]

D. E. Winterbone and R. J. Pearson, Theory of Engine Manifold Design, Professional Engineering Publishing, 2000. Google Scholar

show all references

References:
[1]

A. Agrawal, L. Djenidi and R. A. Antonia, Simulation of gas flow in microchannels with a sudden expansion or contraction, Journal of Fluid Mechanics, 530 (2005), 135-144.  doi: 10.1017/S0022112005003691.  Google Scholar

[2]

J. Alastruey, S. M. Moore, K. H. Parker, T. David, J. Peiró and S. J. Sherwin, Reduced modelling of blood flow in the cerebral circulation: Coupling 1-D, 0-D and cerebral auto-regulation models, Internat. J. Numer. Methods Fluids, 56 (2008), 1061-1067.  doi: 10.1002/fld.1606.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. New York, 2000.  Google Scholar

[4]

M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295–314 (electronic). doi: 10.3934/nhm.2006.1.295.  Google Scholar

[5]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41–56 (electronic). doi: 10.3934/nhm.2006.1.41.  Google Scholar

[6]

R. M. Colombo and M. Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media, 1 (2006), 495–511 (electronic). doi: 10.3934/nhm.2006.1.495.  Google Scholar

[7]

R. M. Colombo and M. Garavello, On the $p$-system at a junction, In Control Methods in PDE-dynamical Systems, volume 426 of Cont. Math., pages 193–217. AMS, Providence, 2007. doi: 10.1090/conm/426/08189.  Google Scholar

[8]

R. M. Colombo and M. Garavello, On the Cauchy problem for the $p$-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.  doi: 10.1137/060665841.  Google Scholar

[9]

R. M. Colombo, G. Guerra, M. Herty and V. Sachers, Modeling and optimal control of networks of pipes and canals, SIAM J. Math. Anal., 48 (2009), 2032-2050.   Google Scholar

[10]

R. M. Colombo, M. Herty and V. Sachers, On $2\times2$ conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.  doi: 10.1137/070690298.  Google Scholar

[11]

R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the $p$-system, J. Math. Anal. Appl., 361 (2010), 440-456.  doi: 10.1016/j.jmaa.2009.07.022.  Google Scholar

[12]

J. C. de Almeida, J. A. Velásquez and R. Barbieri, A methodology for calculating the natural gas compressibility factor for a distribution network, Petroleum Science and Technology, 32 (2014), 2616-2624.  doi: 10.1080/10916466.2012.755194.  Google Scholar

[13]

J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa Novel and G. Bastin, Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.  doi: 10.1016/S0005-1098(03)00109-2.  Google Scholar

[14]

E. Dekama and J. Calverta, Pressure losses in sudden transitions between square and rectangular ducts of the same cross-sectional area, Int. J. Heat Fluid Flow, 9 (1988), 2-7.  doi: 10.1016/0142-727X(88)90023-9.  Google Scholar

[15]

M. Á. Fernández, V. Milišić and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs, Multiscale Model. Simul., 4 (2005), 215–236 (electronic). doi: 10.1137/030602010.  Google Scholar

[16]

L. Formaggia, D. Lamponi, A. Veneziani and D. Tuveri, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart, Computer Methods in Biomechanics and Biomedical Engineering, 9 (2006), 273-288.  doi: 10.1080/10255840600857767.  Google Scholar

[17]

L. Formaggia, A. Quarteroni and A. Veneziani, editors, Cardiovascular Mathematics, volume 1 of MS & A, Springer-Verlag Italia, Milan, 2009. doi: 10.1007/978-88-470-1152-6.  Google Scholar

[18]

M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, volume 9 of AIMS Series on Applied Mathematics, AIMS, Springfield, MO, 2016.  Google Scholar

[19]

P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902.  doi: 10.1016/j.anihpc.2004.02.002.  Google Scholar

[20]

G. Guerra, F. Marcellini and V. Schleper, Balance laws with integrable unbounded sources, SIAM J. Math. Anal., 41 (2009), 1164-1189.  doi: 10.1137/080735436.  Google Scholar

[21]

M. Gugat, Nodal control of conservation laws on networks. Sensitivity calculations for the control of systems of conservation laws with source terms on networks, Cagnol, John (ed.) et al., Chapman & Hall/CRC. Lecture Notes in Pure and Appl. Math., 240 (2005), 201–215. doi: 10.1201/9781420027426.ch16.  Google Scholar

[22]

M. Gugat, M. Herty and S. Müller, Coupling conditions for the transition from supersonic to subsonic fluid states, Netw. Heterog. Media, 12 (2017), 371-380.  doi: 10.3934/nhm.2017016.  Google Scholar

[23]

M. Gugat and G. Leugering, Global boundary controllability of the de St. Venant equations between steady states, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 1-11.  doi: 10.1016/S0294-1449(02)00004-5.  Google Scholar

[24]

M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612.  doi: 10.1137/070688535.  Google Scholar

[25]

M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Internat. J. Numer. Methods Fluids, 56 (2008), 485-506.  doi: 10.1002/fld.1531.  Google Scholar

[26]

H. Holden and N. H. Risebro, Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497–515 (electronic). doi: 10.1137/S0036141097327033.  Google Scholar

[27]

G. Leugering and E. J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164–180 (electronic). doi: 10.1137/S0363012900375664.  Google Scholar

[28]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[29]

T.-M. Liou, C.-F. Kao and S.-M. Wu, The flow in a rectangular channel with sudden contraction and expansion, Chinese Institute of Engineers Journal, 10 (1987), 139-146.   Google Scholar

[30]

T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys., 83 (1982), 243-260.  doi: 10.1007/BF01976043.  Google Scholar

[31]

G. Montenegro and A. Onorati, Modeling of silencers for I.C. engine intake and exhaust systems by means of an integrated 1D-multiD approach, volume 1 of SAE Int. J. Engines, pages 466–479. SAE 2008 Int. Congress & Exp., Detroit, Michigan, 2008. doi: 10.4271/2008-01-0677.  Google Scholar

[32]

E. Rathakrishnana and A. K. Sreekanthb, Rarefied flow through sudden enlargements, Fluid Dynamics Research, 16 (1995), 131-145.  doi: 10.1016/0169-5983(95)00006-Y.  Google Scholar

[33]

G. A. Reigstad, T. Flåtten, N. Erland Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbolic Differ. Equ., 12 (2015), 37-59.  doi: 10.1142/S0219891615500022.  Google Scholar

[34]

J. S. Vrentas and J. L. Duda, Flow of a newtonian fluid through a sudden contraction, Flow, Turbulence and Combustion, 28 (1973), 241-260.  doi: 10.1007/BF00413071.  Google Scholar

[35]

G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley-Interscience Publication. doi: 10.1002/9781118032954.  Google Scholar

[36]

D. E. Winterbone and R. J. Pearson, Theory of Engine Manifold Design, Professional Engineering Publishing, 2000. Google Scholar

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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$\Phi_2(a_l, u_l, a_r, u_r)$ 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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