# American Institute of Mathematical Sciences

March  2014, 9(1): 65-95. doi: 10.3934/nhm.2014.9.65

## 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, Norway

Received  September 2013 Revised  November 2013 Published  April 2014

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. Networks & Heterogeneous Media, 2014, 9 (1) : 65-95. doi: 10.3934/nhm.2014.9.65
##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.  Google Scholar [2] 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. doi: 10.3934/nhm.2006.1.295.  Google Scholar [3] M. K. Banda, M. Herty and J.-M. T. Ngnotchouye, Toward a mathematical analysis for drift-flux multiphase flow models in networks, SIAM J. Sci. Comput., 31 (2010), 4633-4653. doi: 10.1137/080722138.  Google Scholar [4] M. K. Banda, M. Herty and J.-M. T. Ngnotchouye, Coupling drift-flux models with unequal sonic speeds, Math. Comput. Appl., 15 (2010), 574-584.  Google Scholar [5] J. Brouwer, I. Gasser and M. Herty, Gas pipeline models revisited: Model hierarchies, nonisothermal models, and simulations of networks, Multiscale Model. Simul., 9 (2011), 601-623. doi: 10.1137/100813580.  Google Scholar [6] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.  Google Scholar [7] 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. doi: 10.3934/nhm.2006.1.495.  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, M. Herty and V. Sachers, On 2 $\times$ 2 conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622. doi: 10.1137/070690298.  Google Scholar [10] R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, J. Hyperbol. Differ. Eq., 5 (2008), 547-568. doi: 10.1142/S0219891608001593.  Google Scholar [11] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar [12] M. Garavello, A review of conservation laws on networks, Netw. Heterog. Media, 5 (2010), 565-581. doi: 10.3934/nhm.2010.5.565.  Google Scholar [13] M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612. doi: 10.1137/070688535.  Google Scholar [14] M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Netw. Heterog. Media, 56 (2008), 485-506. doi: 10.1002/fld.1531.  Google Scholar [15] H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.  Google Scholar [16] S. W. Hong and C. Kim, A new finite volume method on junction coupling and boundary treatment for flow network system analyses, Int. J. Numer. Meth. Fluids, 65 (2011), 707-742. doi: 10.1002/fld.2212.  Google Scholar [17] T. Kiuchi, An implicit method for transient gas flows in pipe networks, Int. J. Heat and Fluid Flow, 15 (1994), 378-383. doi: 10.1016/0142-727X(94)90051-5.  Google Scholar [18] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, 6th edition, Cambridge University Press, 2007. doi: 10.1017/CBO9780511791253.  Google Scholar [19] A. Osiadacz, Simulation of transient gas flows in networks, Int. J. Numer. Meth. Fluids, 4 (1984), 13-24. doi: 10.1002/fld.1650040103.  Google Scholar [20] R. J. Pearson, M. D. Bassett, P. Batten and D. E. Winterbone, Two-dimensional simulation of wave propagation in a three-pipe junction, J. Eng. Gas Turbines Power, 122 (2000), 549-555. doi: 10.1115/1.1290589.  Google Scholar [21] J. Pérez-García, E. Sanmiguel-Rojas, J. Hernández-Grau and A. Viedma, Numerical and experimental investigations on internal compressible flow at T-type junctions, Experimental Thermal and Fluid Science, 31 (2006), 61-74. Google Scholar [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. Google Scholar [23] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43 (1981), 357-372. doi: 10.1016/0021-9991(81)90128-5.  Google Scholar [24] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer-Verlag, 2009. doi: 10.1007/b79761.  Google Scholar

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##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.  Google Scholar [2] 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. doi: 10.3934/nhm.2006.1.295.  Google Scholar [3] M. K. Banda, M. Herty and J.-M. T. Ngnotchouye, Toward a mathematical analysis for drift-flux multiphase flow models in networks, SIAM J. Sci. Comput., 31 (2010), 4633-4653. doi: 10.1137/080722138.  Google Scholar [4] M. K. Banda, M. Herty and J.-M. T. Ngnotchouye, Coupling drift-flux models with unequal sonic speeds, Math. Comput. Appl., 15 (2010), 574-584.  Google Scholar [5] J. Brouwer, I. Gasser and M. Herty, Gas pipeline models revisited: Model hierarchies, nonisothermal models, and simulations of networks, Multiscale Model. Simul., 9 (2011), 601-623. doi: 10.1137/100813580.  Google Scholar [6] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.  Google Scholar [7] 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. doi: 10.3934/nhm.2006.1.495.  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, M. Herty and V. Sachers, On 2 $\times$ 2 conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622. doi: 10.1137/070690298.  Google Scholar [10] R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, J. Hyperbol. Differ. Eq., 5 (2008), 547-568. doi: 10.1142/S0219891608001593.  Google Scholar [11] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar [12] M. Garavello, A review of conservation laws on networks, Netw. Heterog. Media, 5 (2010), 565-581. doi: 10.3934/nhm.2010.5.565.  Google Scholar [13] M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612. doi: 10.1137/070688535.  Google Scholar [14] M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Netw. Heterog. Media, 56 (2008), 485-506. doi: 10.1002/fld.1531.  Google Scholar [15] H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.  Google Scholar [16] S. W. Hong and C. Kim, A new finite volume method on junction coupling and boundary treatment for flow network system analyses, Int. J. Numer. Meth. Fluids, 65 (2011), 707-742. doi: 10.1002/fld.2212.  Google Scholar [17] T. Kiuchi, An implicit method for transient gas flows in pipe networks, Int. J. Heat and Fluid Flow, 15 (1994), 378-383. doi: 10.1016/0142-727X(94)90051-5.  Google Scholar [18] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, 6th edition, Cambridge University Press, 2007. doi: 10.1017/CBO9780511791253.  Google Scholar [19] A. Osiadacz, Simulation of transient gas flows in networks, Int. J. Numer. Meth. Fluids, 4 (1984), 13-24. doi: 10.1002/fld.1650040103.  Google Scholar [20] R. J. Pearson, M. D. Bassett, P. Batten and D. E. Winterbone, Two-dimensional simulation of wave propagation in a three-pipe junction, J. Eng. Gas Turbines Power, 122 (2000), 549-555. doi: 10.1115/1.1290589.  Google Scholar [21] J. Pérez-García, E. Sanmiguel-Rojas, J. Hernández-Grau and A. Viedma, Numerical and experimental investigations on internal compressible flow at T-type junctions, Experimental Thermal and Fluid Science, 31 (2006), 61-74. Google Scholar [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. Google Scholar [23] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43 (1981), 357-372. doi: 10.1016/0021-9991(81)90128-5.  Google Scholar [24] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer-Verlag, 2009. doi: 10.1007/b79761.  Google Scholar
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