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September  2017, 12(3): 371-380. doi: 10.3934/nhm.2017016

Coupling conditions for the transition from supersonic to subsonic fluid states

Martin Gugat 1, , Michael Herty 2,, and  Siegfried Müller 2,

1. 

Lehrstuhl für Angewandte Mathematik 2, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, D-91058 Erlangen Germany
2. 

Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, D-52064 Aachen, Germany

* Corresponding author: Michael Herty

Received  May 2016 Revised  July 2017 Published  September 2017

Fund Project: The first author is supported by the DFG Collaborative Research Centre SFB-TR-154, C03. The second author is supported by NSF RNMS grant No. 1107444, DFG HE5386/13, 14, 15-1 and the DAAD{MIUR project.The third author is supported by the DFG Collaborative Research Center SFB-TR-40, TP A1.

We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas flow is governed by a general p-system. By extending the notion of demand and supply known from traffic flow analysis we obtain a constructive existence result of solutions compatible with the introduced conditions.

Keywords: p-system, coupling, conditions networks.
Mathematics Subject Classification: Primary: 35L60, 35L04; Secondary: 35R02.
Citation: Martin Gugat, Michael Herty, Siegfried Müller. Coupling conditions for the transition from supersonic to subsonic fluid states. Networks & Heterogeneous Media, 2017, 12 (3) : 371-380. doi: 10.3934/nhm.2017016
References:
[1]

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

[2]

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

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[4]

A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: recent results and perspectives, European Mathematical Society-Surveys in Mathematical Sciences, 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[5]

G.-Q. Chen and D. Wang, The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, 1 (2002), 421-543.  doi: 10.1016/S1874-5792(02)80012-X.  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 (electronic).  doi: 10.1137/S0036141004402683.  Google Scholar

[7]

R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

A. de Saint-Venant, Thèorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l'introduction des marèes dans leur lit., C.R. Acad. Sci. Paris, 73 (1871), 147-154.   Google Scholar

[12]

M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models.  Google Scholar

[13]

E. Godlewski and P. -A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[14]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.  doi: 10.1137/05062617X.  Google Scholar

[15]

M. Herty and M. Seaïd, Assessment of coupling conditions in water way intersections, Internat. J. Numer. Methods Fluids, 71 (2013), 1438-1460.  doi: 10.1002/fld.3719.  Google Scholar

[16]

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

[17]

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

[18]

S. Joana, M. Joris and T. Evangelos, Technical and Economical Characteristics of Co2 Transmission Pipeline Infrastructure, Technical report, JRC Scientic and Technical Reports, European Commission. Google Scholar

[19]

J.-P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45.   Google Scholar

[20]

F. Murzyn and H. Chanson, Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps, Experiments in Fluids, 45 (2008), 513-521.  doi: 10.1007/s00348-008-0494-4.  Google Scholar

[21]

A. Osiadacz, Simulation of transient flow in gas networks, Int. Journal for Numerical Methods in Fluid Dynamics, 4 (1984), 13-23.  doi: 10.1002/fld.1650040103.  Google Scholar

[22]

B. Sultanian, Fluid Mechanics: An Intermediate Approach, CRC Press, 2015. Google Scholar

[23]

R. Ugarelli and V. D. Federico, Transition from supercritical to subcritical regime in free surface flow of yield stress fluids Geophys. Res. Lett. , 34 (2007), L21402. doi: 10.1029/2007GL031487.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[4]

A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: recent results and perspectives, European Mathematical Society-Surveys in Mathematical Sciences, 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[5]

G.-Q. Chen and D. Wang, The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, 1 (2002), 421-543.  doi: 10.1016/S1874-5792(02)80012-X.  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 (electronic).  doi: 10.1137/S0036141004402683.  Google Scholar

[7]

R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

A. de Saint-Venant, Thèorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l'introduction des marèes dans leur lit., C.R. Acad. Sci. Paris, 73 (1871), 147-154.   Google Scholar

[12]

M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models.  Google Scholar

[13]

E. Godlewski and P. -A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[14]

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal., 38 (2006), 595-616.  doi: 10.1137/05062617X.  Google Scholar

[15]

M. Herty and M. Seaïd, Assessment of coupling conditions in water way intersections, Internat. J. Numer. Methods Fluids, 71 (2013), 1438-1460.  doi: 10.1002/fld.3719.  Google Scholar

[16]

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

[17]

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

[18]

S. Joana, M. Joris and T. Evangelos, Technical and Economical Characteristics of Co2 Transmission Pipeline Infrastructure, Technical report, JRC Scientic and Technical Reports, European Commission. Google Scholar

[19]

J.-P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45.   Google Scholar

[20]

F. Murzyn and H. Chanson, Experimental assessment of scale effects affecting two-phase flow properties in hydraulic jumps, Experiments in Fluids, 45 (2008), 513-521.  doi: 10.1007/s00348-008-0494-4.  Google Scholar

[21]

A. Osiadacz, Simulation of transient flow in gas networks, Int. Journal for Numerical Methods in Fluid Dynamics, 4 (1984), 13-23.  doi: 10.1002/fld.1650040103.  Google Scholar

[22]

B. Sultanian, Fluid Mechanics: An Intermediate Approach, CRC Press, 2015. Google Scholar

[23]

R. Ugarelli and V. D. Federico, Transition from supercritical to subcritical regime in free surface flow of yield stress fluids Geophys. Res. Lett. , 34 (2007), L21402. doi: 10.1029/2007GL031487.  Google Scholar

Figure 1.  Junction of $n+m=|\delta^-|+|\delta^+|$ connected pipes
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Figure 2.  The supply function $\rho {\,\mapsto\,} s(\rho;\bar{U})$ in red for given data $\bar{U}$ indicated by a cross for $\bar{\rho }>\rho^*$ (left) and $\bar{\rho }<\rho^*$ (right). Also shown in blue is the curve $L_2^-.$ The state $U^*$ is indicated by a circle.
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Figure 3.  The demand function $\rho {\,\mapsto\,} d(\rho; \bar{U})$ in red for given data $\bar{U}$. Also shown in blue is the curve $L_1.$
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Table 1.  Random initial states $U_{0,k}$ on incoming pipes $k<0$ and outgoing pipes $k>0.$ The initial difference in the sum of the mass fluxes is given by $\Delta = 9.162e+00.$
Pipe $k$$(\rho_{0,k},q_{0,k})$$p(\rho_{0,k})$
-1(5.151e-01, 2.519e+00)2.653e-01
-2(6.317e-01, 2.794e+00)3.991e-01
-3(6.642e-01, 3.905e+00)4.412e-01
1(5.730e-01, -2.648e-01)3.283e-01
2(7.460e-01, -1.523e-01)5.565e-01
3(5.931e-01, -1.280e-01)3.518e-01
4(5.849e-01, 6.020e-01)3.421e-01
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Pipe $k$$(\rho_{0,k},q_{0,k})$$p(\rho_{0,k})$
-1(5.151e-01, 2.519e+00)2.653e-01
-2(6.317e-01, 2.794e+00)3.991e-01
-3(6.642e-01, 3.905e+00)4.412e-01
1(5.730e-01, -2.648e-01)3.283e-01
2(7.460e-01, -1.523e-01)5.565e-01
3(5.931e-01, -1.280e-01)3.518e-01
4(5.849e-01, 6.020e-01)3.421e-01
Table 2.  Terminal states $U_{k}(t,0\pm)$ for $k\in\delta^\pm.$ The difference in the sum of the mass fluxes is zero
Pipe $k$$(\rho_{k},q_{k})$$p(\rho_{k})$
-1(2.089e+00, 5.101e+00)4.364e+00
-2(2.089e+00, 4.868e+00)4.364e+00
-3(2.089e+00, 8.090e+00)4.364e+00
1(2.089e+00, 3.757e+00)4.364e+00
2(2.089e+00, 3.357e+00)4.364e+00
3(2.089e+00, 4.147e+00)4.364e+00
4(2.089e+00, 6.798e+00)4.364e+00
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Pipe $k$$(\rho_{k},q_{k})$$p(\rho_{k})$
-1(2.089e+00, 5.101e+00)4.364e+00
-2(2.089e+00, 4.868e+00)4.364e+00
-3(2.089e+00, 8.090e+00)4.364e+00
1(2.089e+00, 3.757e+00)4.364e+00
2(2.089e+00, 3.357e+00)4.364e+00
3(2.089e+00, 4.147e+00)4.364e+00
4(2.089e+00, 6.798e+00)4.364e+00
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