Figure 1.
Junction of $n+m=|\delta^-|+|\delta^+|$ connected pipes
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2017, 12(3): 371-380.
doi: 10.3934/nhm.2017016
Coupling conditions for the transition from supersonic to subsonic fluid states
| 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 |
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.
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 and Heterogeneous Media,
2017, 12
(3)
: 371-380.
doi: 10.3934/nhm.2017016
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
J.-P. Lebacque,
Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45.
|
| [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. |
| [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. |
| [22] |
B. Sultanian,
Fluid Mechanics: An Intermediate Approach, CRC Press, 2015. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
J.-P. Lebacque,
Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45.
|
| [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. |
| [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. |
| [22] |
B. Sultanian,
Fluid Mechanics: An Intermediate Approach, CRC Press, 2015. |
| [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. |
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.
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 $L_1.$
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 | ||
| -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 |
| Pipe | ||
| -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 | ||
| -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 |
| Pipe | ||
| -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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