Coupling conditions for the transition from supersonic to subsonic fluid states

Martin Gugat; Michael  Herty; Siegfried  Müller

doi:10.3934/nhm.2017016

Article Contents

2017, Volume 12, Issue 3: 371-380. Doi: 10.3934/nhm.2017016

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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

* Corresponding author: Michael Herty
* Corresponding author: Michael Herty

Received: May 2016

Revised: July 2017

Published: October 2017

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L60, 35L04; Secondary: 35R02.

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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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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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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.