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December  2011, 6(4): 665-679. doi: 10.3934/nhm.2011.6.665

Modeling and analysis of pooled stepped chutes

Graziano Guerra 1, , Michael Herty 2, and  Francesca Marcellini 1,

1. 

Dipartimento di Matematica e Applicazioni, Università degli studi di Milano–Bicocca, Via Roberto Cozzi, 53 - 20125 Milano, Italy, Italy
2. 

Mathematik (Kontinuierliche Optimierung), RWTH Aachen University, Templergraben 55 D-52056 Aachen, Germany

Received  October 2010 Revised  August 2011 Published  December 2011

We consider a mathematical model describing pooled stepped chutes where the transport in each pooled step is described by the shallow-water equations. Such systems can be found for example at large dams in order to release overflowing water. We analyze the mathematical conditions coupling the flows between different chutes taken from the engineering literature. For the case of two canals divided by a weir, we present the solution to the Riemann problem for any initial data in the subcritical region, moreover we give a well-posedness result. We finally report on some numerical experiments.
Keywords: Hyperbolic Conservation Laws on Networks, Management of Water..
Mathematics Subject Classification: 35L6.
Citation: Graziano Guerra, Michael Herty, Francesca Marcellini. Modeling and analysis of pooled stepped chutes. Networks & Heterogeneous Media, 2011, 6 (4) : 665-679. doi: 10.3934/nhm.2011.6.665
References:
[1]

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

[2]

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

[3]

G. Bastin, B. Haut, J.-M. Coron and B. D'Andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759 (electronic). doi: 10.3934/nhm.2007.2.751.  Google Scholar

[4]

F. W. Blaisdell, Equation for the free-falling nappe, Proceedings ASCE, no. 482, 80 (1954). Google Scholar

[5]

J. N. Bradley and A. J. Peterka, The hydraulic design of sitlling basins, Journal of the Hydraulics Division, 83 (1957) 1401.1-1401.24. Google Scholar

[6]

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

[7]

M. Chamani and N. Rajaratnam, Jet flow on stepped spillways, Journal of the Hydraulic Engineering, 125 (1994), 254-259. doi: 10.1061/(ASCE)0733-9429(1994)120:2(254).  Google Scholar

[8]

H. Chanson, Comparison of energy dissipation between nappe and skimming flow regimes on stepped chutes, Journal of the Hydraulic Research, 32 (1994), 213-218. doi: 10.1080/00221689409498724.  Google Scholar

[9]

H. Chanson, "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways,'' Pergamon Press, Oxford, England, 1994. Google Scholar

[10]

H. Chanson, "The Hydraulics of Stepped Chutes and Spillways,'' Taylor & Francis, 2002. Google Scholar

[11]

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

[12]

R. M. Colombo and G. Guerra, On general balance laws with boundary, J. Differential Equations, 248 (2010), 1017-1043. doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[13]

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

[14]

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

[15]

R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, Journal of Hyperbolic Differential Equations, 5 (2008), 547-568. doi: 10.1142/S0219891608001593.  Google Scholar

[16]

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

[17]

V. Dos Santos, G. Bastin, J.-M. Coron and B. d'Andréa Novel, Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments, Automatica J. IFAC, 44 (2008), 1310-1318. doi: 10.1016/j.automatica.2007.09.022.  Google Scholar

[18]

R. Dressler, Mathematical solution to the problem of roll-waves in inclined open channels, Communication in Pure and Applied Mathematics, 2 (1949), 149-194. doi: 10.1002/cpa.3160020203.  Google Scholar

[19]

M. El-Kmamash, M. Loewen and N. Rajarantnam, An experimental investigation of jet flow on a stepped chute, Journal of Hydraulic Research, 43 (2005), 31-43. 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, in "Control and Boundary Analysis" (eds. John Cagnol, et al.), Lecture Notes in Pure and Applied Mathematics, 240, Chapman & Hall/CRC, Boca Raton, FL, (2005), 201-215.  Google Scholar

[22]

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).  Google Scholar

[23]

X. Litrico, V. Fromion, J.-P. Baume, C. Arranja and M. Rijo, Experimental validation of a methodology to control irrigation canals based on saint-venant equations, Control Engineering Practice, 13 (2005), 1425-1437. doi: 10.1016/j.conengprac.2004.12.010.  Google Scholar

[24]

A. Marigo, Entropic solutions for irrigation networks, SIAM J. Appl. Math., 70 (2009/10), 1711-1735. doi: 10.1137/09074783X.  Google Scholar

[25]

I. Ohtsu, Y. Yashuda and M. Takahashi, Flow characteristics of skimming flows in stepped channels, Journal of Hydraulic Engineering, 130 (2004), 860-869. doi: 10.1061/(ASCE)0733-9429(2004)130:9(860).  Google Scholar

[26]

W. Rand, Flow geometry at straight drop spillways, Proceedings ASCE, no. 791, 81 (1955). Google Scholar

[27]

H. Rouse, Discharge characteristics of the free overfall, Civil Engineering, 6 (1936), 257-260. Google Scholar

[28]

R. M. Sorenson, Stepped spillway hydraulic model investigation, Journal of Hydraulic Engineering, 111 (1985), 1461-1472. doi: 10.1061/(ASCE)0733-9429(1985)111:12(1461).  Google Scholar

[29]

T. Sturm, "Open Channel Hydraulics,'' McGraw-Hill, 2001. Google Scholar

[30]

J. Thorwarth, "Hydraulisches Verhalten von Treppengerinnen mit eingetieften Stufen - selbstinduzierte Abflussinstationariäten und Energiedissipation,'' Ph.D Thesis, RWTH Aachen University, Department of Civil Engineering. 2008. 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, Networks and Heterogeneous Media, 1 (2006), 295-314. doi: 10.3934/nhm.2006.1.295.  Google Scholar

[2]

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

[3]

G. Bastin, B. Haut, J.-M. Coron and B. D'Andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759 (electronic). doi: 10.3934/nhm.2007.2.751.  Google Scholar

[4]

F. W. Blaisdell, Equation for the free-falling nappe, Proceedings ASCE, no. 482, 80 (1954). Google Scholar

[5]

J. N. Bradley and A. J. Peterka, The hydraulic design of sitlling basins, Journal of the Hydraulics Division, 83 (1957) 1401.1-1401.24. Google Scholar

[6]

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

[7]

M. Chamani and N. Rajaratnam, Jet flow on stepped spillways, Journal of the Hydraulic Engineering, 125 (1994), 254-259. doi: 10.1061/(ASCE)0733-9429(1994)120:2(254).  Google Scholar

[8]

H. Chanson, Comparison of energy dissipation between nappe and skimming flow regimes on stepped chutes, Journal of the Hydraulic Research, 32 (1994), 213-218. doi: 10.1080/00221689409498724.  Google Scholar

[9]

H. Chanson, "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways,'' Pergamon Press, Oxford, England, 1994. Google Scholar

[10]

H. Chanson, "The Hydraulics of Stepped Chutes and Spillways,'' Taylor & Francis, 2002. Google Scholar

[11]

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

[12]

R. M. Colombo and G. Guerra, On general balance laws with boundary, J. Differential Equations, 248 (2010), 1017-1043. doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[13]

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

[14]

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

[15]

R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, Journal of Hyperbolic Differential Equations, 5 (2008), 547-568. doi: 10.1142/S0219891608001593.  Google Scholar

[16]

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

[17]

V. Dos Santos, G. Bastin, J.-M. Coron and B. d'Andréa Novel, Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments, Automatica J. IFAC, 44 (2008), 1310-1318. doi: 10.1016/j.automatica.2007.09.022.  Google Scholar

[18]

R. Dressler, Mathematical solution to the problem of roll-waves in inclined open channels, Communication in Pure and Applied Mathematics, 2 (1949), 149-194. doi: 10.1002/cpa.3160020203.  Google Scholar

[19]

M. El-Kmamash, M. Loewen and N. Rajarantnam, An experimental investigation of jet flow on a stepped chute, Journal of Hydraulic Research, 43 (2005), 31-43. 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, in "Control and Boundary Analysis" (eds. John Cagnol, et al.), Lecture Notes in Pure and Applied Mathematics, 240, Chapman & Hall/CRC, Boca Raton, FL, (2005), 201-215.  Google Scholar

[22]

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).  Google Scholar

[23]

X. Litrico, V. Fromion, J.-P. Baume, C. Arranja and M. Rijo, Experimental validation of a methodology to control irrigation canals based on saint-venant equations, Control Engineering Practice, 13 (2005), 1425-1437. doi: 10.1016/j.conengprac.2004.12.010.  Google Scholar

[24]

A. Marigo, Entropic solutions for irrigation networks, SIAM J. Appl. Math., 70 (2009/10), 1711-1735. doi: 10.1137/09074783X.  Google Scholar

[25]

I. Ohtsu, Y. Yashuda and M. Takahashi, Flow characteristics of skimming flows in stepped channels, Journal of Hydraulic Engineering, 130 (2004), 860-869. doi: 10.1061/(ASCE)0733-9429(2004)130:9(860).  Google Scholar

[26]

W. Rand, Flow geometry at straight drop spillways, Proceedings ASCE, no. 791, 81 (1955). Google Scholar

[27]

H. Rouse, Discharge characteristics of the free overfall, Civil Engineering, 6 (1936), 257-260. Google Scholar

[28]

R. M. Sorenson, Stepped spillway hydraulic model investigation, Journal of Hydraulic Engineering, 111 (1985), 1461-1472. doi: 10.1061/(ASCE)0733-9429(1985)111:12(1461).  Google Scholar

[29]

T. Sturm, "Open Channel Hydraulics,'' McGraw-Hill, 2001. Google Scholar

[30]

J. Thorwarth, "Hydraulisches Verhalten von Treppengerinnen mit eingetieften Stufen - selbstinduzierte Abflussinstationariäten und Energiedissipation,'' Ph.D Thesis, RWTH Aachen University, Department of Civil Engineering. 2008. Google Scholar

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