# American Institute of Mathematical Sciences

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

## Modeling and analysis of pooled stepped chutes

 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.
Citation: Graziano Guerra, Michael Herty, Francesca Marcellini. Modeling and analysis of pooled stepped chutes. Networks and 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. [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. [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. [4] F. W. Blaisdell, Equation for the free-falling nappe, Proceedings ASCE, no. 482, 80 (1954). [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. [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. [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). [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. [9] H. Chanson, "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways,'' Pergamon Press, Oxford, England, 1994. [10] H. Chanson, "The Hydraulics of Stepped Chutes and Spillways,'' Taylor & Francis, 2002. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [24] A. Marigo, Entropic solutions for irrigation networks, SIAM J. Appl. Math., 70 (2009/10), 1711-1735. doi: 10.1137/09074783X. [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). [26] W. Rand, Flow geometry at straight drop spillways, Proceedings ASCE, no. 791, 81 (1955). [27] H. Rouse, Discharge characteristics of the free overfall, Civil Engineering, 6 (1936), 257-260. [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). [29] T. Sturm, "Open Channel Hydraulics,'' McGraw-Hill, 2001. [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.

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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, Networks and Heterogeneous Media, 1 (2006), 295-314. doi: 10.3934/nhm.2006.1.295. [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. [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. [4] F. W. Blaisdell, Equation for the free-falling nappe, Proceedings ASCE, no. 482, 80 (1954). [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. [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. [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). [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. [9] H. Chanson, "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways,'' Pergamon Press, Oxford, England, 1994. [10] H. Chanson, "The Hydraulics of Stepped Chutes and Spillways,'' Taylor & Francis, 2002. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [24] A. Marigo, Entropic solutions for irrigation networks, SIAM J. Appl. Math., 70 (2009/10), 1711-1735. doi: 10.1137/09074783X. [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). [26] W. Rand, Flow geometry at straight drop spillways, Proceedings ASCE, no. 791, 81 (1955). [27] H. Rouse, Discharge characteristics of the free overfall, Civil Engineering, 6 (1936), 257-260. [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). [29] T. Sturm, "Open Channel Hydraulics,'' McGraw-Hill, 2001. [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.
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