# American Institute of Mathematical Sciences

December 2018, 13(4): 663-690. doi: 10.3934/nhm.2018030

## Fluvial to torrential phase transition in open canals

 1 Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Rome, Italy 2 Department of Mathematical Sciences, Rutgers University-Camden, 311 N. 5th Street Camden, NJ 08102, USA

Received  April 2018 Revised  August 2018 Published  November 2018

Network flows and specifically water flow in open canals can be modeled bysystems of balance laws defined ongraphs.The shallow water or Saint-Venant system of balance laws is one of the most used modeland present two phases: fluvial or sub-critical and torrential or super-critical.Phase transitions may occur within the same canal but transitions relatedto networks are less investigated.In this paper we provide a complete characterization of possible phase transitionsfor a case study of a simple scenariowith two canals and one junction.However, our analysis allows the study of more complicate networks.Moreover, we provide some numerical simulations to show the theory at work.

Citation: Maya Briani, Benedetto Piccoli. Fluvial to torrential phase transition in open canals. Networks & Heterogeneous Media, 2018, 13 (4) : 663-690. doi: 10.3934/nhm.2018030
##### References:
 [1] G. Bastin, A. M. Bayen, C. D'Apice, X. Litrico and B. Piccoli, Open problems and research perspectives for irrigation channels, Netw. Heterog. Media, 4 (2009), ⅰ-ⅴ. [2] G. Bastin and J.-M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, Progress in Nonlinear Differential Equations and their Applications, 88 (2016), xiv+307 pp, Birkhäuser/Springer, [Cham]. Subseries in Control. doi: 10.1007/978-3-319-32062-5. [3] G. Bastin, J.-M. Coron and B. d'Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187. doi: 10.3934/nhm.2009.4.177. [4] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000. [5] A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2. [6] M. Briani, B. Piccoli and J.-M. Qiu, Notes on RKDG methods for shallow-water equations in canal networks, Journal of Scientific Computing, 68 (2016), 1101-1123. doi: 10.1007/s10915-016-0172-2. [7] B. Cockburn and C.-W. Shu, Tvb runge-kutta local projection discontinuous galerkin finite element method for conservation laws ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435. doi: 10.2307/2008474. [8] B. Cockburn and C.-W. Shu, Runge-kutta discontinuous galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16 (2001), 173-261. doi: 10.1023/A:1012873910884. [9] R. M. Colombo, M. Herty and V. Sachers, On 2×2 conservation laws at a junction, SIAM Journal on Mathematical Analysis, 40 (2008), 605-622. doi: 10.1137/070690298. [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, fourth edition, 2016. doi: 10.1007/978-3-662-49451-6. [11] R. Dasgupta and R. Govindarajan, The hydraulic jump and the shallow-water equations, International Journal of Advances in Engineering Sciences and Applied Mathematics, 3 (2011), 126-130. doi: 10.1007/s12572-011-0047-6. [12] M. Garavello and B. Piccoli, Traffic Flow on Networks, volume 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [13] M. Garavello and B. Piccoli, Riemann solvers for conservation laws at a node, proceeding of the Hyperbolic problems: Theory, Numerics and Applications, Proc. Sympos. Appl. Math., Amer. Math. Soc., Providence, RI, 67 (2009), 595-604. doi: 10.1090/psapm/067.2/2605255. [14] M. S. Goudiaby and G. Kreiss, A riemann problem at a junction of open canals, Journal of Hyperbolic Differential Equations, 10 (2013), 431-460. doi: 10.1142/S021989161350015X. [15] Q. Gu and T. Li, Exact boundary controllability of nodal profile for unsteady flows on a tree-like network of open canals, J. Math. Pures Appl. (9), 99 (2013), 86-105. doi: 10.1016/j.matpur.2012.06.004. [16] M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, Adv. Model. Optim, 7 (2005), 9-37. [17] M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 257-270. doi: 10.1016/j.anihpc.2008.01.002. [18] M. Gugat, G. Leugering and E. J. P. G. Schmidt, Global controllability between steady supercritical flows in channel networks, Mathematical Methods in the Applied Sciences, 27 (2004), 781-802. doi: 10.1002/mma.471. [19] F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial application, In Industrial mathematics and complex systems, Ind. Appl. Math., (2017), pages 77-122. Springer, Singapore. [20] M. Herty and M. Seaïd, Assessment of coupling conditions in water way intersections, International Journal for Numerical Methods in Fluids, 71 (2013), 1438-1460. doi: 10.1002/fld.3719. [21] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences. Springer Berlin Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2. [22] G. Leugering and J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM journal on control and optimization, 41 (2002), 164-180. doi: 10.1137/S0363012900375664. [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. [24] A. Marigo, Entropic solutions for irrigation networks, SIAM Journal on Applied Mathematics, 70 (2010), 1711-1735. doi: 10.1137/09074783X. [25] P. C. D. Milly, J. Betancourt, M. Falkenmark, R. M. Hirsch, Z. W. Kundzewicz, D. P. Lettenmaier and R. J. Stouffer, Stationarity is dead: Whither water management?, Science, 319 (2008), 573-574. [26] C. Prieur and J. J. Winkin, Boundary feedback control of linear hyperbolic systems: Application to the Saint-Venant-Exner equations, Automatica J. IFAC, 89 (2018), 44-51. doi: 10.1016/j.automatica.2017.11.028.

show all references

##### References:
 [1] G. Bastin, A. M. Bayen, C. D'Apice, X. Litrico and B. Piccoli, Open problems and research perspectives for irrigation channels, Netw. Heterog. Media, 4 (2009), ⅰ-ⅴ. [2] G. Bastin and J.-M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, Progress in Nonlinear Differential Equations and their Applications, 88 (2016), xiv+307 pp, Birkhäuser/Springer, [Cham]. Subseries in Control. doi: 10.1007/978-3-319-32062-5. [3] G. Bastin, J.-M. Coron and B. d'Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187. doi: 10.3934/nhm.2009.4.177. [4] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000. [5] A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2. [6] M. Briani, B. Piccoli and J.-M. Qiu, Notes on RKDG methods for shallow-water equations in canal networks, Journal of Scientific Computing, 68 (2016), 1101-1123. doi: 10.1007/s10915-016-0172-2. [7] B. Cockburn and C.-W. Shu, Tvb runge-kutta local projection discontinuous galerkin finite element method for conservation laws ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435. doi: 10.2307/2008474. [8] B. Cockburn and C.-W. Shu, Runge-kutta discontinuous galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16 (2001), 173-261. doi: 10.1023/A:1012873910884. [9] R. M. Colombo, M. Herty and V. Sachers, On 2×2 conservation laws at a junction, SIAM Journal on Mathematical Analysis, 40 (2008), 605-622. doi: 10.1137/070690298. [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, fourth edition, 2016. doi: 10.1007/978-3-662-49451-6. [11] R. Dasgupta and R. Govindarajan, The hydraulic jump and the shallow-water equations, International Journal of Advances in Engineering Sciences and Applied Mathematics, 3 (2011), 126-130. doi: 10.1007/s12572-011-0047-6. [12] M. Garavello and B. Piccoli, Traffic Flow on Networks, volume 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [13] M. Garavello and B. Piccoli, Riemann solvers for conservation laws at a node, proceeding of the Hyperbolic problems: Theory, Numerics and Applications, Proc. Sympos. Appl. Math., Amer. Math. Soc., Providence, RI, 67 (2009), 595-604. doi: 10.1090/psapm/067.2/2605255. [14] M. S. Goudiaby and G. Kreiss, A riemann problem at a junction of open canals, Journal of Hyperbolic Differential Equations, 10 (2013), 431-460. doi: 10.1142/S021989161350015X. [15] Q. Gu and T. Li, Exact boundary controllability of nodal profile for unsteady flows on a tree-like network of open canals, J. Math. Pures Appl. (9), 99 (2013), 86-105. doi: 10.1016/j.matpur.2012.06.004. [16] M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, Adv. Model. Optim, 7 (2005), 9-37. [17] M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 257-270. doi: 10.1016/j.anihpc.2008.01.002. [18] M. Gugat, G. Leugering and E. J. P. G. Schmidt, Global controllability between steady supercritical flows in channel networks, Mathematical Methods in the Applied Sciences, 27 (2004), 781-802. doi: 10.1002/mma.471. [19] F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial application, In Industrial mathematics and complex systems, Ind. Appl. Math., (2017), pages 77-122. Springer, Singapore. [20] M. Herty and M. Seaïd, Assessment of coupling conditions in water way intersections, International Journal for Numerical Methods in Fluids, 71 (2013), 1438-1460. doi: 10.1002/fld.3719. [21] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences. Springer Berlin Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2. [22] G. Leugering and J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM journal on control and optimization, 41 (2002), 164-180. doi: 10.1137/S0363012900375664. [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. [24] A. Marigo, Entropic solutions for irrigation networks, SIAM Journal on Applied Mathematics, 70 (2010), 1711-1735. doi: 10.1137/09074783X. [25] P. C. D. Milly, J. Betancourt, M. Falkenmark, R. M. Hirsch, Z. W. Kundzewicz, D. P. Lettenmaier and R. J. Stouffer, Stationarity is dead: Whither water management?, Science, 319 (2008), 573-574. [26] C. Prieur and J. J. Winkin, Boundary feedback control of linear hyperbolic systems: Application to the Saint-Venant-Exner equations, Automatica J. IFAC, 89 (2018), 44-51. doi: 10.1016/j.automatica.2017.11.028.
Shocks, rarefaction and critical curves 10-14 on the plane $(h, v)$ (up) and on the plane $(h, q)$ (down)
Graph of $\phi_l$ and $\phi_r$ defined in 15 and 16 respectively
Graph of $q = \tilde{\phi}_l(h)$ for different values of left state $u_l$ and its intersections with critical curves $q = \tilde{\mathcal{C}}^+(h)$ and $q = \tilde{\mathcal{C}}^-(h)$. The left state $u_l$ have been chosen such that: $F_l>1$ (dotted green line), $|F_l| < 1$ (blue dashed line) and $-2\leq F_l<-1$ (red dotted line)
Left-half Riemann problem, Section 4.1. Region $\mathcal{N}^A(u_l) = \mathcal{I}^A_1\bigcup\mathcal{I}^A_2\bigcup\mathcal{I}^A_3$ defined by 29-31. Following our notation $\tilde{\mathcal{S}}_2(u^-_{l, \mathcal{S}};h) = h\mathcal{S}_2(h^-_{l, \mathcal{S}}, \mathcal{C}^-(h^-_{l, \mathcal{S}});h)$
Left-half Riemann problem, Section 4.1. Region $\mathcal{N}^B(u_l) = \mathcal{I}^{*, A}_1\bigcup\mathcal{I}^{*, A}_2\bigcup\mathcal{I}^{*, A}_3$ given in 32
Left-half Riemann problem, Section 4.1. Region $\mathcal{N}^C(u_l)$ bounded by $q = \tilde{\mathcal{S}}_2(u^{-}_{l, \mathcal{R}};h)$ and $q = \tilde{\mathcal{C}}^-(h)$ as defined in 33
Right-half Riemann problem, Section 4.2. Region $\mathcal{P}^A(u_r) = \mathcal{O}^A_1\bigcup\mathcal{O}^A_2\bigcup\mathcal{O}^A_3$ defined by 35-37 where $u_r$ is such that $|\tilde{\mathcal{F}}_r|<1$
Right-half Riemann problem, Section 4.2. Region $\mathcal{P}^B(u_r)$ bounded by $q = \tilde{\mathcal{S}}_2(u^{-}_{l, \mathcal{R}};h)$ and $q = \tilde{\mathcal{C}}^+(h)$ as defined in 38 where $u_r$ is such that $\tilde{\mathcal{F}}_r>1$
Right-half Riemann problem, Section 4.2. Region $\mathcal{P}^C(u_r) = \mathcal{O}^{*, A}_1\bigcup\mathcal{O}^{*, A}_2\bigcup\mathcal{O}^{*, A}_3$ given in 39 where $u_r$ is such that $\tilde{\mathcal{F}}_r<-1$.
Case Fluvial $\rightarrow$ Fluvial, system 42. In this case curves $\tilde{\phi}_l$ and $\tilde{\phi}_r$ intersect inside the subcritical region. The solution is the intersection point $u^b$
Case Fluvial $\rightarrow$ Fluvial, system 42: curves $\tilde{\phi}_l$ and $\tilde{\phi}_r$ have empty intersection inside the subcritical region and $h_r<h_l$. The solution is the critical point $u^+_{l, \mathcal{R}}$
Case Fluvial $\rightarrow$ Fluvial, system 42: curves $\tilde{\phi}_l$ and $\tilde{\phi}_r$ have empty intersection inside the subcritical region and $h_l<h_r$. The solution is the critical point $u^-_{r, \mathcal{R}}$
Case Torrential $\rightarrow$ Fluvial, system 45. In this case the curve $h\mathcal{S}_1(h^*, v^*;h)$ and $\tilde{\phi}_r(h)$ intersect inside the subcritical region. The solution is the intersection point $u^b$.
Case Torrential $\rightarrow$ Fluvial, system 45. In this case the curve $h\mathcal{S}_1(h^*, v^*;h)$ and $\tilde{\phi}_r(h)$ have empty intersection inside the subcritical region. This configuration is an example in which system 45 does not admit a solution.
Case Torrential $\rightarrow$ Fluvial, system 45. In this case the curve $h\mathcal{S}_1(h^*, v^*;h)$ and $\tilde{\phi}_r(h)$ have empty intersection inside the subcritical region. The solution is the point $u^-_{r, \mathcal{R}}$
Case Torrential $\rightarrow$ Torrential. In this case the two admissible regions $\mathcal{N}^B$ and $\mathcal{P}^B$ have empty intersection
Numerical test case for the configuration given in Fig. 9
Numerical test case for the configuration given in Fig. 16
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