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. BastinA. M. BayenC. D'ApiceX. 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. BastinJ.-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. BressanS. CanicM. GaravelloM. 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. BrianiB. 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. ColomboM. 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. GugatG. 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. LitricoV. FromionJ.-P. BaumeC. 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. MillyJ. BetancourtM. FalkenmarkR. M. HirschZ. W. KundzewiczD. 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. BastinA. M. BayenC. D'ApiceX. 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. BastinJ.-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. BressanS. CanicM. GaravelloM. 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. BrianiB. 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. ColomboM. 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. GugatG. 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. LitricoV. FromionJ.-P. BaumeC. 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. MillyJ. BetancourtM. FalkenmarkR. M. HirschZ. W. KundzewiczD. 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.

Figure 1.  Shocks, rarefaction and critical curves 10-14 on the plane $(h, v)$ (up) and on the plane $(h, q)$ (down)
Figure 2.  Graph of $\phi_l$ and $\phi_r$ defined in 15 and 16 respectively
Figure 3.  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)
Figure 4.  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 $\tilde{\mathcal{S}}_2(u^-_{l, \mathcal{S}};h) = h\mathcal{S}_2(h^-_{l, \mathcal{S}}, \mathcal{C}^-(h^-_{l, \mathcal{S}});h)$
Figure 5.  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
Figure 6.  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
Figure 12.  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 $|\tilde{\mathcal{F}}_r|<1$
Figure 13.  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 $\tilde{\mathcal{F}}_r>1$
Figure 14.  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 $\tilde{\mathcal{F}}_r<-1$.
Figure 7.  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 $u^b$
Figure 8.  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 $u^+_{l, \mathcal{R}}$
Figure 9.  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 $u^-_{r, \mathcal{R}}$
Figure 15.  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 $u^b$.
Figure 16.  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.
Figure 17.  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 $u^-_{r, \mathcal{R}}$
Figure 18.  Case Torrential $\rightarrow$ Torrential. In this case the two admissible regions $\mathcal{N}^B$ and $\mathcal{P}^B$ have empty intersection
Figure 10.  Numerical test case for the configuration given in Fig. 9
Figure 11.  Numerical test case for the configuration given in Fig. 16
[1]

Aimin Huang, Roger Temam. The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2005-2038. doi: 10.3934/cpaa.2014.13.2005

[2]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593

[3]

Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799

[4]

Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327

[5]

Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks & Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211

[6]

Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69

[7]

Tatsien Li, Wancheng Sheng. The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 737-744. doi: 10.3934/dcds.2002.8.737

[8]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[9]

Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115

[10]

Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165

[11]

Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134

[12]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237

[13]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[14]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[15]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375

[16]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

[17]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103

[18]

Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127

[19]

Stefano Fasani, Sergio Rinaldi. Local stabilization and network synchronization: The case of stationary regimes. Mathematical Biosciences & Engineering, 2010, 7 (3) : 623-639. doi: 10.3934/mbe.2010.7.623

[20]

Yu Zhang, Yanyan Zhang. Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1523-1545. doi: 10.3934/cpaa.2019073

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (11)
  • HTML views (40)
  • Cited by (0)

Other articles
by authors

[Back to Top]