# 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
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##### References:
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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