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# Fluvial to torrential phase transition in open canals

• 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.

Mathematics Subject Classification: Primary: 35L60, 35L67; Secondary: 65N08.

 Citation: • • 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

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