American Institute of Mathematical Sciences

Advanced
June  2009, 4(2): 325-357. doi: 10.3934/nhm.2009.4.325

Modal decomposition of linearized open channel flow

Xavier Litrico 1, and  Vincent Fromion 2,

1. 

Cemagref, UMR G-EAU, 361 rue JF Breton, F-34196 Montpellier Cedex 5
2. 

INRA, Unité Mathématique Informatique et Génome, UR1077, INRA-MIG, F-78350 Jouy-en-Josas

Received  September 2008 Revised  February 2009 Published  June 2009

Open channel flow is traditionally modeled as an hyperbolic system of conservation laws, which is an infinite dimensional system with complex dynamics. We consider in this paper an open channel represented by the Saint-Venant equations linearized around a non uniform steady flow regime. We use a frequency domain approach to fully characterize the open channel flow dynamics. The use of the Laplace transform enables us to derive the distributed transfer matrix, linking the boundary inputs to the state of the system. The poles of the system are then computed analytically, and each transfer function is decomposed in a series of eigenfunctions, where the influence of space and time variables can be decoupled. As a result, we can express the time-domain response of the whole canal pool to boundary inputs in terms of discharges. This study is first done in the uniform case, and finally extended to the non uniform case. The solution is studied and illustrated on two different canal pools.
Keywords: hyperbolic systems, shallow water equations, frequency domain approach..
Mathematics Subject Classification: Primary: 35B37, 93C20, 93C80, 35L65; Secondary: 35C05, 35Q3.
Citation: Xavier Litrico, Vincent Fromion. Modal decomposition of linearized open channel flow. Networks & Heterogeneous Media, 2009, 4 (2) : 325-357. doi: 10.3934/nhm.2009.4.325
[1]

Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119
[2]

Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134
[3]

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
[4]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[5]

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
[6]

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
[7]

Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, 2019, 6 (1) : 1-37. doi: 10.3934/jcd.2019001
[8]

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
[9]

Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097
[10]

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

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
[12]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331
[13]

Weike Wang, Yucheng Wang. Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6379-6409. doi: 10.3934/dcds.2020284
[14]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
[15]

Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 197-212. doi: 10.3934/dcdss.2021036
[16]

N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079
[17]

George Avalos, Roberto Triggiani. Rational decay rates for a PDE heat--structure interaction: A frequency domain approach. Evolution Equations & Control Theory, 2013, 2 (2) : 233-253. doi: 10.3934/eect.2013.2.233
[18]

Marie-Odile Bristeau, Jacques Sainte-Marie. Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 733-759. doi: 10.3934/dcdsb.2008.10.733
[19]

Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225
[20]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy