Hydro-Electric Open Channel

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Description

Motivation

The so-called Saint-Venant equations are largely used in the hydraulic domain to model the dynamics of an open channel flow. These equations consist of two nonlinear hyperbolic PDEs. In the considered benchmark, under mild simplifying assumptions detailed in , the St Venant PDE equations describing the height variation $h$ of the river as a function of the inflow $q_i$ and outflow $q_o$ variations, at location $x$ ($x\in[0\,\,L]$, $L\in\mathbb R_+$), obtained around some flow and height linearisation point, can be formulated as follows:

$h(x,s) = \mathbf{G_i}(x,s)q_i(s) - \mathbf{G_o}(x,s)q_o(s) = \mathbf H(x,s) u(s).$

The $\mathbf G_i$ and $\mathbf G_o$ functions are irrational and read

$\mathbf{G_i}(x,s)= \dfrac{\lambda_1(s)e^{\lambda_2(s)L+\lambda_1(s)x}-\lambda_2(s)e^{\lambda_1(s)L+\lambda_2(s)x}}{B_0s(e^{\lambda_1(s)L}-e^{\lambda_2(s)L})}$

and

$\mathbf{G_o}(x,s)= \dfrac{\lambda_1(s)e^{\lambda_1(s)x}-\lambda_2(s)e^{\lambda_2(s)x}}{B_0s(e^{\lambda_1(s)L}-e^{\lambda_2(s)L})}$

Considere data

The benchmark contains the above irrational model description together with the numerical data as used in