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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 <ref name= Dalmas2016>V. Dalmas, G. Robert, C. Poussot-Vassal, I. Pontes-Duff and C. Seren, "From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity", in Proceedings of the European Control Conference (ECC), Aalborg, Denmark, July, 2016, pp. 1982-1987.</ref>, the St Venant PDE equations describing the height variation <math>h</math> of the river as a function of the inflow <math>q_i</math> and outflow <math>q_o</math> variations, at location <math>x</math> (<math>x\in[0\,\,L]</math>, <math>L\in\mathbb R_+</math>), obtained around some flow and height linearisation point, can be formulated as follows: |
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 <ref name= Dalmas2016>V. Dalmas, G. Robert, C. Poussot-Vassal, I. Pontes-Duff and C. Seren, "From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity", in Proceedings of the European Control Conference (ECC), Aalborg, Denmark, July, 2016, pp. 1982-1987.</ref>, the St Venant PDE equations describing the height variation <math>h</math> of the river as a function of the inflow <math>q_i</math> and outflow <math>q_o</math> variations, at location <math>x</math> (<math>x\in[0\,\,L]</math>, <math>L\in\mathbb R_+</math>), obtained around some flow and height linearisation point, can be formulated as follows: |
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:<math> |
:<math> |
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| − | 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) |
+ | 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). |
| + | </math> |
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| + | |||
| + | The <math>\mathbf G_i</math> and <math>\mathbf G_o</math> functions are irrational and read |
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| + | :<math> |
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| + | \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})} |
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| + | </math> |
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| + | and |
||
| + | :<math> |
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| + | \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})} |
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</math> |
</math> |
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Revision as of 18:22, 27 January 2021
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Description
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 [1], the St Venant PDE equations describing the height variation 
of the river as a function of the inflow 
and outflow 
variations, at location 
(![x\in[0\,\,L]](/morwiki/images/math/9/2/2/922897af4e1e015582f11b61e69808c9.png)
, 
), obtained around some flow and height linearisation point, can be formulated as follows:
The 
and 
functions are irrational and read
and
Origin
Data
Dimensions
Citation
References
Contact
- ↑ V. Dalmas, G. Robert, C. Poussot-Vassal, I. Pontes-Duff and C. Seren, "From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity", in Proceedings of the European Control Conference (ECC), Aalborg, Denmark, July, 2016, pp. 1982-1987.