Solute Transport Control at Channel Junctions Using Adjoint Sensitivity

Geovanny Gordillo,Mario Morales-Hernández,Pilar García-Navarro

doi:10.3390/math10010093

Geovanny Gordillo, Mario Morales-Hernández + Show 1 more

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https://doi.org/10.3390/math10010093

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Abstract

Water quality control and the control of contaminant spill in water in particular are becoming a primary need today. Gradient descent sensitivity methods based on the adjoint formulation have proved to be encouraging techniques in this context for river and channel flows. Taking into account that most channels and rivers include junctions with other branches, the objective of this study is to explore the adjoint technique on a channel network to reconstruct the upstream boundary condition of the convection-reaction equation. For this purpose, the one-dimensional shallow water equations and the transport equation for a reactive solute are considered. The control is formulated through the gradient-descent technique supplied with a first-order iterative process. Both the physical and the adjoint equations are supplied with suitable internal boundary conditions at the junction and are numerically solved using a finite volume upwind scheme. The results reveal that the adjoint technique is capable of reconstructing the inlet solute concentration boundary condition in an acceptable number of iterations for both steady state and transient configurations using a downstream measurement location. It was also observed that the reconstruction of the boundary condition tends to be less effective the further away the measurement station is from the target.

Highlights

Simulation tools based on hydrodynamic models combined with solute transport have become an essential tool to help decision makers [1], with efficiency and accuracy being both the fundamental keys of any mathematical model
This study reveals that the transport equation model can serve as a first step to understand the relationships between the channel network and the control of the concentration of a solute downstream junction
The results showed that the accuracy of the boundary condition reconstruction depends largely on the shape of the signal to be reconstructed

Summary

Introduction

Simulation tools based on hydrodynamic models combined with solute transport have become an essential tool to help decision makers [1], with efficiency and accuracy being both the fundamental keys of any mathematical model. The numerical simulation of water flow at channel junctions has been addressed by several authors. In [2], it was concluded that it is possible to model the flow in a junction when the Froude numbers are low assuming the same water stage at the junction for every channel. Hsu et al [3] derived an analytical approach through the junction over subcritical flows and uniform beds. The validation of their model was supported by three experimental tests with different junction angles, showing a good correlation between the numerical data and experimental values. The flow propagation in open-channel junctions was analyzed in [9], showing acceptable numerical results for supercritical transitions with small junction angles

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