Inverse Computational Fluid Dynamics: Influence of Discretization and Model Errors on Flows in Water Network Including Junctions

We address the reconstruction of relevant two-dimensional (2D) flows in drinking water networks, especially in key elements such as pipe junctions, in view of representative water quality simulations. From the optimal control theory, a specific inverse technique using few sensors and computational fluid dynamics (CFD) models has been developed. First, we determine the boundary velocities, i.e., the control parameters, by minimizing a data misfit functional. Then, knowing the boundary velocities, a direct solve of the flow model is performed to get the space–time cartography of the water flow. To reduce the number of control parameters to be determined and thus restrict the number of sensors, the spatial shape of the boundary velocities is considered as an a priori information given by the water pipes engineering literature. Thus, only the time evolution of the boundary velocities has to be determined. The whole numerical procedure proposed in this paper easily fits in a general purpose finite element software, featuring user's friendliness for a wide engineering audience. Two ways are investigated to reduce the computation time associated to the flow reconstruction. The adjoint framework is used in the minimization process. The reconstruction of the flow using coarse discretizations and simple flow models, instead of 2D Navier–Stokes equations, is studied. The influence of the flow modeling and of the dicretization on the quality of the reconstructed velocity is studied on two examples: a water pipe junction and a 200 m subsection from a French water network. In the water pipe junction, we show that at a Reynolds number of 200 a hybrid approach combining an unsteady Stokes reconstruction and a single direct Navier–Stokes simulation outperforms the algorithms based on a single model. In the network subsection, we obtain an L² error less than 1% between the reference velocity based on Navier–Stokes equations (Reynolds number of 200) and the velocity reconstructed from Stokes equation. In this case, the reconstruction lasts less than 1 min. Stokes based reconstruction of a Navier–Stokes flow in junctions at Reynolds number up to 100 yields the same accuracy and proves fast.