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Research Papers: Techniques and Procedures

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

[+] Author and Article Information
Julien Waeytens

Université Paris-Est, IFSTTAR,
Laboratory on Instrumentation,
Simulation and Scientific Informatics,
14-20 Boulevard Newton,
Marne-la-Vallée F-77447, France
e-mail: julien.waeytens@ifsttar.fr

Patrice Chatellier

Université Paris-Est, IFSTTAR,
Laboratory on Instrumentation,
Simulation and Scientific Informatics,
14-20 Boulevard Newton,
Marne-la-Vallée F-77447, France
e-mail: patrice.chatellier@ifsttar.fr

Frédéric Bourquin

Université Paris-Est, IFSTTAR,
Components and Systems Department,
14-20 Boulevard Newton,
Marne-la-Vallée F-77447, France
e-mail: frederic.bourquin@ifsttar.fr

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 25, 2014; final manuscript received April 8, 2015; published online May 19, 2015. Assoc. Editor: John Abraham.

J. Fluids Eng 137(9), 091401 (May 19, 2015) (17 pages) Paper No: FE-14-1467; doi: 10.1115/1.4030358 History: Received August 25, 2014

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.

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Figures

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Fig. 1

Schematic representation of the system when nc = 4

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Fig. 2

2D junction geometry and velocity sensor position

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Fig. 3

Superposition principle: Stokes problem I (left) and Stokes problem III (right)

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Fig. 4

Numerical values of the cost functional J (contour plot), control velocities (cross marker) obtained by the adjoint method at iteration 0, 1, 2 and exact control velocities (circle marker)

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Fig. 5

Fine mesh—6.4×104 DOF

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Fig. 6

Coarse mesh—4.4×103 DOF

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Fig. 7

Temporal evolution of (Vc1)ex and (Vc3)ex

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Fig. 8

Comparison of reference and reconstructed control boundary velocities for the unsteady Stokes equations—fine discretization (6.4×104 DOF, 256 time steps)—T-junction at Re = 2000

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Fig. 9

Norm of the reference velocity ||vex|| at t=T/2 obtained with the Navier–Stokes equations—very fine discretization (2.5×105 DOF, 2560 time steps)—T-junction at Re = 2000

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Fig. 10

Norm of the reconstructed velocity ||vex|| at t=T/2 obtained with the Navier–Stokes equations—coarse discretization (4.4×103 DOF, 64 time steps)—T-junction at Re = 2000

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Fig. 11

Norm of the reference velocity ||vex|| at t=T/2 obtained with Navier–Stokes equations (left) and a very fine discretization (2.5×105 DOF, 2560 time steps) and Norm of the reconstructed velocity ||v|| at t=T/2 using the hybrid approach: unsteady Stokes reconstruction and a single Navier–Stokes solve (right) with a fine discretization (6.4×104 DOF, 256 time steps)—T-junction at Re = 200

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Fig. 12

Norm of the reconstructed velocity ||v|| at t=T/2 using Navier–Stokes or simple flow models (left), see Fig. 9 for the reference solution and Norm of the error at T/2 between the reference velocity and the reconstructed velocity (right)—fine mesh (6.4×104 DOF, 256 time steps)—T-junction at Re = 2000

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Fig. 13

Studied water network: three control boundary velocities and two virtual velocity sensors in the x direction. Diameter of pipes: main street: 150 mm, impasse: 80 mm, and residence street: 60 mm.

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Fig. 15

Maximum norm of the error between the reference velocity and the reconstructed velocity using different flow models and different meshes

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Fig. 14

Norm of the reference velocity in junction 2 of Versailles networks (top), Norm of the reconstructed velocity using different flow models and different meshes (left), and Norm of the error between the reference velocity and the reconstructed velocity (right)

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