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Research Papers: Techniques and Procedures

Impacts of Discretization Error, Flow Modeling Error, and Measurement Noise on Inverse Transport-Diffusion-Reaction in a T-Junction

[+] Author and Article Information
Julien Waeytens

Université Paris-Est, IFSTTAR,
14-20 Boulevard Newton,
Marne-la-Vallée F-77447, France
e-mail: julien.waeytens@ifsttar.fr

Patrice Chatellier

Université Paris-Est, IFSTTAR,
14-20 Boulevard Newton,
Marne-la-Vallée F-77447, France
e-mail: patrice.chatellier@ifsttar.fr

Frédéric Bourquin

Université Paris-Est, IFSTTAR,
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 July 4, 2016; final manuscript received January 6, 2017; published online March 21, 2017. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 139(5), 051402 (Mar 21, 2017) (10 pages) Paper No: FE-16-1416; doi: 10.1115/1.4035806 History: Received July 04, 2016; Revised January 06, 2017

By combining a physical model and sensor outputs in an inverse transport-diffusion-reaction strategy, an accurate concentration cartography may be obtained. The paper addresses the influence of discretization errors, flow uncertainties, and measurement noise on the concentration field reconstruction process. We consider a key element of a drinking water network, i.e., a pipe junction, where Reynolds and Peclet numbers are approximately 2000 and 1000, respectively. We show that a 10% error between the reference concentration field and the reconstructed concentration field may be obtained using a coarse discretization. Nevertheless, to keep the error below 10%, a fine concentration discretization is required. We also detail the influence of the flow approximation on the concentration reconstruction process. The flow modeling error obtained when the exact Navier–Stokes flow is approximated by a Stokes flow may lead to a 40% error in the reconstructed concentration. However, if the flow field is obtained from the full set of Navier–Stokes equations, we show that the error may be less than 5%. Then, we observe that the quality of the reconstructed concentration field obtained with the proposed inverse technique is not deteriorated when sensor outputs have a normal distribution noise variance of few percents. Finally, a good engineering practice would be to stop the reconstruction process according to an extended discrepancy principle including modeling and measurement errors. As shown in the paper, the quality of the reconstructed field declines after reaching the threshold of the modeling error.

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Figures

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Fig. 1

Two-dimensional study case geometry and sensor placement (dimensions in millimeter)

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Fig. 2

Temporal evolution of (vc1max) and (vc3max)

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Fig. 3

Temporal evolution of (u1)ex and (u3)ex

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Fig. 6

Influence of flow uncertainty on the chlorine reconstruction—sensors 1 and 2—fine discretization (2.8 × 104 DOF 1280 time steps) for chlorine concentration. εC represents the L2 error between the exact concentration and the reconstructed concentration; nit is the number of iterations of the inverse problem.

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Fig. 7

Influence of the flow model error and the flow discretization error on the chlorine reconstruction—sensors 1 and 2—fine discretization (2.8 × 104 DOF 1280 time steps) for chlorine concentration. εC represents the L2 error between the exact concentration and the reconstructed concentration; εv represents the L2 error between the exact velocity and the approximate velocity; nit is the number of iterations of the inverse problem.

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Fig. 8

Norm of the approximate velocities ||vh|| at t = T/2, which are obtained using sophisticated or simple flow models in the velocity reconstruction process—fine mesh (6.4 × 104 DOF 256 time steps)

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Fig. 9

Reference concentration at t = 3T/5 when considering reference velocity obtained with a very fine discretization—concentration discretization (1.1 × 105 DOF 2560 time steps)

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Fig. 10

Reconstructed concentration at t = 3T/5 when considering approximate velocities obtained from the velocity reconstruction process—sensors 1 and 2—concentration fine discretization (2.8 × 104 DOF 1280 time steps)

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Fig. 11

Fluctuating boundary velocity

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Fig. 12

Influence of flow model error and flow discretization error on the chlorine reconstruction—sensors 1 and 2—coarse discretization (1.9 × 103 DOF 256 time steps) for chlorine concentration. εC represents the L2 error between the exact concentration and the reconstructed concentration; εv represents the L2 error between the exact velocity and the approximate velocity; nit is the number of iterations of the inverse problem.

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Fig. 13

Influence of flow model error and flow discretization error on the chlorine reconstruction—sensors 1 and 2—coarse discretization (1.9 × 103 DOF 256 time steps) for chlorine concentration—representative flow in “distribution mains.” εC represents the L2 error between the exact concentration and the reconstructed concentration; εv represents the L2 error between the exact velocity and the approximate velocity; nit is the number of iterations of the inverse problem.

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Fig. 14

Sensor outputs with noise N(0,σ2) : sensor S1 on the left and sensor S2 on the right

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Fig. 15

Influence of the noise variance σ2 on the chlorine reconstruction—sensors 1 and 2—fine discretization (2.8 × 104 DOF, 1280 time steps)

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Fig. 16

Influence of the noise normal distribution mean μ on the chlorine reconstruction—sensors 1 and 2—fine discretization (2.8 × 104 DOF, 1280 time steps)

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