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TECHNICAL PAPERS

A Reconstruction Method for the Flow Past an Open Cavity

[+] Author and Article Information
B. Podvin, Y. Fraigneau, F. Lusseyran

LIMSI-CNRS UPR3251, Université Paris-Sud, 91403 Orsay Cedex, France

P. Gougat

LIMSI-CNRS UPR3251, Université Paris-Sud, 91403 Orsay Cedex, Francepodvin@limsi.fr

J. Fluids Eng 128(3), 531-540 (Oct 26, 2005) (10 pages) doi:10.1115/1.2175159 History: Received July 14, 2005; Revised October 26, 2005

In this paper we propose a method to reconstruct the flow at a given time over a region of space using partial instantaneous measurements and full-space proper orthogonal decomposition (POD) statistical information. The procedure is tested for the flow past an open cavity. 3D and 2D POD analysis are used to characterize the physics of the flow. We show that the full 3D flow can be estimated from a 2D section at an instant in time provided that some 3D statistical information—i.e., the largest POD modes of the flow— is made available.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Box configuration

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Figure 2

(a) Mean flow in the x−y plane at z=100mm. (b) Mean flow in the y−z plane at x=120mm (scale is six times in part (a)).

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Figure 3

(a) POD spectrum at U=0.9,1.2,2.0m∕s; (b) blow-up

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Figure 4

Longitudinal section of the nth 3D POD structure at z=100mm; (a) n=1; (b) n=2; (c) n=3; (d) n=4; (e) n=5

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Figure 5

Cross section of the nth 3D POD structure at x=190mm; (a) n=1, (b) n=2, (c) n=3, (d) n=4, (e) n=5

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Figure 6

Cross section of the nth 3D POD structure at x=250mm; (a) n=1, (b) n=2, (c) n=3, (d) n=4, (e) n=5

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Figure 7

(a) Total vorticity in first eigenfunction ∥∇×ϕ̱1∥=∣Ω∣=15s−1. (b) Streamwise vorticity in first eigenfunction ∥(∇×ϕ̱1)u∥=∣ωx∣=8s−1 (dark colors mean positive values, and light colors negative ones). (c) Total vorticity in third eigenfunction ∥∇×ϕ̱3∥=∣Ω∣=15s−1. (d) Streamwise vorticity in third eigenfunction ∥(∇×ϕ̱3)u∥=∣ωx∣=8s−1 (dark colors mean positive values, and light colors negative ones).

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Figure 8

POD coefficients an(t) for n=1 to 5 from bottom to top: (a) an(t), (b) time spectra ∣ân(f)∣

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Figure 9

Numerical nth 2D POD structure; (a) n=1, (b) n=2, (c) n=3, (d) n=4, (e) n=5

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Figure 10

2D POD coefficients time spectra—the sampling rate is 30Hz: ∣ân(f)∣ with n=1 to 5 from bottom to top

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Figure 11

Matrix coefficients (in absolute value). Lighter colors correspond to higher values.

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Figure 12

Amount of flow energy captured by the exact and the estimated reconstructions; solid line: ∣∑nanϕ̱n(x)̱∣2∕∫∣u̱(x̱)∣2dx̱ where an is the true POD coefficient; dashed line ∣∑naestnϕ̱n(x)̱∣2∕∫∣u̱(x̱)∣2dx̱ where aestn was estimated from a linear system with 80 modes; dot-dashed line: ∣∑naestnϕ̱n(x)̱∣2∕∫∣u̱(x̱)∣2dx̱ where aestn was estimated from a linear system with 60 modes

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Figure 13

Extract and estimated 3D POD coefficients an(t), with n=1 to 5 from bottom to top: (a) using 80 modes in the estimation, (b) using 60 modes in the estimation

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