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Research Papers: Fundamental Issues and Canonical Flows

On Self-Similarity in the Inner Wall Layer of a Turbulent Channel Flow

[+] Author and Article Information
Bérengère Podvin, Yann Fraigneau

 LIMSI-CNRS, BP133, Orsay 91403, France

Julien Jouanguy, J.-P. Laval

 LML, Lille, Bd. Paul Langevin, Villeneveuve d'Asq Cedex 59655, France

J. Fluids Eng 132(4), 041202 (Apr 16, 2010) (15 pages) doi:10.1115/1.4001385 History: Received September 15, 2009; Revised February 11, 2010; Published April 16, 2010; Online April 16, 2010

We use proper orthogonal decomposition (POD) to estimate the flow in the near-wall region based on information from the outer buffer layer. Our goal is to assess how the flow structures in the inner wall region are connected to those further away from the wall, and to investigate the nature of the coupling between the inner and the outer region in the POD framework. Reconstructions are carried out for numerical simulations of a plane channel flow at two different Reynolds numbers. We show that elongated structures with a spanwise wavelength smaller than a critical value tend to be concentrated in the inner layer. The critical wavelength is shown to scale with the inner layer height, and interactions between the inner and the outer layer appear to take place predominantly over a self-similar, height-dependent, range of wavenumbers, in agreement with Townsend’s attached eddy hypothesis. The reconstructed field appears to capture an adequate energy content and to remain correlated with the real field even close to the wall, which reflects the persistence of energetic structures over the extent of the buffer layer.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Numerical domain-the arrow indicates the mean flow direction. The limit between the inner and the outer wall region is materialized by the dark planes.

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Figure 2

Energy of the real and reconstructed POD modes

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Figure 3

Correlation coefficient between the real and the reconstructed POD modes

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Figure 4

Plane reconstruction of a flow realization at y+=45: (a) real u, (b) reconstructed uR, (c) real v, (d) reconstructed vR, (e) real w, and (f) reconstructed wR

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Figure 5

Plane reconstruction of a flow realization at y+=10: (a) real u, (b) reconstructed uR, (c) real v, (d) reconstructed vR, (e) real w, and (f) reconstructed wR

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Figure 6

Spectral comparison of a flow realization with its reconstruction and its projection: (a) streamwise component at y+=45, (b) streamwise component at y+=10, (c) wall-normal component at y+=45, (d) wall-normal component at y+=10, (e) spanwise component at y+=45, and (f) spanwise component at y+=10—the spectra are averaged in the streamwise direction

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Figure 7

Diagonal term of the correlation matrix Blk11

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Figure 8

Spatial distribution of the first-order POD eigenfunctions as a function of the wavenumber—arrows indicate the possible direction of the average energy transfer across the horizontal plane. Energy transfer to higher-order wall-normal modes is not represented.

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Figure 9

Amount of energy in the first wall-normal mode-log scale representation: (a) real POD mode |alk1|2, and (b) reconstructed POD mode |ãlk1|2

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Figure 10

Correlation coefficient between the real and the reconstructed first wall-normal POD mode alk1

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Figure 11

Comparison of the reconstructed and projected wall-normal velocity at y+=40—contour lines are at −0.08U,−0.04U,0,0.04U, where U is the streamwise velocity at the center of the channel: (a) vp and (b) vR

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Figure 12

Comparison of the reconstructed and projected wall-normal velocity at y+=10—contour lines are at −4.10−5U,−2.10−5U,0,2.10−5U, where U is the streamwise velocity at the center of the channel: (a) vp and (b) vR

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Figure 13

Correlation coefficient between the real and the reconstructed first wall-normal mode alk1: (a) y1=50, (b) y1=100, and (c) y1=160

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Figure 14

Diagonal coefficient Blknn when y1=100—(a) n=1 and (b) n=2

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Figure 15

Reconstruction of the streamwise velocity at y+=90 when y1=100: (a) real field u, (b) projected field with 25 wall-normal modes up, and (c) reconstructed field uR with 25 wall-normal modes-contours from −0.15−0.1−0.05U

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Figure 16

Correlation coefficient between the reconstructed and the projected field for 25 modes: (a) y1=50, (b) y1=100, and (c) y1=160—the dashed line represents the lower limit y1 of the information domain

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Figure 17

Energy of the first wall-normal mode ⟨|alk1|2⟩

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Figure 18

Energy of the reconstructed POD mode ⟨|ãlk1|2⟩ when Blk is approximated by its diagonal: (a) y1=50, (b) y1=100, and (c) y1=160

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Figure 19

Energy of the reconstructed POD mode ⟨|ãlk1|2⟩ taking into account the effect of higher-order wall-normal modes: (a) y1=50, (b) y1=100, and (c) y1=160

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