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Wavelet tools to study intermittency: application to vortex bursting

Published online by Cambridge University Press:  25 September 2009

JORI RUPPERT-FELSOT ,
MARIE FARGE  and
PHILIPPE PETITJEANS
JORI RUPPERT-FELSOT
Affiliation:
LMD–IPSL–CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France PMMH, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
MARIE FARGE*
Affiliation:
LMD–IPSL–CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France
PHILIPPE PETITJEANS
Affiliation:
PMMH, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
*
†Email address for correspondence: farge@lmd.ens.fr
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Abstract

This paper proposes statistical tools adapted to study highly unsteady and inhomogeneous flows, such as vortex bursting. For this, we use the wavelet representation in which each coefficient keeps track of both location and scale, in contrast to Fourier representation which requires keeping the phase of all coefficients to preserve the spatial structure of the flow. Based on the continuous wavelet transform, we propose several diagnostics, such as the local spectrum and the local intermittency measure. We also use the orthogonal wavelet transform to split each flow realization into coherent and incoherent contributions, which are then analysed independently and from which we define the coherency measure. We apply these wavelet tools to analyse the bursting of a three-dimensional stretched vortex immersed in a steady laminar channel flow. The time evolution of the velocity field is measured by particle image velocimetry during several successive bursts.

JFM classification

Vortex Flows: Vortex breakdown
Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 427 - 453
Copyright
Copyright © Cambridge University Press 2009

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References

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Ruppert-Felsot et al. supplementary movie

Movie 1a. Water channel in which a stretched vortex is visualized by injecting dyes of 4 different colors. The vortex evolution is recorded in real time at 25 images per second. The vortex is initiated by a step at the bottom of the test section and stretched by pumping water out throught two oulets located on each lateral wall. For this flow realization the flow rates have been chosen such as the vortex is permanent and does not burst.

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Video 5.2 MB

Ruppert-Felsot et al. supplementary movie

Movie 1b. Time evolution during three successive vortex life cycles of a stretched vortex immersed in a steady laminar channel flow visualized by injecting dyes of 4 different colors and recorded in real time at 25 frames per second from above the experimental setting. For this flow realization the flow rates have been chosen such as the vortex becomes unstable and bursts.

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Video 4.9 MB

Ruppert-Felsot et al. supplementary movie

Movie 2a. Time evolution during three successive vortex life cycles of the velocity modulus and streamlines. These diagnostics correspond to the velocity components in the observation plane measured by particle image velocimetry (PIV).

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Video 4.9 MB

Ruppert-Felsot et al. supplementary movie

Movie 2b. Time evolution during three successive vortex life cycles of the vorticity and streamlines. These diagnostics correspond to the velocities in the observation plane measured by particle image velocimetry (PIV) and to the vorticity component perpendicular to this plane.

Download Ruppert-Felsot et al. supplementary movie(Video)
Video 5 MB

Ruppert-Felsot et al. supplementary movie

Movie 2c. Time evolution during three successive vortex life cycles of the velocity modulus (left), of the vorticity modulus (right), of the time trace of the velocity modulus (top) measured in the location indicated by a cross on the velocity modulus inset, together with the energy spectrum computed either in the whole domain (solid line) or in a square subdomain (dashed line). During each vortex life cycle the slope of the energy spectrum varies between -2 (thick dashed line) and -1 (thin dashed line), while the slope of its time average is -5/3 (dotted line). These diagnostics correspond to the velocities in the observation plane measured by PIV and to the vorticity component perpendicular to this plane.

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Video 4.9 MB

Ruppert-Felsot et al. supplementary movie

Movie 3a. Time evolution during three successive vortex life cycles of the vorticity modulus. In order to compare with the coherent vorticity shown on movie 3b, the color scale is adapted to the extrema of the coherent vorticity whose values are given in figure 3b. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 1.6 MB

Ruppert-Felsot et al. supplementary movie

Movie 3b. Time evolution during three successive vortex life cycles of the coherent vorticity modulus, obtained by orthogonal wavelet decomposition. The color scale is adapted to the extrema of the coherent vorticity, whose values are given in figure 3b. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 1.6 MB

Ruppert-Felsot et al. supplementary movie

Movie 3c. Time evolution during three successive vortex life cycles of the incoherent vorticity modulus obtained by orthogonal wavelet decomposition. The color scale is adapted to the extrema of the incoherent vorticity, whose values are given in figure 3c. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 3.9 MB

Ruppert-Felsot et al. supplementary movie

Movie 4a. Spatial distribution during three successive vortex life cycles of the variance (top right), skewness (bottom left) and flatness (bottom right) of the total (green), coherent (red) and incoherent (blue) vorticities. The star moving on each curve indicates the same instant as the time evolution of the total, coherent and incoherent vorticity (top left). These diagnostics correspond to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 1.5 MB

Ruppert-Felsot et al. supplementary movie

Movie 4b. Time evolution during three successive vortex life cycles of the percent of wavelet coefficients kept to describe the coherent vorticity (top right), of the fraction of the total vorticity variance retained by the coherent (red) and incoherent (blue) vorticities (bottom left), of the coherency measure (CM) given by the signal to noise ratio (black) between the coherent and incoherent vorticity variances (bottom right). The star moving on each curve indicates the same instant as the time evolution of the total, coherent and incoherent vorticity (top left). These diagnostics correspond to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 1.5 MB

Ruppert-Felsot et al. supplementary movie

Movie 5a. Time evolution during three successive vortex life cycles of the scatter plot of the vorticity versus the stream function for the total (green), the coherent (red) and the incoherent (blue) flows, together with the corresponding time evolution of the total, coherent and incoherent vorticities. These diagnostics correspond to the vorticity component perpendicular to the observation plane computed from the PIV velocity measurements.

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Video 2.9 MB

Ruppert-Felsot et al. supplementary movie

Movie 5b. Time evolution during three successive vortex life cycles of the probablility density function (PDF) of vorticity for the total (green), the coherent (red) and the incoherent (blue) flows, together with the corresponding time evolution of the total, coherent and incoherent vorticities. These diagnostics correspond to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 2.9 MB

Ruppert-Felsot et al. supplementary movie

Movie_5c. Time evolution during three successive vortex life cycles of the enstrophy spectrum, for the total (green), the coherent (red) and the incoherent (blue) flows, together with the corresponding time evolution of the total, coherent and incoherent vorticities. These diagnostics correspond to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 2.9 MB

Ruppert-Felsot et al. supplementary movie

Movie 6a. Time evolution during one vortex life cycle of the continuous wavelet transform (CWT) modulus of vorticity visualized in three-dimensional wavelet space with a vertical cut at abscissa x=63. The isosurfaces values and the corresponding color scale are given in the caption of figure 6. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 558.1 KB

Ruppert-Felsot et al. supplementary movie

Movie 6b. Time evolution during three successive vortex life cycles of a two-dimensional cut, in space (horizontal axis) and scale (vertical axis with the smallest scales upwards), of the continuous wavelet transform (CWT) modulus of vorticity. The location of the cut varies in time to track the maximum of the vorticity modulus (top left). These diagnostics correspond to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 5 MB

Ruppert-Felsot et al. supplementary movie

Movie 7a. Time evolution during one vortex life cycle of the local intermittency measure (LIM) of vorticity visualized in three-dimensional wavelet space with a vertical cut at abscissa x=63. The isosurfaces values and the corresponding color scale are given in the caption of figure 7. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 670.8 KB

Ruppert-Felsot et al. supplementary movie

Movie 7b. Time evolution during three successive vortex life cycles of the spatial distribution of the scale averaged local intermittency measure (LIM) of vorticity. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 3.6 MB

Ruppert-Felsot et al. supplementary movie

Movie 8a. Time evolution during one vortex life cycle of a stretched vortex immersed in a steady laminar channel flow visualized by fluorescein and recorded at 1000 frames per second in a vertical plane located in the middle of the channel.

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Video 683 KB

Ruppert-Felsot et al. supplementary movie

Movie 8b. Time evolution during one vortex life cycle of the continuous wavelet transform (CWT) modulus of vorticity visualized from a side in three-dimensional wavelet space, with a rainbow color scale where the strong, intermediate and weak values are respectively in read, green and blue. This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 1.7 MB

Ruppert-Felsot et al. supplementary movie

Movie 8c. Time evolution during one vortex life cycle of the continuous wavelet transform (CWT) modulus of vorticity visualized from the top in three-dimensional wavelet space, with a rainbow color scale where the strong, intermediate and weak values are respectively in read, green and blue. . This diagnostic corresponds to the vorticity component perpendicular to the observation plane computed from the velocity measured by PIV.

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Video 1.3 MB