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Wavelet series method for reconstruction and spectral estimation of laser Doppler velocimetry data

An application to a screeching rectangular jet LDV measurements

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Abstract

Many techniques have been developed in order to obtain spectral density function from randomly sampled data, such as the computation of a slotted autocovariance function. Nevertheless, one may be interested in obtaining more information from laser Doppler signals than a spectral content, using more or less complex computations that can be easily conducted with an evenly sampled signal. That is the reason why reconstructing an evenly sampled signal from the original LDV data is of interest. The ability of a wavelet-based technique to reconstruct the signal with respect to statistical properties of the original one is explored, and spectral content of the reconstructed signal is given and compared with estimated spectral density function obtained through classical slotting technique. Furthermore, LDV signals taken from a screeching jet are reconstructed in order to perform spectral and bispectral analysis, showing the ability of the technique in recovering accurate information’s with only few LDV samples.

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Notes

  1. http://ldvproc.nambis.de/benchmark/index.html.

Abbreviations

i :

Denotes the ith signal sample

j :

Denotes the jth resolution level of the analyzing wavelet, or the jth signal sample

k :

Corresponds to the kth translated version of the analyzing wavelet, or slot k in slotting technique

ψ(t):

Analyzing wavelet

ψ jk (t):

Discretized mother wavelet

a :

Scale

\(\tilde{u}(a,t)\) :

Continuous wavelet coefficient

\(\Updelta \tilde{u}_{jk}\) :

Increment of wavelet coefficient

e :

Error of reconstruction

u(t):

Original velocity signal

\(\hat{u}(t)\) :

Reconstructed velocity signal

S(f):

Spectral density function at frequency f

b 2(f 1, f 2):

Bicoherence function

\(\hat{R}(k \Updelta \tau)\) :

Slotted autocovariance function

G uu (f):

Spectral density function

\(\dot{N}\) :

Mean data rate of LDV signal

N s :

Number of LDV samples available

N min :

Minimum available LDV samples needed on a given support time to calculate a wavelet coefficient

τ = βa :

Time support on which N min original samples must exist \(\beta \in [0;1]\)

M :

Samples number of the evenly sampled grid

M j :

Number of translated wavelet coefficient to be computed at scale j

J :

Maximum number of resolution level to be computed

f e :

Sampling frequency

f s :

Screeching frequency

\(\Updelta t\) :

Time interval between successive samples

References

  • Adrian R, Yao C (1987) Power spectra of fluid velocities measured by laser doppler velocimetry. Exp Fluids 5:17–28

    Google Scholar 

  • Buchhave P, George WK Jr, Lurnley JL (1979) The measrument of turbulence with laser doppler anemometer. Annu Rev Fluid Mech 11:443–503

    Article  Google Scholar 

  • Collis WB, White PR, Hammond JK (1998) High order spectra: the bispectrum and trispectrum. Mech Syst Signal Process 12:375–394

    Article  Google Scholar 

  • Debauchies I (1990) Ten lectures on wavelets. SIAM

  • Farge M (1992) Wavelet transforms and their applications to turbulence. Annu Rev Fluid Mech 24:395–457

    Article  MathSciNet  Google Scholar 

  • Ford C, Etter D (1998) Wavelet basis reconstruction of non uniformly sampled data. Bull AMS 79:61–78

    Google Scholar 

  • Gaster M, Roberts J (1977) Spectral analysis of randomly sampled records by direct transform. Proc R Soc Lond 354:27–58

    Article  MathSciNet  Google Scholar 

  • Jaunet V (2010) Etude d’un jet rectangulaire supersonique à nombre de mach 1.45 vectorise par actionneur fluidique. PhD thesis, University of Poitiers, FRANCE

  • Jaunet V, Aymer D, Collin E, Bonnet JP, Lebedv A, Fourment C (2010) 3d effects in a supersonic rectangular jet vectored by flow separation control, a numerical and experimental study. AIAA paper 2010–4976

  • Kaleva O, Ihalaien H, Saarenrinne P (2000) Wavelet based method for the estimation of the power spectrum from irregularly sampled data. In: Proceedings of the 10th International Symposium on Applications of Laser Techniques to Fluid Mechanics. Lisbon, Portugal, July 10–13

  • Mayo WT (1974) A discussion of limitations and extensions of power spectrum estimation with burst counter ldv systems. In: Proceedings of Second International Workshop on Laser Velocimetry, pp 90–1024

  • Nita LC (2000) Analyse spectrale de signaux aleatoires a temps continus echantillonnes non uniformement. PhD thesis, University of Paris Sud, France

  • Nobach H (2002) Local estimation for the slotted correlation function of randomly sampled lda data. Exp Fluids 32:337–345

    Article  Google Scholar 

  • Nobach H, Muller E, Tropea C (1994) Refined reconstruction techniques for lda data analysis. In: Proceedings 7th International Symposiym Applications Laser Technol, vol 36.2. Fluid Mech., Lisbon

  • Nobach HE, Tropea C (1998) Efficient estimation of power spectral density for laser doppler velocimetry data. Exp in Fluids 24:499–509

    Article  Google Scholar 

  • Raman G (1999) Supersonic jet screech: half-century from powell to the present. J Sound Vib 225:543–571

    Article  Google Scholar 

  • Rioul O, Duhamel P (1992) Fast algorithms for discrete and continuous wavelet transforms. Tran Inf Theory 38(N2):569–586

    Article  MATH  MathSciNet  Google Scholar 

  • Torrence C, Compo G (1998) A practical guide to wavelet analysis. Bull AMS 79:61–78

    Google Scholar 

  • Tummers MJ, Passchier MD (1996) Spectral estimation using a variable window and the slotting technique with local normalisation. Meas Sci Technol 7:1541–1546

    Article  Google Scholar 

  • Tummers MJ, Passchier MD (2001) Spectral analysis of biased lda data. Meas Sci Technol 12:1641–1650

    Article  Google Scholar 

  • van Maanen HRE, Nobach H, Benedict LH (1999) Improved estimator for the slotted autocorrelation function of randomly sampled lda data set. Meas Sci Technol 10:L4–L7

    Article  Google Scholar 

  • Walker H, Thomas F (1997) Experiments characterizing nonlinear shear layer dynamics in a supersonic rectangular jet undergoing screech. Phys Fluids 9:2562–2579

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the Region Poitou-Charentes, the CNRS and the DGA-Spae for supporting part of this study.

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Authors and Affiliations

  1. PPRIME Institute, UPR 3346, CEAT, 43, route de l’aerodrome, 86036, Poitiers Cedex, France

    Vincent Jaunet, Erwan Collin & Jean-Paul Bonnet

Authors
  1. Vincent Jaunet
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  2. Erwan Collin
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  3. Jean-Paul Bonnet
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Correspondence to Vincent Jaunet.

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Jaunet, V., Collin, E. & Bonnet, JP. Wavelet series method for reconstruction and spectral estimation of laser Doppler velocimetry data. Exp Fluids 52, 225–233 (2012). https://doi.org/10.1007/s00348-011-1222-z

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  • DOI: https://doi.org/10.1007/s00348-011-1222-z

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