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Review of Some Fundamentals of Data Processing

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Springer Handbook of Experimental Fluid Mechanics

Part of the book series: Springer Handbooks ((SHB))

Abstract

This chapter is devoted to reviewing some fundamental transforms and analysis procedures commonly used for both signal and data processing in fluid mechanics measurements. The chapter begins with a brief review of the Fourier transform and its digital counterpart the discrete Fourier transform. In particular its use for estimating power spectral density is discussed in detail. This is followed by an introduction of the correlation function and its relation to the Fourier transform. The Hilbert transform completes the introductory topics. The chapter then turns to a rigorous presentation of the proper orthogonal decomposition (POD) in the context of the approximation theory and as an application of singular value decomposition (SVD). The relationship between POD and SVD is discussed and POD is described in a statistical setting using an averaging operation for use with turbulent flows. The different POD approaches are briefly introduced, whereby the main differences between the classical POD and the snapshot POD are highlighted. This section closes with a presentation of the POD as a generalization of the classical Fourier analysis to inhomogeneous directions. The chapter continues with a discussion of conditional averages and stochastic estimation as a means of studying coherent structures in turbulent flows before moving in a final section to a comprehensive discussion of wavelets as a combination of data processing in time and frequency domain. After first introducing the continuous wavelet transform and orthogonal wavelet transform their application in experimental fluid mechanics is illustrated through numerous examples.

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Abbreviations

3-D:

three-dimensional

CFT:

continuous Fourier transform

CS:

coherent structures

DFT:

discrete Fourier transform

FFT:

fast Fourier transform

LES:

large-eddy simulation

LIM:

local intermittency measure

MRA:

multiresolution analysis

ODE:

ordinary differential equations

PDE:

partial differential equations

PDF:

probability density function

PIV:

particle image velocimetry

POD:

proper orthogonal decomposition

PSD:

particle size distribution

PSD:

power spectral density

RIC:

relative information content

SVD:

singular value decomposition

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Authors and Affiliations

  1. Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077, Göttingen, Germany

    Holger Nobach

  2. Fachgebiet Strömungslehre und Aerodynamik, Technische Universität Darmstadt, Petersenstraße 30, 64287, Darmstadt, Germany

    Cameron Tropea Dr.

  3. Laboratoire dʼEtudes Aérodynamiques, CNRS, Université de Poitiers, 43, route de lʼaérodrome, 86036, Poitiers, France

    Laurent Cordier

  4. Laboratoire dʼEtudes Aérodynamiques, Université de Poitiers, ENSMA, CNRS, 43 rue de lʼaérodrome, 86036, Poitiers, France

    Jean-Paul Bonnet Dr.

  5. Laboratoire dʼEtudes Aérodynamiques, CNRS, Université de Poitiers – CEAT, 47, route de lʼAérodrome, 86036, Poitiers, France

    Joël Delville Ph.D.

  6. Mechanical Engineering, Syracuse University, 149 Link Hall, 13244, Syracuse, NY, USA

    Jacques Lewalle

  7. Laboratoire de Météorologie Dynamique (IPSL-CNRS), Ecole Normale Supérieure, 24, rue Lhomond, 75231, Paris, France

    Marie Farge Dr.

  8. Centre de Mathématiques et dʼInformatique, Université de Provence, 39, rue F. Joliot-Curie, 13453 Cedex 13, Marseille, France

    Kai Schneider Dr.

  9. Ira A. Fulton School of Mechanical and Aerospace Engineering, Arizona State University, P.O. Box 876106, 85210, Tempe, AZ, USA

    Ronald Adrian Prof.

Authors
  1. Holger Nobach
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  2. Cameron Tropea Dr.
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  3. Laurent Cordier
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  4. Jean-Paul Bonnet Dr.
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  5. Joël Delville Ph.D.
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  6. Jacques Lewalle
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  7. Marie Farge Dr.
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  8. Kai Schneider Dr.
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  9. Ronald Adrian Prof.
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Corresponding authors

Correspondence to Holger Nobach , Cameron Tropea Dr. , Laurent Cordier , Jean-Paul Bonnet Dr. , Joël Delville Ph.D. , Jacques Lewalle , Marie Farge Dr. , Kai Schneider Dr. or Ronald Adrian Prof. .

Editor information

Editors and Affiliations

  1. Fachgebiet Strömungslehre und Aerodynamik, Technische Universität Darmstadt, Petersenstraße 30, 64287, Darmstadt, Germany

    Cameron Tropea Dr.

  2. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor St., 60607-7022, Chicago, IL, USA

    Alexander L. Yarin Dr.

  3. Mechanical Engineering, Michigan State University, A118 Engineering Research Complex, 48824-1326, East Lansing, MI, USA

    John F. Foss Dr.

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Nobach, H. et al. (2007). Review of Some Fundamentals of Data Processing. In: Tropea, C., Yarin, A.L., Foss, J.F. (eds) Springer Handbook of Experimental Fluid Mechanics. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30299-5_22

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