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Experimental estimation of fluctuating velocity and scalar gradients in turbulence

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Abstract

The effect of numerical differentiation is investigated in the context of evaluating fluctuating velocity and scalar quantities in turbulent flows. In particular, 2-point forward-difference and 3-, 5-, 7-, and 9-point centred-difference schemes are investigated. The spectral technique introduced by Wyngaard (in J Sci Instr 1(2):1105–1108, 1968) for homogeneous turbulence is used to quantify the effects of the schemes. Numerical differentiation is shown to attenuate gradient spectra over a range of wavenumbers. The spectral attenuation, which varies with the order of the scheme, results in a reduction in the measured mean-squared gradients. High-order schemes (e.g. 7- or 9-point) are shown to significantly decrease the attenuation at all wavenumbers and as a result produce more accurate gradients. Hot-wire measurements and direct numerical simulations of decaying homogeneous, isotropic turbulence are found to be in good agreement with the predictions of the analysis, which suggests that high-order schemes can be used to improve empirical gradient estimates. The shape of the probability density functions is also found to be sensitive to the choice of numerical differentiation scheme. The effect of numerical differentiation is also discussed with respect to particle image velocimetry (PIV) measurements of a nominally two-dimensional planar mixing layer. It is found that the relatively low signal-to-noise ratio inherent in typical PIV measurements necessitates the use of low-order schemes to prevent excessive noise amplification, which increases with the order of the scheme. The results of the present work demonstrate that high-order numerical differentiation schemes can be employed to more accurately resolve gradients measured at a given resolution provided the measurements have an adequate signal-to-noise ratio.

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Acknowledgments

RJH and PL would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada and the Ontario Government. BG and ORHB acknowledge financial support from EPSRC. The authors would also like to thank Dr. Burattini for kindly providing access to his DNS data.

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Authors and Affiliations

  1. Institute for Aerospace Studies, University of Toronto, Toronto, ON, M3H 5T6, Canada

    R. J. Hearst & P. Lavoie

  2. Center for Aeromechanics Research, The University of Texas at Austin, Austin, TX, 78712-1085, USA

    O. R. H. Buxton

  3. Aerodynamics and Flight Mechanics Research Group, University of Southampton, Southampton, SO17 1BJ, UK

    B. Ganapathisubramani

Authors
  1. R. J. Hearst
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  2. O. R. H. Buxton
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  3. B. Ganapathisubramani
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  4. P. Lavoie
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Correspondence to P. Lavoie.

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Hearst, R.J., Buxton, O.R.H., Ganapathisubramani, B. et al. Experimental estimation of fluctuating velocity and scalar gradients in turbulence. Exp Fluids 53, 925–942 (2012). https://doi.org/10.1007/s00348-012-1318-0

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  • DOI: https://doi.org/10.1007/s00348-012-1318-0

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