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Improvements on the accuracy of derivative estimation from DPIV velocity measurements

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Abstract

Accuracy of out-of-plane vorticity estimation from in-plane experimental velocity measurements is investigated with particular application to digital particle image velocimetry (DPIV). Simulations of known flow fields are used to quantify errors associated with amplification of the velocity measurement noise and method bias error due to spatial sampling resolution. A novel, adaptable, hybrid estimation scheme combining implicit compact finite difference and Richardson extrapolation schemes is proposed for improved vorticity estimation. The scheme delivers higher-order truncation error with less noise amplification than an explicit second order finite difference scheme. Finally, a complete framework for predicting, a priori, the random, bias, and total error of the vorticity estimation on the basis of the error of the resolved velocities and the choice of differentiation scheme is developed and presented.

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Abbreviations

A, A k :

Constants for Richardson extrapolation

α, β, a, b, c :

Constants for the compact schemes

i, j, k :

Indices

θ , r, z :

Coordinate axes (subscripts)

r :

Radius

Δ, Δx :

Spatial sampling resolution

Γ:

Circulation (set to 1,000 mm2/s for Oseen vortex solution)

Vref :

Characteristic velocity scale

L :

Characteristic length scale (set to 130 mm for Oseen vortex solution)

U :

Velocity in i direction

V :

Velocity in j direction

ω:

Out-of-plane vorticity

σ:

Standard deviation

δ:

Uncertainty

λ o :

Noise transmission ratio

K :

Coefficient (numerator) of the noise transmission ratio

γ, ϕ:

Constants for power law

ɛ:

Noise amplification coefficient (probabilistic quadratic approach)

χ:

Constants from velocity derivative schemes evaluated in probabilistic quadratic approach

FD2:

Second-order central finite difference scheme

Chapra-4:

Fourth-order Chapra scheme

comp-4:

Fourth-order compact scheme

comp-6:

Sixth-order compact scheme

*:

Denotes noise optimization

Rich-4*:

Noise optimized fourth-order Richardson extrapolation scheme

Rich-6:

Sixth-order Richardson extrapolation scheme

CR4*:

Noise optimized fourth-order hybrid compact-Richardson extrapolation scheme

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Acknowledgments

The authors would like to thank Dr. Demetri Telionis for his guidance and support, as well as the National Institute of Standards and Technology (NIST), the National Aeronautics and Space Administration (NASA), and Aeroprobe Corp.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Virginia Tech, 215 Norris Hall, Blacksburg, VA, 24061, USA

    Ali Etebari & Pavlos P. Vlachos

Authors
  1. Ali Etebari
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  2. Pavlos P. Vlachos
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Corresponding author

Correspondence to Ali Etebari.

Additional information

A portion of this work was presented at ASME IMECE 2003 conference

An erratum to this article is available at http://dx.doi.org/10.1007/s00348-005-0085-6.

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Etebari, A., Vlachos, P.P. Improvements on the accuracy of derivative estimation from DPIV velocity measurements. Exp Fluids 39, 1040–1050 (2005). https://doi.org/10.1007/s00348-005-0037-1

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  • DOI: https://doi.org/10.1007/s00348-005-0037-1

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