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Development and evaluation of gappy-POD as a data reconstruction technique for noisy PIV measurements in gas turbine combustors

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Abstract

Low signal-to-noise in particle image velocimetry (PIV) measurements in systems such as high pressure gas turbine combustors can result in significant data gaps that negatively affect subsequent analysis. Here, gappy proper orthogonal decomposition (GPOD) is evaluated as a method of filling such missing data. Four GPOD methods are studied, including a new method that utilizes a median filter (MF) to adaptively select whether a local missing data point is updated after each iteration. These methods also are compared against local Kriging interpolation. The GPOD methods are tested using PIV data without missing vectors that were obtained in atmospheric pressure swirl flames. Parameters studied include the turbulence intensity, amount of missing data, and the amount of noise in the valid data. Two criteria to check for GPOD convergence also were investigated. The MF method filled in the missing data with the lowest error across all parameters tested, with approximately one-third the computational cost of Kriging. Furthermore, the accuracy of MF GPOD was relatively insensitive to the quality of the convergence criterion. Therefore, compared to the three other GPOD methods and Kriging interpolation, the MF GPOD method is an effective method for filling missing data in PIV measurements in the studied gas turbine combustor flows.

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Abbreviations

N :

Total number of snapshots in a data set

P :

Number of POD/GPOD modes

\(a_{i}, b_{i}\) :

POD temporal coefficients

\(\phi _{{i}}\) :

POD spatial eigenmodes

\(\varvec{R}\) :

Correlation matrix

\(\lambda _{n,j,i}\) :

POD eigenvalues

\(\mathbf {u}\) :

Velocity

\(\bar{\xi }_{n,j}\) :

Approximate filled field

\(\breve{\bar{\xi }}_{n,j}\) :

POD approximation of field

\(\hat{\xi }_{n}\) :

GPOD reconstructed field at the end of main iteration n (ES and Gunes methods)

\(\hat{\hat{\xi }}_{n}\) :

GPOD reconstructed field at the end of main iteration n (FS and MF methods)

\(\tilde{\xi }_{n,j}\) :

POD approximation of field using coefficients \(b_i\)

\(F_{{s,n}}\) :

Field smoothness parameter (FS method)

\(R_{n}\) :

Residual parameter for the Median Filter method

\(\varepsilon\) :

Convergence tolerance for sub iterations

E :

Reconstruction error over entire data set

\(\psi\) :

Reconstruction error over a single snapshot

\(K_{P_{n}}\) :

Kinetic energy calculated from POD eigenvalues of main iteration n

\(K_{P_0}\) :

Kinetic energy calculated from POD eigenvalues of initial guess

G :

Gappiness percentage

q :

Turbulent kinetic energy calculated at center of jet

\(\dot{m}_A\) :

Air flow rate (g/min)

\(\dot{m}_{\mathrm{CH}_4}\) :

Fuel flow rate (g/min)

\(P_\mathrm{th}\) :

Thermal power (kW)

\(P^*\) :

Main iteration (GPOD mode) at which \(E = E^*\)

\(E^*\) :

Minimum reconstruction error

\(V_{\mathrm{mag}}\) :

Velocity magnitude

n:

GPOD main iteration

j:

GPOD sub iteration

d:

Valid data

g:

‘Real’ gaps

cc:

Convergence checking gaps

i:

Index of POD modes

cp:

Center point

ad:

Points adjacent to center point cp

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Acknowledgments

This work was supported by NSERC Canada and GE under Grant CRDPJ 47740.

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Authors and Affiliations

  1. Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Toronto, ON, M3H 5T6, Canada

    Pankaj Saini & Adam M. Steinberg

  2. Institute of Combustion Technology, German Aerospace Center, Pfaffenwaldring 38-40, 70569, Stuttgart, Germany

    Christoph M. Arndt

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  1. Pankaj Saini
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  2. Christoph M. Arndt
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  3. Adam M. Steinberg
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Correspondence to Adam M. Steinberg.

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Saini, P., Arndt, C.M. & Steinberg, A.M. Development and evaluation of gappy-POD as a data reconstruction technique for noisy PIV measurements in gas turbine combustors. Exp Fluids 57, 122 (2016). https://doi.org/10.1007/s00348-016-2208-7

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