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Spatiotemporal superresolution measurement based on POD and sparse regression applied to a supersonic jet measured by PIV and near-field microphone

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Abstract

The present study proposed the framework of the spatiotemporal superresolution measurement based on the sparse regression with dimensionality reduction using the proper orthogonal decomposition (POD). The non-time-resolved particle image velocimetry (PIV) and the time-resolved near-field acoustic measurements using microphones were simultaneously performed for a Mach 1.35 supersonic jet. POD is applied to PIV and microphone data matrices, and the sparse linear regression model of the reduced-order data is calculated using the least absolute shrinkage and selection operator regression. The effects of the hyperparameters of the superresolution measurement were quantitatively evaluated through randomized cross-validation. The superresolved velocity field indicated the smooth convection of the velocity fluctuations associated with the screech tone, while the convection of the large-scale structures at the downstream side was not observed. The proposed framework can reconstruct the unsteady fluctuation with multiple frequency phenomena, although the reconstruction is limited to the phenomena that are associated with the microphone output.

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Abbreviations

D :

Nozzle exit diameter

E :

Reconstruction error

f :

Frequency

\(f_s\) :

Peak frequency of screech tone

\(L_{sh}\) :

Shock cell length

m :

Number of data points in space

M :

Mach number

n :

Number of data points in time

\(n_{td}\) :

Number of data points for time-delay

N :

Total number of PIV snapshots

\(p_{j}\) :

Microphone data acquired by jth microphone

PR:

Projection ratio

r :

Number of POD modes for the dimensionality reduction

St:

Strouhal number

\(t_{i}\) :

Discrete-time

u, v :

Streamwise and radial velocity

\(u_c\) :

Convection velocity

\(U_{j}\) :

Streamwise velocity at the nozzle exit derived under the assumption of the isentropic flow

\(\kappa\) :

Sampling-rate ratio of the acoustic and PIV measurements

\(\lambda\) :

Regularization parameter for group-LASSO regression

\({\mathbf {M}}\) :

Time-delay embedded microphone data matrix

\({\mathbf {S}}\) :

Diagonal matrix of singular values

\({\mathbf {u}}\) :

Column vector of streamwise velocity components

\({\mathbf {U}}\) :

Orthogonal spatial modes matrix

\({\mathbf {v}}\) :

Column vector of radial velocity components

\({\mathbf {V}}\) :

Orthogonal temporal modes matrix

\({\mathbf {X}}\) :

Data matrix

\({\mathbf {Z}}\) :

POD coefficients matrix

\(\mathbf {Z_{{\Phi }}}\) :

Projection of the \(\mathbf {Z_{\mathrm {PIV}}}\) Onto the \({\Phi }\) Space

\({\Psi }\) :

Regression coefficients matrix

\({\Phi }\) :

Regression coefficients matrix consists of the nonzero components of \({\Psi }\)

\(\bar{}\) :

Time-averaged component

\(\widetilde{}\) :

Fluctuation component

\(\hat{}\) :

Estimated component

\('\) :

Downsampled component

\(\mathrm{MIC}\) :

Microphone data

\(\mathrm{PIV}\) :

PIV data

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Acknowledgements

This work was partially supported by JSPS KAKENHI Grant Numbers JP18H03809, JP19KK0361, and JP20H00278, and the research grants from Shimadzu Science Foundation. T. Nagata was supported by Japan Science and Technology Agency, CREST Grant Number JPMJCR1763.

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  1. Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Miyagi, Japan

    Yuta Ozawa, Takayuki Nagata & Taku Nonomura

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  1. Yuta Ozawa
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  2. Takayuki Nagata
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Correspondence to Yuta Ozawa.

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Ozawa, Y., Nagata, T. & Nonomura, T. Spatiotemporal superresolution measurement based on POD and sparse regression applied to a supersonic jet measured by PIV and near-field microphone. J Vis 25, 1169–1187 (2022). https://doi.org/10.1007/s12650-022-00855-6

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