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Extracting dominant turbulent structures in supersonic flow using two-dimensional Fourier transform

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Abstract

A new image process for quantifying both convection velocities (U C) and scales (λ d) of turbulent structures captured in a fast-framing schlieren movie is presented. We obtained 90 time-series schlieren images of a transverse jet into a Mach 2 supersonic flow with 1-MHz sampling. The schlieren images captured not only the shock and expansion waves but also the turbulent structures within the jet and the boundary layer. The image intensities were extracted along the outer edges of the jet and the boundary layer and were remapped as a time–space intensity map. The time–space map exhibited swept stripe patterns, indicating that stable turbulent structures were periodically generated and convected downstream. The angle and interval of the stripe patterns were efficiently extracted using the two-dimensional Fourier transform, which corresponded to U C and λ d of the dominant structures. The zero-padding fast Fourier transform and the sub-pixel estimation of the spectral peak positions in the Fourier domain improved the accuracy for evaluating the angle and interval of the stripes, which resulted in the accurate evaluation of U C and λ d. The proposed method was validated by comparing U C obtained using the proposed method to those obtained via schlieren image velocimetry for both the transverse jet and the supersonic boundary layer.

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Abbreviations

D :

Injector diameter, mm

f :

Frequency, Hz

f d :

Formation frequency of dominant structure, Hz

f s :

Frame rate of camera, fps

I :

Original image

\(\hat{I}\) :

Spatial-averaged intensity

\(I\text{n}\) :

Normalized image

\(\overline{{I\text{n}}}\) :

Time-averaged image

\(I\text{n}^{\prime }\) :

Fluctuating image

J :

Jet-to-freestream momentum flux ratio

k :

Wave number, m−1 (=1/λ)

k d :

Wave number of dominant structure, m−1

N :

Total number of frames

N S :

Data length in space

N T :

Data length in time

px, py :

Total number of pixels in streamwise and heightwise directions

r k :

Distance from center to high-amplitude point in frequency-wave-number domain

s :

Distance along relevant trajectory of motion

U C :

Convection speed of turbulent structure, m/s

U ∞ :

Freestream velocity (=507 m/s)

V j :

Jet exit velocity (=874 m/s)

x, y, z :

Streamwise, heightwise, and spanwise directions, mm

δ 99 :

Boundary-layer thickness, determined by 99% of freestream velocity

Δx :

Spatial resolution of charge-coupled device sensor (=0.28 mm/pixel)

Δs :

Spatial resolution of time–space map, mm/pixel

λ :

Wavelength of turbulent structure, mm

\(\hat{\sigma }\) :

Spatial-standard-deviation intensity

\(\overline{{\sigma \text{n}}}\) :

Standard-deviation image

θ :

Angle of stripe pattern on time–space map, °

θ k :

Angle to high-amplitude point in two-dimensional power spectrum, °

n :

Frame number

i, j :

Streamwise and heightwise pixel numbers of schlieren image

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Acknowledgements

This work was supported by the Ministry of Education, Culture, Sports, Science and Technology in Japan, Grant-in-Aid for Young Scientist (B), 20760105 and JSPS KAKENHI Grant-in-Aid for Scientific Research (B) 15H04199.

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Authors and Affiliations

  1. Department Mechanical and System Engineering, Okayama University, Tsushimanaka 3-1-1, Kita, Okayama, Okayama, 700-8530, Japan

    Toshinori Kouchi & Shinichiro Yanase

  2. Department Mechanical and Aerospace Engineering, Tohoku University, Aoba 6-6-01, Aramaki, Aoba, Sendai, Miyagi, 980-8579, Japan

    Goro Masuya

Authors
  1. Toshinori Kouchi
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  2. Goro Masuya
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  3. Shinichiro Yanase
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Correspondence to Toshinori Kouchi.

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Kouchi, T., Masuya, G. & Yanase, S. Extracting dominant turbulent structures in supersonic flow using two-dimensional Fourier transform. Exp Fluids 58, 98 (2017). https://doi.org/10.1007/s00348-017-2377-z

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  • DOI: https://doi.org/10.1007/s00348-017-2377-z

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