Shock Waves

, Volume 24, Issue 5, pp 489–500 | Cite as

Statistical behavior of post-shock overpressure past grid turbulence

  • Akihiro SasohEmail author
  • Tatsuya Harasaki
  • Takuya Kitamura
  • Daisuke Takagi
  • Shigeyoshi Ito
  • Atsushi Matsuda
  • Kouji Nagata
  • Yasuhiko Sakai
Original Article


When a shock wave ejected from the exit of a 5.4-mm inner diameter, stainless steel tube propagated through grid turbulence across a distance of 215 mm, which is 5–15 times larger than its integral length scale \(L_{u}\), and was normally incident onto a flat surface; the peak value of post-shock overpressure, \(\Delta P_{\mathrm{peak}}\), at a shock Mach number of 1.0009 on the flat surface experienced a standard deviation of up to about 9 % of its ensemble average. This value was more than 40 times larger than the dynamic pressure fluctuation corresponding to the maximum value of the root-mean-square velocity fluctuation, \(u^{\prime }= 1.2~\hbox {m}/\hbox {s}\). By varying \(u^{\prime }\) and \(L_{u}\), the statistical behavior of \(\Delta P_{\mathrm{peak}}\) was obtained after at least 500 runs were performed for each condition. The standard deviation of \(\Delta P_{\mathrm{peak}}\) due to the turbulence was almost proportional to \(u^{{\prime }}\). Although the overpressure modulations at two points 200 mm apart were independent of each other, we observed a weak positive correlation between the peak overpressure difference and the relative arrival time difference.


Turbulence Overpressure modulation 


\(a_{\infty }\)

Speed of sound in upstream flow


Correlation coefficient


Grid thickness


Laplace transform of \(f (t)\)

\(f (t)\)

Recovered signal


Laplace transform of \(g (t)\)

\(g (t)\)

Recorded signal


Laplace transform of \(h\)

\(h(\tau )\)

Device transfer function


Serial run number


Imaginary unit




Grid mesh size


Integral length scale


Mach number


Turbulent Mach number, \(=u^{\prime }/a_\infty \)






Number of runs performed for the same condition

\(P_{\infty }\)

Static pressure of upstream flow


Shock wave propagation distance, Fig. 1b

\(R_{\lambda }\)

Reynolds number based on Taylor’s microscale


Complex parameter in Laplace transform



\(U_{\infty }\)

Upstream flow speed

\(u^{\prime }\)

Root-mean-square velocity fluctuation


Coordinate in the direction of main flow


Coordinate in the span-wise direction


Coordinate in the vertical direction

\(\beta \)

\(\Delta P_{\mathrm{peak}}\) / \(P_\infty \), normalized peak overpressure

\(\beta _{\mathrm{w}/\mathrm{o grid}}\)

\(\beta \) without grid

\(\tilde{\beta }\left( {U_\infty } \right) \)

\(\beta \left( {U_\infty } \right) /\overline{\beta _{\mathrm{w}/\mathrm{o grid}} \left( {U_\infty } \right) }\)

\(\chi \)

Heaviside function

\(\Delta \,P\)

Post-shock overpressure

\(\Delta \,P_{0}\)

Post-shock overpressure behind the reflected shock wave that is theoretically obtained from the shock–tube relation, constant value

\(\Delta \, P_{\mathrm{peak}}\)

Peak value of \(\varDelta P\)

\(\nu \)

Kinematic viscosity

\(\lambda _{u}\)

Taylor’s microscale

\(\theta \)

Half apex angle, Fig. 1b

\(\rho _{\infty }\)

Density of upstream flow

\(\sigma \)

Solidity (blockage ratio)

\(\sigma _{\mathrm{IH}}\)

Standard deviation of \(\tilde{\beta }\) due to an apparent overpressure fluctuation before shock wave arrival

\(\sigma _{\mathrm{SG}}\)

Standard deviation of \(\tilde{\beta }\) due to reproducibility of a shock wave generator

\(\sigma _{\mathrm{TF}}\)

Standard deviation of \(\tilde{\beta }\) due to turbulence

\(\tau \)

Arrival time of a shock wave, or time notation used in Appendix

\(\tau _{\mathrm{BA}}\)

\(\tau _{\mathrm{B}} -\tau _{\mathrm{A}}\), relative arrival time of a shock wave

\(\xi \)

\(({0.5\rho _\infty u{\prime }^{2}/\overline{\beta (0)} P_\infty })^{0.5}\)

\(\overline{\left\{ \hbox { } \right\} }\)

Ensemble average

\(\overline{\overline{\left\{ \hbox { } \right\} }}\)

Time average



The first author thanks Dr. Osami Kitoh, Professor Emeritus, Nagoya Institute of Technology, for his valuable suggestion that motivated us to use grid turbulence for shock wave interaction. The authors would like to express their gratitude to Messrs. Katsuyoshi Kumazawa and Takao Sumi from Technical Division and Dr. Shigeru Yokota and Mr. Hiroki Saito, all affiliated with Nagoya University, for their valuable technical assistance. We appreciate the technical assistance of Mr. Kakuei Suzuki in the numerical processing of overpressure signals. This research was supported by the Japan Aerospace Exploration Agency as project No. 27-J-J6711 and as Grant-in-Aid for Scientific Research (S) 22226014 from the Japan Society for the Promotion of Science.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Akihiro Sasoh
    • 1
    Email author
  • Tatsuya Harasaki
    • 1
  • Takuya Kitamura
    • 2
  • Daisuke Takagi
    • 1
  • Shigeyoshi Ito
    • 2
  • Atsushi Matsuda
    • 3
  • Kouji Nagata
    • 2
  • Yasuhiko Sakai
    • 2
  1. 1.Department of Aerospace EngineeringNagoya UniversityNagoyaJapan
  2. 2.Department of Mechanical Science and EngineeringNagoya UniversityNagoyaJapan
  3. 3.Department of Mechanical EngineeringMeijo UniversityNagoyaJapan

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