Shock Waves

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

Statistical behavior of post-shock overpressure past grid turbulence

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

Abstract

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.

Keywords

Turbulence Overpressure modulation 

Nomenclature

\(a_{\infty }\)

Speed of sound in upstream flow

\(C\)

Correlation coefficient

\(d\)

Grid thickness

\(F(s)\)

Laplace transform of \(f (t)\)

\(f (t)\)

Recovered signal

\(G(s)\)

Laplace transform of \(g (t)\)

\(g (t)\)

Recorded signal

\(H(s)\)

Laplace transform of \(h\)

\(h(\tau )\)

Device transfer function

\(i\)

Serial run number

\(j\)

Imaginary unit

\(k\)

Integer

\(L\)

Grid mesh size

\(L_{u}\)

Integral length scale

\(M\)

Mach number

\(M_\mathrm{t}\)

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

\(m\)

Integer

\(n\)

Integer

\(N\)

Number of runs performed for the same condition

\(P_{\infty }\)

Static pressure of upstream flow

\(R\)

Shock wave propagation distance, Fig. 1b

\(R_{\lambda }\)

Reynolds number based on Taylor’s microscale

\(s\)

Complex parameter in Laplace transform

\(t\)

Time

\(U_{\infty }\)

Upstream flow speed

\(u^{\prime }\)

Root-mean-square velocity fluctuation

\(x\)

Coordinate in the direction of main flow

\(y\)

Coordinate in the span-wise direction

\(z\)

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Akihiro Sasoh
    • 1
  • 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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