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Shock wave diffraction in micro-shock tubes with sudden expansion

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Abstract

The present study investigates the shock wave propagation and diffraction characteristics in a micro-shock tube with sudden expansion and compares with the well-established classical shock wave diffraction in macro-length scale sudden expansions using computational techniques. The Knudsen number for the present micro-shock tube falls in the slip regime and therefore the fluid flow is simulated using the continuum-based Navier–Stokes equation with Maxwell’s slip jump boundary condition. It is found that the shock wave attenuates rapidly in micro-shock tube compared to the shock wave propagation in macro-shock tube. The shock wave diffraction in micro-steps shows similar characteristics compared to macro-steps, such as reflection of the diffracted shock wave from the outer wall and the subsequent transition from regular reflection to Mach reflection, the vortex formation at the step corner, Mach reflection shock structure in the shock-processed gas exiting from the shock tube. However, the secondary shock wave formed due to the interaction of the reflected shock wave with the corner vortex is not seen for the micro-step case compared to the macro-step case. This can be attributed to the reduction in shock strength produced by the thick boundary layer in micro-shock tubes. Different step sizes have been compared for the micro-shock tube with sudden expansion ranging from the step size 1.5 to 3. Also, a detailed comparison has been done between micro- and macro-shock tube with sudden expansion. It is also found that the use of slip velocity increases the shock wave propagation speed compared to the no-slip boundary condition.

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Abbreviations

Kn:

Knudsen number

L :

Characteristic length

L T :

Total shock tube length

λ :

Mean free path

P4:

Driver gas pressure

P1:

Driven gas pressure

T:

Centre line temperature

T1 :

Initial temperature

X0 s :

Starting position

Xs :

Shock location

a1 :

Speed of sound through the medium

H :

Diameter of expanding tube

h :

Diameter of smaller tube

\({u}_{\mathrm{face}}\) :

Fluid velocity at the wall face centre

\({u}_{\mathrm{cell}}\) :

Fluid velocity at the wall cell centre

\({T}_{\mathrm{face}}\) :

Static temperature at the wall face centre

\({T}_{\mathrm{cell}}\) :

Static temperature at the wall cell centre

\({\alpha }_{\mathrm{m}}\) :

Momentum coefficient

\({\alpha }_{\mathrm{t}}\) :

Thermal coefficient

\(\gamma\) :

Specific heat ratio of gas

Pr:

Prandtl number

K b :

Boltzmann constant

\(\sigma\) :

Lennard Jones characteristic

P:

Static pressure

X:

Actual distance travelled by shock

x o :

Starting point of the shock (diaphragm)

x n :

End point of the shock tube

X*:

Non-dimensionalized shock position, X* = (x − x0)/(xn − x0)

References

  1. Wing, T., Lifshitz, A.: Shock tube techniques in chemical kinetics. Annu. Rev. Phys. Chem. 41, 559–599 (1990)

    Article  Google Scholar 

  2. Glass, I.I., Hall, J.G.: Handbook of supersonic aerodynamics. Section 18. Shock tubes. Bureau Of Naval Weapons Washington DC pp 451–498 (1959)

  3. Jagadeesh, G., Takayama, K.: Novel applications of micro-shock waves in biological sciences. J. Indian Inst. Sci. 82(1), 49–57 (2002)

    Google Scholar 

  4. Yasunaga, K., Takahiro Mikajiri, S., Sarathy, M., Koike, T., Gillespie, F., Nagy, T., Simmie, J.M., Curran, H.J.: A shock tube and chemical kinetic modeling study of the pyrolysis and oxidation of butanols. Combust. Flame 159(6), 2009–2027 (2012)

    Article  Google Scholar 

  5. Kuwahara, M., Ioritani, N., Kambe, K., Orikasa, S., Takayama, K.: Anti-miss-shot control device for selective stone disintegration in extracorporeal shock wave lithotripsy. Shock Waves 1, 145–148 (1991)

    Article  Google Scholar 

  6. Rakesh, S.G., Gnanadhas, D.P., Allam, U.S., Nataraja, K.N., Barhai, P.K., Jagadeesh, G., Chakravortty, D.: Development of micro-shock wave assisted dry particle and fluid jet delivery system. Appl. Microbiol. Biotechnol. 96, 647–662 (2012)

    Article  Google Scholar 

  7. Subburaj, J., Datey, A., Gopalan, J., Chakravortty, D.: Insights into the mechanism of a novel shockwave-assisted needle-free drug delivery device driven by in situ-generated oxyhydrogen mixture which provides efficient protection against mycobacterial infections. J. Biol. Eng. 11(1), 1–13 (2017)

    Article  Google Scholar 

  8. Datey, A., Subburaj, J., Gopalan, J., Chakravortty, D.: Mechanism of transformation in Mycobacteria using a novel shockwave assisted technique driven by in-situ generated oxyhydrogen. Sci. Rep. 7(1), 1–11 (2017)

    Article  Google Scholar 

  9. Delius, N.I.: Medical applications and bioeffects of extracorporeal shock waves. Shock Waves 4, 55–72 (1994)

    Article  Google Scholar 

  10. Lynch, P.T., Troy, T.P., Ahmed, M., Tranter, R.S.: Probing combustion chemistry in a miniature shock tube with synchrotron VUV photo ionization mass spectrometry. Anal. Chem. 87(4), 2345–2352 (2015)

    Article  Google Scholar 

  11. Tranter, R.S., Lynch, P.T.: A miniature high repetition rate shock tube. Rev. Sci. Instrum. 84(9), 094102 (2013)

    Article  Google Scholar 

  12. Bisht, A., Kumar, L., Subburaj, J., Jagadeesh, G., Suwas, S.: Effect of stacking fault energy on the evolution of microstructure and texture during blast assisted deformation of FCC materials. J. Mater. Process. Technol. 271, 568–583 (2019)

    Article  Google Scholar 

  13. Ramachandran, R.C., Raman, G., Janardhanraj, S., G. Jagadeesh, G.: Miniature shock tube actuators for flow control applications. In 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, p 1259 (2010)

  14. Arun Kumar, R., Kim, H.D.: Computational study of the unsteady flow characteristics of a micro shock tube. J. Mech. Sci. Technol. 27(2), 451–459 (2013)

    Article  Google Scholar 

  15. Singh, S.K., Arun Kumar, R.: A parametric study on the fluid dynamics and performance characteristic of micro nozzle flows. J. Fluids Eng. 144(3), 031208 (2022)

    Article  Google Scholar 

  16. Mirshekari, G., Brouillette, M.: One-dimensional model for microscale shock tube flow. Shock Waves 19(1), 25–38 (2009)

    Article  MATH  Google Scholar 

  17. Brouillete, M.: Shock waves at microscales. Shock Waves 13, 3–12 (2003)

    Article  Google Scholar 

  18. Mirshekari, G., Brouillette, M.: Microscale shock tube. J. Microelectromech. Syst. 21(3), 739–748 (2012)

    Article  MATH  Google Scholar 

  19. Zeitoun, D.E., Burtschell, Y.: Navier–Stokes computations in micro shock tubes. Shock Waves 15, 241–246 (2006)

    Article  MATH  Google Scholar 

  20. Zeitoun, D.E., Burtschell, Y., Graur, I.A., Ivanov, M.S., Kudryavtsev, A.N., Bondar, Y.A.: Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches. Shock Waves 19, 307–316 (2009)

    Article  MATH  Google Scholar 

  21. Arun Kumar, R., Kim, H.D., Setoguchi, T.: Computational analysis of the wave motions in micro-shock tube flow. J. Aerosp. Eng. 228(4), 594–610 (2014)

    Google Scholar 

  22. Arun Kumar, R., Kim, H.D., Setoguchi, T.: Effect of finite diaphragm rupture process on micro shock tube flows. J. Fluids Eng. 135(8), 081203 (2013)

    Article  Google Scholar 

  23. Oh, C.K., Oran, E.S., Sinkovits, R.S.: Computations of high-speed high Knudsen number microchannels flows. J. Thermophys. Heat Transfer 11, 497–505 (1997)

    Article  Google Scholar 

  24. Skews, B.W.: The shape of a diffracting shock wave. J. Fluid Mech. 29(2), 297–304 (1967)

    Article  Google Scholar 

  25. Skews, B.W.: The perturbed region behind a diffracting shock wave. J. Fluid Mech. 29(4), 705–719 (1967)

    Article  Google Scholar 

  26. Kim, H.D., Setoguchi, T.: Study of the discharge of weak shocks from an open end of a duct. Journal of Sound Vibration. 226(5), 1011–1028 (1999)

    Article  Google Scholar 

  27. Abe, A., Takayama, K.: Numerical simulation and density measurement of a shock wave discharged from the open end of a shock tube. Jpn. Soc. Mech. Eng. Int. J. 33, 216 (1990)

    Google Scholar 

  28. Chang, K.S., Kim, J.K.: Numerical investigation of inviscid wave dynamics in an expansion tube. Shock Wave. 5, 33–45 (1995)

    Article  MATH  Google Scholar 

  29. Jiang, Z., Takayama, K., Babinsky, H., Meguro, T.: Transient shock wave flows in tubes with a sudden change in cross section. Shock Waves 7, 151–162 (1997)

    Article  Google Scholar 

  30. BeskokAluru, K.: Microflows and Nanoflows-Fundamentals and Simulation. Springer, New York (2005)

    Google Scholar 

  31. Janardhanraj, S., Abhishek, K., Jagadeesh, G.: Insights into the shockwave attenuation in miniature shock tubes. J. Fluid Mech. 910, A3 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the financial support for this project from the INSPIRE Grant (Grant No. DST/INSPIRE/04/2018/001780), Department of Science and Technology (DST), India.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, Jodhpur, Rajasthan, 342037, India

    Aswin Suresh, Rajat Raj & Arun Kumar Rajagopal

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  1. Aswin Suresh
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  2. Rajat Raj
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Appendix

Appendix

$${\text{Knudsen}}\;{\text{Number}}, {\text{Kn}} = \frac{{{\text{Mean}}\;{\text{Free}}\;{\text{Path}}, \lambda }}{{{\text{Characteristic}}\;{\text{Dimension}}, D}}$$
$${\text{Mean}}\;{\text{free}}\;{\text{path}}, \lambda = \frac{kT}{{\sqrt 2 \pi d^{2} P}}$$
$${\text{where}}\; k = {\text{Boltzmann}}\;{\text{constant}} = 1.38 \times 10^{ - 23 } JK^{ - 1}$$
$$T = {\text{Temperature}}, \;{\text{K}}$$
$$P = {\text{Pressure}}, {\text{Pa}}$$
$$d = {\text{Diameter}}\;{\text{of}}\;{\text{gas }}\;{\text{molecule}}$$
$${\text{Considering }}\;{\text{case}} - 2\;{\text{of}}\;{\text{micro}}\;{\text{shock}}\;{\text{tube,}}\;{\text{for}}\;{\text{the }}\;{\text{driver}}\;{\text{section}},$$
$$T = 300 \;{\text{K}}$$
$$P = 10000 {\text{Pa}}$$
$$d = {\text{Diameter }}\;{\text{of }}\;{\text{air}}\;{\text{molecule}} = 4 \times 10^{ - 10} {\text{m}}$$
$$\lambda = 5.8239 \times 10^{ - 7} {\text{m}}$$
$${\text{Characteristic dimension}}, \;D = {\text{Half }}\;{\text{of}}\;{\text{cell}}\;{\text{size}}\;{\text{adjacent}}\;{\text{to }}\;{\text{all}} = 2.7536 \times 10^{ - 4} {\text{m}}$$
$${\text{Kn}} = \frac{{5.8239 \times 10^{ - 7} }}{{2.7536 \times 10^{ - 4} }} = 0.002115$$

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Suresh, A., Raj, R. & Rajagopal, A.K. Shock wave diffraction in micro-shock tubes with sudden expansion. Acta Mech 234, 2599–2615 (2023). https://doi.org/10.1007/s00707-023-03518-1

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