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Computational study of shock wave control by pulse energy deposition

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Abstract

We have developed a computational code based on the axisymmetric Navier–Stokes equations with thermochemical kinetics for assessing wave drag reduction and other effects in pulse-energy deposition ahead of a bow shock by means of full simulations from generation of a laser-induced blast wave to interaction with the bow shock. Thermochemical nonequilibrium computations can reproduce the process of blast wave formation with laser ray tracing, and the computed low-density core inside the blast wave has a teardrop-like shape, depending on the laser input condition. The flowfield interacting with a bow shock formed in Mach 5 flow was computed. The result suggests that the shape of the low-density core affects the resultant wave drag, and parameters of an incident laser beam should be taken into account in exploring the optimal condition of the proposed wave-drag scheme.

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Acknowledgments

This study was supported by the Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and a Grant-in-Aid from the Japan Society for the Promotion of Science (JSPS) Fellows. The authors appreciate the anonymous reviewers who provided valuable comments for revision.

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Author notes

  1. M. Tate

    Present address: JFE Steel Corporation, 1-1 Ohgishima, Kawasaki-ku, Kawasaki, 210-0868, Japan

Authors and Affiliations

  1. Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8579, Japan

    N. Ohnishi, M. Tate & Y. Ogino

Authors
  1. N. Ohnishi
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  2. M. Tate
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  3. Y. Ogino
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Corresponding author

Correspondence to N. Ohnishi.

Additional information

Communicated by E. Timofeev.

Appendices

Appendix: A validation of laser-induced blast wave

Laser-induced blast waves have been studied in many experiments in terms of shock wave propagation and plasma-surface evolution. Mori et al. [32] measured the positions of the shock wave front and the plasma front in their laser blast experiment using a CO\(_2\) laser. The experiment setup was similar to our computational conditions in this paper, so our simulation of laser-induced blast waves in air can be validated by comparison with their measurements.

Axisymmetric simulation conditions were set by following their experiment. The input laser profile consists of a 120-ns pulse part and an exponential decay part like Fig. 2. Input energy is 10 J with maximum power of 24 WM and is numerically divided into 160 rays. The incident laser is focused with 10.6 \(\upmu \)m wavelength and \(f\)-number of 2.2. The computational domain covers a 32-mm\(\times \)12-mm rectangular area with 161\(\times \)61 grids.

Figure 8 depicts the time evolution of the shock front and the plasma front toward the incident laser at the axis of symmetry, compared with experiment data [32]. The focal position is adjusted for a later shock position, since the precise focal point is difficult to obtain in the experiment. In the simulation, the shock was detected by density jump, and the plasma front was detected by vibrational temperature jump. Agreement of shock speed and plasma evolution with the experiment, especially in the later time, suggests that laser-energy deposition is predicted well by the developed code.

In the very early time, plasma development is slightly faster than the experiment. This may be due to the fact that the initial breakdown process cannot be described by the present code. However, detailed prediction in the early time is not necessary in the context of the interaction with a bow shock wave, because the blast wave develops sufficiently in the time of interest and approaches the measured trajectory at a later time.

Appendix: B Grid dependency of the interaction between bow shock and blast wave

In order to check the grid dependency of the flowfield induced by interaction between the bow shock and the blast wave, three different grids were prepared to represent the computational domain used in Sect. 4. The number of original grids is 181\(\times \)151, and those of the two denser grids are 261\(\times \)226 and 341\(\times \)301.

Fig. 8
figure 8

Time evolution of the shock front and the plasma front in the laser-induced blast wave obtained by our code (lines) and past experiment [32] (symbols)

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Computed drag and reduction energy were compared among simulations with the three grids. In Fig. 9a, the time variation of drag is almost the same among them, while some differences appear in small oscillations of drag. The minute structure induced by the interaction in the shock layer depends on grid density and may produce such differences. However, large oscillations and their amplitudes agree among the different grids.

Fig. 9
figure 9

Time histories of drag and reduction energy with three different grids. a Drag. b Reduction energy

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Small oscillations in drag history are not important in drag-reduction analysis. Actually, reduction energy results in similar values within 5% difference (Fig. 9b). Since the main objective is to present the effects of realistic plasma deposition and heating environment in the drag reduction scheme of pulse energy deposition, an accurate prediction with reduced numerical error less than the above level is beyond the scope of the present paper.

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Ohnishi, N., Tate, M. & Ogino, Y. Computational study of shock wave control by pulse energy deposition. Shock Waves 22, 521–531 (2012). https://doi.org/10.1007/s00193-012-0407-6

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