# Numerical Study of Shock Wave Attenuation in Two-Dimensional Ducts Using Solid Obstacles: How to Utilize Shock Focusing Techniques to Attenuate Shock Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

#### 2.1. Overture

#### 2.1.1. Solvers in Overture

#### 2.1.2. Adaptive Mesh Refinement

#### 2.2. Logarithmic Spiral Shape

#### 2.3. Simulation Setup

**Figure 2.**Procedure to place 15 square-shaped obstacles along a logarithmic spiral. (

**a**) Mark points along the logarithmic spiral curve; (

**b**) Place a square obstacle at the focal region; (

**c**) Complete obstacle arrangement.

**Table 1.**Summary of simulated cases with details on obstacle size and total area covered by the obstacles.

Case | Name | Remark | Obstacle Size | Obstacle Area | Normalized |
---|---|---|---|---|---|

(mm) | (mm${}^{2}$) | Obstacle Area (–) | |||

1 | NC | Non-staggered cylinder | 8.8 | 973.14 | 1 |

2 | NS | Non-staggered squares | 8.8 | 1239.04 | 1.27 |

3 | NBT | Non-staggered backward triangles | 8.8 | 536.52 | 0.55 |

4 | NFT | Non-staggered forward triangles | 8.8 | 536.52 | 0.55 |

5 | SC | Staggered cylinders | 8.8 | 973.14 | 1 |

6 | SS | Staggered squares | 8.8 | 1239.04 | 1.27 |

7 | LSS | Logarithmic spiral squares | 7.45 | 832.54 | 0.86 |

8 | SLS | Logarithmic spiral squares | 6.40 | 696.32 | 0.72 |

9 | LCS | Logarithmic spiral cylinders | 7.45 | 653.87 | 0.67 |

## 3. Results and Analysis

**Figure 4.**Schlieren contour for shock wave propagation past non-staggered cylindermatrix and squares along a logarithmic spiral under (

**a**) inviscid Euler equations and (

**b**) full Navier–Stokes equations.

**Figure 5.**Dimensionless pressure, ${p}_{n}$, along the centerline for Navier–Stokes and Euler simulations at time instant $t=500$ μs for non-staggered cylinders (NC) and logarithmic spiral squares (LSS). (

**a**) NC; (

**b**) LSS.

#### 3.1. Numerical Results for Obstacle Arrays

**Figure 6.**Top to bottom: NS, NC, NBT, NFT, SS, SC, LSS, SLS and LCS schlieren contours taken at $t=500$ μs after the shock first impacts the obstacle array. The locations of the incident shock wave and the reflected shock wave are marked with arrows. Note: the first six cases, from top to bottom, are reproduced from [15].

**Figure 7.**Plot of dimensionless pressure, ${p}_{n}$, at (

**a**) upstream probe $(x,y)=(-50,0)$ and (

**b**) downstream probe $(x,y)=(169.4,0)$, as functions of time for all cases.

**Table 2.**Comparison of time-averaged overpressure ${\overline{\mathcal{P}}}_{r}$, ${\overline{\mathcal{P}}}_{t}$, shock arrival time at the upstream probe, ${t}_{pr}$, and downstream probe, ${t}_{pt}$, and the difference ${E}_{\overline{\mathcal{P}}}$. ${}^{\u2605}$ Results reported in [15].

Case | ${\overline{\mathcal{P}}}_{r}$ | ${\overline{\mathcal{P}}}_{t}$ | ${t}_{\mathrm{pr}}$ (μs) | ${t}_{\mathrm{pt}}$ (μs) | ${E}_{\overline{\mathcal{P}}}$ (%) |
---|---|---|---|---|---|

NS | 2.25 | 0.61 (0.58 ${}^{\u2605}$) | 174 | 377 | 5.2 |

NC | 2.18 | 0.54 (0.56 ${}^{\u2605}$) | 182 | 395 | −3.6 |

NBT | 2.22 | 0.62 (0.55 ${}^{\u2605}$) | 203 | 389 | 12.7 |

NFT | 2.33 | 0.38 (0.37 ${}^{\u2605}$) | 174 | 412 | 2.7 |

SS | 2.47 | 0.38 (0.24 ${}^{\u2605}$) | 174 | 406 | 58.3 |

SC | 2.18 | 0.54 (0.55 ${}^{\u2605}$) | 182 | 398 | −1.8 |

LSS | 2.06 | 0.27 (−) | 203 | 409 | – |

SLS | 1.79 | 0.43 (−) | 209 | 383 | – |

LCS | 1.62 | 0.66 (−) | 211 | 374 | – |

**Table 3.**Impulse, I, recorded by the upstream (reflected shock) and downstream (transmitted shock) probes for three time intervals. Units in Pa·s.

Case | Reflected Shock | Transmitted shock | |||||
---|---|---|---|---|---|---|---|

${I}_{100}$ | ${I}_{200}$ | ${I}_{300}$ | ${I}_{30}$ | ${I}_{60}$ | ${I}_{90}$ | ||

NS | 1.87 | 3.74 | 5.61 | 0.24 | 0.55 | 1.24 | |

NC | 1.75 | 3.51 | 5.27 | 0.18 | 0.45 | 1.11 | |

NBT | 1.77 | 3.61 | 5.46 | 0.20 | 0.50 | 1.22 | |

NFT | 1.88 | 3.88 | 5.94 | 0.16 | 0.35 | 0.92 | |

SS | 2.12 | 4.36 | 6.55 | 0.13 | 0.32 | 0.91 | |

SC | 1.75 | 3.51 | 5.27 | 0.23 | 0.50 | 1.17 | |

LSS | 0.84 | 2.13 | 4.75 | 0.08 | 0.23 | 0.76 | |

SLS | 0.72 | 1.44 | 3.60 | 0.11 | 0.32 | 0.96 | |

LCS | 0.67 | 1.31 | 2.88 | 0.25 | 0.60 | 1.29 |

#### 3.2. Effects of Incident Shock Mach Number

**Figure 8.**Dimensionless pressure, ${p}_{n}$, along the centerline of (

**a**) LSS and (

**b**) NS for ${M}_{s}=1.3,1.4$ and $1.5$ at time instant $t=500$ μs.

**Table 4.**The ratio of the peak pressure measured behind the transmitted and reflected shocks along the centerline of the shock tube at time $t=500$ μs to the initial high pressure behind the incident shock, ${q}_{pt}$ and ${q}_{pr}$, where the subscript t denotes transmitted shock and r denotes reflected shock.

Case | LSS | NS | |||||
---|---|---|---|---|---|---|---|

${M}_{s}=1.3$ | ${M}_{s}=1.4$ | ${M}_{s}=1.5$ | ${M}_{s}=1.3$ | ${M}_{s}=1.4$ | ${M}_{s}=1.5$ | ||

${q}_{pt}$ | 0.757 | 0.728 | 0.678 | 1.23 | 0.861 | 0.985 | |

${q}_{pr}$ | 1.75 | 2.04 | 2.32 | 1.50 | 1.69 | 1.90 |

#### 3.3. Effective Flow Area and Sensitivity of Obstacle Size

**Table 5.**The effective flow area. For matrix arrangements, case NC–SS, Equation (12) is used to calculate ${\u03f5}_{1}$, and for logarithmic spiral cases, LSS–SLS, Equation (13) is used to calculate ${\u03f5}_{2}$.

Case | ${\u03f5}_{1}$ or ${\u03f5}_{2}$ | ${\u03f5}_{3}$ |
---|---|---|

NC | 0.25 | – |

NS | 0.25 | – |

SC | 0.25 | – |

NBT | 0.25 | – |

NFT | 0.25 | – |

SC | 0.25 | – |

SS | 0.25 | – |

LSS | 0.0091 | 0.080 |

LCS | 0.14 | 0.21 |

SLS | 0.036 | 0.23 |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Perry, R.W.; Kantrowitz, A. The production and stability of converging shock waves. J. Appl. Phys.
**1951**, 22, 878–886. [Google Scholar] [CrossRef] - Balasubramanian, K.; Eliasson, V. Numerical investigations of the porosity effect on the shock focusing process. Shock Waves
**2013**, 23, 583–594. [Google Scholar] [CrossRef] - Eliasson, V.; Apazidis, N.; Tillmark, N. Controlling the form of strong converging shocks by means of disturbances. Shock Waves
**2007**, 17, 29–42. [Google Scholar] [CrossRef] - Kjellander, M.; Tillmark, N.; Apazidis, N. Experimental determination of self-similarity constant for converging cylindrical shocks. Phys. Fluids
**2011**, 23, 116103. [Google Scholar] [CrossRef] - Takayama, K.; Kleine, H.; Grönig, H. An experimental investigation of the stability of converging cylindrical shock waves in air. Exp. Fluids
**1987**, 5, 315–322. [Google Scholar] [CrossRef] - Takayama, K.; Onodera, O.; Hoshizawa, Y. Experiments on the stability of converging cylindrical shock waves. Theor. Appl. Mech.
**1984**, 32, 117–127. [Google Scholar] - Watanabe, M.; Onodera, O.; Takayama, K. Shock wave focusing in a vertical annular shock tube. In Shock Waves @ Marseille IV; Springer: Berlin, Germany, 1995; pp. 99–104. [Google Scholar]
- Dosanjh, D.S. Interaction of Grids with Traveling Shock Waves; NACA Technical Note 3680; Johns Hopkins University: Washington, DC, USA, 1956. [Google Scholar]
- Shi, H.; Yamamura, K. The interaction between shock waves and solid spheres arrays in a shock tube. Acta Mech. Sin.
**2004**, 20, 219–227. [Google Scholar] - Britan, A.; Karpov, A.V.; Vasilev, E.I.; Igra, O.; Ben-Dor, G.; Shapiro, E. Experimental and Numerical Study of Shock Wave Interaction with Perforated Plates. J. Fluids Eng.
**2004**, 126, 399. [Google Scholar] [CrossRef] - Britan, A.; Igra, O.; Ben-Dor, G.; Shapiro, H. Shock wave attenuation by grids and orifice plates. Shock Waves
**2006**, 16, 1–15. [Google Scholar] [CrossRef] - Naiman, H.; Knight, D. The effect of porosity on shock interaction with a rigid, porous barrier. Shock Waves
**2007**, 16, 321–337. [Google Scholar] [CrossRef] - Seeraj, S.; Skews, B.W. Dual-element directional shock wave attenuators. Exp. Therm. Fluid Sci.
**2009**, 33, 503–516. [Google Scholar] [CrossRef] - Berger, S.; Sadot, O.; Ben-Dor, G. Experimental investigation on the shock-wave load attenuation by geometrical means. Shock Waves
**2009**, 20, 29–40. [Google Scholar] [CrossRef] - Chaudhuri, A.; Hadjadj, A.; Sadot, O.; Ben-Dor, G. Numerical study of shock-wave mitigation through matrices of solid obstacles. Shock Waves
**2013**, 23, 91–101. [Google Scholar] [CrossRef] - Chaudhuri, A.; Hadjadj, A.; Sadot, O.; Glazer, E. Computational study of shock-wave interaction with solid obstacles using immersed boundary methods. Int. J. Numer. Meth. Eng.
**2012**, 89, 975–990. [Google Scholar] [CrossRef] - Henshaw, W.D.; Schwendeman, D.W. An adaptive numerical scheme for high-speed reactive flow on overlapping grids. J. Comput. Phys.
**2003**, 191, 420–447. [Google Scholar] [CrossRef] - Henshaw, W.D.; Schwendeman, D.W. Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow. J. Comput. Phys.
**2006**, 216, 744–779. [Google Scholar] [CrossRef] - Inoue, O.; Takahashi, N.; Takayama, K. Shock wave focusing in a log-spiral duct. AIAA J.
**1993**, 31, 1150–1152. [Google Scholar] [CrossRef] - Inoue, O.; Imuta, S.; Milton, B.; Takayama, K. Computational study of shock wave focusing in a log-spiral duct. Shock Waves
**1995**, 5, 183–188. [Google Scholar] [CrossRef] - Wang, C.; Eliasson, V. Shock wave focusing in water inside convergent structures. Int. J. Multiphys.
**2012**, 6, 267–282. [Google Scholar] [CrossRef] - Wang, C.; Qiu, S.; Eliasson, V. Quantitative pressure measurement of shock waves in water using a schlieren-based visualization technique. Exp. Techn.
**2013**. [Google Scholar] [CrossRef] - Milton, B.; Archer, R. Generation of implosions by area change in a shock tube. AIAA J.
**1969**, 7, 779–780. [Google Scholar] - Whitham, G. Linear and Nonlinear Waves; Wiley-Interscience: New York, NY, USA, 1974. [Google Scholar]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wan, Q.; Eliasson, V.
Numerical Study of Shock Wave Attenuation in Two-Dimensional Ducts Using Solid Obstacles: How to Utilize Shock Focusing Techniques to Attenuate Shock Waves. *Aerospace* **2015**, *2*, 203-221.
https://doi.org/10.3390/aerospace2020203

**AMA Style**

Wan Q, Eliasson V.
Numerical Study of Shock Wave Attenuation in Two-Dimensional Ducts Using Solid Obstacles: How to Utilize Shock Focusing Techniques to Attenuate Shock Waves. *Aerospace*. 2015; 2(2):203-221.
https://doi.org/10.3390/aerospace2020203

**Chicago/Turabian Style**

Wan, Qian, and Veronica Eliasson.
2015. "Numerical Study of Shock Wave Attenuation in Two-Dimensional Ducts Using Solid Obstacles: How to Utilize Shock Focusing Techniques to Attenuate Shock Waves" *Aerospace* 2, no. 2: 203-221.
https://doi.org/10.3390/aerospace2020203