# Two-Dimensional Geometrical Shock Dynamics for Blast Wave Propagation and Post-Shock Flow Effects

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

**original GSD model**consisting of Equations (1) and (2), an explicit expression for $\frac{{A}^{\prime}}{A}=\frac{1}{A}\frac{dA}{dn}$ is needed. In fact, ${A}^{\prime}/A$ is the curvature of the shock front, $\kappa $. The proof starts from one assumption of GSD theory, that the rays are normal to the surface, which leads to

## 3. Application of GSD to an Expanding Blast Wave

#### 3.1. Original GSD Model

#### 3.2. First-Order Complete GSD Model

**first-order complete GSD model**is obtained as follows, which contains three coupled ordinary differential equations for three unknowns, namely, $\mathit{x}\left(t\right)$, $M\left(t\right)$, and ${Q}_{1}\left(t\right)$:

#### 3.3. Modified GSD Model

**modified GSD model**[10] is obtained as follows

#### 3.4. Point-Source GSD Model

**point-source GSD model (PGSD)**, will be introduced next. Considering that Bach and Lee’s analytical solution to point-blast propagation [25], outlined in Appendix C, already describes an accurate blast behavior, a modification to the original GSD model to include this essential blast property was proposed by Yoo and Butler [33]. Using Taylor’s similarity law [31] and Equation (11) together with expressions for shock front speed, ${\dot{R}}_{s}=\frac{d{R}_{s}}{dt}={a}_{0}M$, and shock front acceleration, ${\ddot{R}}_{s}={a}_{0}\frac{dM}{dt}$, the following expression is obtained

#### 3.5. Post-Shock Flow Effect

#### 3.6. Interaction of Two Cylindrical Blast Waves Using PGSD

## 4. Modifying PGSD Using Shock–Shock Approximate Theory

#### 4.1. Shock–Shock Approximate Theory for Planar Shock Diffraction by a Wedge

#### 4.2. Shock–Shock Approximate Theory for Cylindrical Shock Diffraction by a Wedge with Continuously Decreasing Tilt Angle

#### 4.3. Point-Source GSD Model with the Shock–Shock Approximate Theory

**PGSD model with the shock–shock approximate theory (PGSDSS)**. This forms an alternative framework for studying the symmetric interaction between initially separated blasts.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFL | Courant–Friedrichs–Lewy |

GPU | Graphics processing unit |

GSD | Geometrical shock dynamics |

PGSD | Point-source GSD |

PGSDSS | Point-source GSD with the shock–shock approximate theory |

CPU | Central processing unit |

## Appendix A. Grid Independence Study for Two-Dimensional Simulations Using the Euler Equations

#### Appendix A.1

**Figure A1.**Schematic illustration of the 2D simulation domain for the grid independence study with locations of probes A and B denoted by orange diamonds. The initial blast wave radius was ${R}_{0}=1.5$ mm.

**Figure A2.**Time history of pressure recorded at probe A, which is located halfway between the explosion center and the wall, and a zoomed in view of the pressure during the initial shock passage.

## Appendix B. Taylor’s Similarity Law

#### Appendix B.1

## Appendix C. Bach and Lee’s Analytical Solution

#### Appendix C.1

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**Figure 1.**M-R plots of the propagation of a single cylindrical blast in air. Initial conditions: ${E}_{0}=8000$ J/m for the Euler and analytical solutions, and initial conditions ${R}_{0}=10$ mm with either ${M}_{01}=11.81$ or ${M}_{02}=9.76$ for GSD solutions.

**Figure 2.**Geometrical terms and post-shock flow terms are plotted as functions of the blast radius. Initial conditions: ${R}_{0}=10$ mm and ${M}_{0}=9.76$ for the first-order complete GSD and modified GSD solutions.

**Figure 3.**M-R plots of the propagation of a single cylindrical blast in air for different GSD models. Initial conditions: ${E}_{0}=8000$ J/m for the analytical solution; ${R}_{0}=10$ mm and ${M}_{0}=9.76$ for the GSD, $1\mathrm{st}$-order complete GSD, modified GSD, and PGSD solutions.

**Figure 4.**Schematic illustration of the experiments of [34], in which the exploding wire centers (represented by red stars) are located 60 mm apart from each other. The wedge angle is represented by ${\theta}_{w}$.

**Figure 5.**Ratio of maximum pressure at the Mach stem, ${P}_{\mathrm{m}}$, to ambient pressure, ${P}_{\mathrm{a}}$, as a function of time. Modified GSD data are reproduced from [10], which also provided the initial conditions to the current PGSD simulations referred to as I.C.1.

**Figure 6.**Time history of the shock front as a function of the radius using initial conditions I.C.2.

**Figure 7.**Ratio of maximum pressure at the Mach stem, ${P}_{m}$, to ambient pressure, ${P}_{a}$, as a function of time. Experimental data reproduced from [34] with permission from Springer.

**Figure 8.**Schematic illustration of Whitham’s application of GSD to diffraction of a planar shock by a straight wedge. Triple points are represented by red dots.

**Figure 9.**Schematic illustration of the diffraction of a planar shock interaction with a straight surface. Triple points are represented by red dots.

**Figure 10.**Schematic illustration of the diffraction of a cylindrical shock interaction with a straight surface. Triple points are represented by red dots.

**Figure 11.**Comparison of the trajectory of the triple point for the interaction between two identical cylindrical blasts from the Euler (solid line) and PGSDSS (dashed line with circles) solutions. Initial conditions: ${E}_{0}=10,000$ J/m for the Euler solution; ${R}_{0}=5$ mm and ${M}_{0}=26.7$ for the PGSDSS solution.

**Figure 12.**PGSDSS results showing the evolution of the shock front formed by the interaction between two identical cylindrical blast waves. Only half of the shock fronts shown with triple points represented by red circles, and the explosion center indicated by a red star. Initial conditions: ${R}_{0}=5$ mm and ${M}_{0}=26.7$.

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**MDPI and ACS Style**

Liu, H.; Eliasson, V.
Two-Dimensional Geometrical Shock Dynamics for Blast Wave Propagation and Post-Shock Flow Effects. *Aerospace* **2023**, *10*, 838.
https://doi.org/10.3390/aerospace10100838

**AMA Style**

Liu H, Eliasson V.
Two-Dimensional Geometrical Shock Dynamics for Blast Wave Propagation and Post-Shock Flow Effects. *Aerospace*. 2023; 10(10):838.
https://doi.org/10.3390/aerospace10100838

**Chicago/Turabian Style**

Liu, Heng, and Veronica Eliasson.
2023. "Two-Dimensional Geometrical Shock Dynamics for Blast Wave Propagation and Post-Shock Flow Effects" *Aerospace* 10, no. 10: 838.
https://doi.org/10.3390/aerospace10100838