# Simulation of Underwater Explosions Initiated by High-Pressure Gas Bubbles of Various Initial Shapes

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## Abstract

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## 1. Introduction

## 2. Mathematical Models and Methods

#### 2.1. Governing Equations of the Flow Field

#### 2.2. Gas–Water Interface Treatment

- (I)
- On one side of the interface, an adjacent point closest to the interface is selected. Then, a corresponding point on the other side of the interface is found that has a minimum dot product of the normal vectors. Then, a pair of Riemann points is constructed.
- (II)
- Use the parameter values (density, normal velocity, pressure) of the Riemann points on both sides of the interface to construct a Riemann problem. Then, an Approximate Riemann Problem Solver (ARPS) [39,40] is used to predict the interface values such as the density and the pressure. This ARPS is developed based on a double-shock characteristics structure, which can achieve a second-order accuracy in solving the gas–water Riemann problem [39]. Then, the predicted interface values (pressure, normal velocity and entropy) are taken to replace the values of the adjacent point on the real fluid side.
- (III)
- The above processes (I) and (II) are repeated until all the adjacent points on one side of the interface are re-defined. Then, the values at these adjacent points are extrapolated to the other side of the interface where the “ghost points” are distributed, by iteratively solving the following extrapolation equation [26]:$$\frac{\partial I}{\partial t}\pm ({n}_{x}\frac{\partial I}{\partial x}+{n}_{y}\frac{\partial I}{\partial y}+{n}_{z}\frac{\partial I}{\partial z})=0.$$
- (IV)
- Advancement of the flow field for one time step for both sides of the interface, so that two sets of the solutions to the Euler equations are obtained.
- (V)
- Advancement of the location of the interface for one time step by solving the level-set equation:$$\frac{\partial \varphi}{\partial t}+u\frac{\partial \varphi}{\partial x}+v\frac{\partial \varphi}{\partial y}+w\frac{\partial \varphi}{\partial z}=0.$$This level-set method has proved to be capable of capturing the multi-medium interface successfully and also the interaction between a shock wave and the interface [33]. However, when the simulation proceeds, the level-set function $\varphi $ tends to be distorted, and thus cannot maintain the property as a signed distance function. Therefore, a re-initialization process is needed [41], which is performed after every time step in the present study. After locating the new position of the interface, we use the updated level-set function to determine which solution obtained in process (IV) should be chosen.
- (VI)
- Proceed to the next time step.

#### 2.3. Initial and Boundary Conditions

## 3. Results and Discussion

#### 3.1. Scenario 1 (Sphere)

#### 3.2. Scenario 2 (Cylinder)

#### 3.3. Scenario 3 (Cuboidal)

#### 3.4. Scenario 4 (Bullet-Like)

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

- Scenario 1 (Sphere): $\varphi =0.1-\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$.
- Scenario 2 (Cylinder):
$d=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$ z $\varphi $ $d\ge 0.1$ $-0.067\le z<0.067$ $\varphi =0.1-d$ $d<0.1$ $z\ge 0.067$ $\varphi =0.067-z$ $d<0.1$ $z<-0.067$ $\varphi =z+0.067$ $d\ge 0.1$ $z\ge 0.067$ $\varphi =-\sqrt{{(\sqrt{{x}^{2}+{y}^{2}}-0.1)}^{2}+{(z-0.067)}^{2}}$ $d\ge 0.1$ $z<-0.067$ $\varphi =-\sqrt{{(\sqrt{{x}^{2}+{y}^{2}}-0.1)}^{2}+{(z+0.067)}^{2}}$ $d<0.1$ $-0.067\le z<0.067$ $\varphi $ = $\mathrm{min}\left\{|d-0.1|,|z-0.067|,|z+0.067|\right\}$ - Scenario 3 (Cuboidal):
$\mathit{x}$ $\mathit{y}$ $\mathit{z}$ $\mathit{\varphi}$ $x<-0.1$ $y<-0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-(-0.1))}^{2}+{(z-0.052)}^{2}}$ $x<-0.1$ $y\ge 0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-0.1)}^{2}+{(z-0.052)}^{2}}$ $x\ge 0.1$ $y<-0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-(-0.1))}^{2}+{(z-0.052)}^{2}}$ $x\ge 0.1$ $y\ge 0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-0.1)}^{2}+{(z-0.052)}^{2}}$ $x<-0.1$ $y<-0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-(-0.1))}^{2}+{(z-(-0.052))}^{2}}$ $x<-0.1$ $y\ge 0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-0.1)}^{2}+{(z-(-0.052))}^{2}}$ $x\ge 0.1$ $y<-0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-(-0.1))}^{2}+{(z-(-0.052))}^{2}}$ $x\ge 0.1$ $y\ge 0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-0.1)}^{2}+{(z-(-0.052))}^{2}}$ $x<-0.1$ $y<-0.1$ $-0.052\le z<0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-(-0.1))}^{2}}$ $x<-0.1$ $y\ge 0.1$ $-0.052\le z<0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-0.1)}^{2}}$ $x\ge 0.1$ $y<-0.1$ $-0.052\le z<0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-(-0.1))}^{2}}$ $x\ge 0.1$ $y\ge 0.1$ $-0.052\le z<0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-0.1)}^{2}}$ $-0.1\le x<0.1$ $y<-0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-0.052)}^{2}}$ $-0.1\le x<0.1$ $y\ge 0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-0.052)}^{2}}$ $x<-0.1$ $-0.1\le y<0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-0.052)}^{2}}$ $x\ge 0.1$ $-0.1\le y<0.1$ $z\ge 0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-0.052)}^{2}}$ $-0.1\le x<0.1$ $y<-0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-(-0.052))}^{2}}$ $-0.1\le x<0.1$ $y\ge 0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-(-0.052))}^{2}}$ $x<-0.1$ $-0.1\le y<0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-(-0.1))}^{2}+{(y-(-0.052))}^{2}}$ $x\ge 0.1$ $-0.1\le y<0.1$ $z<-0.052$ $\varphi $ = $-\sqrt{{(x-0.1)}^{2}+{(y-(-0.052))}^{2}}$ $x<-0.1$ $-0.1\le y<0.1$ $-0.052\le z<0.052$ $\varphi $ = $x-(-0.1)$ $x\ge 0.1$ $-0.1\le y<0.1$ $-0.052\le z<0.052$ $\varphi $ = $0.1-x$ $-0.1\le x<0.1$ $y<-0.1$ $-0.052\le z<0.052$ $\varphi $ = $y-(-0.1)$ $-0.1\le x<0.1$ $y\ge 0.1$ $-0.052\le z<0.052$ $\varphi $ = $0.1-y$ $-0.1\le x<0.1$ $-0.1\le y<0.1$ $z<-0.052$ $\varphi $ = $z-(-0.052)$ $-0.1\le x<0.1$ $-0.1\le y<0.1$ $z\ge 0.052$ $\varphi $ = $0.052-z$ $-0.1\le x<0.1$ $-0.1\le y<0.1$ $-0.052\le z<0.052$ $\varphi $ = $\mathrm{min}\left\{\right|x-0.1|,|y-0.1|,|z-0.052|,$ $|x+0.1|,|y+0.1|,|z-(-0.052)|\}$ - Scenario 4 (Bullet-like):
x $d=\sqrt{{y}^{2}+{z}^{2}}$ $\varphi $ $x<-0.071$ $d<-x-0.071$ $\varphi =-\sqrt{{(x+0.071)}^{2}+{d}^{2}}$ ($d\ge -x-0.071$) .AND. ($d\ge x+0.071$) .AND. ($d<-x+0.271$) $\varphi =-0.71d+0.71(x+0.071)$ ($d\ge -x-0.271$) .AND. ($d\ge x+0.071$) $\varphi =-\sqrt{{(d-0.171)}^{2}+{(x-0.1)}^{2}}$ ($d\ge 0.171$) .AND. ($d<x+0.071$) $\varphi =-\sqrt{{(d-0.171)}^{2}+{(x-0.1)}^{2}}$ $x\ge 0.1$ $0.1\le d<0.171$ $\varphi =-\sqrt{{(x-0.1)}^{2}}$ $d\ge -x+0.2$ $d<0.1$ $\varphi =-\sqrt{{(d-0.1)}^{2}+{(x-0.1)}^{2}}$ $d<-x+0.2$ $d<x$ $\varphi =0.71d-0.71x$ $x<0.1$ ($d<x+0.071)$ .AND. ($d\ge x$) $\varphi $ = $\mathrm{min}\left\{\right|x-0.1|,|0.71d-0.71(x+0.071)|,$ $|0.71d-0.71x|\}$

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**Figure 1.**The initial setup of the underwater explosion (UNDEX) simulation with a high-pressure gas bubble in a spherical shape. (

**a**) the initial distribution of the pressure, shown in a slice plot (surfaces $y=0$ and $z=0$). Two recording points, Point 1 with the position (0.5, 0.0, 0.0) and Point 2 (0.0, 0.0, 0.5) were placed at the boundaries of the computational domain; (

**b**) a 2-D plot of the initial level-set function $\varphi $ in the x–z surface ($y=0$).

**Figure 2.**The temporal evolution of the pressure field for an initially spherical bubble in slice plots ($y=0.0$). Note that the range of the legend in each figure is different for a better display.

**Figure 3.**The pressure history recorded at the boundaries of the computational domain for an initially spherical bubble (Scenario 1). Point 1 is located at the position (0.5, 0.0, 0.0), and the position of Point 2 is (0.0, 0.0, 0.5).

**Figure 4.**The initial setup of the UNDEX simulation with a high-pressure gas bubble of a cylindrical shape. The configurations of the figure are similar to Figure 1.

**Figure 5.**The temporal evolution of the pressure field for an initially cylindrical bubble in slice plots ($y=0.0$). Note that the range of the legend in each figure is different for a better display.

**Figure 6.**The pressure history recorded at the boundaries of the computational domain for an initially cylindrical bubble (Scenario 2).

**Figure 7.**The initial setup of the UNDEX simulation with a high-pressure gas bubble in a cuboidal shape. The configurations of the figure are similar to Figure 1.

**Figure 8.**The temporal evolution of the pressure field for an initially cuboidal bubble in slice plots ($y=0.0$).

**Figure 9.**The pressure history recorded at the boundaries of the computational domain for an initially cuboid-shaped bubble (Scenario 3).

**Figure 10.**The initial setup of the UNDEX simulation with a high-pressure gas bubble in a bullet-like shape. The configurations of the figure are similar to Figure 1.

**Figure 11.**The temporal evolution of the pressure field for an initially bullet-like bubble in slice plots ($y=0.0$). Note that in (

**a**,

**b**), the maximum value of the legend is 10,000 and 4000, respectively.

**Figure 12.**The pressure history recorded at the boundaries of the computational domain for an initially bullet-like bubble (Scenario 4).

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**MDPI and ACS Style**

Cao, L.; Fei, W.; Grosshans, H.; Cao, N.
Simulation of Underwater Explosions Initiated by High-Pressure Gas Bubbles of Various Initial Shapes. *Appl. Sci.* **2017**, *7*, 880.
https://doi.org/10.3390/app7090880

**AMA Style**

Cao L, Fei W, Grosshans H, Cao N.
Simulation of Underwater Explosions Initiated by High-Pressure Gas Bubbles of Various Initial Shapes. *Applied Sciences*. 2017; 7(9):880.
https://doi.org/10.3390/app7090880

**Chicago/Turabian Style**

Cao, Le, Wenli Fei, Holger Grosshans, and Nianwen Cao.
2017. "Simulation of Underwater Explosions Initiated by High-Pressure Gas Bubbles of Various Initial Shapes" *Applied Sciences* 7, no. 9: 880.
https://doi.org/10.3390/app7090880