# Hugoniot Relation for a Bow-Shaped Detonation Wave Generated in RP Laser Propulsion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Numerical Models

#### 2.1.1. Analytical Model Reproducing the Experiment Conditions

_{2}laser (10 J/pulse maximum energy, 10.6 µm wavelength) was used. It had a history of laser power and cumulative energy, as shown in Figure 1. The laser cross section has the shape of a 30 × 30 mm square, with different transverse modes in the horizontal and vertical directions. Therefore, the beam was focused by two mirrors, as shown in Figure 2. An aluminum target was placed at the focal point. Then an LSD was generated on it. Numerical calculations reproduced the experiment conditions using these characteristics of the laser and focusing optics. Additionally, the propagation velocity and shape of ionization wavefront and the local laser intensity were given as input parameters based on the measurements. The details will be described later.

#### 2.1.2. Ionization Wavefront Propagation Velocity

^{2}, as portrayed in Figure 3.

#### 2.1.3. Ionization Wavefront Shape

#### 2.1.4. Laser Intensity Calculation

#### 2.2. Governing Equations

^{2}based on the results of experimentation.

#### 2.3. Calculation Conditions

## 3. CFD Code Validation Based on Pressure History Calculated and Measured Values

#### 3.1. Validation Methods

#### 3.2. Validation Results

## 4. Calculation Results by Two-Dimensional Axisymmetric CFD

#### 4.1. Calculation Results of the Propagation History of LSD

#### 4.2. Evaluating Effects of Radial Flow from the Central Axis of LSD

#### 4.3. Hugoniot Relation Considering Radial Flow from the LSD Central Axis

_{1}> 1.

## 5. Discussion

#### 5.1. Differences in the Radial Flows of Mass, Momentum, and Enthalpy

#### 5.2. Comparison of CJ Velocity Calculated Using the Two-Dimensional Axisymmetric CFD with the Measured Propagation Velocity

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**History of laser power and cumulative energy of the TEA CO

_{2}laser used for experimentation.

**Figure 3.**Relation between the laser intensity and measured propagation velocity on the central axis [7].

**Figure 5.**Spatial distribution of (

**a**) actual profile and (

**b**) axisymmetric profile. Here, x and y, respectively, represent the Gaussian and Top-hat directions. (

**c**) Variation of each profile with respect to distance from the center. (

**d**) Cross-sectional shape of each profile at intensity of 0.8.

**Figure 6.**Calculation results and fitting curves of shock wavefront propagation history in the z and r directions. Solid lines represent calculated values. Dashed lines represent fitting curves.

**Figure 8.**Comparison of pressure histories as: (

**a**) calculated and measured and as (

**b**) calculated using the filter and measured.

**Figure 11.**Variation of the ratio of the radial flow of mass, momentum, and enthalpy according to the peak laser intensity for different input energies.

**Figure 12.**p—v diagram for: (

**a**) ${S}_{\mathrm{peak}}=400\mathrm{GW}/{\mathrm{m}}^{2}$, (

**b**) ${S}_{\mathrm{peak}}=300\mathrm{GW}/{\mathrm{m}}^{2}$, and (

**c**) ${S}_{\mathrm{peak}}=200\mathrm{GW}/{\mathrm{m}}^{2}$. The changes on the p—v diagram of the CFD calculation results are written from state 1 to state 2.

**Figure 13.**Spatial distribution of: (

**a**) density, (

**b**) axial velocity, and (

**c**) total enthalpy in shock wave fixed coordinates at ${S}_{\mathrm{peak}}=300\mathrm{GW}/{\mathrm{m}}^{2}$.

**Figure 14.**(

**a**) Measured propagation velocity and CJ velocity considering the effects of radial flow and (

**b**) considering the error in ${\eta}_{\mathrm{bw}}.$

**Table 1.**Peak pressures and plateau pressures of pressure histories as measured and calculated using the filter.

Peak Pressure/atm | Plateau Pressure/atm | |
---|---|---|

Calculated value | 27.04 | 2.43 |

Measured value | 28.57 ± 2.21 | 2.27 ± 0.44 |

**Table 2.**Average values in the range of 200–400 GW/m

^{2}of ${\eta}_{\mathrm{mass}}$, ${\eta}_{\mathrm{momentum}}$, and ${\eta}_{\mathrm{enthalpy}}$ in a steady state.

${\mathit{\eta}}_{\mathit{m}\mathit{a}\mathit{s}\mathit{s}}$ | ${\mathit{\eta}}_{\mathit{m}\mathit{o}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{u}\mathit{m}}$ | ${\mathit{\eta}}_{\mathit{e}\mathit{n}\mathit{t}\mathit{h}\mathit{a}\mathit{l}\mathit{p}\mathit{y}}$ | |
---|---|---|---|

Average value | 0.82 | 0.13 | 0.17 |

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**MDPI and ACS Style**

Sugamura, K.; Kato, K.; Komurasaki, K.; Sekine, H.; Itakura, Y.; Koizumi, H.
Hugoniot Relation for a Bow-Shaped Detonation Wave Generated in RP Laser Propulsion. *Aerospace* **2023**, *10*, 102.
https://doi.org/10.3390/aerospace10020102

**AMA Style**

Sugamura K, Kato K, Komurasaki K, Sekine H, Itakura Y, Koizumi H.
Hugoniot Relation for a Bow-Shaped Detonation Wave Generated in RP Laser Propulsion. *Aerospace*. 2023; 10(2):102.
https://doi.org/10.3390/aerospace10020102

**Chicago/Turabian Style**

Sugamura, Kenya, Kyohei Kato, Kimiya Komurasaki, Hokuto Sekine, Yuma Itakura, and Hiroyuki Koizumi.
2023. "Hugoniot Relation for a Bow-Shaped Detonation Wave Generated in RP Laser Propulsion" *Aerospace* 10, no. 2: 102.
https://doi.org/10.3390/aerospace10020102