# Multiparameter Optimization of Thrust Vector Control with Transverse Injection of a Supersonic Underexpanded Gas Jet into a Convergent Divergent Nozzle

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Optimization Procedure

- The parametrization of the investigated model based on ideas about the ongoing physical phenomena. This step allows one to automatically generate a multiblock or unstructured mesh in the specified area;
- The solution of a series of optimization problems for this model. The tasks are set based on the possible modes of operation of the system under study;
- The systematization of the best obtained options for the formation of equations describing the optimal state of the system in the specified range of the boundary conditions.

## 4. Geometric Model

## 5. Computational Mesh

## 6. Numerical Method

## 7. Results and Discussion

#### 7.1. Flow in the Propellant Channel

#### 7.2. Flow Regimes

#### 7.3. Distributions of Flow Quantities

#### 7.4. Optimization Results

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

A | Area |

${A}_{a}$ | Area of outlet section |

${A}_{*}$ | Area of critical section |

${d}_{j}$ | Diameter of injection nozzle |

${d}_{*}$ | Critical diameter |

e | Total energy |

F | Flux vector |

h | Channel radius |

k | Turbulent kinetic energy |

l | Distance between inlet section and secondary nozzle |

L | Length of channel |

n | Pressure ratio |

p | Pressure |

Pr | Prandtl number |

${\mathrm{Pr}}_{t}$ | Turbulent Prandtl number |

q | Heat flux |

Q | Vector of conservative variables |

Re | Reynolds number |

t | Time |

T | Temperature |

${v}_{x},{v}_{y},{v}_{z}$ | Velocity components |

${v}_{w}$ | Injection velocity |

$x,y,z$ | Cartesian coordinates |

$\alpha $ | Angle of nozzle inclination |

$\gamma $ | Ratio of specific heat capacities |

$\epsilon $ | Dissipation rate |

$\lambda $ | Thermal conductivity |

$\mu $ | Dynamic viscosity |

$\rho $ | Density |

${\tau}_{ij}$ | Components of viscous stress tensor |

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**Figure 4.**Computational mesh (

**a**), mesh near injection port (

**b**) and mesh in cross section of the injection port (

**c**) with ${l}_{j}=0.3$, ${\alpha}_{j}={90}^{\circ}$, ${d}_{2}/{d}_{1}=1$, ${A}_{*}/{A}_{a}=1$.

**Figure 5.**Computational mesh (

**a**), mesh near injection port (

**b**) and mesh in cross section of the injection port (

**c**) with ${l}_{j}=0.5$, ${\alpha}_{j}={110}^{\circ}$, ${d}_{2}/{d}_{1}=1$, ${A}_{*}/{A}_{a}=1$.

**Figure 6.**Computational mesh (

**a**), mesh near injection port (

**b**) and mesh in cross section of the injection port (

**c**) with ${l}_{j}=0.7$, ${\alpha}_{j}={130}^{\circ}$, ${d}_{2}/{d}_{1}=1$, ${A}_{*}/{A}_{a}=1$.

**Figure 7.**Computational mesh (

**a**), mesh near injection port (

**b**) and mesh in cross section of the injection port (

**c**) with ${l}_{j}=0.7$, ${\alpha}_{j}={130}^{\circ}$, ${d}_{2}/{d}_{1}=4.5$, ${A}_{*}/{A}_{a}=1$.

**Figure 8.**Computational mesh (

**a**), mesh near injection port (

**b**) and mesh in cross section of the injection port (

**c**) with ${l}_{j}=0.7$, ${\alpha}_{j}={130}^{\circ}$, ${d}_{2}/{d}_{1}=0.6$, ${A}_{*}/{A}_{a}=1$.

**Figure 9.**Computational mesh (

**a**), mesh near injection port (

**b**) and mesh in cross section of the injection port (

**c**) with ${l}_{j}=0.7$, ${\alpha}_{j}={130}^{\circ}$, ${d}_{2}/{d}_{1}=0.6$, ${A}_{*}/{A}_{a}=0.1$.

**Figure 11.**Sound line shape at various pressure drops at nozzle pressure ratios of 2 (line 1), 2.5 (Line 2), 3 (Line 3) and 4 (Line 4).

**Figure 12.**Flow regimes in the channel with fluid injection and comparison of the velocity profiles predicted with the inviscid model (dashed lines), the viscous model (solid lines) and the experiments from [2] (symbols •).

**Figure 13.**Contours of the Mach number for ${\alpha}_{j}={90}^{\circ}$ (

**a**), ${100}^{\circ}$ (

**b**), ${130}^{\circ}$ (

**c**) and ${150}^{\circ}$ (

**d**).

**Figure 14.**Contours of static pressure for: $n=0.4$, ${p}_{n}=0.5$ atm (

**a**); $n=0.2$, ${p}_{n}=1.0$ atm (

**b**); $n=0.2$, ${p}_{n}=1.5$ atm (

**c**); $n=0.1$, ${p}_{n}=2.0$ atm (

**d**).

**Figure 16.**Distributions of static pressure for: $n=0.4$, ${p}_{n}=0.5$ atm (

**a**); $n=0.2$, ${p}_{n}=1.0$ atm (

**b**); $n=0.2$, ${p}_{n}=1.5$ atm (

**c**); $n=0.1$, ${p}_{n}=2.0$ atm (

**d**).

**Figure 17.**Length of the recirculation region (

**a**) and the depth of jet penetration into the main flow (

**b**).

**Figure 19.**Dependance of the thrust coefficient (

**a**) and thrust (

**b**) on the angle and location of jet injection.

Parameter | Location of Injection | Angle of Injection | Shape of Nozzle | Area Ratio |
---|---|---|---|---|

${\mathit{l}}_{\mathit{j}}/{\mathit{L}}_{\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{j}}$, deg | ${\mathit{d}}_{2}/{\mathit{d}}_{1}$ | ${\mathit{A}}_{*}/{\mathit{A}}_{\mathit{a}}$ | |

Interval | 0.1–0.9 | 30–160 | 0.5–5.5 | 0.1–1 |

Mesh | ${\mathit{l}}_{\mathit{j}}$ | ${\mathit{\alpha}}_{\mathit{j}}$, deg | ${\mathit{d}}_{2}/{\mathit{d}}_{1}$ | ${\mathit{A}}_{*}/{\mathit{A}}_{\mathit{a}}$ | Number of Nodes, mln | yplus |
---|---|---|---|---|---|---|

Mesh 1 | 0.3 | 90 | 1 | 1 | 1.2 | 1.9 |

Mesh 2 | 0.5 | 110 | 1 | 1 | 1.32 | 1.92 |

Mesh 3 | 0.7 | 130 | 1 | 1 | 1.45 | 1.92 |

Mesh 4 | 0.7 | 130 | 4.5 | 1 | 1.62 | 2.1 |

Mesh 5 | 0.7 | 130 | 0.6 | 1 | 1.92 | 1.82 |

Mesh 6 | 0.7 | 130 | 0.6 | 0.1 | 2.02 | 2.21 |

Shape of Outlet Section | Ratio of Axis | Thrust Coefficient |
---|---|---|

Ellipse | 0.88 | 3.05 |

Circle | 1.0 | 3.12 |

Ellipse | 2.0 | 3.18 |

Ellipse | 4.6 | 3.32 |

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**MDPI and ACS Style**

Emelyanov, V.; Yakovchuk, M.; Volkov, K.
Multiparameter Optimization of Thrust Vector Control with Transverse Injection of a Supersonic Underexpanded Gas Jet into a Convergent Divergent Nozzle. *Energies* **2021**, *14*, 4359.
https://doi.org/10.3390/en14144359

**AMA Style**

Emelyanov V, Yakovchuk M, Volkov K.
Multiparameter Optimization of Thrust Vector Control with Transverse Injection of a Supersonic Underexpanded Gas Jet into a Convergent Divergent Nozzle. *Energies*. 2021; 14(14):4359.
https://doi.org/10.3390/en14144359

**Chicago/Turabian Style**

Emelyanov, Vladislav, Mikhail Yakovchuk, and Konstantin Volkov.
2021. "Multiparameter Optimization of Thrust Vector Control with Transverse Injection of a Supersonic Underexpanded Gas Jet into a Convergent Divergent Nozzle" *Energies* 14, no. 14: 4359.
https://doi.org/10.3390/en14144359