# Numerical Investigation on the Thrust Vectoring Performance of Bypass Dual Throat Nozzle

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Nozzle Configuration

_{b}/h

_{t}) from 0.13 to 0.25. For NPR = 2–10, all these configurations are studied to investigate the thrust vectoring performance for under-expanded and over-expanded jets. A total of 160 simulation cases were conducted to analyze how these selected variables affected BDTN’s performance.

#### FTV Performance Parameters Considered

## 3. Computational Method

#### 3.1. Governing Equation

#### 3.2. Boundary Conditions

#### 3.3. Grid Independence Test

^{+}= 5 value is selected for the meshes using grid stretching with 0.3 mm as the initial spacing. As per the Quality Metric Criteria, the minimum quality of mesh obtained is greater than 0.95. On all three grids, the static pressure distribution is monitored. The static pressure distribution for all three grids on the lower wall of the nozzle is presented in Figure 3a. In Figure 3a, the “0” location is the throat of the nozzle and the location after “0” provides the description of the shock position in the bottom wall of the nozzle to the nozzle’s exit. The results indicate the differential values in static pressure distribution for the coarse grid and the other two grids. However, static pressure distributions for medium and fine grids demonstrated a difference of 1%. As a result, the medium mesh was selected to conduct this study.

#### 3.4. Computational Validation

## 4. Results and Discussion

#### 4.1. Effect of Bypass Angle and Bypass Width

#### 4.1.1. BDTN Performance under Different NPR

#### 4.1.2. BDTN Performance under Different Bypass Width

#### 4.1.3. Combined Effect of Bypass Width and Bypass Angle

_{b}= 2–5 mm. Increasing the bypass angle and decreasing the bypass width resulted in a decrease in vectoring angle. At ${\mathrm{h}}_{\mathrm{b}}$ = 5 mm, Model 1 had the highest vectoring angle. With h

_{b}= 2.6 mm, model 6 showed the lowest vectoring angle. Vectoring efficiency, on the other hand, revealed a decreasing trend. Model 6 with ${\mathrm{h}}_{\mathrm{b}}$ = 5 mm showed the lowest vectoring efficiency, while model 1 with ${\mathrm{h}}_{\mathrm{b}}$ = 2.6 mm showed the highest. The thrust and discharge coefficients for ${\mathsf{\theta}}_{2}$ = 35–90° and ${\mathrm{h}}_{\mathrm{b}}$ = 2–5 mm are presented in Figure 10b. As the bypass angle and width increase the thrust coefficient decreases. Among the models analyzed, model 6 achieved the highest thrust coefficient with ${\mathrm{h}}_{\mathrm{b}}$ = 2.6 mm, while model 1 with ${\mathrm{h}}_{\mathrm{b}}$ = 5 mm achieved the lowest thrust coefficient value. The discharge coefficient shows a similar trend. Increasing the bypass width along with bypass angle resulted in decreasing discharge coefficient with highest values found at 2.6 mm, while minimum values at the bypass width of 5 mm. Among all the investigated models, model 6 achieved the highest discharge coefficient at ${\mathrm{h}}_{\mathrm{b}}$ = 2.6 mm, and model 1 reported the lowest discharge coefficient with ${\mathrm{h}}_{\mathrm{b}}$ = 5 mm.

#### 4.2. Effect of Nozzle Convergence Angle and Bypass Width

#### 4.2.1. BDTN Performance under Different NPR

_{1}) is varied from 22° to 37° for NPR = 2–10 at a constant width ratio of 0.185. Figure 11a depicts the thrust vectoring angles for different configurations at different NPR. Increasing the nozzle convergence angle from 22° to 37° decreases the vectoring angle. The vectoring angles obtained for all models under NPR = 2 was higher than those obtained at NPR = 3–10. Of all the investigated models, model 7 produced the highest vectoring angle. A 1.5% difference in vectoring angle was reported with changing the convergence angle from 22° to 37°. Therefore, the vectoring angle was not significantly affected by the nozzle convergence angle. As illustrated in Figure 11d, vectoring efficiency also shows a similar pattern. Model 7 has the highest vectoring efficiency of 3.87, whereas model 11 has the lowest vectoring efficiency of 2.31. Model 7 has the overall best vectoring efficiency among the other models. Figure 11b shows the thrust obtained for different convergence angles. Among the other models, model 7 reported the highest thrust coefficient of 0.948 at NPR = 5. In Figure 11c, the discharge coefficient increases till NPR = 4 and remains almost linear with further increasing the NPR. Model 7 reported higher discharge coefficient values. Figure 12 shows the Mach contours for the nozzle convergence angle configuration for NPR = 4, 6, 8, and 10.

#### 4.2.2. BDTN Performance under Different Bypass Width

#### 4.2.3. Combined Effect of Bypass Width and Nozzle Convergence Angle

## 5. Conclusions

- NPR significantly affects the thrust vectoring performance of BDTN. As NPR increases, the squeezing effect of the vortex in the cavity reduces, which reduces the supersonic region within the nozzle. Because the vortex size and supersonic region are reduced, BDTN has a lower thrust vectoring performance.
- As bypass width influences vectoring angle, increasing bypass width increases the vectoring angle due to increased mass flow. However, a reduction in vectoring efficiency, thrust, and discharge coefficient is obtained to reach a higher vectoring angle. It is found that a bypass width of 3.7 mm is an optimal choice for effective vectoring performance.
- The bypass angle is an important factor in generating effective vectoring angles. Increasing the bypass angle and decreasing the bypass width resulted in an increase in the thrust and discharge coefficient and a decrease in vectoring angle. Optimal vectoring performance is achieved with a bypass angle of 35°.
- BDTN’s performance is not significantly affected by nozzle convergence angle. An increase of 1.5% in vectoring performance is obtained with increasing convergence angle. Increasing the convergence angle and bypass width increases the vectoring angle while decreases the vectoring efficiency, thrust, and discharge coefficient of the nozzle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

δ | Thrust vectoring angle |

${\mathrm{m}}_{\mathrm{p}}$ | Primary mass flow |

${\mathrm{C}}_{\mathrm{d}}$ | Discharge coefficient |

${\mathrm{C}}_{\mathrm{f}}$ | Thrust coefficient |

η | Thrust vectoring efficiency |

${\mathrm{F}}_{\mathrm{A}}$ | Axial force |

${\mathrm{F}}_{\mathrm{N}}$ | Normal force |

${\mathrm{F}}_{\mathrm{S}}$ | Side force |

${\mathrm{P}}_{0}$ | Stagnation pressure |

${\mathrm{P}}_{\mathrm{e}}$ | Exit pressure |

${\mathrm{P}}_{\mathrm{s}}$ | Static Pressure |

${\mathrm{P}}_{\mathrm{t}}$ | Total pressure |

${\mathrm{T}}_{0}$ | Stagnation temperature |

${\mathrm{T}}_{\mathrm{t}}$ | Total temperature |

FTV | Fluidic thrust vectoring |

MTV | Mechanical thrust vectoring |

BDTN | Bypass dual throat |

DTN | Dual throat nozzle |

NPR | Nozzle pressure ratio |

SVC | Shock vector control |

TSC | Throat skewing control |

CFTV | Counter flow thrust vectoring |

RNG | Renormalization group |

SPR | Secondary pressure ratio |

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**Figure 3.**Plot (

**a**) static pressure distribution for all three grids. (

**b**) comparing static pressure distribution for the experimental and computational calculations at NPR = 3.

**Figure 4.**Variation of (

**a**) the vectoring angle, (

**b**) thrust coefficient, (

**c**) discharge coefficient, and (

**d**) vectoring efficiency with NPR for ${\mathrm{h}}_{\mathrm{b}}$/${\mathrm{h}}_{\mathrm{t}}$ = 0.185.

**Figure 5.**Mach contours at ${\mathsf{\theta}}_{2}$ = 50°, ${\mathsf{\theta}}_{1}$ = 30° and ${\mathrm{h}}_{\mathrm{b}}$/${\mathrm{h}}_{\mathrm{t}}$ = 0.185 for different NPR.

**Figure 6.**Vortex formation at the upper cavity wall at ${\mathsf{\theta}}_{2}$ = 50°, ${\mathsf{\theta}}_{1}$ = 30° and ${\mathrm{h}}_{\mathrm{b}}$/${\mathrm{h}}_{\mathrm{t}}$ = 0.185 for (

**a**) NPR = 3 (

**b**) NPR = 5 (

**c**) NPR = 7 (

**d**) NPR = 10.

**Figure 8.**Variation of (

**a**) vectoring angle, (

**b**) thrust coefficient, (

**c**) discharge coefficient, and (

**d**) the vectoring efficiency for NPR = 3.

**Figure 9.**Mach contours on the center-plane at NPR = 3, ${\mathsf{\theta}}_{2}$ = 70° and ${\mathsf{\theta}}_{1}$ = 30° for bypass width of (

**a**) 2.7 mm (

**b**) 3.2 mm (

**c**) 3.7 mm (

**d**) 4.2 mm (

**e**) 5 mm.

**Figure 10.**The variation of (

**a**) vectoring angle and vectoring efficiency (

**b**) thrust coefficient and discharge coefficient with bypass angle.

**Figure 11.**Plotting of (

**a**) vectoring angle, (

**b**) thrust coefficient, (

**c**) discharge coefficient, and (

**d**) vectoring efficiency for ${\mathrm{h}}_{\mathrm{b}}$/${\mathrm{h}}_{\mathrm{t}}$ = 0.185.

**Figure 12.**This represents the Mach contours at ${\mathsf{\theta}}_{1}$ = 25°, ${\mathsf{\theta}}_{2}$ = 45°, and ${\mathrm{h}}_{\mathrm{b}}$/${\mathrm{h}}_{\mathrm{t}}$ = 0.185 for different NPR.

**Figure 13.**Variation of (

**a**) the vectoring angle, (

**b**) thrust coefficient, (

**c**) discharge coefficient, and (

**d**) vectoring efficiency with increasing bypass width at NPR = 3.

**Figure 14.**(

**a**) refers to the trend of vectoring angle and vectoring efficiency. (

**b**) refers to the trend of thrust and discharge coefficient with nozzle convergence angle.

**Table 1.**The constant geometric dimensions and parameters adopted from Rui Gu [40].

Parameters | Dimensions |
---|---|

Cavity divergence angle ${\mathsf{\theta}}_{3}$ | 15° |

Cavity convergence angle ${\mathsf{\theta}}_{4}$ | 50° |

Inlet height ${\mathrm{h}}_{\mathrm{i}}$ | 60 mm |

Throat height ${\mathrm{h}}_{\mathrm{t}}$ | 20 mm |

Exit height ${\mathrm{h}}_{\mathrm{e}}$ | 24 mm |

Radius ${\mathrm{r}}_{1}$ | 0.8 mm |

Radius ${\mathrm{r}}_{2}$ | 1 mm |

Radius ${\mathrm{r}}_{3}$ | 0.3 mm |

Length of cavity L | 66.8 mm |

Model | ${\mathsf{\theta}}_{1}$ | ${\mathsf{\theta}}_{2}$ | ${\mathbf{h}}_{\mathbf{b}}$$/{\mathbf{h}}_{\mathbf{t}}$ |
---|---|---|---|

1 | 30° | 35° | 0.13–0.25 |

2 | 30° | 50° | 0.13–0.25 |

3 | 30° | 60° | 0.13–0.25 |

4 | 30° | 70° | 0.13–0.25 |

5 | 30° | 80° | 0.13–0.25 |

6 | 30° | 90° | 0.13–0.25 |

7 | 22° | 45° | 0.13–0.25 |

8 | 25° | 45° | 0.13–0.25 |

9 | 27° | 45° | 0.13–0.25 |

10 | 32° | 45° | 0.13–0.25 |

11 | 37° | 45° | 0.13–0.25 |

**Table 3.**(

**a**) 2D model validation percentage error for various NPR values. (

**b**) 3D model validation percentage error for NPR = 3, 5, and 10.

(a) | ||||||

NPR | Rui Gu [40] | Present Data | Percentage Error | |||

δ | C_{f} | δ | C_{f} | δ | C_{f} | |

2 | 32.02° | 0.933 | 31.8° | 0.912 | 0.68% | 2.25% |

3 | 27.21° | 0.959 | 27.15° | 0.94 | 0.22% | 1.98% |

4 | 24.52° | 0.965 | 24.3° | 0.95 | 0.90% | 1.55% |

5 | 23.12° | 0.963 | 22.9° | 0.951 | 0.95% | 1.25% |

6 | 22.54° | 0.96 2 | 2.13° | 0.948 | 1.82% | 1.25% |

7 | 22.09° | 0.955 | 21.6° | 0.946 | 2.22% | 0.94% |

8 | 21.71° | 0.95 | 21.2° | 0.942 | 2.35% | 0.84% |

(b) | ||||||

NPR | Rui Gu [40] | Present Data | Percentage Error | |||

δ | C_{f} | δ | C_{f} | δ | C_{f} | |

3 | 26.95° | 0.949 | 25.78° | 0.934 | 4.32% | 1.61% |

5 | 21.08° | 0.956 | 21.02° | 0.946 | 0.28% | 1.01% |

10 | 20.27° | 0.934 | 19.52° | 0.923 | 3.70% | 1.17% |

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## Share and Cite

**MDPI and ACS Style**

Afridi, S.; Khan, T.A.; Shah, S.I.A.; Shams, T.A.; Mehmood, K.; Li, W.; Kukulka, D.
Numerical Investigation on the Thrust Vectoring Performance of Bypass Dual Throat Nozzle. *Energies* **2023**, *16*, 594.
https://doi.org/10.3390/en16020594

**AMA Style**

Afridi S, Khan TA, Shah SIA, Shams TA, Mehmood K, Li W, Kukulka D.
Numerical Investigation on the Thrust Vectoring Performance of Bypass Dual Throat Nozzle. *Energies*. 2023; 16(2):594.
https://doi.org/10.3390/en16020594

**Chicago/Turabian Style**

Afridi, Saadia, Tariq Amin Khan, Syed Irtiza Ali Shah, Taimur Ali Shams, Kashif Mehmood, Wei Li, and David Kukulka.
2023. "Numerical Investigation on the Thrust Vectoring Performance of Bypass Dual Throat Nozzle" *Energies* 16, no. 2: 594.
https://doi.org/10.3390/en16020594