# Maximum-Thrust Nozzle Based on Height Constraints

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approaches

#### 2.1. Validation of Numerical Method

_{k}represents the generation of turbulence kinetic energy; G

_{ω}represents the generation of ω; and Γ

_{k}and Γ

_{ω}represent the effective diffusivity of k and ω, respectively. Y

_{k}and Y

_{ω}represent the dissipation of k and ω, whereas D

_{ω}represents the cross-diffusion term. S

_{k}and S

_{ω}are user-defined source terms. In (3), G

_{k}represents the generation of turbulence kinetic energy due to the mean velocity gradients. G

_{b}is the generation of turbulence kinetic energy due to buoyancy. Y

_{M}represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. C

_{μ}, C

_{2}, and C

_{1ε}are constants. σ

_{k}and σ

_{ε}are the turbulent Prandtl numbers for k and ε, respectively. S

_{k}and S

_{ε}are user-defined source terms.

_{t}> 8, the realizable findings of the k–ε model are closer to the experimental results than those of the k–ω SST model. Thus, all numerical simulations in the remainder of this paper use the realizable k–ε turbulence model.

#### 2.2. Grid Resolution Independence Study

_{t}≈ 1.8 via amplification at x/h

_{t}≈ 1.8 and 5.1. Moreover, the maximum relative error between the medium grid and the fine grid is 0.18% at x/h

_{t}≈ 5.1. A medium-scale computational grid is used in the following numerical simulation to comprehensively evaluate the numerical accuracy and cost.

## 3. Nozzle Design Methodology

#### 3.1. Maximum-Thrust Nozzle Theory under Height Constraint

_{DE}, and the mass flow rate constraint, ${\dot{m}}_{DE}$, along the control surface, DE, are as follows:

_{a}are the local static pressure and ambient back pressure, respectively, then the axial thrust, F, acting on the nozzle control surface, DE, is obtained as follows:

_{1}and λ

_{2}as follows:

#### 3.2. MOC

_{+}/C

_{−}) and their corresponding compatibility equations are as follows:

_{0}) and its corresponding compatibility equations are as follows:

#### 3.3. Design Process of SERN

_{ideal}, can be obtained using (18) according to the geometry parameters of the inlet and outlet.

_{exit}, h

_{in}, Ma

_{in}, and γ are the outlet section area, inlet section area, inlet Mach number, and heat capacity ratio, respectively.

_{u}OI

_{d}, in Figure 7 is calculated, labeled as serial number 1.

_{u}CDAI

_{d}O, labeled as serial numbers 2 and 3, is calculated by considering θ

_{u}and θ

_{d}. Here, θ

_{u}and θ

_{d}represent the initial expansion angle of the ramp and flap, respectively. They are shown in Figure 7. When θ

_{u}and θ

_{d}are provided, asymmetrical factor β can be obtained as follows:

_{Ma}, is a design parameter in the proposed design method. The impact of β

_{Ma}on the nozzle is, therefore, examined.

_{u}and θ

_{d}must be provided again. Kernel region I

_{u}CDAI

_{d}O is calculated until the condition satisfied by point E is found.

_{cr}–v/a

_{cr}graph is also shown in Figure 7, where λ is speed coefficient, defined as the ratio of velocity magnitude to critical speed of sound.

## 4. Parametric Study of Design Parameters

#### 4.1. Typical Nozzle Design

_{∞}is 5.

_{in}) and outlet of the nozzle (h

_{exit}) are 157 mm and 400 mm, respectively. The length of the nozzle is set to 800 mm. The flight Mach number Ma

_{∞}is 5. The flight altitude H is 23 km.

_{1}, β

_{Ma}, and β should be provided and are set to 30°, 1, and 0.75, respectively. A comparison of the normalized pressure contours between the inviscid CFD and MOC is shown in Figure 8. The results show the accuracy of the proposed design method.

#### 4.2. Definition of Nozzle Performance Parameters

_{w}is the wall pressure, p

_{a}is the ambient pressure, ṁ is the mass flow rate, T

_{0}is the nozzle inlet total temperature, p

_{0}is the nozzle inlet total pressure, ${\overline{V}}_{in}$ is the average axial velocity of the nozzle inlet, p

_{in}is the nozzle inlet static pressure, and R is the gas constant.

#### 4.3. Influence of β

_{u}CDAI

_{d}O, in Figure 7, and the nozzle geometry. In the design process, θ

_{u}remains constant when the inlet parameters, outlet height constraints, expansion arc radius, and β

_{Ma}remain unchanged. The value of θ

_{d}is determined when β is fixed.

_{fx}and negatively correlated with the lift force. Figure 10 shows that the streamline near the lower wall deviates downward from the axial direction as β decreases. This phenomenon results in a decrease in axial momentum at the outlet, a decrease in axial thrust, and an increase in lift.

#### 4.4. Influence of β_{Ma}

_{Ma}, can control the overall length of the nozzle while still meeting the specified height constraint. The influence of β

_{Ma}on the nozzle profile is shown in Figure 11. The figure shows that increasing β

_{Ma}results in an increase in the ramp length and a decrease in the flap length. This adjustment is necessary to meet the height constraint and β.

_{Ma}values on the thrust coefficient and lift of the nozzle when β = 0.75. The figure shows that an increase in β

_{Ma}results in an increase in the total length of the nozzle and a decrease in the length of the nozzle’s flap. Thus, the positive lift provided by the ramp increases, while the negative lift provided by the flap decreases simultaneously. The increase in the geometric asymmetry of the nozzle leads to greater inhomogeneity of the aerodynamic parameters at the nozzle outlet, resulting in a decrease in the axial thrust coefficient of the nozzle. Moreover, the calculation results show that an excessively high β

_{Ma}causes significant nozzle deformation, which further leads to a severe deterioration of nozzle performance.

_{Ma}in Figure 11 on the nozzle geometry and the nozzle design process in Figure 7 indicates that β

_{Ma}can be considered a method for nozzle truncation design. When β

_{Ma}is unequal to 1, the upper wall, I

_{u}CF, is designed according to Ma = Ma

_{E}instead of Ma = Ma

_{ideal}. Therefore, the flow design of the nozzle when β

_{Ma}is larger than 1 is as follows: (1) The nozzle is designed based on the outlet Mach number Ma

_{E}in the characteristic line design program, and the area ratio of the nozzle’s inlet and outlet should conform to (18). (2) The ramp remains stationary, and the flap is truncated to meet the specified height constraint.

#### 4.5. Adjusting Nozzle Geometry by Design Parameters

_{Ma}while considering the given height constraint. Therefore, the two design parameters can be used to adjust the geometry of the nozzle. The influences of β and β

_{Ma}on the nozzle geometry are further illustrated using three design examples.

_{Ma}= 1. If it is not possible to meet the total length constraint, β

_{Ma}should be adjusted and recalculated to ensure compliance with the constraint. Then, β

_{Ma}is maintained, and β is adjusted to satisfy the constraint of the flap length.

_{Ma}values for each nozzle are shown in Table 4. The normalized pressure contours are shown in Figure 14, whereas the distribution of the Mach number on the outlet section is shown in Figure 15. Additionally, the pressure distribution on the ramp and flap is shown in Figure 16.

## 5. Aerodynamic Performance Comparisons Using Traditional Method

#### 5.1. Performance Comparisons

#### 5.2. Off-Design Performance

_{∞}= 2.5 and Ma

_{∞}= 3 are in the overexpansion state. The operating points Ma

_{∞}= 3.5 and Ma

_{∞}= 4 are in a state of underexpansion. However, the NPR is less than that at the design point. In addition, the transition point of the combined cycle engine’s working mode is typically near Ma

_{∞}= 2.5 during actual flight. Therefore, the off-design working points presented in Table 6 represent the typical performance of the nozzle throughout the entire flight envelope in the ramjet mode.

## 6. Discussion

_{Ma}are set as 0.75 and 1, respectively. In this figure, Method 1 is the method mentioned in Section 3.3. In Methods 2–4, the characteristic line grid is calculated sequentially to obtain the nozzle profile. Point B is calculated using the symmetric unit process. In Method 2, arc I

_{d}D is calculated using asymmetric factor β, which leads to the last characteristic line of flap EF. The location of point F is determined via point G when the nozzle meets the height constraint. In Methods 3 and 4, the generation of the ramp profile is consistent with Method 2. The calculation of arc I

_{d}D is also calculated using β, whereas point E is calculated using the symmetric point unit process. Then, this leads to the last characteristic line of flap EF, ensuring that the conditions of axial flow and evenly distributed parameters are met. However, the height of the nozzle exceeds the given height constraint, so truncation is necessary. To ensure that the flow at the wall outlet is horizontal, either the ramp or the flap should be truncated. Method 3 involves truncating the lower wall, while Method 4 involves truncating the upper wall, as indicated by serial number ⑥. In Method 4, the profile is not designed according to Ma = Ma

_{idea}l = f(h

_{exit}/h

_{in}), that is, β

_{Ma}≠ 1. Therefore, the discussion revolves around Methods 1–3.

_{idea}l = f(h

_{exit}/h

_{in}), the nozzle meets the height constraint. At that time, the nozzle becomes a symmetric nozzle, which does not match the SERN of the research object.

## 7. Conclusions

_{Ma}while adhering to the specified outlet height constraint. The main conclusions are as follows:

- (i).
- β
_{Ma}primarily controls the overall length of the nozzle. The total length of the nozzle increases with an increase in β_{Ma}. The length of the flap decreases, and the positive lift of the nozzle increases in order to meet the height constraint. The increase in β_{Ma}increases the geometric asymmetry of the nozzle while decreasing C_{fx}and increasing C_{L}. In the actual design, an excessively large β_{Ma}leads to a rapid deterioration of nozzle performance when β and the height constraint are maintained. - (ii).
- The influence of β on nozzle geometry is that only the length of the flap changes, while the ramp profile and the height of the flap remain unchanged. The flap length increases with an increase in β. Moreover, the streamline near the flap aligns closely with the axial direction, resulting in an increase in the thrust coefficient and a decrease in lift. In the actual design, the nozzle is designed under the specified geometric constraints by adjusting β
_{Ma}and β. - (iii).
- Compared to the nozzle (Nozzle A) designed using the traditional truncated design method, the thrust and lift coefficients of the nozzle (Nozzle B) designed using the proposed method increase by 11.93% and 138.45%, respectively, at the design point. At the same off-design point, the thrust and lift coefficients of Nozzle B are greater than those of Nozzle A. Compared to the thrust coefficient of Nozzle A, the thrust coefficient of Nozzle B increases by a maximum of 8.79% under various typical off-design working conditions. Moreover, the positive lift force is maintained throughout the entire working condition range.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | speed of sound |

C_{fx} | axial thrust coefficient |

C_{L} | lift coefficient |

e | total energy |

F | force |

h | height |

k | turbulent kinetic energy |

L | length |

Ma | Mach number |

ṁ | mass flow rate |

p | pressure |

Q | rate of volumetric heat addition per unit mass |

R | gas constant |

T | temperature |

V | velocity |

x | x-direction coordinate |

y | y-direction coordinate |

δ | axial/2D switch |

ε | kinetic energy dissipation rate |

φ | slope angle |

γ | heat capacity ratio |

ρ | density |

θ | velocity angle |

τ | shearing stress |

ω | specific dissipation rate |

Subscript | |

a | atmosphere |

cr | critical parameter |

exit | nozzle exit |

in | nozzle entrance |

t | throat |

w | wall |

0 | total parameter |

∞ | free stream |

## References

- Wang, Z.; Liang, J.; Ding, M.; Fan, X.; Wu, J.; Lin, Z. A review on hypersonic airbreathing propulsion system. Adv. Mech.
**2009**, 39, 716–739. [Google Scholar] - Wu, Y.; He, Y.; He, W.; Le, J. Progress in airframe-propulsion integration technology of air-breathing hypersonic vehicle. Acta Aeronaut. Astronaut. Sin.
**2015**, 36, 245–260. [Google Scholar] - Jin, J.; Chen, M.; Liu, Y.; Du, G. Turbine Based Combined Cycle Engine; National Defense Industry Press: Beijing, China, 2019; pp. 154–196. [Google Scholar]
- Chen, Y.; Yu, K.; Xu, J. New design method for scramjet nozzles with strong geometric constraints. Acta Aeronaut. Astronaut. Sin.
**2021**, 42, 12459. [Google Scholar] - Rao, G.V.R. Exhaust nozzle contour for optimum thrust. Jet Propuls.
**1958**, 28, 377–382. [Google Scholar] [CrossRef] - Lu, X.; Yue, L.; Xiao, Y.; Chen, L.; Chang, X. Design of Scramjet Nozzle Employing Streamline Tracing Technique. In Proceedings of the 16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conference, Bremen, Germany, 19–22 October 2009. [Google Scholar]
- Lv, Z.; Xu, J.; Yu, Y.; Mo, J. A new design method of single expansion ramp nozzles under geometric constraints for scramjets. Aerosp. Sci. Technol.
**2017**, 66, 129–139. [Google Scholar] [CrossRef] - Liu, Y.; Li, P.; Chen, H.; Yang, J.; Ren, X. A Big Area Ratio Liquid Rocket Engine Nozzle with Full-Flow in Low Altitude. J. Propuls. Technol.
**2022**, 43, 205–212. [Google Scholar] - Yu, K.; Chen, Y.; Huang, S.; Xu, J. Inverse Design Method on Scramjet Nozzles Based on Maximum Thrust Theory. Acta Astronaut.
**2020**, 166, 162–171. [Google Scholar] [CrossRef] - Yu, K.; Chen, Y.; Huang, S.; Xu, J. Inverse Design Method on Scramjet Nozzle with Full Geometrical Constraints for Nozzle-Afterbody Integration. J. Aerosp. Eng.
**2021**, 34, 04021004. [Google Scholar] [CrossRef] - Wang, Z.; Liu, A.; Cai, Y. Design method and flow field simulation of single expansion ramp nozzle. Gas Turbine Exp. Res.
**2007**, 20, 8–12. [Google Scholar] - Argrow, B.M.; Emanuel, G. Comparison of minimum length nozzles. J. Fluids Eng.
**1988**, 110, 283–288. [Google Scholar] [CrossRef] - Shyne, R.; Keith, T., Jr. Analysis and design of optimized truncated scarfed nozzles subject to external flow effects. In Proceedings of the 26th Joint Propulsion Conference, Orlando, FL, USA, 16–18 July 1990. [Google Scholar]
- Hoffman, J.D. Design of compressed truncated perfect nozzles. AIAA Pap.
**2015**, 3, 150–156. [Google Scholar] - Lan, Q. The Method of Characteristics for the Inverse Problem of Three-Dimensional Hypersonic Aerodynamics. Master’s Thesis, National University of Defense Technology, Changsha, China, 2017. [Google Scholar]
- Liu, H.; Zhao, Y. Aerodynamic Inverse Design via Characteristics Tracing. J. Propuls. Technol.
**2017**, 38, 289–297. [Google Scholar] - Bare, E.A.; Capone, F.J. Static Internal Performance of Convergent Single-Expansion-Ramp Nozzles with Various Combinations of Internal Geometric Parameters; NASA: Washington, DC, USA, 1989. [Google Scholar]
- Zucrow, M.J.; Hoffman, J.D. Gas Dynamics; vol. 2–Multidimensional Flow; John Wiley & Sons, Inc.: New York, NY, USA, 1977. [Google Scholar]
- Mo, J.; Xu, J.; Quan, Z.; Yu, K.; Lv, Z. Design and cold flow test of a scramjet nozzle with nonuniform inflow. Acta Astronaut.
**2015**, 108, 92–105. [Google Scholar] [CrossRef]

**Figure 5.**Influence of different grid numbers on normalized pressure distribution on the ramp of nozzle.

**Figure 7.**Schematic of MOC design procedure. ①: Mach region. ② and ③: Kernel region. ④: Upper turning region. ⑤: Lower turning region.

**Figure 22.**Several expressions of the design method for maximum thrust SERN based on height constraint. For Methods 2–4: ①: Mach region. ② and ④: Kernel region. ③: Upper turning region. ⑤: Kernel region (Method 2), lower turning region (Methods 3 and 4). ⑥: Lower turning region (Method 2), truncation (Methods 3 and 4).

Boundary Condition Type | Introduction |
---|---|

Pressure inlet | Defines the total pressure, total temperature, and static pressure at the inlet. In Section 2.1, the total pressure and static pressure are set as 8571 Pa, and the total temperature is set as 600 K. |

Pressure far field | Defines the static pressure, static temperature, free-stream Mach number, and x and y flow components of the free stream, only when the density is calculated using the ideal gas law. In Section 2.1, the static pressure, static temperature, free-stream Mach number, and flow direction of the free stream are set as 1000 Pa, 300 K, 0.01, 1, and 0, respectively. |

Pressure outlet | Defines the gauge pressure at the outlet. When the flow is supersonic, the pressure is solved under upstream conditions. In Section 4.3, the gauge pressure at the outlet is set as 3466.9 Pa. |

Wall (ramp/flap) | In Section 2.1, the ramp and flap are set as no-slip and adiabatic. |

Ma_{in} | p_{in} (Pa) | T_{in} (K) | H (km) | h_{in} (mm) | h_{exit} (mm) | L (mm) | Ma_{∞} |
---|---|---|---|---|---|---|---|

1.76 | 45,455 | 1822 | 23 | 157 | 400 | 800 | 5 |

Nozzle No. | Flap Length (m) | Ramp Length (m) |
---|---|---|

Nozzle C | 0.8 | 0.3 |

Nozzle D | 1 | 0.25 |

Nozzle E | 0.7 | 0.4 |

Nozzle No. | β_{Ma} | β |
---|---|---|

Nozzle C | 1 | 0.61 |

Nozzle D | 1.064 | 0.87 |

Nozzle E | 0.963 | 0.625 |

Nozzle No. | C_{fx} | C_{L} |
---|---|---|

Nozzle A | 0.641 | 1.178 |

Nozzle B | 0.716 | 1.214 |

Ma_{∞} | p_{a} (Pa) | p_{0} (Pa) | T_{0} (K) | Ma_{in} |
---|---|---|---|---|

2.5 | 11,554.39 | 131,300 | 1500 | 1.37 |

3 | 8054.66 | 133,000 | 1871 | 1.19 |

3.5 | 5886.79 | 160,000 | 2203 | 1.3 |

4 | 4515.75 | 210,000 | 2409 | 1.4 |

Ma_{∞} | C_{fx} | C_{L} |
---|---|---|

2.5 | 0.671 | −0.007 |

3 | 0.868 | 0.681 |

3.5 | 0.840 | 0.859 |

4 | 0.767 | 0.908 |

Ma_{∞} | C_{fx} | C_{L} |
---|---|---|

2.5 | 0.517 | −1.010 |

3 | 0.801 | 0.302 |

3.5 | 0.777 | 0.705 |

4 | 0.708 | 0.866 |

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**MDPI and ACS Style**

Huang, B.; Xu, J.; Yu, K.
Maximum-Thrust Nozzle Based on Height Constraints. *Aerospace* **2023**, *10*, 976.
https://doi.org/10.3390/aerospace10120976

**AMA Style**

Huang B, Xu J, Yu K.
Maximum-Thrust Nozzle Based on Height Constraints. *Aerospace*. 2023; 10(12):976.
https://doi.org/10.3390/aerospace10120976

**Chicago/Turabian Style**

Huang, Bowen, Jinglei Xu, and Kaikai Yu.
2023. "Maximum-Thrust Nozzle Based on Height Constraints" *Aerospace* 10, no. 12: 976.
https://doi.org/10.3390/aerospace10120976