# Transient Characteristics of Fluidic Pintle Nozzle in a Solid Rocket Motor

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Principle of FPN

_{e}is the pressure of motor outlet and P

_{a}is the pressure of the external environment. This equation takes the volume formed by the inner wall and the outlet section of the nozzle as the control body to integrate, which is not affected by the propellant type and the internal geometric mechanism of the motor, and can be calculated only by obtaining the exit interface parameters.

## 3. Models and Numerical Methods

#### 3.1. Geometric Models and Boundary Conditions

^{−7}ms. In the simulation of the pintle motion, the time step is set to 10

^{−6}ms, and the total duration is 66.6 ms. The pintle’s range of motion spans from 100% to 0% opening, with additional clearance both before and after. The displacement is 10 mm, and both forward and backward movements occur at a speed of 150 mm/s, as illustrated in Figure 2. After the cessation of the pintle motion, the calculation continues until the combustion chamber pressure stabilizes. At 4.67 ms, the geometric throat area (A

_{t}) equals the nozzle throat area (A

_{nt}), and after 66.67 ms, the geometric throat area no longer undergoes variations. Figure 4 illustrates the relationship between time and the actual geometric throat area (A

_{t}) during the pintle advancing process.

_{p}denotes the radius of the pintle, while L

_{i}designates the location of the injection port, defined as the distance from the head of the pintle to the injection port. The dimensionless parameter L

_{i}/R

_{p}serves to characterize the relative position of the injection port.

#### 3.2. Numerical Methods

- (1)
- Monophasic flow, excluding the consideration of solid particulates introduced into the propellant;
- (2)
- The gas is treated as an ideal gas, adhering to the equations of state for an ideal gas;
- (3)
- No consideration for radiative heat transfer, neglect of gravity, and absence of volume forces;
- (4)
- Adiabatic boundaries, devoid of thermal exchange between the external environment and the entire flow field.

_{k}represents the generation term for turbulent kinetic energy (induced by mean velocity gradients), G

_{ω}is the generation term for the specific rate of dissipation ω, and ${\mathsf{\Gamma}}_{k}$ and ${\mathsf{\Gamma}}_{\omega}$ are the effective diffusion coefficients for k and ω, respectively. ${Y}_{k}$ and ${Y}_{\omega}$ represent the turbulent dissipation due to turbulence production for k and ω, respectively. ${S}_{k}$ and ${S}_{\omega}$ are user-defined source terms, and ${D}_{\omega}$ represents the cross-diffusion term.

^{®}. The numerical method is grounded in the finite volume method. Gradient interpolation is predicated on the least squares cell method, accompanied by standard pressure interpolation. The density, momentum, and energy equations are instantiated in the first-order upwind format. The pressure–velocity coupling is facilitated through the implementation of the SIMPLEC algorithm. The transient term in time is discretized using a first-order implicit scheme.

#### 3.3. Dynamic Mesh

#### 3.4. Calculation Verification

## 4. Results and Discussion

#### 4.1. Transient Process of Secondary Flow Opening and Closing

#### 4.1.1. Opening

_{p}= 0%, injection angle α = 90°, and modified flow ratio f

_{w}= 0.3. Figure 11 illustrates the variation in combustion chamber pressure and wall pressure distribution during the opening process. From the visual representations, the post-secondary injection response process can be broadly categorized into three phases, namely the pressure propagation stage, pressure oscillation stage, and equilibrium stability stage.

_{0}~t

_{1}). Subsequently, the secondary flow flows downstream while adhering to the pintle, forming a secondary shear layer. Due to the compression by secondary flow, the effective throat area decreases, leading to an increase in combustion chamber pressure. The momentary high pressure causes the secondary shear layer to thin, increasing the effective throat area and reducing pressure. The transient low pressure thickens the secondary shear layer, decreasing the effective throat area. This is a repetitive process, and although the overall combustion chamber pressure is increasing, the thickness of the secondary shear layer and the combustion chamber pressure exhibit oscillatory characteristics until the combustion chamber pressure no longer shows a decreasing trend. Throughout this process, the thicknesses of the secondary shear layer and pressure oscillate. As the lower-temperature secondary flow gradually fills the vortex region at the head of the pintle, the overall temperature in this area begins to decrease, while the area of the recirculation zone does not change significantly. This stage is referred to as the pressure oscillation stage (Figure 10, approximately 0.023 ≤ t ≤ 0.33 ms; Figure 11b, t

_{1}~t

_{2}). The primary flow and secondary flow gradually begin to reach equilibrium, and the rate of increase in the combustion chamber pressure gradually decreases until it stabilizes. The flow also tends to stabilize during this stage, known as the equilibrium stability stage (Figure 10, approximately 0.33 ≤ t ≤ 5.7 ms; Figure 11b, approximately t

_{2}~5.7 ms). Research suggests that the response process of injection is related to the free volume [25].

#### 4.1.2. Closing

_{w}= 0.3.

#### 4.2. Coupling of Pintle Movement and Secondary Flow

#### 4.2.1. Forward Movement of the Pintle (Pressure Increase Process)

_{w}= 0.3. Figure 16 delineates the evolution of pressure and thrust across the entire progression.

_{w}= 0”, and “opening = 0%, f

_{w}= 0.3”, the thrust is increased by 80.95% (from 417.35 N to 755.18 N).

#### 4.2.2. Backward Movement of the Pintle (Pressure Decrease Process)

_{w}= 0.3 in FPN during the backward movement, and Figure 18 illustrates the changes in pressure and thrust.

_{w}= 0.3” and “opening = 100%, f

_{w}= 0”, the thrust is decreased by 44.87% (from 775.23 N to 416.37 N).

#### 4.3. Effect of Injection Angle and Injection Port Position

_{w}= 0.3. The pintle velocity and starting/stopping positions are consistent with those in the previous section. The results are presented in Figure 20 and Figure 21.

_{i}/R

_{p}set at 0, 0.5, and 1. The injection angle is set at 90°, and the modified flow ratio is denoted as f

_{w}= 0.3. The pintle velocity and starting/stopping positions are consistent with those in the previous section. The results are presented in Figure 22 and Figure 23.

_{i}/R

_{p}(port closer to the pintle head) corresponds to a higher combustion chamber pressure and thrust. When L

_{i}/R

_{p}= 0, after 55 ms, there is a minimal increase in both pressure and thrust. This is attributed to the pintle entering the straight section of the nozzle throat, and as the pintle motion ceases, the geometry near the nozzle throat remains unchanged. Consequently, there is minimal alteration in the flow field, leading to a relatively stable combustion chamber pressure. For L

_{i}/R

_{p}equal to 0.5 and 1, after the pintle ceases its motion, the pressure and thrust gradually stabilize. This indicates that even when the pintle stops, the injection port remains at the upstream arc of the nozzle throat. At this point, the flow field near the throat is still not entirely stable, requiring some time after the pintle stops to achieve pressure stability. Comparatively, L

_{i}/R

_{p}= 0.5 achieves both the maximum thrust control range and exhibits good responsiveness.

## 5. Conclusions

- (1)
- The injection process in FPN can be roughly divided into three stages: the pressure propagation stage (combustion chamber pressure remains constant), pressure oscillation stage (combustion chamber pressure undergoes oscillations), and equilibrium stability stage (the combustion chamber pressure steadily rises), accounting for approximately 0.4%, 5.39%, and 94.21% of the total time, respectively.
- (2)
- During the forward movement of the pintle, the combustion chamber pressure rapidly increases, with the rate of increase gradually decreasing (related to the upstream arc of the nozzle throat). Compared with the condition with maximum throat opening and no secondary flow, the thrust of the condition with minimum throat opening and 0.3-flow-ratio secondary flow is increased by 80.95%. In the backward movement of the pintle, the combustion chamber pressure gradually decreases, with the rate of decrease gradually increasing.
- (3)
- Under the condition of a limited flow ratio, the injection angle of the secondary flow has little influence on the dynamic thrust control, but the control effect of reverse injection is more obvious when the throat opening is smaller. The closer the injection port is to the pintle head, the better the thrust control effect is, albeit at the cost of weakening the thermal protection of the low-temperature secondary flow.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**Streamlines, temperature, and Mach number contours after injection, opening = 0%, α = 90°, f

_{w}= 0.3. (

**a**) t = 0.0 ms. (

**b**) t = 0.02 ms. (

**c**) t = 0.04 ms. (

**d**) t = 0.06 ms. (

**e**) t = 0.08 ms. (

**f**) t = 0.2 ms. (

**g**) t = 0.4 ms. (

**h**) t = 1.0 ms. (

**i**) t = 3.0 ms.

**Figure 11.**Pressure variation after injection, opening = 0%, α = 90°, f

_{w}= 0.3. (

**a**) Pressure distribution along the nozzle wall. (

**b**) Combustion chamber pressure.

**Figure 13.**Pressure variation and wall pressure distribution after closing the secondary flow, opening = 0%, α = 90°, f

_{w}= 0.3. (

**a**) Pressure distribution along the nozzle wall. (

**b**) Combustion chamber pressure.

**Figure 14.**Temperature and Mach number contours during the forward movement process of the pintle motor, α = 90°, f

_{w}= 0. (

**a**) t = 0.0 ms. (

**b**) t = 4.6 ms. (

**c**) t = 15 ms. (

**d**) t = 30 ms. (

**e**) t = 45 ms. (

**f**) t = 66.6 ms.

**Figure 15.**Temperature and Mach number contours during the forward movement process of FPN, α = 90°, f

_{w}= 0.3. (

**a**) t = 0.0 ms. (

**b**) t = 4.6 ms. (

**c**) t = 15 ms. (

**d**) t = 30 ms. (

**e**) t = 45 ms. (

**f**) t = 66.6 ms.

**Figure 17.**Temperature and Mach number contours during the backward movement process of pintle motor, α = 90°, f

_{w}= 0. (

**a**) t = 0.0 ms. (

**b**) t = 6.6 ms. (

**c**) t = 20 ms. (

**d**) t = 35 ms. (

**e**) t = 50 ms. (

**f**) t = 66.6 ms.

**Figure 18.**Temperature and Mach number contours during the backward movement process of FPN, α = 90°, f

_{w}= 0.3. (

**a**) t = 0.0 ms. (

**b**) t = 6.6 ms. (

**c**) t = 20 ms. (

**d**) t = 35 ms. (

**e**) t = 50 ms. (

**f**) t = 66.6 ms.

**Figure 19.**Variation of combustion chamber pressure and thrust during the backward movement process.

**Figure 21.**Temperature contours for different injection angles, t = 45 ms. (

**a**) α = 60°. (

**b**) α = 90°. (

**c**) α = 120°.

**Figure 22.**Variation in combustion chamber pressure and thrust with different injection port positions. (

**a**) Combustion chamber pressure. (

**b**) Thrust.

**Figure 23.**Temperature contours for different injection port positions at t = 45 ms. (

**a**) L

_{i}/R

_{p}= 0.0. (

**b**) L

_{i}/R

_{p}= 0.5. (

**c**) L

_{i}/R

_{p}= 1.0.

Component | Value |
---|---|

Combustion chamber diameter | 60 mm |

Nozzle throat diameter | 14 mm |

Pintle diameter | 10 mm |

Nozzle outlet diameter | 28 mm |

Convergent half angle | 45° |

Half-angle expansion | 15° |

Expansion ratio | 4 |

Free volume of the cavity | $3.2\times {10}^{4}\mathrm{m}{\mathrm{m}}^{3}$ |

Component | Parameter | Value |
---|---|---|

Propellant | Burning rate coefficient | 6.7 |

Pressure index | 0.24 | |

Density | 1700 kg/m^{3} | |

Burning area | 1.53 × 10^{−2} mm^{2} | |

Gas | Molar mass | 26.3157 g/mol |

Specific heat | 1.63 kJ/(kg·K) | |

Thermal conductivity | 0.285 w/(m·k) | |

Temperature of primary flow | 3000 K | |

Temperature of secondary flow | 1789 K | |

Mass flow of secondary flow | 0.07968 kg/s |

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**MDPI and ACS Style**

Yan, D.; Zhao, Z.; Song, A.; Li, F.; Ye, L.; Zhao, G.; Ma, S.
Transient Characteristics of Fluidic Pintle Nozzle in a Solid Rocket Motor. *Aerospace* **2024**, *11*, 243.
https://doi.org/10.3390/aerospace11030243

**AMA Style**

Yan D, Zhao Z, Song A, Li F, Ye L, Zhao G, Ma S.
Transient Characteristics of Fluidic Pintle Nozzle in a Solid Rocket Motor. *Aerospace*. 2024; 11(3):243.
https://doi.org/10.3390/aerospace11030243

**Chicago/Turabian Style**

Yan, Dongfeng, Zehang Zhao, Anchen Song, Fengming Li, Lu Ye, Ganchao Zhao, and Shan Ma.
2024. "Transient Characteristics of Fluidic Pintle Nozzle in a Solid Rocket Motor" *Aerospace* 11, no. 3: 243.
https://doi.org/10.3390/aerospace11030243