1. Introduction
In recent years, with the continuous increase in space launch missions around the world, the demand for low – cost rockets and flight – based transportation services has become more and more urgent. Reusable liquid rocket engines have emerged as a crucial development direction for future space exploration. Both domestic and international research institutions have carried out extensive studies on reusable liquid rocket engines. During the “12th Five – Year Plan” and “13th Five – Year Plan” periods in China, the Beijing Aerospace Propulsion Institute collaborated with Beihang University to research the reusability of liquid rocket engines. In 2023, the 130 – ton – class reusable liquid oxygen – kerosene staged – combustion cycle engine designed by the Xi’an Aerospace Propulsion Institute successfully passed the test run. Abroad, SpaceX’s Falcon 9 rocket first – stage vertical take – off and landing technology has been maturely applied, and it is rapidly iterating the reusable Super Heavy Starship, with continuous breakthroughs and maturation of relevant reusable technologies.
Turbopumps are the core components of liquid rocket engines, and their reliability plays a vital role in the overall reliability of the engine. Practical experiences at home and abroad have shown that many failures in space shuttles and rocket launches are closely related to turbopump malfunctions. For example, in 1999, the failure of Japan’s H – IIA rocket launch was due to the resonance of the guide vane caused by the pressure pulsation of the turbopump cavitation, resulting in vane fracture. In 2014, the presence of titanium and silicon debris in the liquid oxygen turbopump of the American AJ – 26 engine led to a rocket explosion.
Compared with conventional aircraft engine turbines, the service environment of liquid rocket engine turbine hot components is extremely harsh. Rocket engine turbines not only bear greater steady – state mechanical loads and thermal loads but also face more dangerous force – thermal shocks and gas corrosion. These factors seriously threaten the normal operation of rocket engines. For instance, a certain liquid rocket engine operates in an extremely complex force – thermal environment, which induces the two – nodal diameter vibration of the tuning – fork – type turbine blisk, causing cumulative fatigue damage to the turbine blisk and leading to multiple impeller crack failures during the test run assessment. Therefore, it is of great significance to conduct research on the low – cycle fatigue life prediction and reliability of rocket engine tuning – fork – type turbines.
Currently, research on turbine low – cycle fatigue and reliability mainly focuses on aircraft engine turbines. Although some research has been carried out on liquid rocket engine turbines, there are still relatively few discussions on the low – cycle fatigue reliability of liquid rocket engine tuning – fork – type integral blisks. This article aims to fill this gap by conducting in – depth research on this topic.
2. Numerical Model
2.1 Geometric Model
This study focuses on a tuning – fork – type integral blisk of a liquid rocket engine. The structure of the turbine is shown in Figure 1. When the turbine operates, high – temperature and high – pressure gas flows in through the nozzle blade S1, driving the first – stage moving blade R1 and the second – stage moving blade R2 to rotate and do work. S2 and OGV are the inter – stage guide vane and the outlet guide vane, respectively. For a single – stage turbine, due to its cyclic – symmetric structure, a single blisk can be taken as the basic calculation area. However, for the tuning – fork – type integral blisk in this study, the number of turbine blades in the two stages is different. Therefore, the turbine disk and two sets of full – circle blades need to be meshed.
[Insert Figure 1: Schematic diagram of turbine impeller structure]
2.2 Analysis Process
During the start – up and shutdown processes of ground tests, no abnormalities were observed. However, cracks frequently occurred in the tuning – fork – type turbine blisk during the steady – state operation, resulting in turbopump test failures. Thus, this article mainly conducts flow, strength, fatigue, and reliability analyses on the turbopump during the steady – state operation stage.
The analysis process for the turbine fatigue life reliability is presented in Figure 2. The rotational speed n, density \(\rho\), and elastic modulus E are selected as random input variables. First, flow field calculations, load analyses, and static strength analyses are carried out to calculate the low – cycle fatigue life under standard conditions. Then, Kriging surrogate models for the maximum stress \(\sigma\) and maximum strain \(\varepsilon\) of the turbine are established for sampling simulations to obtain the probability distribution of the low – cycle fatigue life. Finally, fatigue reliability analyses are performed based on the above results.
[Insert Figure 2: Flow chart of fatigue life reliability analysis]
3. Results and Discussion
3.1 Load Analysis
The loads acting on the tuning – fork – type integral blisk mainly include centrifugal force loads, aerodynamic loads, and temperature loads. The centrifugal force load is a volume force generated by the rotation of the turbine and can be directly applied to the entire turbine blisk according to the turbine rotational speed. The aerodynamic load is obtained through the calculation of the turbine cascade flow and is mainly the pressure difference between the pressure surface and the suction surface. Therefore, this section focuses on exploring the internal flow of the cascade and the characteristics of the aerodynamic load.
The upstream of the turbine S1 blade is set as a pressure – inlet boundary condition, with the inlet total pressure and static temperature specified. The turbine OGV blade is set as a pressure – outlet boundary condition, with the outlet static pressure given. Under the design conditions, the turbine operating speed \(n = 18600r/min\), the inlet total pressure \(p_{t0}=4.61MPa\), the inlet temperature \(T_{t0}=862K\), and the outlet static pressure \(p_{1}=0.37MPa\). The SST \(k – \omega\) turbulence model is selected.
The calculation grid is divided using polyhedron grids. In the grid – independence verification, four groups of grids with 1.1 million, 2.1 million, 2.94 million, and 3.5 million elements are considered. The efficiency \(\eta\) of the 2.94 – million – grid is close to that of the 3.5 – million – grid, as shown in Figure 3. Therefore, the 2.94 – million – grid is selected for experimental verification.
[Insert Figure 3: Grid independence verification]
Five ground hot – test conditions of a certain liquid rocket turbopump are selected for flow simulation, and the corresponding boundary conditions are shown in Table 1. The design operating speed of the turbopump is 18600r/min, and its operating time is the longest. These five test conditions cover the performance of the turbine from under – speed to over – speed conditions. The verification results are shown in Figure 4. The maximum error between the simulation efficiency \(\eta_{1}\) and the experimental efficiency \(\eta_{2}\) is within 5%. It can be seen that the simulation results are in good agreement with the experimental data, indicating that the flow simulation model in this article can accurately describe the internal flow of the turbine.
| n (r/min) | \(T_{t0}\) (K) | \(p_{t0}\) (MPa) | \(p_{1}\) (MPa) | \(\eta_{1}\) | \(\eta_{2}\) |
|---|---|---|---|---|---|
| 16672 | 839.0 | 4.04 | 0.292 | 0.320 | 0.334 |
| 17093 | 768.8 | 4.21 | 0.33 | 0.337 | 0.338 |
| 18188 | 805.6 | 4.56 | 0.355 | 0.348 | 0.366 |
| 19122 | 875.0 | 5.32 | 0.400 | 0.351 | 0.363 |
| 19394 | 833 | 4.92 | 0.355 | 0.361 | 0.372 |
| Table 1: Boundary conditions and efficiency for verification |
[Insert Figure 4: Turbine efficiency at different conditions]
Figure 5 shows the velocity vector diagram, pressure cloud diagram at the 50% height position of the turbine cascade, and the wall – pressure distribution of each stage of blades. High – temperature and high – pressure fuel – rich gas enters the turbine from the left. In the S1 stage, the gas expands and accelerates. The airflow changes from a sub – sonic state to a supersonic state and reaches a maximum Mach number of 2.74 at the outlet. The supersonic airflow impacts the R1 blade, forming a shock – wave structure at the blade leading edge, and decelerates in the R1 stage to enter a sub – sonic state. Subsequently, the airflow adjusts its direction in S2 and enters R2, achieving a small – scale expansion and acceleration. Finally, it is discharged through OGV, and a small – range flow separation occurs in a local area of the pressure surface.
The pressure cloud diagram of the turbine cascade in Figure 5(b) shows that the high – pressure gas undergoes a significant pressure drop in the S1 stage, which is related to the expansion and acceleration of the airflow mentioned above. The wall – pressures of R1 and R2 are used as the aerodynamic loads for subsequent strength analyses. The wall – pressure distributions of the blades at the 50% height position of each stage are shown in Figure 5(c). It can be seen that there is a sharp pressure peak at the leading edge of R1, which is formed by the stagnation of the supersonic airflow at the blade leading edge. The pressure curves of R2 cross, resulting in a relatively small resultant force on the blade. Compared with the centrifugal load, the aerodynamic loads on R1 and R2 are smaller and contribute less to the blade stress. The wall – temperatures of R1 and R2 are used as the temperature loads for subsequent strength analyses. The temperature distributions of a single blade in the R1, S2, and R2 cascades are shown in Figure 5(d).
[Insert Figure 5: Flow fields of the turbine at 50% span location under design condition]
3.2 Strength Analysis
The tuning – fork – type integral blisk is meshed using hexahedron grids and prism – shaped grids. The total number of elements is approximately 7.26 million, with hexahedron grids accounting for 93% and the rest being prism – shaped grids. The turbine integral blisk is made of GH4169 material, with a density of \(8240kg/m^{3}\), an elastic modulus of 168GPa, a thermal conductivity of \(21.2W/(m\cdot^{\circ}C)\), and a linear expansion coefficient of \(1.48\times 10^{-5}/^{\circ}C\). The axial two – end faces at the bottom of the integral blisk are set with fixed – boundary conditions.
The rationality of the finite – element model of the tuning – fork – type integral blisk can be verified by comparing the modal results of the turbine. The analysis results of the first five – order free – vibration modes of the turbine are shown in Table 2. Here, 1ND represents the first – nodal diameter mode, and 1NC represents the first – nodal circle mode. It can be seen that the error between the calculated results of the natural frequencies of the turbine’s free – vibration and the measured results is small. The natural frequency of the two – nodal diameter vibration of the turbine is close to the abnormal signal frequency during the previous turbopump test run.
| Item | 1ND | 2ND | 1NC | 3ND | 4ND |
|---|---|---|---|---|---|
| FEM | 525 | 1276 | 2178 | 2475 | 3768 |
| Exp. | 523 | 1278 | 2235 | 2505 | 3820 |
| Table 2: The first five modes of free vibration (Hz) |
The temperature distribution obtained from the cascade flow analysis is mapped onto the surface of the turbine blade. The thermal load of the integral blisk is caused by blade heat conduction. In the finite – element analysis of the turbine integral blisk, the temperature distribution inside the turbine integral blisk is determined using the thermal – mechanical coupling method. Further, the pressure distribution obtained from the cascade flow analysis is mapped onto the surface of the turbine blade, and the centrifugal load (i.e., the design operating speed of 18600r/min) is applied. Thus, the stress distribution and strain distribution of the turbine integral blisk under the combined action of the centrifugal force load, aerodynamic load, and temperature load are obtained.
The stress distribution of the turbine integral blisk is shown in Figure 6. It can be seen that stress concentrations occur in the middle part of the tuning – fork, the fillet of the disk, and the root of the R1 blade, with a maximum stress of 649.08MPa. When the turbine operates, the two blades are in a cantilever state. Under the action of the centrifugal load, the two blades will squeeze towards the middle, generating an equivalent couple moment at the middle position, resulting in stress concentration in the middle part of the tuning – fork. At the same time, the fillet of the disk is subjected to tensile bending, generating a large tensile stress. At the connection part of the integral blisk, due to the sudden change in the cross – sectional area, stress concentration also occurs under the action of the equivalent couple. The maximum strain is located at the fillet of the disk, as shown in Figure 7, and the maximum strain is \(9.15\times 10^{-}\).
[Insert Figure 6: Stress distribution of turbine under design condition] [Insert Figure 7: Strain distribution of turbine under design condition]
3.3 Low – Cycle Fatigue Life Analysis
From the strength analysis, it is known that the position of the maximum stress of the integral blisk is the failure position. In this article, the Smith – Watson – Topper (SWT) modified model of the Manson – Coffin equation is used to evaluate the fatigue life of the integral blisk. The expression is as follows:
\(\sigma_{max}\frac{\varepsilon_{max}}{2}=\frac{(\hat{\sigma}_{f})^{2}}{E}(2N_{f})^{2b}+\hat{\varepsilon}_{f}\hat{\sigma}_{f}(2N_{f})^{b + c}\)
where \(\sigma_{max}\) is the maximum stress of the turbine, \(\varepsilon_{max}\) is the maximum strain of the turbine, \(\hat{\sigma}_{f}\) is the material fatigue strength coefficient, \(\hat{\varepsilon}_{f}\) is the material fatigue plasticity coefficient, b is the material fatigue strength index, c is the material fatigue plasticity index, \(N_{f}\) is the turbine fatigue life, and E is the elastic modulus.
The integral blisk is made of GH4169 alloy material. The temperature at the failure position is close to \(600^{\circ}C\). The low – cycle fatigue life parameters at \(650^{\circ}C\) are used for extrapolation and interpolation, and the specific parameters are shown in Table 3.
| Parameter | Value |
|---|---|
| \(\hat{\sigma}_{f}\) (GPa) | 1476 |
| \(\hat{\varepsilon}_{f}\) | 0.108 |
| b | – 0.09 |
| c | – 0.58 |
| Table 3: Fatigue parameters of GH4169 |
Substituting the fatigue parameters in Table 3 and the maximum stress of 649.1MPa and maximum strain of \(6.72\times 10^{-}\) obtained from the static strength analysis into the above formula, the low – cycle fatigue life of the turbine under the design condition is calculated to be 15199 cycles.
3.4 Reliability Analysis
3.4.1 Surrogate Model and Its Verification
Haubert et al. pointed out that geometric parameters have a relatively small impact on the fatigue life of aircraft engine turbine blades. In addition, the tuning – fork – type turbines of liquid rockets adopt integrated precision manufacturing technology, and the processing size deviation of the integral blisk is extremely small. Therefore, combined with engineering experience and relevant literature, in this article, the turbine density, elastic modulus, and rotational speed are taken as random parameters (normally distributed), and the relevant coefficients are shown in Table 4. The Latin hyper – cube sampling method is used for 49 sampling times to establish Kriging surrogate models for the maximum stress and maximum strain of the turbine. Further, 361 sampling – point data are used to verify the surrogate model. The verification results of the finite – element analysis (FEA) are shown in Figure 8. The relative root – mean – square errors of the maximum stress and maximum strain are 1.2514‰ and 0.0051‰, respectively, indicating that the established Kriging model is effective.
| Parameter | Average value | Coefficient of variation |
|---|---|---|
| \(\rho\) (\(kg\cdot m^{-3}\)) | 8240 | 0.05 |
| E (GPa) | 168 | 0.025 |
| n (\(r\cdot min^{-1}\)) | 18600 | 0.01 |
| Table 4: Random distribution parameters of turbine |
3.4.2 Probability Distribution of Low – Cycle Fatigue Life
Based on the Latin hyper – cube sampling, a large – sample sampling (\(10^{5}\) times) is carried out on the sample space. The maximum stress and maximum strain of the turbine in the large – sample are predicted using the established maximum stress surrogate model and maximum strain surrogate model. The SWT modified model mentioned above is used to obtain the low – cycle fatigue life of the large – sample.
The frequency distribution of the turbine low – cycle fatigue life is shown in Figure 9. It can be clearly seen that the low – cycle fatigue life of the turbine presents an obvious right – skewed distribution. The distribution is respectively fitted with the Exponential Weibull (Exponweib) distribution, Erlang distribution, Gamma distribution, Logistic distribution, and Generalized Logistic (Gen – Logistic) distribution. Through the Kolmogorov – Smirnov test, it is found that the Exponential Weibull distribution is the most consistent with the simulation sampling results. Its probability density function is:
\(f(x)=\alpha\beta\left[1 – e^{(-x^{\beta})}\right]^{\alpha – 1}e^{(-x^{\beta})}x^{\beta – 1}\)
The cumulative probability density function is:
\(F(x)=\left[1 – e^{(-x^{\beta})}\right]^{\alpha}\)
where \(\alpha\) and \(\beta\) are shape parameters, \(x>0\), \(\alpha>0\), \(\beta>0\).
Although the random parameters in the turbine reliability analysis are given in the normal distribution form as shown in Table 4, the low – cycle fatigue distribution of the tuning – fork – type integral blisk shown in Figure 9 is in a right – skewed form, which is approximately a Weibull distribution. This is consistent with current literature reports and practical experience. Liu Shijie et al. pointed out that the probability life models of the main engine components (bearings, impellers, blades, etc.) of the US space shuttle are Weibull distribution or log – normal distribution. Compared with the normal distribution form, the right – skewed distribution of the low – cycle fatigue life is not conducive to the overall life of the tuning – fork – type integral blisk.
[Insert Figure 9: Curve – fitting for low – cycle fatigue life probability density]
3.4.3 Low – Cycle Fatigue Life Reliability
The fatigue life reliability of the tuning – fork – type integral blisk refers to the probability that its life can reach the target life under certain working conditions, that is:
\(R(N_{d}) = P(N_{f}\geq N_{d})\)
where \(N_{d}\) is the life index, and \(N_{f}\) is the turbine fatigue life that satisfies a certain random distribution.
It can also be expressed as:
\(R(N_{d}) = 1 – P(N_{f}<N_{d})\)
where \(P(N_{f}<N_{d})\) is the cumulative failure probability of the turbine, that is, the unreliability F.
By integrating the right – skewed low – cycle fatigue life distribution curve of the tuning – fork – type integral blisk shown in Figure 9, the cumulative probability density curve F of the low – cycle fatigue life and the reliability curve R can be obtained, as shown in Figure 10. According to the R curve, it can be known that in the high – reliability region, the maximum number of cycles to ensure a reliability of 99.999% is 285 cycles.
[Insert Figure 10: Cumulative probability density curve and reliability curve of low – cycle fatigue]
4. Conclusion
This article comprehensively analyzes the flow field and static strength of a tuning – fork – type integral blisk in a liquid rocket engine turbopump, and obtains the turbine low – cycle fatigue life reliability based on the local stress – strain method and the Kriging surrogate model. The main conclusions are as follows:
(1) A numerical simulation model for the flow of the tuning – fork – type integral blisk cascade is constructed based on the CFD method. The performance parameters of the tuning – fork – type integral blisk obtained are in good agreement with the test – run data. The high – pressure fuel – rich gas forms a supersonic flow field in the turbine, resulting in a greater aerodynamic load on the first – stage moving blade.
(2) A strength vibration model of the tuning – fork – type integral blisk is established based on the finite – element method. The free – mode results of the turbine are consistent with the experimental results. The high – speed rotation condition causes the maximum stress in the middle fillet of the tuning – fork – type integral blisk and the fillets on both sides of the disk. The low – cycle fatigue life obtained by combining with the SWT model is consistent with the preliminary design target.
(3) Through Latin hyper – cube sampling, taking the rotation speed, density, and elastic modulus with normal distribution as random variables, the turbine stress – strain surrogate model is established. It is found that the low – cycle fatigue life distribution of the turbine is right – skewed and approximately follows the Weibull distribution.
(4) A low – cycle fatigue life reliability model for the tuning – fork – type integral blisk of a liquid rocket engine is established, and the relationship between the low – cycle fatigue life reliability and the number of cycles is obtained.
These research results can provide important theoretical support and technical guidance for the design, optimization, and reliability improvement of liquid rocket engine turbopumps. In future research, more complex working conditions and influencing factors can be considered to further improve the accuracy of the research results. For example, the influence of manufacturing process errors, material inhomogeneity, and long – term service degradation on the low – cycle fatigue life and reliability of the integral blisk can be studied. At the same time, with the continuous development of computational technology, more advanced numerical algorithms and simulation methods can be introduced to improve the efficiency and accuracy of the analysis.
5. Future Research Directions
5.1 Incorporating More Complex Working Conditions
In real – world applications, liquid rocket engine turbopumps operate under a variety of complex conditions. Future research could focus on including factors such as variable operating temperatures, pressure fluctuations, and off – design operating points in the analysis. These additional factors can have a significant impact on the low – cycle fatigue life and reliability of the tuning – fork – type integral blisk. For example, sudden changes in temperature during engine startup and shutdown can cause thermal stress, which may interact with mechanical stress and accelerate fatigue damage. By considering these complex working conditions, more accurate fatigue life predictions and reliability assessments can be achieved.
5.2 Considering Material Inhomogeneity and Degradation
Materials used in liquid rocket engine components, such as the GH4169 alloy for the integral blisk, may have inherent inhomogeneities. These inhomogeneities can lead to local variations in material properties, affecting the stress and strain distribution within the blisk. Moreover, during long – term service, the material may degrade due to factors like high – temperature oxidation, corrosion, and fatigue – induced micro – damage accumulation. Future research could aim to develop models that account for material inhomogeneity and degradation over time. This would enable a more realistic evaluation of the blisk’s performance and remaining life during its service life.
5.3 Applying Advanced Numerical Algorithms
The development of computational technology offers new opportunities for improving the analysis of low – cycle fatigue life and reliability. Advanced numerical algorithms, such as machine – learning – based surrogate models and high – fidelity multi – physics simulations, can be explored. Machine – learning algorithms can potentially capture complex relationships between input variables and fatigue life more accurately than traditional methods. High – fidelity multi – physics simulations, which couple fluid dynamics, heat transfer, and structural mechanics more precisely, can provide a more detailed understanding of the physical processes occurring in the turbopump. By applying these advanced algorithms, the efficiency and accuracy of the analysis can be enhanced, leading to more reliable design and operation of liquid rocket engines.
6. Significance of the Research
6.1 Improving Rocket Engine Reliability
The research on the low – cycle fatigue life and reliability of tuning – fork – type integral blisks in liquid rocket engine turbopumps is crucial for enhancing the overall reliability of rocket engines. Turbopumps are key components of liquid rocket engines, and their failure can lead to catastrophic consequences during rocket launches. By accurately predicting the fatigue life and reliability of the integral blisk, engineers can identify potential failure modes in advance and take appropriate measures to improve the design. This includes optimizing the structure, selecting better materials, or adjusting the manufacturing process. Ultimately, this research contributes to reducing the risk of rocket engine failures and ensuring the success of space missions.
6.2 Supporting the Development of Reusable Rocket Technology
The development of reusable rocket technology requires components with high durability and reliability. The results of this research can provide valuable information for the design and improvement of reusable liquid rocket engine turbopumps. Understanding the low – cycle fatigue behavior of the integral blisk helps in developing more robust components that can withstand multiple launch and landing cycles. This is essential for reducing the cost of space transportation and making space exploration more sustainable. As the demand for reusable rockets continues to grow, the research findings can play a significant role in meeting the challenges associated with component durability and reliability in reusable systems.
7. Comparison with Previous Studies
7.1 Research Focus
Previous studies on turbine low – cycle fatigue and reliability mainly concentrated on aircraft engine turbines. While some research has been conducted on liquid rocket engine turbines, the focus on tuning – fork – type integral blisks in liquid rocket engine turbopumps is relatively new. This article fills this research gap by specifically targeting the unique structure and working conditions of tuning – fork – type integral blisks. In contrast to previous studies that may have overlooked the specific characteristics of this type of blisk, this research provides in – depth analysis and understanding of its low – cycle fatigue life and reliability.
7.2 Research Methods
The methods used in this study also have some differences from previous research. By using the Latin hyper – cube sampling method to establish Kriging surrogate models for maximum stress and maximum strain, and considering multiple random input variables such as rotational speed, density, and elastic modulus, a more comprehensive and accurate assessment of the low – cycle fatigue life probability distribution and reliability is achieved. Some previous studies may have used simpler methods or considered fewer random factors. The approach in this article takes into account the complex uncertainties in the system, resulting in more reliable research results.
8. Conclusion and Outlook
In conclusion, this article has conducted a systematic study on the low – cycle fatigue life and reliability of tuning – fork – type integral blisks in liquid rocket engine turbopumps. Through numerical simulations and theoretical analyses, important conclusions have been drawn, which are of great significance for improving the design and reliability of liquid rocket engines. However, there is still room for further improvement and exploration in this field. Future research should continue to focus on more complex engineering problems, combine new materials and manufacturing technologies, and use advanced numerical methods to continuously improve the understanding and prediction of the low – cycle fatigue life and reliability of liquid rocket engine components. This will contribute to the continuous development and progress of space exploration technology.
