Study of Fluid Flow Characteristics and Mechanical Properties of Aviation Fuel-Welded Pipelines via the Fluid–Solid Coupling Method

Abstract

The welded pipeline structure of aircraft fuel is a complex and diverse entity, significantly influenced by fluid–solid coupling. The refined aviation fuel-welded pipeline model plays a pivotal role in the investigation of its fluid–solid coupling mechanical properties. However, the mechanical analyses of pipelines with welded structures frequently simplify or ignore the influence of the weld zone (WZ). Consequently, these analyses fail to reveal the complex interactions between different weld zones in detail. In this study, a comprehensive and precise fuel-welded pipeline refinement model is developed through the acquisition of microstructural dimensions and mechanical parameters of the weld zone via metallographic inspection and microtensile testing. Additionally, the influence of clamps and brackets under airborne conditions is fully considered. Furthermore, the numerical simulation results are compared and verified using modal and random vibration tests. This paper addresses the impact of diverse fluid characteristics on the velocity field, pressure field, and stress in disparate areas, and it also conducts an investigation into the random vibration characteristics of the pipeline. The results demonstrate that the fluid pressure and velocity exert a considerable influence on the fluid flow state and structural stress distribution within the pipeline. An increase in flow velocity and alteration to the pipeline geometry will result in a change to the local velocity distribution, which in turn affects the distribution of the fluid pressure field. The highest stresses are observed in the weld zone, particularly at the junction between the weld zone and the heat-affected zone (HAZ). In contrast, the stresses in the bend region exhibit a corrugated distribution in both the axial and circumferential directions. An increase in fluid pressure has a significant impact on the natural frequency of the pipeline. This study enhances our comprehension of the mechanical properties of aircraft fuel lines with fluid–solid coupling and provides a foundation and guidance for the optimal design of fuel-welded lines.

1. Introduction

The aviation industry is experiencing a period of significant growth, driven by the rapid expansion of the global economy and the acceleration of industrialization. As aircraft design and manufacturing processes become increasingly complex and precise, the incidence of accidents caused by fuel piping system failures is also rising [1,2,3,4]. As a pivotal element of the entire fuel system, the fuel-welded line is constrained not only by the spatial limitations of the fuselage but also by the intricate flow-solid coupling vibration transmission law, which arises from its intricate configuration. The disparate mechanical properties of the WZ give rise to the complex flow-solid coupling vibration characteristics [5,6,7]. The reliability of the WZ has become a key concern in ensuring the service performance and structural safety of piping systems [8,9,10]. Coherent welding offers significant advantages in terms of strength, stability, and force performance [11,12], making it an effective solution for connecting multi-section piping problems. It is a technique widely used in the aerospace industry. Scholars in various countries have also conducted research on the process parameters of the WZ.

The weld area is divided into three distinct zones: the WZ, the HAZ, and the base metal zone (BMZ). These zones have been extensively studied by scholars in various countries, employing both experimental analysis and numerical simulation. Yi et al. [13] conducted tests to assess the mechanical properties of weld structures, utilizing hardness and tensile testing techniques. Chu et al. [14] employed nanoindentation tests to determine the slight softening of the HAZ resulting from recrystallisation. Ai et al. [15] investigated the issue of weld area prediction under diverse process conditions, employing techniques such as image recognition and machine learning. Lu et al. [16] employed the thermoelastic plasticity method to simulate the welding temperature field and residual stress field, establishing a solid model of a butt joint of aluminum alloy with a double-pass weld and subsequently verifying it with corresponding tests. Wu et al. [17] employed the friction stir welding (FSW) technique to create a joint in aluminum alloy 6061 and investigated the factors influencing the mechanical properties of the weld based on the observed macrostructure and fracture surface of the joint. Although the mechanical property parameters of the weld area are observed in their experimental study, there are some limitations to the study of the effect of the interaction between different zones of the weld.

The numerical simulation method allows for the simultaneous exploration of the flow characteristics of the fluid domain and the mechanical properties of the solid domain of the pipeline, thereby revealing the flow characteristics of the fluid domain and the mechanical mechanism of the solid domain of the pipeline [18]. The method has the advantages of high efficiency, low cost, and rapid calculation speed. Furthermore, the internal flow field characteristics of the pipeline are intuitively and clearly simulated, and the results are more accurate than those obtained from experimental tests [19,20,21]. Consequently, numerical simulation is also frequently employed in the investigation of pipe flow-solid coupling characteristics. Lazareff et al. [22] employed a numerical simulation to examine the fluid–solid coupling interface conditions throughout a comprehensive transient flight cycle and put forth an adaptive condition that integrated the Dirichlet–Robin and Neumann–Robin interface conditions for the first time. Xu et al. [23] employed the “L”-type heat pipe network as the subject of their investigation, utilizing the fluid–structure-heat coupling method to numerically simulate the heat transfer and flow of the medium within the pipe network. This approach enabled the researchers to ascertain the pressure, temperature, and the equivalent force exerted on the solid structure of the flow field under varying conditions. Feng et al. [24] employed the flow-heat-solid coupling method, based on the ANSYS Workbench platform, to simulate the fluid flow and heat transfer inside a large-diameter buried pipeline and conducted a stability analysis to examine the structural strength of the pipeline. In a high-pressure pipeline within a nuclear power plant, Solomon et al. [25] investigated the phenomenon of water hammer and analyzed the mechanical response of the pipeline using the fluid–solid coupling method. The results of this analysis were then compared with those of the experimental study to verify the correctness of the simulation. Mohammed et al. [26] investigated the flow-solid coupling law of plug flow in a gas–liquid section of a horizontal pipeline, and they validated the accuracy of their model through experimentation. Talemi et al. [27] employed a fluid–solid coupling model to simulate a dynamic toughness fracture in steel pipes and validated the reliability of their model through comparison with experimental data. Despite the fact that numerous scholars have conducted numerical simulations of the fluid–solid coupling characteristics of pipelines, the results of these studies have demonstrated high accuracy and feasibility. However, it is evident that the current research is still limited due to the combined effects of the weld area interaction, service load complexity, and other factors on the fuel-welded pipelines of aircraft. It is therefore necessary to investigate the fluid–solid coupling vibration characteristics of fuel-welded pipelines under complex flow fields.

Due to the accelerated advancement of the numerical simulation technology, scholars across the globe have commenced recognizing the pivotal role that model precision plays in numerical simulation. It is therefore imperative that the weld structure is modeled in detail in order to accurately assess the mechanical properties of the weld area and the overall mechanical behavior of the pipeline. Jin et al. [28] put forth a technique for modeling the partition of a weld, with the objective of enhancing the precision of fatigue life simulations in the weld region. This was achieved through the utilization of metallographic and hardness tests. Chu et al. [29] put forth three methods for weld modeling, demonstrating that reasonable weld modeling is of paramount importance for ensuring the accuracy of simulations, particularly in the prediction of wall thickness variation in the vicinity of the weld. Jin et al. [30] and Chiocca et al. [31] investigated the effect of weld residual stresses on the fatigue life of joints using a fine weld model, demonstrating the efficacy of the model in assessing fatigue durability. Ghafouri et al. [32] and Kim et al. [33] accurately predicted the mechanical properties and residual stresses of welded joints by means of a finite element model. The results demonstrated that the method was capable of accurately capturing the weld distortion and stress distribution. It is therefore of great significance to establish a refined model that includes the WZ, HAZ, and BMZ, in accordance with the differences in mechanical properties observed among these various zones. Furthermore, a detailed analysis of the stress distribution, deformation characteristics, and mechanical response of the welded zones under various working conditions is required.

In conclusion, the majority of current studies on fluid–solid coupling are concentrated on relatively simple piping structures. Conversely, there is a paucity of research investigating the fluid–solid coupling mechanical properties of aircraft fuel lines with welded seams. In order to elucidate the flow-solid coupling vibration law and mechanical properties of coherently fuel-welded lines, it is advantageous to employ numerical simulation techniques to investigate the fluid flow behavior within the lines, which is challenging to observe through experimentation. In this paper, the microstructural dimensions and mechanical parameters of the weld area are obtained through metallographic testing and microtensile testing. Additionally, the role of clamps and brackets under actual working conditions is considered to construct a full-solid, high-precision fuel-welded pipeline refinement model. Furthermore, a combination of numerical simulations of the fluid–solid coupling and experimental validation of pipeline dynamics is employed to investigate the effects of different fluid velocities and pressures on the velocity field, pressure field, and stress distribution of the different areas inside the pipeline. The impact of varying fluid velocities and pressures on the velocity and pressure fields within the pipeline and the stress distribution across different regions is examined. The impact of fluid pressure on the random vibration characteristics of the pipeline in response to random load excitation is examined. This study contributes to the advancement of fuel piping system design and performance optimization, while also facilitating a more profound comprehension of the fluid–solid coupling dynamics inherent to pipelines.

2. Methodology

2.1. Mathematical Model of Fluid–Solid Coupling

2.1.1. Solid State Control Equation

In the fluid–solid coupling analysis of fuel lines, the finite element method is employed to discretize the intricate solid structure into a multitude of diminutive cells, which are interconnected by nodes. Each cell is postulated to be an elastomer, and the displacement interpolation function at the nodes is utilized to express the displacement alterations within the cells [34]. The initial step is to establish the pipeline’s solid field equations without considering structural damping, as illustrated in Equation (1).

$${\sigma }_{ij,j}+f_i={\rho }_s{\ddot{x}}_i$$

(1)

${\sigma }_{ij,j}$ is the piping structure stress component; $f_i$ is the piping volume force component; ${\rho }_s$ is the piping density; and ${\ddot{x}}_i$ is the piping structure vibration acceleration component.

The pipeline force and displacement boundary conditions are presented in Equations (2) and (3), respectively.

$${\sigma }_{ij}{n}_{sj}={\overline{T}}_i$$

(2)

$$u_i={\overline{u}}_i$$

(3)

${\overline{T}}_i$ represents the known surface force component of the pipeline, while ${\overline{u}}_i$ denotes the known structural displacement component of the pipeline.

2.1.2. Fluid Control Equation

In practical engineering applications, the fluid flow in fuel lines is typically at or near steady state. Therefore, it is assumed that the flow is subjected to small perturbations near steady state [35]. By associating the continuity equation, the equation of motion and the equation of state of the fluid flow, based on the assumptions, are obtained as shown in Equation (4).

$$p=-k{u}_{i,i}$$

(4)

The boundary equation for the rigid fixed surface of the fuel is shown in Equation (5).

$${\displaystyle \frac{\partial p}{\partial n_f}}=0$$

(5)

The variable n represents the direction of the boundary normal, while the variable $n_f$ denotes the vector of normal conditions situated outside the fuel boundary unit.

The fuel-free surface boundary equation is specified in Equation (6).

$${\displaystyle \frac{\partial p}{\partial z}}+{\displaystyle \frac{1}{g}}\ddot{p}=0$$

(6)

2.1.3. Fluid–Solid Coupling Control Equation

In order to guarantee the transfer of force and motion between the fluid and the solid, it is essential that the velocity and pressure of the fluid and the elastomer at the coupled interface are identical. By applying these boundary conditions to the fluid–solid coupling equations, it is possible to combine the equations of motion of the solid with the fluid dynamics equations in order to obtain equations which describe the dynamic behavior of the entire coupled system.

As illustrated in the preceding analysis, the typical velocity and force on the coupled interface should be uninterrupted, as evidenced by Equations (7) and (8).

$$v_{f_n}=v_f\cdot n_f=v_s\cdot n_f=-v_s\cdot n_s=v_{sn}$$

(7)

$${\sigma }_{ij}{n}_{sj}={\tau }_{ij}{n}_{fj}=-{\tau }_{ij}{n}_{sj}$$

(8)

${\sigma }_{ij}$ represents the component of the structural stress tensor, while ${\tau }_{ij}$ denotes the component of the fluid stress tensor.

Following the integration of the aforementioned equations and conditions, the fluid–solid coupled vibration problem of the pipeline is subjected to further analysis using the finite element software (ANSYS Workbench 2022).

2.2. Numerical Simulations

2.2.1. Physical Model, Boundary, and Initial Conditions

Welded Joint Parameters

In the aluminum alloy welding process, the WZ has an inhomogeneous structure. Influenced by the welding rod material, the welding method, welding current of different welding layers and other parameters, including the mechanical properties of the final molded WZ, show significant differences in different zones. To carry out metallographic testing for aluminum alloy-welded joints, the joints are first finely cut and ground to ensure that the details of each zone can be clearly observed under the microscope. Subsequently, the specimens are viewed using a high-resolution metallographic microscope to determine the boundary parameters and specific metallographic organization types of each zone in the welded joint. The metallographic diagrams of the specific test pieces and welding zones are shown in Figure 1. Through this test, the dimensional parameters of each part of the WZ of the T-type coherent-welded test piece are obtained, and the specific dimensional labeling and values are shown in Figure 2 and

Table 1. Welding area size parameters.

Position	Parameters	Size/mm
Weld thickness	l4	2.60
HAZ of right-angle weld branch pipe	l5	7.81
HAZ of right-angle weld supervisor	l6	8.02
Outside residual height	hC	0.72
Outside width	l3	6.61
Height of point A	hA	0.20
Distance of point A from the center line	LA	1.76
Height of point B	hB	0.31
Distance of point B from the center line	lB	3.15
Height of HAZ of flat fillet weld branch pipe	l1	4.74
Height of HAZ of flat fillet weld supervisor	l2	6.06

. In order to measure the mechanical property parameters of the WZ and the HAZ, tensile tests are generally used to obtain the stress–strain curves of the specimens and to determine the mechanical property parameters of the materials based on the test data. However, due to the small size of each individual zone in the WZ studied in this paper, specimen handling is very difficult, and it is difficult to directly apply the traditional tensile test method to specimen measurements. Therefore, a microtensile test method is needed to measure the specimens in the WZ and HAZ. Through the analysis and processing of the microtensile test data, the parameters of the WZ and HAZ can be obtained, mainly for the measurement of the modulus of elasticity and Poisson’s ratio, as shown in

Table 2. Fuel-welded line zones material properties.

Name	Density/(kg/m3)	E/GPa	υ
BMZ	2.713 × 103	75	0.35
WZ	2.713 × 103	81.8	0.32
HAZ	2.713 × 103	74	0.29

.

Pipeline Physical Model

The present study concerns an aircraft fuel welding pipeline with an elbow, which is positioned at an angle of 90° with respect to the main pipe and the branch pipe. On the basis of the findings of the metallurgical testing, a more detailed model of the welded section of the pipeline was constructed. The fundamental geometric characteristics of the pipeline are defined by parameters such as the outer diameter of the main pipeline and the branch pipeline D_w, the length of the main pipeline L_u, and the vertical distance of the end face of the main pipeline from the axis of the branch pipeline L_z, among others. The specific structural parameters are presented in

Table 3. Pipeline structural parameters.

Parameter Symbol	Parameter Meaning	Design Data/mm
Dw	Outside diameter of main and branch lines	31.75
Dn	Inside diameter of main and branch lines	29.97
Lu	Length of main line	550
Lz	Vertical distance between the end face of the main line and the axis of the branch line	350
Li	Straight length of branch line	150
Rz	Bending radius of branch line	116
Bj	Width of clamp to line contact	15
Lk	Distance of branch pipe clamp end face from main line end face	30
Ld	Distance of straight pipe clamp end face from Branch line elbow end face	20

. In accordance with the aforementioned parameters, a comprehensive, high-resolution fuel welding pipeline refinement model is constructed, as illustrated in Figure 3. In practice, the aircraft fuel-welded line is typically constrained by clamps. However, previous studies [36,37] have not conducted a comprehensive analysis of the clamps, instead adopting a simplified approach to constraint processing. Accordingly, in order to accurately reflect the onboard environment of the pipeline, this paper will utilize the P-type clamp to restrain the pipeline and analyze the bracket in conjunction with it, thereby fully considering the impact of the bracket quality on the vibration characteristics of the pipeline system. The P-type clamp is connected to the bracket by bolts and nuts. Consequently, an assembly model including bolts and nuts, clamps, bracket, and fuel-welded piping is constructed, as illustrated in Figure 4. Furthermore, in order to conduct a comprehensive analysis of the fluid flow characteristics and the coupling effect between the fluid and the structure, a fluid domain model of the fuel-welded pipeline was constructed, upon which the subsequent flow field study was based.

Boundary and Initial Condition Settings

In the actual piping system, there is minimal relative motion between the clamp and the piping. Consequently, the linear contact mode is adopted, and the influence of the friction effect is excluded. The clamp and the piping are set as bound contact, and the specific boundary conditions are set as shown in Figure 5.

In the fluid analysis, the medium within the tube is identified as aviation fuel, with a density of 779.6 kg/m³ and a dynamic viscosity of 0.001154 kg/(m s). In order to ascertain the type of fluid flow, the specific Reynolds number (Re) is calculated using Formula (9), which is based on the value of the Reynolds number.

$$Re={\displaystyle \frac{{\rho }_{f}V{D}_n}{\mu }}$$

(9)

The variables are defined as follows: ${\rho }_f$ represents the density of the fluid, $V$ denotes the flow rate, $D_n$ signifies the inner diameter of the pipeline, and $\mu$ stands for the dynamic viscosity of the fluid.

In accordance with the aforementioned parameters, the Reynolds number of the fuel is 20,246, which is considerably greater than 2300. Therefore, the flow state of the fluid within the pipeline is turbulent. In this paper, the Realizable $k-\epsilon$ turbulence model is employed for analysis, with the coupled coupling algorithm utilized for its solution.

During the actual operation of the aircraft, the flow of fuel in the line undergoes alterations in accordance with the varying stages of operation. During ground operations, the engine’s fuel demand is relatively low, resulting in a reduced fuel flow through the line. During the takeoff and climb phase, the fuel demand of the engine reaches its maximum, resulting in an increase in fuel flow within the pipeline. In this study, four distinct flow velocity gradients (1, 9, 17, and 25 m/s) and four pressure gradients (100, 500, 1000, and 2000 KPa) are selected for subsequent analysis to investigate the fluid–solid coupling mechanical properties of the fuel line.

2.2.2. Grid Independence Verification

Since the local density of the network has a significant influence on the stress values at the points of sudden change in the cross-section, such as the throat of a branch and weld as a geometric notch [38], higher local mesh densities are used in this study in the above critical zones to improve the accuracy of the stress analysis. The pipeline fluid domain, solid domain, clamp bracket, etc., were meshed. Given that the mesh quality is a key factor affecting the accuracy and convergence of simulation calculations, the ANSYS Workbench 2022 software commonly employs four methods, namely, cell quality, aspect ratio, orthogonal quality, and deviation, to assess the mesh quality [39]. The mesh model quality of the fuel weld lines, clamps, and brackets were evaluated based on these methods, as illustrated in

Table 4. Grid quality of pipeline areas and components.

Name	Unit Quality	Aspect Ratio	Orthogonal Quality	Deviation
Solid domain	0.648	2.438	0.876	0.438
Fluid domain	0.793	1.977	0.915	0.151
Clamp	0.876	1.646	0.953	0.003
Nuts and bolts	0.928	1.379	0.702	0.219
Brackets	0.825	1.883	0.991	0.244

. The specific mesh models for the fuel weld line fluid domain, solid domain, and clamp and bracket assembly are illustrated in Figure 6. The mesh independence of the pipeline was verified using a multi-parameter and multi-node approach, with the inlet velocity of the fluid domain set to 9 m/s and the outlet pressure set to 0.5 MPa, respectively. The critical point A, located at the outer surface of the inlet end of the pipeline stub, was selected for comparison of the stresses exhibited at varying mesh numbers. The critical point B, situated at the weld end of the fluid domain, was selected for comparison of the flow velocity at varying grid numbers. The random vibration stress values at different grid numbers were compared at key point C, which is located at the outlet end of the inner surface of the pipeline. Figure 7 illustrates the variation in stress, flow velocity, and random vibration stress with the number of grids at specific key point locations. It was determined that the impact of parameter value fluctuations is inconsequential when the number of grids reaches 200,000. Accordingly, 213,932 meshes were employed in the present study for analysis purposes. The subsequent meshing of other models were verified using the same method.

2.3. Monitoring Position Setting

2.3.1. Fluid Domain

The fluid domain of the pipeline is divided into five areas, including the inlet elbow, inlet straight pipe, welded joints, and the left and right sides of the outlet straight pipe, in order to accurately analyze the flow rate and pressure changes in each area.

In the entire fluid domain, 19 monitoring surfaces were set up, specifically at the inlet bend region of the monitoring surfaces A, A₁, B, B₁, and C, which are in the elbow section at the 0°, 22.5°, 45°, 67.5°, and 90° position; inlet straight pipe region were used to set up the monitoring surfaces C₁, D, and D₁ and were located at a distance of 40, 75, and 115 mm from the I₁; the outlet at the left side of the straight pipe of the region sets up the monitoring surfaces E, F, G, H, E₁, and I, which are located at 0, 50, 100, 150, 200 and 250 mm from II₁; monitoring surfaces J, F₁, K, G₁, and L, are located at 0, 50, 100, 150, and 200 mm from the position of III₁ in the region to the right of the outlet straight pipe. In the subsequent fluid domain analysis, the focus will be on the above monitoring locations. At the same time, the state of the flow field at the monitoring locations A–L will be studied in detail, as shown in Figure 8.

2.3.2. Solid Domain

Three monitoring positions were selected to be explored in the welding and bending areas of the fuel weld line. For the welding region, A_1aA_1b on the upper side of the welding region, A_2aA_2b on the left side, and A_3aA_3b on the lower side of the welding region, which is symmetrical with the upper side, are selected, and for the bending region, B_1aB_1b on the inner side of the bending region, B_2aB_2b on the left side, and B_3aB_3b on the lower side are selected; meanwhile, to further investigate the effects of different fluid parameters on the pipeline’s random vibration characteristics, two random vibration stress response monitoring points are created at the critical areas of the pipeline, respectively, at the junction of the HAZ and the WZ at the weld. This was performed to set up monitoring point I, and at the junction of the inlet elbow and the straight pipe, it was performed to set up monitoring point II, see Figure 9 for details.

2.4. Pipeline Natural Frequency Analysis

In the study of coupled dynamics of fuel-welded pipelines, it is essential to first analyze the inherent characteristics of the pipeline. This was achieved through the execution of dry, wet, and prestressed modal analyses, resulting in the acquisition of the first 20 orders of the inherent frequency values. These values are presented as comparative histograms, as illustrated in Figure 10. It was determined that the fuel-welded pipeline exhibited the highest natural frequency for each order under the prestressed modal analysis. This result can be attributed to the non-uniform pressure distribution generated by the fluid flow on the inner wall of the pipeline, which enhances the pipeline’s rigidity to a certain extent. In the wet mode analysis, the overall mass matrix of the system is increased due to the effect of the fluid mass, which in turn results in a decrease in the natural frequency. The natural frequency of the dry mode is situated between that of the wet mode and the prestressed mode, exhibiting a value comparable to the initial eight orders of the natural frequency observed in the prestressed mode. Nevertheless, as the order increases, the frequency difference between the two gradually increases.

It is therefore crucial to consider both the impact of fluid mass on the structural mass matrix and the influence of fluid flow on the structural stiffness matrix when conducting a modal analysis of a fuel-welded line. This approach enables a more precise evaluation of the line’s dynamic characteristics under actual operating conditions.

2.5. Validation and Analysis of the Numerical Model

In order to evaluate the static dynamic behavior of the welded pipeline in the aircraft fuel system, a pipeline system test platform was constructed to carry out the free mode, constrained mode, and random vibration tests of the fuel-welded pipeline, respectively, as shown in Figure 11. In this test system, the pressure of the control system is precisely controlled by adjusting the opening size of the spool in the proportional valve. Sensors within the system are responsible for collecting and controlling flow and pressure data, as well as vibration response data at the pipeline monitoring points. The system consists of several key components such as the main pump, asynchronous motor, manual relief valve, accumulator, lines to be tested, proportional valve, filter, and cooler. The main pump is a 7-piston, constant pressure variable axial piston pump with a displacement of 92 L/min and an operating pressure of up to 35 MPa. At the same time, in order to enhance the safety of the test system, the pump port is equipped with a group of special safety relief valves in order to prevent the system from overpressure.

The simulation-calculated natural frequencies are compared with the modal test results, respectively, and according to Figure 11a,b, the maximum error rate between the free mode simulation and the test is found to be 7.92%, and the maximum error rate between the constrained mode simulation and the test is found to be 8.09%.

Tests were conducted for the pipeline under X-axial excitation, mainly comparing the results of the pipeline’s response in the X-direction under X-axial excitation. The flow rate of the test system was set to 9 m/s and the pressure to 500 KPa by adjusting the pump displacement and the back pressure proportional valve. Subsequently, the appropriate sampling frequency and duration were set to ensure that comprehensive and accurate data were collected from each sensor. The data were imported into MATLAB for the necessary filtering and conversion processes. The data are converted to the frequency domain via fast Fourier transformation for comparison with the simulation results. The established random vibration frame diagram and the test set up are shown in Figure 11c. From Figure 11d, it can be seen that the simulation and test results have high agreement at the 1^st-, 2^nd-, and 3^rd-order solid frequencies, specifically the response results at the 1^st-order solid frequency are in good conformity. At the 2^nd-order natural frequency, there is a certain degree of deviation between the experimental results and the simulation, but in general, the two trends are more or less the same.

Combining the three tests mentioned above, it is found that there are some differences between the results of the tests and the simulations. This is due to the fact that the piping used in the tests will produce certain errors during the manufacturing and installation process, resulting in differences between the actual piping model and the simulation model. The rubber hose used for connecting the piping in the random vibration test increases the damping of the system, and the measurement noise in the data acquisition process will lead to additional response frequency points in the frequency domain curve, presenting noise clutter, which will affect the accuracy. The errors of the two comparisons are within a reasonable range, and the simulation is considered to be a good fit to the test data, verifying the accuracy of the model and parameter settings.

3. Results and Discussion

3.1. Velocity Field Analysis with Different Fluid Properties

3.1.1. Fluid Pressure

As illustrated in Figure 12a,b, the overall distribution trend of the velocity field at varying outlet pressures exhibits minimal variation, suggesting that outlet pressure has a negligible impact on fluid velocity distribution. However, the velocity field distribution characteristics exhibit notable differences in specific localized regions. In the inlet bend region, the fluid in proximity to the wall is subject to significant viscous shear, resulting in a low velocity flow. Conversely, the center of the bend exhibits a high velocity flow due to the reduction in shear [40]. In the inlet straight pipe region, the velocity of the flow increases gradually in a longitudinal direction, accompanied by an increase in velocity stratification. In the region of the welded joints, there is a notable disparity in the flow velocity distribution between the inlet and outlet ends, with the latter exhibiting a more intricate pattern. As illustrated in the D and E velocity slices, the complex geometry of the welded region perturbs the flow when the fluid flows through the welded region. In the outlet straight pipe region, the overall velocity distribution is almost symmetrical. As the fluid flows, the velocity distribution at different locations within the region varies but gradually tends to become uniform, thereby increasing the stability of the flow state.

To gain a more detailed understanding of the velocity distribution characteristics of the fluid domain, a detailed analysis was conducted on the cross-sectional velocity distributions observed in different parts of the pipeline. The results of this analysis are presented in Figure 12c,d. The results demonstrate that the overall distribution trend of the cross-sectional velocity field of the pipeline is consistent with the increase in fluid pressure. At the inlet bend, the velocity distribution is relatively uniform, with the highest velocity observed in the center region, and a gradual decrease is observed along the pipe wall direction. This is attributed to the viscous force near the pipe wall that reduces the velocity near the wall. In the region of the inlet straight pipe, the fuel is subject to its own gravity, resulting in a velocity gradient perpendicular to the flow direction. This produces significant stratification. In the region of the welded joint, the tee structure gives rise to abrupt, localized changes, which result in flow separation and the formation of backflow regions. The velocity distribution in the region of the outlet straight pipe is gradually uniform, indicating that the flow state tends to stabilize after the fluid flows through the complex structure. This is evidenced by a reduction in turbulence intensity and an increase in regularity of the flow state.

The aforementioned velocity cloud analysis allows for the observation of the spatial distribution of fluid flow rates, as well as the inference of the mechanical action of the fluid within the pipeline and its flow characteristics. This provides a novel perspective for the reduction in flow separation, the enhancement of conveying efficiency, and the optimization of pipeline control.

3.1.2. Fluid Velocity

As illustrated in Figure 13a,b, the velocity stratification gap between the inner and outer sides of the inlet bend diminishes as the flow velocity increases. In the case of low flow velocity, the bend induces a shift in the flow direction, resulting in the formation of a secondary flow that is evident in the distinct velocity differences between the inner and outer sides of the bend. Conversely, in the case of high flow velocity, the fluid’s inertia effect is amplified, enabling it to overcome the centrifugal force generated by the bend and reduce the impact of the secondary flow, thereby promoting a more uniform velocity distribution. The inlet straight pipe experiences a lower flow rate due to the combined effect of gravity and the lower fluid density. As the flow rate increases, the impact of gravity is reduced, resulting in an enhanced inertia effect and a more uniform distribution of flow. In the region of the weld, with the increase in flow velocity, the high-speed zone extends to the side of the pipeline, and the flow is affected by the tee pipe, resulting in a significant disturbance in the velocity distribution.

Similarly, as illustrated in Figure 13c,d, the velocity field distribution is largely consistent, with the exception of the inlet straight pipe region. In the case of low flow velocity, the interaction between the fluid and the pipe wall is greater, resulting in a larger high-speed flow region on the lower side of the pipeline. Conversely, in the case of high flow velocity, the fluid inertia effect increases, while the influence of gravity is diminished, leading to a more concentrated high-speed region and the formation of a narrow band of high-flow velocity region. In the regions of the welded joint and the outlet straight pipe, the flow velocity is predominantly influenced by the geometry of the piping.

The aforementioned analysis demonstrates that the augmentation in flow velocity predominantly influences the velocity distribution within the inlet straight region, whereas the impact on bends, welded joints, and the outlet straight region is comparatively minimal. The augmentation of flow velocity facilitates the overcoming of the geometric constraints and viscous resistance intrinsic to the pipeline, thereby engendering a more stable fluid flow condition. Further improvements to the conveying efficiency and stability of the piping system can be achieved by implementing modifications to the piping design, optimizing the degree of bending, and adjusting the flow conditions.

3.2. Pressure Field Analysis with Different Fluid Properties

Given the pivotal role of pressure distribution in ensuring the structural integrity of the pipeline, comprehensive research has been conducted to elucidate the pressure dynamics within the fluid domain of the pipeline. During the course of an aircraft’s flight, the fuel piping system frequently regulates the flow rate and pressure in accordance with the prevailing flight conditions. Fluctuations in the flow rate within the pipeline will impact the pressure field distribution within the pipeline. Therefore, it is crucial to examine the pressure field alterations under diverse pressure and flow rate circumstances to gain a more profound comprehension and enhance the pipeline design.

As illustrated in Figure 14a,b, in the bend region, the fluid is subjected to the effects of centrifugal force, resulting in a notable increase in pressure on the outer wall surface. With the rise in outlet pressure, the overall pressure rises, whereas pressure gradient along the radius direction remains consistent. In the region of the inlet straight pipe, the fluid pressure distribution is more uniform, which is a consequence of the fluid moving axially along the pipeline without generating additional centrifugal force. In the region of the welded joint situated at the bottom center, the pressure exhibits a gradual decrease from the central point towards the perimeter. In accordance with Bernoulli’s theorem, this result can be attributed to the abrupt alteration in the line’s configuration at the welded joint, which gives rise to a reduction in flow rate, an increase in pressure at the center region, a concomitant rise in flow rate, and a decline in pressure around the perimeter. In the region of the outlet straight pipe, the pressure distribution tends to be uniform, which indicates that the fluid passes through the welded joint to readjust the pressure, thereby achieving a more stable flow.

As illustrated in Figure 14c,d, the pressure field distribution within the pipe is relatively uniform in the low flow rate case, suggesting that the dynamic effect of low velocity fluid on the pipe wall is limited and does not result in a significant pressure difference. In contrast, the value of the ultimate pressure difference increases significantly in the high flow rate case. In accordance with Bernoulli’s principle, an augmentation in velocity gives rise to an escalation in dynamic pressure, which in turn generates a more substantial pressure disparity across disparate regions. Concurrently, under conditions of elevated flow velocity, the impact of intricate structural elements, such as bends and joints, on the fluid dynamics is amplified, leading to phenomena such as fluid acceleration and separation as well as notable alterations in the pressure field within specific regions.

In conclusion, the distribution of fluid pressure within a system is contingent upon the interplay between flow velocity and the configuration of the lines in question. In the design of fuel lines, it is essential to give due consideration to the hydrodynamic properties in order to ensure structural safety and efficiency. By optimizing the curvature of bends, abrupt changes at welded joints can be reduced, ensuring that straight sections are of sufficient length to allow for a stable flow; the aforementioned methods can be employed to mitigate the adverse effects of pressure gradients and enhance the safety of piping systems.

3.3. Stress Analysis of Different Zones via Fluid Properties

3.3.1. Stress Analysis of Fluid Pressure on Welded and Bent Pipe Areas

A study of the stress distribution characteristics in the welded region under various working conditions, as illustrated in Figure 15a,b, reveals that the stress distribution in the welded region remains largely unchanged under different fluid pressures. As fluid pressure increases, the maximum stress also rises in accordance with the pressure gradient. When the fluid pressure increased from 100 KPa to 2000 KPa, the maximum stress in the WZ increased by 99.1474 KPa, which is an increase of 1248.23% compared to 100 KPa. This is due to the fact that an increase in fluid pressure within a pipeline results in a proportional increase in the force acting on the pipe wall. Additionally, due to the discontinuities in the microstructure of the welded area of the pipeline and the disparate material properties observed in each area, the maximum stress in the welded region is concentrated at the junction between the weld zone and the HAZ. Further analysis indicates that the stress gradient in the vicinity of the weld region is considerable, and the elevated stress gradient can result in the fracturing of the material at the micro-scale. This has a notable impact on the structural integrity and service life of the pipeline. Specifically, in fuel-welded pipelines, microcracks may serve as the initial point of pipeline failure, which in turn causes crack propagation and ultimately pipeline rupture [41]. In conclusion, it is essential to consider the stress distribution and material property alterations in the weld region when designing and fabricating fuel-welded pipelines, with the objective of reducing stress concentration in this region.

The stress distribution in the elbow region is studied for various operating conditions, as shown in Figure 15c,d. Under different pressures, the stress trends of the pipeline are generally consistent. As the fluid pressure in the pipe increases from 100 KPa to 2000 KPa, the value of maximum stress in the bend region increases by 1041.03% to 429.38 KPa. Specifically, the stress distribution in the whole region shows a corrugated change along the axial and circumferential directions of the bend, which is related to the flow state of the fluid inside the bend.

In order to conduct a comprehensive examination of the stress trend with length and varying fluid pressures at distinct points along the welded pipeline, stress plots of stress change with length at A_1aA_1b on the upper side of the welded area, A_2aA_2b on the left side, and A_3aA_3b on the lower side under different pressures were constructed. As illustrated in Figure 16a–c, it is evident that the stress levels at disparate locations within the weld region exhibit a pronounced increase with rising fluid pressure. With regard to the upper, left, and lower sides of the weld region, the maximum stresses are observed at the junction between the WZ and the HAZ. In fuel-welded pipelines, the regions exhibiting high variability in stress levels represent potentially hazardous areas of the pipeline. In conclusion, the design of the weld region of the pipeline must consider the continuity of material properties, which can be prevented through the use of transition materials or appropriate heat treatment to avoid significant changes in material properties.

In order to gain a deeper understanding of the relationship between pressure change and length at different points along the bend region, as well as the influence of varying fluid pressures, stress plots were constructed for the inner B_1aB_1b, left B_2aB_2b, and outer B_3aB_3b regions of the bend. These plots illustrate the stress change with length at each of these points, and they were created at different pressures. As illustrated in Figure 16d–f, an increase in fluid pressure is accompanied by a notable rise in stress levels at all positions within the weld region. The maximum stresses in both the interior and the left side of the bend region are concentrated within a range of 110–130 mm. Concurrently, the stresses in the inner and left regions of the bend exhibit an initial increase, followed by a subsequent decrease, while the stresses in the outer region of the bend demonstrate a persistent increase.

3.3.2. Stress Analysis of Fluid Velocity on Welded and Bent Pipe Areas

In order to analyze the variation in stress at different sides of the welded region of the pipeline with respect to the length and different fluid velocities, stress plots are established as illustrated in Figure 17a–c. As the flow velocity increases, the stress level rises in the majority of locations within the welded region. This is primarily attributable to the fact that an increase in flow rate gives rise to an elevation in dynamic pressure, which in turn results in a load being placed on the pipe wall. Concurrently, the stress fluctuations in the curve mirror the intricate flow dynamics of the fluid within the pipeline. The nonlinear stress characteristics in this region are predominantly attributable to the interaction between the dynamic behavior of the fluid and the geometry of the pipeline. In particular, when a sudden change occurs in the welded region, the fluid may separate from the pipe wall, resulting in a localized pressure decrease and a sudden pressure increase near the fluid reattachment point. This phenomenon gives rise to peaks and valleys in the stress distribution [42].

Similarly, in order to analyze the trend of pressure change with length and different flow rates at different sides of the bend region, stress plots are established at the inner, left, and outer regions of the bend region, respectively. From Figure 17d–f, the overall trend of the influence of flow velocity and fluid pressure on the stress in the inner, left, and outer regions of the bend is consistent. According to the fluid–solid coupling mechanism, it can be observed that when the fluid flows inside the pipeline, its velocity change will be converted into pressure on the pipeline wall through the form of fluid dynamic pressure, which in turn generates stresses on the structure. Therefore, in the design of pipe bends, we need to consider not only the structural integrity of the pipeline and the maximum pressure value that it can withstand but also the flow velocity changes that lead to dynamic pressure within the stress distribution.

3.4. Modal Analysis of Pipelines with Different Fluid Characteristics

This section examines the impact of fluid pressure and fluid velocity on the natural frequency of the pipeline. To this end, the inlet velocity of the fluid domain was set at 9 m/s, while the outlet pressures were set at 100, 500, 1000, and 2000 KPa. This allowed for an analysis of the first 20 orders of natural frequency of the fuel-welded pipeline under different pressures. Furthermore, the outlet pressure of the fluid domain was set to 500 KPa, while the inlet velocities were 1, 9, 17, and 25 m/s. This allowed for an analysis of the first 20 orders of natural frequency of the fuel-welded pipeline at four flow rates.

3.4.1. Fluid Pressure Influence

As illustrated in Figure 18, an increase in fluid pressure results in a rise in the natural frequency of the pipeline, particularly in the high-order modes. This result can be attributed to the fact that an increase in fluid pressure causes the pipeline to generate prestress which enhances its resilience to external forces. Consequently, the pipeline exhibits a heightened degree of rigidity, which in turn amplifies the pipeline’s equivalent stiffness, leading to an elevated natural frequency of the fluid–solid coupling mode. Accordingly, when examining the dynamics of the fuel line, it is essential to consider the influence of fluid pressure fluctuations on the fluid–solid coupling effect, particularly in terms of its impact on the frequency of higher-order modes.

3.4.2. Fluid Velocity Influence

As illustrated in Figure 19, the higher-order natural frequency of the pipeline demonstrates an increase with the rise in flow rate, whereas the variation in the lower-order frequency is relatively minimal. This suggests that at low frequencies, the structural vibration is predominantly influenced by the structural stiffness, whereas at high frequencies, the impact of fluid dynamics becomes more pronounced. However, in comparison to the impact of fluid pressure on the natural frequency, the natural frequency is subject to a relatively minor influence from fluid velocity. Consequently, in the subsequent analysis of the random vibration response characteristics of the pipeline, our primary focus will be on the effect of fluid pressure on the random vibration characteristics of the pipeline.

3.5. Impact Analysis of Random Vibration Characteristics of Pipelines

3.5.1. X-Axis Excitation Effect

Since in practice the aircraft is mainly subjected to vibrations along the axial direction of the pipeline stubs, the study of the random vibration characteristics of the pipeline is carried out in the X-direction. The inlet velocity is 9 m/s, and the outlet pressure is 500 KPa. This is to investigate the random vibration characteristics of the pipeline under X-direction excitation. The three Sigma Mises stress characteristics of the specific welded and bend regions are shown in Figure 20a,b, respectively. It was determined that the maximum dynamic stress in the welded region of the pipeline under X-direction excitation is concentrated at the junction of the WZ and the HAZ, and its stress distribution is not uniform. This is attributable to the disparities in the microstructure of disparate regions within the welded joint and the discontinuities among the material properties. Concurrently, the bend region is subjected to unequal loading, which is susceptible to localized stress concentration, and its maximum dynamic stress manifests on the inner side of the bend. This result can be attributed to the geometrical irregularities introduced by the bending of the pipeline and the resulting fluid flow dynamics.

The results of the dynamic response analysis of the welded region of the pipeline indicate that the node with the largest stress response under X-axial excitation represents a critical point of concern. To gain further insights, the stress–frequency response curves of this critical point in the X-direction under X-axial random load excitation were obtained. As illustrated in Figure 21a, the welded region exhibits four distinct resonance response peaks under X-direction excitation. These peaks occur at the first-, third-, seventh-, and twelfth-order intrinsic frequencies, respectively. The maximum stress response is observed in the third-order mode in the X direction.

Similarly, the results of the aforementioned analysis of the dynamic response of the bend region were used to generate the response plot shown in Figure 21b. The results indicate that the resonance peaks are predominantly concentrated at the first- and third-order natural frequencies under random load excitation in the X-direction. Consequently, the first-order resonance peak in this direction occurs at approximately 330 Hz, which suggests that the piping system is subjected to an elevated risk of fatigue at this frequency.

3.5.2. Fluid Pressure Effect

The monitoring points I and II were selected as the research objects. According to the aforementioned set up conditions, the stress–frequency response curves of random vibrations in the X-direction under X-axial excitation were analyzed and obtained for each monitoring point under different fluid pressures. These results are presented in Figure 22. As illustrated in Figure 22a–d, the presence of four resonance peaks at monitoring point I within 0–2000 Hz range is evident upon the increase in fluid pressure. These peaks exhibit a tendency to shift towards the right. In the initial and third resonance regions, the stress response amplitude typically increases with rising fluid pressure. This suggests that the resonance response of the pipeline at these frequency points is amplified under an elevated pressure, potentially due to its geometry and material properties [43]. In the second and fourth resonance regions, the overall decrease in stress response amplitude with an increase in fluid pressure is attributed to the fact that the overall stiffness of the pipeline increases with an increase in fluid pressure. Consequently, the pipeline produces less deformation when subjected to the same external force. Consequently, an increase in fluid pressure results in a reduction in the stress response of the pipeline when subjected to an identical vibration load. A similar trend is observed in the non-resonant region as a whole. The peak stress response is observed to be the greatest at the second resonance region, which indicates that the pipeline is more susceptible to fatigue damage at this frequency point.

As the fluid pressure increases, the maximum stress response amplitude decreases by about 9.94%, indicating that the stress level of the pipeline at the high-frequency resonance point can be reduced by increasing the internal fluid pressure in this direction of excitation, which contributes to the safety of the pipeline.

As illustrated in Figure 22e–g, the emergence of five resonance peaks at monitoring point II within the frequency range of 0–2000 Hz was observed with the increase in fluid pressure. Additionally, a rightward shift in the position of these peaks was noted. In all resonance zones, the stress response amplitude typically increases with rising fluid pressure. The trends observed in the non-resonance regions are analogous, with the stress response peaks occurring at the greatest magnitude within the initial resonance region. This indicates that an increase in fluid pressure is associated with elevated line stress levels at low-frequency vibration. This phenomenon may be attributed to the ease with which low-frequency vibrations propagate through piping systems, coupled with the increased fluid pressure, which amplifies the piping–fluid interaction, thereby resulting in a heightened stress response at the resonance frequency [44]. Fluid pressure has a considerable impact on the maximum stress response magnitude. As pressure increases, the stress response magnitude of the pipeline also rises, with an approximate increase of 22.48%.

In conclusion, the amplitude of the stress response exhibits two distinct trends with increasing fluid pressure across different resonance frequency intervals. In some resonance intervals, the amplitude of the stress response increases with the rise in fluid pressure. Conversely, in other resonance intervals, the amplitude decreases with the increase in pressure. This suggests that the variation in amplitude is influenced by a multitude of factors, including the fluid force acting on the structure and the alteration in structural stiffness. It is thus evident that the impact of fluid pressure on the response characteristics must be taken into account during the design and analysis of pipelines in order to guarantee the safety of pipelines operating under a range of service pressures.

4. Conclusions

This paper adopts a combination of simulation analyses and experimental verification methods, mainly focusing on the aircraft fuel-welded pipeline to carry out the study of fluid–solid coupling mechanical properties. The specific research results and conclusions are as follows:

(1) The microstructural dimensions and mechanical property parameters of the welded area are obtained through metallographic inspection and microtensile testing, based on which a high-precision fuel-welded pipeline finite element model is constructed. It should be noted that although the finite element model with the welded area fully considering the influence of clamps and brackets under the state of airborne loads was established, the geometric deformation caused by the welding process was not investigated. The results of free mode test, constrained mode test, and random vibration test are compared with the simulation to verify the accuracy of the model and parameter settings;

(2) For the analysis of the mechanical behavior and random vibration characteristics of the pipeline under different fluid pressures and velocities, this study shows that fluid pressure and velocity have a significant effect on the fluid flow state and structural stress distribution in the pipeline. Although the flow velocity has a small effect on the velocity field distribution, the increase in flow velocity and the change in pipeline geometry will change the local velocity distribution, which in turn will change the distribution of fluid pressure. Based on the analysis of the structural stresses, it was found that the stresses in the welded area, especially at the junction of the weld zone and the heat-affected zone, were the highest, while the stresses in the bend region varied in a corrugated manner along the axial and circumferential distributions. In addition, the increase in fluid pressure significantly increases the natural frequency of the pipeline, while the flow velocity has less effect on it. The variation in vibration amplitude is affected by a number of factors such as fluid forces on the structure and changes in structural stiffness. It is worth noting that the cyclic variation in stresses in aircraft fuel-welded lines during service may lead to fatigue at the location of the maximum variable stress component, which will need to be addressed in future research;

(3) In the design and manufacture of fuel-welded pipelines, we need to consider the stress distribution in the welded area and the changes to the material properties to reduce the stress concentration in the area. In the case of bends, not only must the structural integrity of the line and the maximum pressure it can withstand be considered but also the effect of dynamic pressure on the stress distribution due to changes in flow velocity. This study provides a basis for the design of aircraft fuel-welded lines, which is of great engineering significance to ensure the stability and safety of the fuel system.