In low-pressure tubes, where gas pressure is subnormal, gas density diminishes as tube pressure lowers, resulting in a more significant rarefaction impact of the gas. The degree of gas rarefaction is generally measured by the Knudsen number [14], expressed as follows:(1) Kn=λL

In the equation, The mean free route of air molecules, denoted by λ, denotes the average distance traveled by a molecule between collisions. L represents the characteristic length of the flow, which is taken as the height of the train body, 0.32 m. Gas flow is classified into four primary regimes according to the Knudsen number [15]: the continuum regime (Kn < 0.01), the slip flow regime (0.01 < Kn < 0.1), the transition regime (0.1 < Kn < 10), and the free molecular flow regime (Kn > 10). In the continuum regime, gas flow can be accurately characterized using a continuum medium model.

Under the simulated operating parameters of this study, the minimum internal pressure of the tube is 0.01 atm, and the internal environmental temperature is 300 K. Under these conditions, the maximum Knudsen number is:(2) Knmax≈2.35×10−5≪0.01

Therefore, under all simulated operating conditions in this study, the fluid within the tube may be categorized as a continuum medium.

At high speeds within a low-pressure tube, the adjacent fluid demonstrates intricate turbulent dynamics. In fluid dynamics calculations, the fluid’s compressibility must be considered when the Mach number surpasses 0.3 [16]. The operating speed of the low-pressure tube maglev train studied in this paper ranges from 600 to 1000 km/h, corresponding to a Mach number range of 0.49 to 0.82. Therefore, the flow field is characterized by the three-dimensional compressible Navier-Stokes equations, with air treated as an ideal gas. The k-epsilon two-equation turbulence model, introduced by Shih et al. [17], is widely employed in industrial computational fluid dynamics. The realizable k-epsilon model, which delivers enhanced accuracy and superior alignment with the physical properties of turbulence relative to the standard k-epsilon model, yields more dependable precision in simulating intricate flow fields, including rotating flows, boundary layer flows with significant pressure gradients, and secondary flows. Therefore, this work selected the realizable k-epsilon model for flow field computations to tackle the low-pressure, high-speed flow conditions in low-pressure tubes [18].

The principal sources of aerodynamic noise in the flow field are monopole, dipole, and quadrupole sources [19,20]. This study’s numerical calculations, utilizing wind tunnel models, assume the train body’s surface is rigid. In real situations, train surfaces often do not undergo much displacement [21]. Therefore, this study primarily analyzes the dipole and quadrupole acoustic sources, excluding the influence of monopole sources to noise. The broadband noise source model for near-field noise prediction employs flow field data derived from Reynolds-Averaged Navier-Stokes (RANS) simulations to ascertain the position and magnitude of major noise sources. This method is advantageous due to its shorter computation time and higher efficiency This approach is beneficial owing to its reduced computing time and enhanced efficiency.

The Curle model assesses the local acoustic power of dipole noise sources. Dipole noise sources arise from the interaction of an object surface in a flow field with an air stream. The radiated power of these noise sources is proportional to the sixth power of the flow velocity:(3) wd∝ρ0D2v6c03

In the equation, w_d represents the radiated power of the dipole noise source, ρ₀ indicates the density of the flow field, and D denotes the characteristic length of the noise source. The total sound power radiated by a solid surface, P_SA, can be approximated as [22]:(4) PSA=∫Ac(y)12ρ0πc03[∂p∂t]2dS(y)=∫I(y)dS(y)

In the equation, A_c represents the associated region, S denotes the solid surface, and I (y) indicates the dipole noise sources on the solid surface in a flow field.

The surface sound power level is expressed as:(5) LSAP=10logPSAPref

In the formula, L_SAP represents the surface sound power level, and P_ref denotes the reference sound power, where P_ref = 10⁻¹² W/m².

The Proudman model evaluates the local power of quadrupole noise sources. Quadrupole noise sources are caused by viscous stresses resulting from the interaction between fluids. The radiated power of these noise sources is proportional to the eighth power of the flow velocity:(6) wq∝ρ0D2v8c05

In the formula, w_q represents the radiated power of the quadrupole noise source.

The sound power P_A produced per unit volume by quadrupole noise sources in isotropic turbulence is expressed as [23]:(7) PA=αρ0(u3l)u5c05

In the formula, u represents the turbulence velocity, l shows the turbulence length scale, while α denotes the model constant.

The sound power level is expressed as:(8) LAP=10logPAPref

In the equation, L_AP represents the sound power level, and P_ref denotes the reference sound power, where P_ref = 10⁻¹² W/m³.

This study utilizes a 1:10 scale model train, simplifying complex components such as the pantograph, windshields, and bogies due to computing limitations. The model consists of a three-car configuration (head, middle, and tail) and measures 5 m in length, with a maximum width of 0.37 m and a height of 0.32 m (refer to Fig. 1). A cylindrical geometric model, positioned at the middle height of the train body, features a blockage ratio of 0.11 with a radius (R) of 0.65 m. The total length of the tube is set to 40 m, with the front of the train positioned 10 m from the tube entrance. To avert backflow, a space of 30 m is preserved between the train’s front and the tunnel outlet [18].The levitation height of the train is 0.037 m, and the train-tube model is depicted in Fig. 2. A monitoring line (Line 1) is positioned along the longitudinal centerline of the train body, at the midline above the train’s surface, to evaluate the distribution of sound power levels released by the train, as seen in Fig. 3.

This study analyzes a steady-state flow field. Assuming that the train remains stationary during the numerical simulation, the inlet boundary velocity is modified to simulate the train’s continuous motion through the tube. The free-flow boundary (non-reflective Riemannian boundary) is employed at the pipe’s intake and outlet to facilitate the natural exit of fluid while preserving continuity and stability in the fluid dynamics of the computational domain. The train is depicted as a stationary no-slip surface, but the pipeline is characterized as a dynamic no-slip wall, with its velocity corresponding to the airflow speed in the wind tunnel model. Additionally, both the train and pipeline walls experience adiabatic thermal boundary conditions.

This study examines train speeds ranging from 600 to 1000 km/h within a low-pressure pipeline. The low-pressure conditions inside the pipeline are simulated by adjusting the internal air pressure, which ranges from 0.1 to 0.01 atm. The initial ambient temperature within the pipeline is set at 300 K.

The numerical simulation employs an unstructured hybrid mesh. Triangular meshes facilitate surface reconstruction for both the train and the pipeline, whereas hexahedral meshes are utilized for discretizing the computational domain of the train. High-resolution meshing is essential near the train body during aerodynamic noise numerical calculations to accurately represent the flow characteristics within the boundary layer. Specifically, the train surface is discretized with a detailed boundary layer mesh consisting of 12 layers, a growth factor of 1.2, and the total boundary layer thickness is set to 4 mm. Additionally, the surface mesh is further improved around the train body, with multiple refinement blocks located near the front and rear cars. The volumetric mesh is additionally improved in certain areas. Fig. 4 depicts the mesh configuration for a maglev train-pipeline model, distinguished by a blockage ratio of 0.11.

To achieve mesh independence, three mesh sets with consistent methods but differing densities were created by modifying the mesh resolution of the computational domain. The meshes consisted of 7.03, 16.70, and 26.30 million elements, respectively. Table 1 compares the aerodynamic drag values for the complete vehicle with the dipole noise source value at a specific location on the intermediate car surface for these three mesh sizes. The results demonstrate that the aerodynamic drag values derived from the medium-density mesh show negligible differences when compared to those obtained from the fine mesh. After evaluating computational efficiency and correctness, the medium-density mesh was selected for the further calculations. This meshing approach preserves the y+ values across most of the train’s surface within the range of 1 to 30, conforming to the specifications of the realizable k-epsilon two-layer model. Fig. 5 illustrates the distribution of the y+ values.

This study confirms the numerical approach and mesh accuracy by comparing simulation results with surface pressure data from wind tunnel measurements on a maglev train, given the close link between dipole and quadrupole noise sources and the train’s surface pressure [24,25]. Fig. 6 presents the comparison results of the train surface pressure coefficient. The formula used to determine the surface pressure coefficient of the train is as follows:

(9) Cp=P0.5ρ0v2

In the above-mentioned formula, P represents the surface pressure on the train, C_p denotes the pressure coefficient, ρ₀ stands for the air density, and v signifies the speed of the train.

The numerical simulation results, as illustrated in the Fig. 6, closely align with the wind tunnel test data, exhibiting variances typically within 5%, thereby satisfying the engineering accuracy standards. Consequently, the numerical simulation method employed in this work is confirmed to be valid.

To enhance the validation of the noise source model’s accuracy, the aerodynamic noise produced by a high-speed train traversing the tunnel beneath Chengdu Tianfu Airport Station was chosen as a reference case. The train was moving at a velocity of 350 km/h through a tunnel with a cross-sectional area of approximately 100 m². The measurement points, as illustrated in Fig. 7, were positioned at three locations: Point A (on the windward side of the driver’s cabin), Point B (on the left sidewall of the sightseeing portion), and Point C (on the left rear sidewall of the passenger compartment). A numerical simulation model was created under conditions that mirrored the experiment. Table 2 delineates a comparison between the experimental data and the results of the numerical simulation. The table indicates that, owing to surface effects throughout the experiment, the numerical simulation results were generally lower than the experimental measurements, with a maximum divergence of 1.76 dB. This inconsistency indicates that the noise model utilized in this research is rather precise.

Aerodynamic noise sources are derived from flow field data, rendering precise flow field data crucial for noise simulation. A simulation was conducted for a train model functioning at a low pressure of 0.05 atm, with a blockage ratio of 0.11, and an operation speed of 600 km/h. Fig. 8 depicts the distribution of the flow field along the longitudinal portion of the pipeline. The velocity distribution reveals a stagnation point at the front of the lead train, whereas the flow field surrounding the tail car is more intricate, displaying turbulence. As the airflow traverses the streamlined intermediate car section, the cross-sectional area diminishes, resulting in a progressive increase in velocity. Ultimately, as the airflow moves through the streamlined posterior segment, the cross-sectional area expands, which leads to a reduction of flow velocity. The pressure distribution illustrated in Fig. 8b reveals that the peak positive pressure is located at the car’s nose tip. Thereafter, the pressure progressively diminishes at the interface between the streamlined head car and the main body. The pressure variation is minimal along the middle car. As the cross-sectional area enlarges near the tail, the flow speed decreases, and the pressure increases.

The results of the steady-state flow field study were utilized to implement the broadband noise source model for forecasting the distribution of noise sources on the train body surface. The computed data are represented in terms of surface sound power level and overall sound power level. The surface sound power level quantifies the intensity of dipole noise sources per a unit area of the train surface, generated by boundary layer turbulence. The sound power level measures the intensity of quadrupole noise sources per unit volume of the flow field, originating from isotropic turbulence around the train.

Fig. 9 illustrates the distribution of surface sound power levels on the train. The figure clearly indicates substantial fluctuations in surface sound power levels within the streamlined regions of the head and tail cars, reflecting heightened turbulence in these areas. The changes in surface sound power level are less pronounced in the middle car. The surface sound power level in the streamlined area of the head car is more than the tail car, with the maximal surface sound power level attaining 77 dB for the front car.

Fig. 10 illustrates the dispersion of the sound power level of the train. The sound power levels in the streamlined regions of the head and tail cars exhibit considerable variation, while the middle car shows less variation. The head car exhibits a maximum acoustic power level of 79 dB, somewhat exceeding the surface sound power level. The pipeline’s presence induces compression and expansion of airflow at the front and rear of the train, leading to elevated sound power levels in the train’s wake region.

Fig. 11 depicts the distribution of sound sources along the longitudinal centerline (Line 1) of the train at different speeds, with an internal low-pressure level of 0.05 atm and a blockage ratio of 0.11. The distribution patterns of sound sources exhibit similarities at different speeds, with notable discrepancies in the streamlined areas of the head and tail cars, whilst the middle car demonstrates minimal variance. This phenomenon arises from the pipeline, which induces compression and expansion of airflow around the train’s head and tail, leading to pronounced turbulence in these regions. As speed increases, both dipole and quadrupole noise sources of the train intensify. When the train travels at speeds of less than 1000 km/h, the sound source intensity in the streamlined area of the head car is greater than that of the tail car. As velocity escalates, the turbulence effects in the wake region intensifies, leading to higher sound source intensity at the tail. When the speed of the train is up to 1000 km/h, the sound intensity in the streamlined region of the tail car surpasses that in the head car. The dipole noise sources for the middle automobile are 77, 96, and 121 dB at varying speeds, but the quadrupole noise sources are 79, 108, and 132 dB, respectively. The intensity of quadrupole noise sources surpasses that of dipole noise sources, suggesting that At train operating speeds greater than 600 km/h, quadrupole noise sources progressively dominates in low-pressure tube maglev trains.

Fig. 12 shows the distribution of sound sources along the longitudinal centerline (Line 1) of the train at different internal low-pressure levels, with a blockage ratio of 0.11 and a speed of 600 km/h. The distribution patterns of sound power levels remain consistent across different low-pressure levels, with notable fluctuations observed in the streamlined regions of the head and tail cars, whilst the middle car shows minimal variation. When the low-pressure level decreases from 0.2 to 0.01 atm, the air pressure diminishes, the gas density within the tube reduces, and the turbulence strength in the flow field lessens, resulting in a downward trend in the train’s sound source. The average surface sound power levels of the middle car at different low-pressure levels are 103, 77, and 60 dB, respectively, while the sound power levels are 115, 79, and 40 dB, respectively. When the internal low pressure level ranges from 0.2 to 0.05 atm, the average sound power level of the middle car surpasses the surface sound power level, signifying that the intensity of quadrupole noise sources exceeds that of dipole noise sources. As the low-pressure level decreases, the disparity between quadrupole and dipole noise sources diminishes. When the low pressure level decreases to 0.01 atm, the sound power level of the train operating at 600 km/h becomes in general lower than the surface sound power level, and the intensity of quadrupole noise sources significantly weakens. The pipeline’s presence amplifies the fluid’s turbulent motion around the train body, while decreasing the internal low pressure significantly mitigates aerodynamic noise in low-pressure tube trains.

This study concentrates on the average noise source values of the middle car for quantitative impact analysis, as the substantial changes in noise from the head and tail cars render them inadequate as reference points. According to Reference [26], there is a discernible relationship between the sound power level of the train L and the logarithm of its speed, and the relationship can be expressed as:(10) L=a+blog⁡(v/v0)+c[log⁡(v/v0)]2

In the equation, v represents the operational speed of the train, v₀ denotes the reference speed, which is specified as 600 km/h in this study, and a, b, and c are the regression coefficients.

Table 3 provides the average surface sound power level and sound power level of the middle car at various train speeds, with a low-pressure level of 0.05 atm and a blockage ratio of 0.11. By substituting these results into Eq. (10) and applying the least squares approach to ascertain the unknown parameters, the values of a, b, and c may be individually determined. This enables the derivation of the relationship between the dipole noise source L_SA and quadrupole noise sources L_A and the operating speed of the train.

(11) LSA=77+92.50log⁡(v/v0)+477.35[log⁡(v/v0)]2 (12) LA=79+223.46+log⁡(v/v0)+69.85[log⁡(v/v0)]2

The fitted formula indicates that at a train speed of v = 700 km/h, the dipole sound source level L_SA is 85 dB and the quadrupole sound source level L_A is 94 dB. Numerical simulations, in contrast, yield a dipole sound source level of 82 dB and a quadrupole sound source level of 91 dB for the middle car, with an error margin of less than 3.6%. The comparison with the results presented in Fig. 13 demonstrates that the fitting is highly precise. Therefore, the formula can be effectively used to predict noise levels within the range of 600 to 1000 km/h, providing a valuable reference for pertinent study.

Alterations in the low-pressure level within the pipeline will result in modifications to the gas density and sound speed. The relationships between the pressure, sound speed, and density of an ideal gas are described by the following equations:(13) p=ρ0RT (14) c02=Bρ0

In these equations, R denotes the molar gas constant, T represents the gas temperature, and B signifies the bulk modulus of the gas.

Combining Eqs. (3) and (6), the relationship between the sound power level of the train L and the logarithm of the internal pipeline pressure can be shown as follows:(15) L=d+elog⁡(p/p0)+f[log⁡(p/p0)]2

In the equation, p represents the internal pipeline pressure, p₀ denotes the reference pressure (set to 0.01 atm in this study), and d, e, and f are the regression coefficients.

The average surface sound power level and sound power level of the middle car at different low-pressure levels within the pipeline, with a train speed of 600 km/h and a blockage ratio of 0.11, are displayed in Table 4. By entering these values into Eq. (15) and applying the least squares approach, the unknown parameters d, e, and f can be determined. This allows for the derivation of the relationship between the dipole noise sources L_SA and quadrupole noise sources L_A and the low-pressure level inside the tube.

(16) LSA=60+15.17log⁡(p/p0)+13.75[log⁡(p/p0)]2 (17) LA=40+55.30log⁡(p/p0)+1.80[log⁡(p/p0)]2

According to the fitted formula, at an internal pipeline low pressure level of p = 0.1 atm, the dipole sound source level L_SA is 89 dB, whereas the quadrupole sound source level L_A is 97 dB. Numerical simulations show that the dipole and quadrupole sound source levels for the intermediate cars are 89 and 99 dB, respectively, with a margin of error of 2%, and the comparison results are shown in Fig. 14. The fitting is precise and serves as a significant reference, facilitating accurate predictions of aerodynamic noise for low-low pressure tube maglev trains.