Numerical Study on the Aeroacoustic Performance of Different Diversion Strategies in the Pantograph Area of High-Speed Trains at 400 km/h

: The speed increase in high-speed trains is a critical procedure in the promotion of high-speed railway technology. As an indispensable and complex structure of high-speed trains, the pantograph’s aerodynamic drag and noise is a signiﬁcant limitation in the speed increase process of high-speed trains. In the present study, the hybrid method of large eddy simulation (LES) and Ffowcs Williams-Hawkings (FW-H) acoustic analogy is applied to analyze the aerodynamic and aeroacoustic performances of pantograph installed in different ways, i.e., sinking platform and fairing. The results of simulation show that the application of pantograph fairing can reduce the aerodynamic drag greatly. In addition, compared with the pantographs installed alone on the train roof, the installation of the sinking platform brings about 2 dBA reduction in sound pressure level (SPL). Meanwhile, the utilization of the pantograph fairing mainly decreases the noise in the frequency band above 1000 Hz and the largest SPL reduction is up to 3 dBA among the monitoring points. Further analysis shows that the inﬂuence of different diversion strategies on the spectral characteristics actually attenuates the dominant frequency of the panhead. In the horizontal plane, the noise directivity of the pantograph installed with a fairing is similar to the pantograph installed alone on the train roof.


Introduction
High-speed trains (HSTs) stand out in the competition of many means of transportation by virtue of their high efficiency, safety, comfort, and environmental friendliness. With the deepening of theoretical research and the gradual maturity of technical development, China's 350 km/h HSTs have been put into operation, and the development of higherspeed trains has become the pursuit goal of researchers in related industries at home and abroad [1,2]. However, there are significant aerodynamic problems in the process of speed increase in HSTs. Aerodynamic noise is proportional to the sixth power of the train running speed. When the train running speed exceeds 300 km/h, aerodynamic noise will be the main noise source, and the proportion of aerodynamic drag will also be significantly increased. Strong annoying noise not only affects the comfort of passengers, but also interferes with the normal life of surrounding residents [3,4]. Therefore, it is essential to control the aerodynamic noise.
Talotte [5] considered that the critical aspect of any acoustical study is to ascertain noise sources, and revealed that the predominant aerodynamic noise sources of HSTs are the pantograph with smoother insulators had no prominent effect in terms of noise, while the pantograph with smoother base frame had lower drag and noise. Shaltout et al. [20] compared different cross-sectional panheads and showed that at 250 km/h, the elliptical cross-sectional panhead could reduce the sound pressure level (SPL) of the pantograph by over 20%.
The above numerical studies for pantograph noise reduction can be divided into changing the installation type or shape of pantograph [21][22][23]. Moreover, the variation of aerodynamic drag under the new type of pantograph is rarely considered at the same time. Furthermore, changing the rod type of the pantograph may have an impact on its receiving current. Studies have shown that the pantograph cover becomes a feasible solution to achieve the drag reduction requirements at higher speeds. However, its effect on aerodynamic noise is still unclear and remains to be specified, and this is the purpose of this paper.

Numerical Simulation Methodology
The numerical methods of aerodynamic noise are categorized into direct noise simulation and hybrid method. Due to the large difference in scale and energy between the acoustic and flow field, a discrete format with high order accuracy, low dissipation, and low dispersion is needed to improve the solution accuracy, which greatly reduces the feasibility of the direct method. Particularly for HSTs and their components with complex geometry, the hybrid method is usually used to predict the aerodynamic noise. The hybrid solution divides the numerical simulation of aeroacoustic performance into near-field and far-field solutions, i.e., sound source simulation and sound propagation solution.
In the present study, CFD simulations are proceeded by ANSYS Fluent software. The LES turbulence model is used to solve the flow-field structure near the pantograph first, and then the FW-H analogy equation is applied to predict the far-field noise of the pantograph area.

LES Method
Turbulent flows are characterized by disordered, eddying fluid motions over a variety of length scales. The size of largest eddies is almost comparable to the characteristic length of the mean flow, and the dissipation of turbulence kinetic energy can be ascribed to the smallest scales. In LES, the basic idea is to establish a mathematical filter function to separate and filter large-scale eddies and small-scale eddies in turbulent flows, where the turbulent motion of large-scale eddies is resolved directly by the Navier-Stokes equation (N-S equation), and small-scale eddies are modeled by a closed model to show the dissipation relation for large-scale quantities [24].
The Filter function equation is as follows: where φ is the transient turbulent fluctuation volume and V denotes the occupied geometric space of the governing volume. The N-S equation for a compressible fluid filtered by a function takes the form of: ∂ρ ∂t where τ ij is the subgrid-scale stresses term and τ ij is defined as τ ij = ρu i u j − ρu i ·u j . Moreover, u is the flow velocity, ρ is the density, and p is the filtering pressure. Using the filtering operation, the subgrid-scale stresses are introduced and require modeling. To close the filtered N-S Equations (2) and (3), the subgrid-scale (SGS) model based on Boussinesq hypothesis is employed widely. The corresponding SGS model is: where µ t is the subgrid-scale turbulent viscosity, τ kk is the isotropic part of the subgrid-scale stresses, and S ij is the rate-of-strain tensor for the resolved scale defined by Equation (5): The Smagorinsky-Lilly model is chosen here and µ t is defined as µ t = ρL 2 s 2S ij S ij , where L s represents the mixing length for subgrid scales.

Ffowcs Williams-Hawkings Acoustics Analogy
In 1969, Ffowcs Williams-Hawkings applied the generalized function method to extend Curle's result to the sound generation problem of arbitrary moving surfaces in an unbounded fluid. The FW-H equation is essentially an inhomogeneous wave equation that can be derived by manipulating the continuity equation and N-S equation [25,26]. It can be written as: where u i and v i represent the flow velocity and surface velocity component in the x i direction, u n and v n represent the flow velocity and surface velocity normal to the surface, δ(f ) is the Dirac delta function, and H(f ) is the Heaviside function. In addition, p denotes the sound pressure at the far-field (p = p − p 0 ), a 0 is the far-field sound speed, and T ij is the Lighthill stress tensor, defined as: where P ij is the compressive stress tensor. Equation (6) contains the effects of all the sound sources in the flow field: The first term on the right reflects the quadrupole sound caused by the turbulent flow, the second term is the dipole sound caused by the solid surface force, and the third term is the monopole sound caused by the solid boundary motion.
In the present study, the pantograph surfaces are set as integration surfaces and can be regarded as arbitrary rigid bodies; namely, the fluctuating volume quantity turns to zero. Therefore, there is no need to consider the monopole sources. Meanwhile, the Mach number (M) is set as 0.33 and the corresponding incoming flow velocity is 400 km/h. Moreover, the authors of [14] noted that the ratio of quadrupole source intensity to the dipole source intensity in flow field is proportional to M 2 . The noise intensity induced by quadrupole sources is significantly less than the dipole source, and thus the quadrupole sources can be ignored.

Validation of the Methods
Herein, we considered the lack of aerodynamic noise experiment data for real pantographs and the inevitable simplification of pantographs for numerical simulations. For the purpose of verifying the reliability of the computational method, a classical finite length cylindrical flow is selected for numerical simulation in this paper. The rod-airfoil experiment was carried out by Jacob et al. [27] and was introduced as a validation case in a previous study on pantograph noise [13]. The diameter of the rod (D) obtained for the experiment is 10 mm, and the spanwise length is 30 D. The incoming flow velocity is 72 m/s and the free flow turbulence intensity is set as 0.8%. The Reynolds number based on the diameter is about 4.8 × 10 4 , based on rod diameter. Figure 1 displays the computational domain for the validation. The size of 40 D in the stream-wise direction and 20 D in the corresponding normal direction is employed. In addition, considering the computational resources due to the large amount of mesh brought about by the experimental spanwise length, here the spanwise length is chosen as πD according to references [13,[28][29][30]].

Validation of the Methods
Herein, we considered the lack of aerodynamic noise experiment data for real pantographs and the inevitable simplification of pantographs for numerical simulations. For the purpose of verifying the reliability of the computational method, a classical finite length cylindrical flow is selected for numerical simulation in this paper. The rod-airfoil experiment was carried out by Jacob et al. [27] and was introduced as a validation case in a previous study on pantograph noise [13]. The diameter of the rod (D) obtained for the experiment is 10 mm, and the spanwise length is 30 D. The incoming flow velocity is 72 m/s and the free flow turbulence intensity is set as 0.8%. The Reynolds number based on the diameter is about 4.8 × 10 4 , based on rod diameter. Figure 1 displays the computational domain for the validation. The size of 40 D in the stream-wise direction and 20 D in the corresponding normal direction is employed. In addition, considering the computational resources due to the large amount of mesh brought about by the experimental spanwise length, here the spanwise length is chosen as πD according to references [13,[28][29][30]. The computational mesh is shown in Figure 2. Two refinement zones are used to refine the volume in the wake flow region of the rod. To make y+ values meet the requirement and y + < 1 in this paper; the first layer thickness is 0.004 mm and the growth ratio of 1.1 is given in the wall-normal direction. The radial grid near the wall is generated according to reference [13,28]. The total number of grid cells is about 1 million. Moreover, the size of grid cells in the spanwise direction is about 1 mm and the spanwise direction is uniformly distributed with about 33 grid cells. In a previous study, the authors of [31] found that the two-point pressure correlation length is about 3 D, indicating that the spanwise space in this study is physically reasonable. To be consistent with the study by Yu et al. [13], the velocity inlet and pressure outlet boundary conditions are prescribed. The incoming free flow velocity is 72 m/s, which is consistent with the experiment. Of note, both the velocity inlet and pressure far-field The computational mesh is shown in Figure 2. Two refinement zones are used to refine the volume in the wake flow region of the rod. To make y+ values meet the requirement and y+ < 1 in this paper; the first layer thickness is 0.004 mm and the growth ratio of 1.1 is given in the wall-normal direction. The radial grid near the wall is generated according to reference [13,28]. The total number of grid cells is about 1 million. Moreover, the size of grid cells in the spanwise direction is about 1 mm and the spanwise direction is uniformly distributed with about 33 grid cells. In a previous study, the authors of [31] found that the two-point pressure correlation length is about 3 D, indicating that the spanwise space in this study is physically reasonable.
Herein, we considered the lack of aerodynamic noise experiment data for real pantographs and the inevitable simplification of pantographs for numerical simulations. For the purpose of verifying the reliability of the computational method, a classical finite length cylindrical flow is selected for numerical simulation in this paper. The rod-airfoil experiment was carried out by Jacob et al. [27] and was introduced as a validation case in a previous study on pantograph noise [13]. The diameter of the rod (D) obtained for the experiment is 10 mm, and the spanwise length is 30 D. The incoming flow velocity is 72 m/s and the free flow turbulence intensity is set as 0.8%. The Reynolds number based on the diameter is about 4.8 × 10 4 , based on rod diameter. Figure 1 displays the computational domain for the validation. The size of 40 D in the stream-wise direction and 20 D in the corresponding normal direction is employed. In addition, considering the computational resources due to the large amount of mesh brought about by the experimental spanwise length, here the spanwise length is chosen as πD according to references [13,[28][29][30]. The computational mesh is shown in Figure 2. Two refinement zones are used to refine the volume in the wake flow region of the rod. To make y+ values meet the requirement and y + 1 in this paper; the first layer thickness is 0.004 mm and the growth ratio of 1.1 is given in the wall-normal direction. The radial grid near the wall is generated according to reference [13,28]. The total number of grid cells is about 1 million. Moreover, the size of grid cells in the spanwise direction is about 1 mm and the spanwise direction is uniformly distributed with about 33 grid cells. In a previous study, the authors of [31] found that the two-point pressure correlation length is about 3 D, indicating that the spanwise space in this study is physically reasonable. To be consistent with the study by Yu et al. [13], the velocity inlet and pressure outlet boundary conditions are prescribed. The incoming free flow velocity is 72 m/s, which is consistent with the experiment. Of note, both the velocity inlet and pressure far-field To be consistent with the study by Yu et al. [13], the velocity inlet and pressure outlet boundary conditions are prescribed. The incoming free flow velocity is 72 m/s, which is consistent with the experiment. Of note, both the velocity inlet and pressure far-field conditions are compared here and the results show few discrepancies. The latter is used in the following simulations of the pantograph. In addition, the turbulence intensity is set to 0.8%. For the spanwise and normal directions, symmetry boundary conditions are used. Contrarily to the periodic condition, the symmetry condition only makes the velocity components in the z-direction disappear and does not correlate the aerodynamic flow field variables in the existing planes. In addition, the two-point spanwise pressure correlation on the separate rod was in good agreement with the experimental results, which is very important for acoustic calculations and cannot be acquired with the periodic condition. Therefore, the symmetry condition should be a better choice here [32]. Table 1 presents the comparison of the drag coefficients. Of note, the average drag coefficient and its fluctuating values are in good agreement with those reference experimental results, indicating the rationality of the computation method in this study. Herein, we considered the effect of spanwise correlation, according to reference [35]. At the sound monitoring point, which is located at a distance of 185 D in the normal direction of the flow, the far-field SPL versus the Strouhal number is displayed in Figure 3. It is shown that the base level of the spectrum is captured. Specifically, the acoustic peak of nearly 90 dB appears at St = 0.197, indicating the shedding frequency of Karman vortex. Meanwhile, the second and third modes can also be accurately predicted by the present numerical simulations. This performance indicates that the current computational methodology is capable of qualitatively predicting aerodynamic noise.
Contrarily to the periodic condition, the symmetry condition only makes the velocity components in the z-direction disappear and does not correlate the aerodynamic flow field variables in the existing planes. In addition, the two-point spanwise pressure correlation on the separate rod was in good agreement with the experimental results, which is very important for acoustic calculations and cannot be acquired with the periodic condition. Therefore, the symmetry condition should be a better choice here [32]. Table 1 presents the comparison of the drag coefficients. Of note, the average drag coefficient and its fluctuating values are in good agreement with those reference experimental results, indicating the rationality of the computation method in this study. Herein, we considered the effect of spanwise correlation, according to reference [35]. At the sound monitoring point, which is located at a distance of 185 D in the normal direction of the flow, the far-field SPL versus the Strouhal number is displayed in Figure 3. It is shown that the base level of the spectrum is captured. Specifically, the acoustic peak of nearly 90 dB appears at St = 0.197, indicating the shedding frequency of Karman vortex. Meanwhile, the second and third modes can also be accurately predicted by the present numerical simulations. This performance indicates that the current computational methodology is capable of qualitatively predicting aerodynamic noise.

Computational Models
The pantograph is a current receiving device, in which the train receives current from the contact wire of the catenary. It is installed on the roof of the train, and when the pantograph is raised, the current can be introduced into the train carriage through the contact

Computational Models
The pantograph is a current receiving device, in which the train receives current from the contact wire of the catenary. It is installed on the roof of the train, and when the pantograph is raised, the current can be introduced into the train carriage through the contact between the components and the contact wire. The pantograph used in this study is a single-arm pantograph, which consists of upper arm bar, lower arm bar, balance bar, bow head, base frame, insulator, camera, and other parts as shown in Figure 4. In addition, the height of H is from the bottom of the train to the top of the pantograph, which is defined as the characteristic length in the numerical simulation, as depicted in Figure 5. When a sinking platform occurs, the height of the pantograph remains the same. The sinking height is 0.15 m, which is relatively small compared to the characteristic length H. single-arm pantograph, which consists of upper arm bar, lower arm bar, balance bar, bow head, base frame, insulator, camera, and other parts as shown in Figure 4. In addition, the height of H is from the bottom of the train to the top of the pantograph, which is defined as the characteristic length in the numerical simulation, as depicted in Figure 5. When a sinking platform occurs, the height of the pantograph remains the same. The sinking height is 0.15 m, which is relatively small compared to the characteristic length H.

Computational Domain and Numerical Setup
The computational model used in this paper is the pantograph placed on the train roof model with three cars to ensure the reasonability of the inflow and reduce the mesh amount. In addition, the computational domain is sufficiently large to eliminate the effects of the surrounding boundaries in the numerical simulation [36]. The origin of the used coordinate is at the bottom center of the train roof model and the inlet flow is due to the negative direction of x-axis. Moreover, the size of the computational domain is 60 × 20 × 10 H, where H = 2.85 m, as depicted in Figure 5. To ensure that the inflow develops fully before reaching the head car, the upstream length is 10 H. Similarly, the downstream length is longer than 20 H to ensure that the vortex shedding in the wake flow is welldeveloped. The computational domain is shown in Figure 6. This study mainly focuses on the effects of pantograph diversion strategies on train aerodynamic and aeroacoustic performance. Three configurations designed here, including the pantograph installed on the train roof alone, on the train roof sinking platform, and the pantograph fairing adopted on the train roof are described in Figure 7 and Table 2. single-arm pantograph, which consists of upper arm bar, lower arm bar, balance bar, bow head, base frame, insulator, camera, and other parts as shown in Figure 4. In addition, the height of H is from the bottom of the train to the top of the pantograph, which is defined as the characteristic length in the numerical simulation, as depicted in Figure 5. When a sinking platform occurs, the height of the pantograph remains the same. The sinking height is 0.15 m, which is relatively small compared to the characteristic length H.

Computational Domain and Numerical Setup
The computational model used in this paper is the pantograph placed on the train roof model with three cars to ensure the reasonability of the inflow and reduce the mesh amount. In addition, the computational domain is sufficiently large to eliminate the effects of the surrounding boundaries in the numerical simulation [36]. The origin of the used coordinate is at the bottom center of the train roof model and the inlet flow is due to the negative direction of x-axis. Moreover, the size of the computational domain is 60 × 20 × 10 H, where H = 2.85 m, as depicted in Figure 5. To ensure that the inflow develops fully before reaching the head car, the upstream length is 10 H. Similarly, the downstream length is longer than 20 H to ensure that the vortex shedding in the wake flow is welldeveloped. The computational domain is shown in Figure 6. This study mainly focuses on the effects of pantograph diversion strategies on train aerodynamic and aeroacoustic performance. Three configurations designed here, including the pantograph installed on the train roof alone, on the train roof sinking platform, and the pantograph fairing adopted on the train roof are described in Figure 7 and Table 2.

Computational Domain and Numerical Setup
The computational model used in this paper is the pantograph placed on the train roof model with three cars to ensure the reasonability of the inflow and reduce the mesh amount. In addition, the computational domain is sufficiently large to eliminate the effects of the surrounding boundaries in the numerical simulation [36]. The origin of the used coordinate is at the bottom center of the train roof model and the inlet flow is due to the negative direction of x-axis. Moreover, the size of the computational domain is 60 × 20 × 10 H, where H = 2.85 m, as depicted in Figure 5. To ensure that the inflow develops fully before reaching the head car, the upstream length is 10 H. Similarly, the downstream length is longer than 20 H to ensure that the vortex shedding in the wake flow is well-developed. The computational domain is shown in Figure 6. This study mainly focuses on the effects of pantograph diversion strategies on train aerodynamic and aeroacoustic performance. Three configurations designed here, including the pantograph installed on the train roof alone, on the train roof sinking platform, and the pantograph fairing adopted on the train roof are described in Figure 7 and Table 2.        Table 2. Three configurations and their so-called names.

/ Configurations
Case 1 Pantograph installed on the train roof alone Case 2 Pantograph installed on the sinking platform Case 3 Pantograph faring installed on the train roof To preclude the influence of grid, a grid independence study is carried out. Three grid resolutions including coarse, medium, and fine grids are set and the total drag coefficient of case 1 is compared, as shown in Table 3. It can be found that the calculated drag coefficient of case 1 shows a larger difference when the coarse and medium grids are adopted. In addition, the relative error in the calculated results between the medium and fine grid models is less than 2%. This indicates that the flow field computation results are reliable when the computational domain reaches the medium grid scale. The medium computational mesh is depicted in Figure 8. It is discretized by the poly-hexcore unstructured grid. The grid size of the pantograph surface is less than 8 mm to guarantee the prediction accuracy of the pantograph aerodynamic noise calculation. In addition, the boundary layer grids are generated near the pantograph surface, obtaining the first layer thickness of 0.12 mm which is normal to the wall, to ensure y+ ≈ 1 with a growth ratio of 1.2. To capture the flow details, the refinement zone is carried out on the leeward side of the pantograph and the tail flow area of the roof model.
The pressure far-field condition and the pressure outlet condition are prescribed for the inlet and outlet boundary here, respectively. Both side planes of the computational model and the top planes are set as symmetry boundaries. To simulate the ground effects, the ground is set to be the moving wall condition with an incoming velocity of 400 km/h. The physical time step of large eddy simulation is considered as 5 × 10 −5 s, and the corresponding aerodynamic noise for analysis frequency is 10 kHz, according to the Nyquist sampling theorem [37]. Moreover, 25 sub-steps are iterated within each time step and a total of 8000 time steps are calculated. The first 2500 time steps are calculated to ensure that the turbulent flow field is fully developed, and the remaining 5500 time steps are used to extract the noise source information. coefficient of case 1 shows a larger difference when the coarse and medium grids are adopted. In addition, the relative error in the calculated results between the medium and fine grid models is less than 2%. This indicates that the flow field computation results are reliable when the computational domain reaches the medium grid scale. The medium computational mesh is depicted in Figure 8. It is discretized by the poly-hexcore unstructured grid. The grid size of the pantograph surface is less than 8 mm to guarantee the prediction accuracy of the pantograph aerodynamic noise calculation. In addition, the boundary layer grids are generated near the pantograph surface, obtaining the first layer thickness of 0.12 mm which is normal to the wall, to ensure y + ≈1 with a growth ratio of 1.2. To capture the flow details, the refinement zone is carried out on the leeward side of the pantograph and the tail flow area of the roof model.  The pressure far-field condition and the pressure outlet condition are prescribed for the inlet and outlet boundary here, respectively. Both side planes of the computational model and the top planes are set as symmetry boundaries. To simulate the ground effects, the ground is set to be the moving wall condition with an incoming velocity of 400 km/h. The physical time step of large eddy simulation is considered as 5 10 s, and the corresponding aerodynamic noise for analysis frequency is 10 kHz, according to the Nyquist sampling theorem [37]. Moreover, 25 sub-steps are iterated within each time step and a total of 8000 time steps are calculated. The first 2500 time steps are calculated to ensure that the turbulent flow field is fully developed, and the remaining 5500 time steps are used to extract the noise source information.

Drag Coefficient of Different Pantograph Diversion Strategies
The following tables show the drag coefficients of the three calculated models, where Table 4 shows the drag coefficients in total and Table 5 shows the drag coefficients of each pantograph component. It can be found that the drag coefficients of the whole computational model decrease to different degrees when the sinking platform is installed on the train roof or after the pantograph fairing is installed. As shown in Table 4, compared with the pantograph exposed on the roof alone, the drag reduction rates of the above two cases are 2.93% and 5.42%, respectively. To obtain a better idea of how the total drag coefficients changed, it is necessary to analyze the drag of each pantograph part. As shown in Table 5, the drag coefficients of the base frame, insulator 1, and camera are dropped when the sinking platform is installed on the train roof, due to the reduction in their windward area. When the pantograph fairing is installed, most of the components are shielded by the fairing, especially the base frame, insulators, and camera; therefore, their drag coefficient decreases sharply. Although the drag coefficient of the fairing is comparable to the entire pantograph, as shown in Table 4, the total drag coefficient of the train model with three cars roof is greatly reduced after the installation of the fairing. As a consequence, the pantograph fairing could be a good alternative in terms of drag optimization for higher-speed train.   Figure 9 shows the normalized Q-criterion-based instantaneous iso-surface vorticity for different pantograph diversion strategies, colored by magnitude velocity. As seen in Figure 9, the incoming airflow directly impacts the pantograph surface and then separates, forming a clear separated vortex on the leeward side of the panhead, balance arm, and base frame. When the pantograph is installed on the train roof alone, the airflow first forms a series of small, flocculated vortices on the train surface while in front of the camera. The large vortex is formed on the pantograph surface with a large velocity at the beginning, and its velocity magnitude gradually decreases as the vortex moves backward away from the pantograph. When there is a sinking platform on the roof, the airflow is separated at the front edge of the sinking platform first, and a series of hairpin-shaped vortices are formed inside the platform. Then, the airflow that entered the sinking platform flows through the pantograph, and will separate again due to the interference of the pantograph, but with a different separated vortices shape when the pantograph is installed on the train roof alone. However, the airflow above the sinking platform is similar to case 1, and there are many fragmentized small-scale vortices. When the pantograph fairing is installed, the airflow first impacts the leading edge of the fairing, generating a small number of low-speed vortices, and then the disturbed incoming airflow continues to impact the pantograph surface, forming a series of hairpin-shaped large-scale vortices, which then impacts the rear of the fairing. These results are consistent with reference [38]. Compared with the former configurations, a large number of hairpin-shaped separated vortices are generated when the airflow enters the cavity due to the existence of the fairing. The high-speed airflow impacts the rear of the pantograph fairing, generating more separated vortices which are larger in scale, and elongate in the downstream.

Aerodynamic Noise Source
This section analyzes the intensity characteristics of the aerodynamic noise dipole source using the root mean square of the time derivative of the surface fluctuating pressure. After completing the instantaneous calculation, the time gradient p' of the fluctuating pressure on the vehicle surface is calculated by Equation (8), and then the root mean square p′rms is computed by Equation (9) as follows: ' / p dp dt = (8)

Aerodynamic Noise Source
This section analyzes the intensity characteristics of the aerodynamic noise dipole source using the root mean square of the time derivative of the surface fluctuating pressure.
After completing the instantaneous calculation, the time gradient p of the fluctuating pressure on the vehicle surface is calculated by Equation (8), and then the root mean square p rms is computed by Equation (9) as follows: Figure 10 shows the p rms distributions for the three pantograph installation methods. As seen in Figure 10, the pantograph sound source intensity is mainly distributed at the panhead, lower arm rod, base frame, etc. When the pantograph is installed on the train roof alone, the sound source intensity is also distributed at the balance arm. Combined with Figure 9 for analysis, the incoming airflow directly impacts the front edge of the panhead without interference, and vortex shedding occurs at the balance arm. When the sinking platform is installed on the train roof, the pressure fluctuation at the balance arm and the leeward side of the camera is weakened. When the pantograph fairing is not installed, the great sound source intensity also appears in the bottom area of the insulators. The installation of the fairing changes the position of vortex shedding when the airflow hits the pantograph. Consequently, the pressure fluctuation at the top of the insulator was stronger, and the strong pressure fluctuation area was also distributed at the front edge of the rear end of the fairing. In general, the pantograph pressure fluctuation is distributed in the area that is seriously affected by vortex shedding.

Spectral Characteristics of SPL
This section studies the effect of different diversion strategies on the far-field radiated noise of pantographs. The standard observing points should be 25 m from the central line of rail track and 3.5 m above the surface of rail track. In view of the train roof model which is used in this study, the far-field acoustic measurement points are arranged at a height of 1.5 m from the ground and 25 m from the center line of rail track. As depicted in Figure  11, 20 monitoring points are equally spaced along the longitudinal direction of the train (x direction), each observer point is 5 m apart, and the origin is at the bottom center of train. Therefore, the range of monitoring points is [−35, 60] m. The pantograph is located near the 6th monitoring point along the x direction. Figure 11. Locations of the standard sound monitoring points along the x-direction. Figure 12 demonstrates the comparison curves of the SPL at monitoring points under different pantograph diversion strategies when the train model and pantograph are set as the sound sources. As shown in Figure 12, along the direction of train running, the aero-

Spectral Characteristics of SPL
This section studies the effect of different diversion strategies on the far-field radiated noise of pantographs. The standard observing points should be 25 m from the central line of rail track and 3.5 m above the surface of rail track. In view of the train roof model which is used in this study, the far-field acoustic measurement points are arranged at a height of 1.5 m from the ground and 25 m from the center line of rail track. As depicted in Figure 11, 20 monitoring points are equally spaced along the longitudinal direction of the train (x direction), each observer point is 5 m apart, and the origin is at the bottom center of train. Therefore, the range of monitoring points is [−35, 60] m. The pantograph is located near the 6th monitoring point along the x direction. Figure 12 demonstrates the comparison curves of the SPL at monitoring points under different pantograph diversion strategies when the train model and pantograph are set as the sound sources. As shown in Figure 12, along the direction of train running, the aerodynamic noise SPL tends to increase first and then decrease, and the maximum SPL appears near the pantograph area. Compared with the SPL when pantograph is installed on the train roof alone, it is clear that the SPL is reduced by about 2 dBA when the pantograph is installed on the sinking platform, indicating a better aeroacoustic performance. In addition, the maximum SPL of cases 1 and 2 occurs at the 6th monitoring point. When the pantograph is installed with the fairing, the location changes, which is the 4th monitoring point. Meanwhile, the SPL at the monitoring point before the 6th monitoring point increases, after which the SPL at almost all monitoring points decreases along the train running direction, with an increasingly lower decrease in degree. This can be explained by the fact that the fairing itself brings great noise, and the airflow is disturbed by the leading edge of the fairing first, resulting in some noise reduction at the front end of the fairing cavity. After a period of time, the turbulent flow continues to impact the rear part of the pantograph fairing area, together with the vortex shedding of the pantograph rod, which weakens the noise reduction effect of the fairing. Importantly, compared with case 1, the fairing mainly decreases the noise near the pantograph, and the largest SPL reduction is up to 3 dBA at the 8th monitoring point.

Spectral Characteristics of SPL
This section studies the effect of different diversion strategies on the far-field radiated noise of pantographs. The standard observing points should be 25 m from the central line of rail track and 3.5 m above the surface of rail track. In view of the train roof model which is used in this study, the far-field acoustic measurement points are arranged at a height of 1.5 m from the ground and 25 m from the center line of rail track. As depicted in Figure  11, 20 monitoring points are equally spaced along the longitudinal direction of the train (x direction), each observer point is 5 m apart, and the origin is at the bottom center of train. Therefore, the range of monitoring points is [−35, 60] m. The pantograph is located near the 6th monitoring point along the x direction. Figure 11. Locations of the standard sound monitoring points along the x-direction. Figure 12 demonstrates the comparison curves of the SPL at monitoring points under different pantograph diversion strategies when the train model and pantograph are set as the sound sources. As shown in Figure 12, along the direction of train running, the aerodynamic noise SPL tends to increase first and then decrease, and the maximum SPL appears near the pantograph area. Compared with the SPL when pantograph is installed on the train roof alone, it is clear that the SPL is reduced by about 2 dBA when the pantograph is installed on the sinking platform, indicating a better aeroacoustic performance. In addition, the maximum SPL of cases 1 and 2 occurs at the 6th monitoring point. When the pantograph is installed with the fairing, the location changes, which is the 4th monitoring point. Meanwhile, the SPL at the monitoring point before the 6th monitoring point increases, after which the SPL at almost all monitoring points decreases along the train running direction, with an increasingly lower decrease in degree. This can be explained by the fact that the fairing itself brings great noise, and the airflow is disturbed by the leading edge of the fairing first, resulting in some noise reduction at the front end of the fairing cavity. After a period of time, the turbulent flow continues to impact the rear part of the pantograph fairing area, together with the vortex shedding of the pantograph rod, which weakens the noise reduction effect of the fairing. Importantly, compared with case 1, the fairing mainly decreases the noise near the pantograph, and the largest SPL reduction is up to 3 dBA at the 8th monitoring point.  Figure 13 illustrates the 1/3 octave band frequency curves at the maximum SPL monitoring points of the abovementioned three cases. As can be seen in Figure 13, the distribution of aerodynamic noise energy in all the cases is concentrated in 630-2500 Hz, and the SPL gradually decays toward the high frequency and low frequency regions. Furthermore, the noise reduction effect of the sinking platform in the frequency band of 200-1000 Hz is not clear, while the magnitude above 1250 Hz is almost less than case 1 by 2 dBA, which is consistent with the behavior in Figure 12. With regard to case 3, the presence of the fairing greatly increases the noise in the frequency band below 500 Hz, and decreases the noise in the frequency band above 1000 Hz. Consequently, a large SPL maximum is obtained for case 3 in Figure 12.  Figure 13 illustrates the 1/3 octave band frequency curves at the maximum SPL monitoring points of the abovementioned three cases. As can be seen in Figure 13, the distribution of aerodynamic noise energy in all the cases is concentrated in 630-2500 Hz, and the SPL gradually decays toward the high frequency and low frequency regions. Furthermore, the noise reduction effect of the sinking platform in the frequency band of 200-1000 Hz is not clear, while the magnitude above 1250 Hz is almost less than case 1 by 2 dBA, which is consistent with the behavior in Figure 12. With regard to case 3, the presence of the fairing greatly increases the noise in the frequency band below 500 Hz, and decreases the noise in the frequency band above 1000 Hz. Consequently, a large SPL maximum is obtained for case 3 in Figure 12.
To further investigate the causes of the difference in SPL energy between the three cases, the spectral characteristics of three critical components of the pantograph, namely, the panhead, the lower arm rod, and the base frame, are analyzed in this section. Figure 14 shows the noise spectrum curves at the maximum SPL monitoring point of each case when the panhead, the lower arm rod, and the base frame are considered as noise sources, respectively. Figure 14a illustrates that when the pantograph is placed on the train roof alone, the main-order and the second-order frequency of 1767 and 3468 Hz occur in the panhead, with the noise spectrum corresponding to SPL of 69.75 and 52.09 dB, respectively. When the pantograph diversion strategies are ameliorated, the main-order and secondorder frequency disappears. In Figure 14b,c, it can be observed that the aerodynamic noise spectrum of the lower arm rod and the base frame in all three cases exhibits a broad frequency characteristic, and there is no clear primary frequency. Moreover, after the installation of the pantograph fairing, the SPL of the lower arm rod and the base frame with energy less than 300 Hz is mainly raised, while the SPL with energy above 1000 Hz is reduced. Clearly, the noise reduction effect of the fairing of the pantograph on the base frame is more salient, which may be due to the fact that the fairing shields part of the incoming flow in the base frame. However, the sinking platform does not have a strong effect on the noise spectrum characteristics of the lower arm rod and the base frame.  Figure 13 illustrates the 1/3 octave band frequency curves at the maximum SPL monitoring points of the abovementioned three cases. As can be seen in Figure 13, the distribution of aerodynamic noise energy in all the cases is concentrated in 630-2500 Hz, and the SPL gradually decays toward the high frequency and low frequency regions. Furthermore, the noise reduction effect of the sinking platform in the frequency band of 200-1000 Hz is not clear, while the magnitude above 1250 Hz is almost less than case 1 by 2 dBA, which is consistent with the behavior in Figure 12. With regard to case 3, the presence of the fairing greatly increases the noise in the frequency band below 500 Hz, and decreases the noise in the frequency band above 1000 Hz. Consequently, a large SPL maximum is obtained for case 3 in Figure 12. To further investigate the causes of the difference in SPL energy between the three cases, the spectral characteristics of three critical components of the pantograph, namely, the panhead, the lower arm rod, and the base frame, are analyzed in this section. Figure  14 shows the noise spectrum curves at the maximum SPL monitoring point of each case when the panhead, the lower arm rod, and the base frame are considered as noise sources, respectively. Figure 14a illustrates that when the pantograph is placed on the train roof alone, the main-order and the second-order frequency of 1767 and 3468 Hz occur in the panhead, with the noise spectrum corresponding to SPL of 69.75 and 52.09 dB, respectively. When the pantograph diversion strategies are ameliorated, the main-order and second-order frequency disappears. In Figure 14b,c, it can be observed that the aerodynamic noise spectrum of the lower arm rod and the base frame in all three cases exhibits a broad frequency characteristic, and there is no clear primary frequency. Moreover, after the installation of the pantograph fairing, the SPL of the lower arm rod and the base frame with energy less than 300 Hz is mainly raised, while the SPL with energy above 1000 Hz is reduced. Clearly, the noise reduction effect of the fairing of the pantograph on the base frame is more salient, which may be due to the fact that the fairing shields part of the incoming flow in the base frame. However, the sinking platform does not have a strong effect on the noise spectrum characteristics of the lower arm rod and the base frame. Next, the characteristics of aerodynamic noise at a wide range of frequencies are analyzed. Figure 15 shows the sound source intensity distribution of different cases at 40, 1600, and 8000 Hz, respectively. As can be seen in Figure 15, when there is no pantograph fairing, the sound source intensity in the pantograph area is more uniformly distributed Next, the characteristics of aerodynamic noise at a wide range of frequencies are analyzed. Figure 15 shows the sound source intensity distribution of different cases at 40, 1600, and 8000 Hz, respectively. As can be seen in Figure 15, when there is no pantograph fairing, the sound source intensity in the pantograph area is more uniformly distributed at high frequencies, and the difference in distribution is larger at low and medium frequencies.
Meanwhile, the sound source intensity is significantly greater at a frequency of 1600 Hz. In addition, at a low frequency, the high sound source intensity mainly occurs around the camera for case 1, while it appears on the rear surface of the fairing for case 3. A similar distribution can be observed at a medium frequency, although the base frame contributes significantly greatly. Overall, the installation of the pantograph fairing remarkably changed the distribution of the sound source intensity at low and medium frequencies, but had less effect on the distribution at high frequencies.
Appl. Sci. 2022, 12, x FOR PEER REVIEW  15 of 19 at high frequencies, and the difference in distribution is larger at low and medium frequencies. Meanwhile, the sound source intensity is significantly greater at a frequency of 1600 Hz. In addition, at a low frequency, the high sound source intensity mainly occurs around the camera for case 1, while it appears on the rear surface of the fairing for case 3.
A similar distribution can be observed at a medium frequency, although the base frame contributes significantly greatly. Overall, the installation of the pantograph fairing remarkably changed the distribution of the sound source intensity at low and medium frequencies, but had less effect on the distribution at high frequencies.

Aerodynamic Noise Directivity
To reveal the difference in aerodynamic noise directivity for different configurations, the pantograph is considered as the sound source and sound monitoring points arranged

Aerodynamic Noise Directivity
To reveal the difference in aerodynamic noise directivity for different configurations, the pantograph is considered as the sound source and sound monitoring points arranged every 10 • on a circle with a radius of 7.5 m are used. As a result, 36 monitoring points (0 • ≤ θ ≤ 360 • ) are set in X-Y plane, and 19 monitoring points (0 • ≤ θ ≤ 180 • ) are set in X-Z and Y-Z planes, respectively. The overall sound pressure level (OASPL) in the three planes is calculated and compared in Figure 16. It can be observed that in the X-Y plane, the noise directivity of cases 1 and 3 is similar, and distinct variations that the OASPL first decreases and then increases appear between 50 • ≤ θ ≤ 90 • , which might be caused by the asymmetry of insulator 1. When the pantograph is installed on the sinking platform, the noise directivity is different from the other two cases, and its variation range is within 60 • ≤ θ ≤ 90 • . In the X-Z plane, as shown in Figure 16b, the OASPL within 90 • ≤ θ ≤ 180 • is higher than within 0 • ≤ θ ≤ 90 • for all cases, probably due to the closer proximity to the insulators in this region. In the Y-Z plane, the noise directivity of the three cases is almost symmetrical, and the higher OASPL is mainly distributed above the pantograph, which is 80 • ≤ θ ≤ 100 • . every 10° on a circle with a radius of 7.5 m are used. As a result, 36 monitoring points (0° ≤ θ ≤ 360°) are set in X-Y plane, and 19 monitoring points (0° ≤ θ ≤ 180°) are set in X-Z and Y-Z planes, respectively. The overall sound pressure level (OASPL) in the three planes is calculated and compared in Figure 16. It can be observed that in the X-Y plane, the noise directivity of cases 1 and 3 is similar, and distinct variations that the OASPL first decreases and then increases appear between 50° ≤ θ ≤ 90°, which might be caused by the asymmetry of insulator 1. When the pantograph is installed on the sinking platform, the noise directivity is different from the other two cases, and its variation range is within 60° ≤ θ ≤ 90°. In the X-Z plane, as shown in Figure 16b, the OASPL within 90° ≤ θ ≤ 180° is higher than within 0° ≤ θ ≤ 90° for all cases, probably due to the closer proximity to the insulators in this region. In the Y-Z plane, the noise directivity of the three cases is almost symmetrical, and the higher OASPL is mainly distributed above the pantograph, which is 80° ≤ θ ≤ 100°.
Of note, in the planes of Y-Z and X-Z, for all monitoring points, the highest OASPL is found for the pantograph alone on the roof, followed by the configuration with the sinking platform on the roof, and the lowest OASPL is found for the pantograph with the fairing. This performance signifies that the fairing effectively reduces the noise induced by the pantograph itself. Meanwhile, the pantograph fairing could play a certain sound barrier role, as well. However, as mentioned above, the noise caused by the fairing itself makes a great contribution and should be considered in a practical scene.

Conclusions
In this study, large eddy simulation and Ffowcs Williams-Hawkings analogy are used to study the aerodynamic drag and noise characteristics of the pantograph area of a train roof model with three cars at a speed of 400 km/h with different diversion strategies. Specifically, the flow field characteristics and pressure fluctuation, noise spectrum Of note, in the planes of Y-Z and X-Z, for all monitoring points, the highest OASPL is found for the pantograph alone on the roof, followed by the configuration with the sinking platform on the roof, and the lowest OASPL is found for the pantograph with the fairing. This performance signifies that the fairing effectively reduces the noise induced by the pantograph itself. Meanwhile, the pantograph fairing could play a certain sound barrier role, as well. However, as mentioned above, the noise caused by the fairing itself makes a great contribution and should be considered in a practical scene.

Conclusions
In this study, large eddy simulation and Ffowcs Williams-Hawkings analogy are used to study the aerodynamic drag and noise characteristics of the pantograph area of a train roof model with three cars at a speed of 400 km/h with different diversion strategies. Specifically, the flow field characteristics and pressure fluctuation, noise spectrum characteristics, and noise directivity of the computational models are analyzed. The main conclusions are as follows: