Comparative study of aeroacoustic performance of 1/8 and 1/1 pantographs coupled with cavity

Abstract

The technology of pantograph sinking in the cavity is generally adopted in the new generation of high-speed trains in China for aerodynamic noise reduction in this region. This study takes a high-speed train with a 4-car formation and a pantograph as the research object and compares the aerodynamic acoustic performance of two scale models, 1/8 and 1/1, using large eddy simulation and Ffowcs Williams–Hawkings integral equation. It is found that there is no direct scale similarity between their aeroacoustic performance. The 1/1 model airflow is separated at the leading edge of the panhead and reattached to the panhead, and its vortex shedding Strouhal number (St) is 0.17. However, the 1/8 model airflow is separated directly at the leading edge of the panhead, and its St is 0.13. The cavity’s vortex shedding frequency is in agreement with that calculated by the Rooster empirical formula. The two scale models exhibit some similar characteristics in distribution of sound source energy, but the energy distribution of the 1/8 model is more concentrated in the middle and lower regions. The contribution rates of their middle and lower regions to the radiated noise in the two models are 27.3% and 87.2%, respectively. The peak frequencies of the radiated noise from the 1/1 model are 307 and 571 Hz. The 307 Hz is consistent with the frequency of panhead vortex shedding, and the 571 Hz is more likely to be the result of the superposition of various components. In contrast, the peak frequencies of the radiated noise from the 1/8 scale model are 280 and 1970 Hz. The 280 Hz comes from the shear layer oscillation between the cavity and the bottom frame, and the 1970 Hz is close to the frequency at which the panhead vortex sheds. This shows that the scaled model results need to be corrected before applying to the full-scale model.

1 Introduction

The pantograph region of the new generation of high-speed train features a pantograph cavity coupling system (PCCS), which is the primary source of aerodynamic noise on the roof. Due to limitations in wind tunnel size, numerical simulation costs, and boundary layer simulation, current studies on aerodynamic noise of this system typically utilize a scaled-down model of the entire vehicle. In aerodynamic studies, if the Reynolds number of the scaled-down model exceeds the critical Reynolds number, then, the flow field of the scaled-down model will be similar to that of the 1/1 model at the same free incoming velocity. This means that relevant aerodynamic studies can be carried out using a scaled-down model [1]. In aerodynamic acoustic research, achieving aerodynamic acoustic similarity requires satisfying two or more parameters simultaneously, such as Reynolds number (Re), Strouhal number (St), Mach number (Ma), and Froude number (Fr), among others. However, this is challenging to implement in practice.

Lauterbach et al. [2] carried out a study on the Reynolds number effect of aerodynamic noise sources on a high-speed train model with a scaling ratio of 1:25, and the results showed that the aerodynamic noise in the pantograph region is the main source of noise sources above 5 kHz in high-speed trains, and the discrete characteristics of its spectrum have a Straw–Haar number similarity. Zhang et al. [3] and Tan et al. [4] used numerical simulations to comprehensively analyze the intensity and frequency characteristics of the aerodynamic noise of a separate pantograph and elaborated on the correlation between the vortex structure scale development and aerodynamic excitation in this region. Zhang’s results show that the radiated noise spectrum of the pantograph is mainly concentrated in 250–1000 Hz, which mainly originates from the bow head of the pantograph, and the main order of the frequency moves to the high frequency, which is linearly related to the velocity. This characteristic is consistent with the simulation results of Liu et al. [5], where the pantograph was mounted on the body model; they believed that the first-order frequency is contributed by the lifting oscillations, which are related to the vortex shedding frequency of the bow head, and that the second-order frequency is mainly caused by the interaction between the bow head tail track and the other structures of the pantograph. Besides, the simulation results of Sun and Xiao [6] on pantographs also show that in the high-frequency band, the far-field aerodynamic noise of pantographs is mainly generated by the pantograph head, which belongs to the single-frequency tones of the far-field aerodynamic noise of pantographs. Iglesias et al. [7] tried to construct a semi-analytic prediction model of pantograph noise by drawing on the relevant experience in the field of aeroacoustics.

The pantograph system mainly consists of rod structures such as slide plates, arms, and push/pull shafts [8]. The main mechanism that induces the complex flow field structure in this region is the flow interference between the cavity flow pattern and the blunt body flow pattern, which includes the supporting frame and arm [9, 10]. The most significant changes in the pulsating flow field of the system brought about by different scaling models are the migration of the flow field morphology due to the change of Reynolds number, the change of the flow field coupling effect due to the change of the spatial distance of different components, and the change of the thickness of the incoming boundary layer.

According to Norberg [11], the main factors affecting the Strouhal number of a blunt body flow include the Reynolds number, profile shape, and angle of attack of the incoming flow, of which the Reynolds number and the cross section aspect ratio are the two main factors affecting the vortex shedding frequency of columns with a rectangular cross section [12]. Okajima et al. [13] investigated the relationship between the Strouhal number and Reynolds number for rectangular columns with various aspect ratios (B/H, where B is the side length of rectangular column and H is its height) in wind tunnel tests. For a column of rectangular section with B/H = 2, the flow field pattern experiences a sudden qualitative change within a specific range of Reynolds numbers. This change results in a sudden disruption in the curve of the Strouhal number. In the region below this Reynolds number, the flow that separates from the leading edge reattaches to both the upper and lower surfaces of the column. However, in the region above this Reynolds number, the flow that separates from the leading edge will fully detach from the column. For columns with B/H = 3 rectangular cross section, two different modes occur alternately and intermittently in the Reynolds number range from 1000 to 3000. As the Reynolds number increases, the flow pattern with small amplitude and high frequency, known as Mode I, begins to grow and persist, while Mode II (i.e., the flow pattern with large amplitude and low frequency) starts to disappear. When the Reynolds number exceeds about 8000, the value of the Strouhal number is close to the value of Mode I, St = 0.16–0.17. This is due to the predominance of the entrainment effect of the turbulence that causes the flow to appear reattached at the trailing edge.

Bai and Alam [14] studied the influence of Reynolds number on the vortex shedding frequency of square column flow. They found that the flow field morphology of the square column flow undergoes a significant migration with the increase in Reynolds number in the region below 1000, whereas the flow field morphology remains unchanged in the range of Reynolds number above 1000. Therefore, the frequency of vortex shedding remains unchanged.

Cavity flow is a complex flow in which a boundary layer of thickness δ separates at the leading edge of the cavity, forming a free shear layer. The development of this layer depends on the conditions upstream of the cavity and within it. These conditions can be described by the dimensionless parameters length-to-depth ratio (L/D), length-to-width ratio (L/W), free-stream Mach number (Ma), cavity length versus boundary layer thickness parameter (L/δ), and the shape factor of the boundary layer h = δ^∗/θ (δ^∗ is the displacement thickness of the boundary layer, and θ is the momentum thickness). It is important to note that the scaling factor has a significant effect on the boundary layer thickness and shape factor.

According to the flow field characteristics, the cavity flow can be categorized as open, transitional or closed at an incoming Mach number of 0.6–0.9. The flow field of open-cavity flow is characterized by a shear layer that separates at the leading edge of the cavity, flows along the length of the cavity, and reattaches to the following edge or back of the cavity. The self-sustained flow oscillations, induced by the interaction of the shear layer with the back wall, cause the opening cavity to exhibit strong periodic pressure fluctuations [15]. In general, this flow field occurs in cavities with a length-to-depth ratio (L/D) of less than 10. The flow field in a closed cavity is characterized by a shear layer that separates at the leading edge of the cavity, flows into the cavity, reattaches and separates again at the bottom of the cavity, and finally reattaches near the following edge. This type of flow field is usually observed in cavities with length-to-depth ratios L/D > 10–13 [16].

In this study, the cross section of the panhead is rectangular with a B/H ratio of 2.3. The length-to-depth ratio of the concave cavity in the pantograph region of the high-speed train is 7.3, indicating that it is an open cavity. According to the literature, it is necessary to explore the flow field morphology of the PCCS with different scaling scales in detail.

2 Aerodynamic noise calculation method

The pantograph model used in this study is a simplified version of the Faiveley CX-PG pantograph, as shown in Fig. 1. The important rod dimensions of the pantograph are provided in Table 1. The characteristic length (L = 54 mm) is taken as the longitudinal diameter of the panhead. Under the 1/1 model, the corresponding Reynolds number is 4.3 × 10⁵ when the incoming flow speed is 350 km/h..

Fig. 1

Pantograph model and geometry of the main rod structure

Table 1 Geometric dimensions of the main rod structures of pantograph (mm)

The shape of the pantograph cavity is shown in Fig. 2. Its length in the flow direction L_cavity = 3.37 m, spreading width W_cavity = 2.26 m, and cavity depth D_cavity = 0.46 m.

Fig. 2

Geometry of the pantograph cavity

The geometric model of the whole vehicle is a high-speed train model with a 4-car formation and a pantograph, which is 105.5 m in length, 3.4 m in width and 4.05 m in height (Fig. 3).

Fig. 3

Simulation of high-speed train geometry model

The inlet of the basin is set as a velocity inlet boundary condition, and the incoming velocity magnitude is equal to the train’s running velocity magnitude. The outlet of the basin is set as a pressure outlet boundary condition, with the relative pressure P_out = 1 atm. Symmetric boundary conditions are applied to both the side and top surfaces of the computational domain. The ground surface is set as a frictionless surface, and the speed is given as 350 km/h, which is equal to the incoming velocity. The dimensions of the computational region and the boundary conditions are shown in Fig. 4.

Fig. 4

Computational domain and boundary pieces of the four-car high-speed train model

The commercial grid generation software ICEM is used to perform non-structural grid generation on the train surface, attached surface layer, and spatial grids, respectively. Triangular grids, triangular prismatic grids, tetrahedral grids, local surface change regions, and flow intense regions are used for encryption. In the top pantograph area of the train, the surface grid size of the pantograph is 2–5 mm; the surface grid size of the nested density boxes enclosing the panhead area/arm knuckle is 10, 20, and 30 mm in order from the inside to the outside; the lower body grid density control box size is 20 mm; the nested density box size of the whole pantograph is 30, 50, 100, and 300 mm in order from the inside to the outside. The surface mesh size of the other areas of the train is controlled in the range of 5–50 mm, and the local curved surface areas and the areas of intense flow are refinemented locally with the mesh size of 30 mm. Considering the spatial distribution characteristics of the surrounding flow field excitation during the train running, multi-layer encryption around the train as well as in the wake flow region is carried out by using a wedge-shaped density box with a gradual widening to the downstream sides. The surface grid size of the three progressively enlarged density boxes encircling the train and wake region is 100, 200 and 400 mm, respectively, and the number of prismatic layers of the vehicle body appendage layer is set to 25, with the thickness of the first layer of grid of 0.05 mm and the growth rate of 1.03, while the ground appendage layer is set to contain 8 layers of prismatic grids, with the thickness of the first layer of 10 mm and the growth rate of 1.1. The total number of grids is approximately 420 million. Figure 5 provides a schematic diagram of the cross section grids. The schematic of the pantograph region grids is shown in Fig. 6.

Fig. 5

Schematic diagram of the section grids: a longitudinal plane of symmetry of the train (y = 0 m); b train lateral plane of symmetry (x = 2 m); c contour cross section at the tip of the nose (z = 1 m)

Fig. 6

Schematic diagram of the pantograph region grids: a localized enlargement of the spatial grids in the pantograph region; b grids distribution on the pantograph surface

Compressible gas modeling is used in the simulation process. The steady-state flow field results are obtained by solving the initial flow field using a pressure-based implicit solution method. The settings for the steady-state and transient numerical solvers for the turbulent pulsation numerical computation method are provided in Table 2.

Table 2 Algorithm parameter settings

The analysis frequency and CFL (Courant–Friedrichs–Lewy) condition are the two factors used to determine the time step Δt. According to the Nyquist sampling theorem, the sampling frequency should be at least twice the highest frequency being analyzed in order to accurately recover the original signal without any added interference. The aerodynamic sound distribution frequency range of the 1/8 scale model is 0–10 kHz, corresponding to Δt ≤ 5×10⁻⁵ s. Therefore, the nonstationary calculation time step Δt for the 1/1 scale model and the 1/8 scale model simulation is 1×10⁻⁴ and 5×10⁻⁵ s, respectively, which meets the requirement of CFL < 1, as shown in Fig. 7.

Fig. 7

CFL distribution for y = 0 cross section of 1/8 model

Appropriate boundary layer grid distribution and spatial grid distribution are two fundamental requirements for grid distribution in high-precision eddy simulations. The y + parameter is the primary factor in determining the distribution of boundary layer grids. Since large-vortex simulations need to capture the proposed order structure, the first information that triggers the proposed order structure is the formation of a low-velocity strip structure in the viscous substratum (y +  < 10). To achieve this, the viscous bottom layer should be divided into an appropriate number of grid layers, with a y + value of less than 1. The distribution of y + on the surface of the train is illustrated in Fig. 8.

Fig. 8

Cloud map of y + distribution on the train surface: a head train; b tail train; c PCCS

The main parameter for the distribution of spatial grids is the scale of the grids. Due to the isotropic nature of small-scale vortices, it is necessary to position the grid scale within the local inertial subregion in order to ensure the validity of the subgrid model for small-scale vortices. The integral scale is used to differentiate between the inertial subregion and the large-scale vortex region. Therefore, the grid scale should be smaller than the local integral scale. The local integral scale (${l}_{\text{t}}$) is defined as follows:

$$ l_{{\text{t}}} = C_{0} k^{3/2} /\varepsilon , $$

(1)

$$ \varepsilon = 2\upsilon_{{\text{t}}} \overline{\user2{S}}_{ij} \overline{\user2{S}}_{ij} , $$

(2)

where ${C}_{0}$ is 0.2, k is the turbulent kinetic energy, $\varepsilon$ is the dissipation rate, ${\upsilon }_{{\text{t}}}$ is coefficient of eddy viscosity for the turbulence, and $\overline{\user2{S}}_{ij}$ is strain rate tensor ($i,j$= 1,2,3…). Equation (3) gives the definition of the grid equivalent length ${l}_{\Delta }$:

$$ l_{{\Delta }} = V^{1/3}, $$

(3)

where $V$ is the grid volume. The ${l}_{\Delta }/{l}_{{\text{t}}}$ distribution of the spatial grids of the 1/1 scale model is shown in Fig. 9.

Fig. 9

Spatial distribution of grid parameter ${l}_{\Delta }/{l}_{{\text{t}}}$

In summary, the grid model established in this section meets the requirements for conducting large eddy simulation experiments. This model ensures the reliability and accuracy of the calculation results, providing reliable data to support the research in this paper.

The coordinates of the far-field radiation point location are (73.0, 25.0, 3.5) m. This means that the measurement point is 25.0 m away from the track centerline, the height from the rail surface is 3.5 m, and it is aligned with the geometrical center of the pantograph in the direction of the train operation. The location of the far-field radiation point is shown in Fig. 10. The aerodynamic noise source data of the PCCS is obtained using a microphone array located on the track side of the train, employing the beam algorithm. The far-field radiated noise signal is obtained by combining this data with the acoustic radiation formula for free-field motion sound sources. Figure 11 compares the numerical simulation results of the far-field radiated noise of the 1/1 PCCS in this paper with the results of a real-vehicle test.

Fig. 10

Location of the far-field radiation point

Fig. 11

Far-field noise spectrum in the pantograph region of 1/1 model simulation results and real-vehicle test results

The sound radiation formula [17] of a point sound source $i$ in a free-field linear motion is as follows:

$$ p_{i} \left( {{\varvec{x}},t} \right) = \left[ {\frac{{Q_{i} }}{{4{\uppi }r_{i} \left| {1 - M_{{r_{i} }} } \right|^{2} }}} \right]_{\tau } , $$

(4)

where ${p}_{i}$ is the radiation sound pressure from the point sound source$i$, ${\varvec{x}}$ is the position vector of the observation point, $t$ is the sound receiving time, ${Q}_{i}$ is the sound source intensity, ${r}_{i}$ is the radiation diameter from the point sound source $i$, $M_{{r_{i} }}$ is the ith component of the Mach number in the direction of the radiation direction $r$, and $\tau$ is the acoustic delay time.

Simulation test results and real-vehicle test results in the sound pressure level (SPL) spectral pattern show a high agreement in the middle and low-frequency bands and a significant difference in the high-frequency band. This is due to the low-frequency noise mainly from the flow separation and vortex development near the wall, where the independence of the sound source and lack of correlation is more apparent. However, high-frequency sound often originates from the turbulent pulsation in the attached layer, which exhibits significant coherent characteristics. The sound source imaging technology in the real test is based on the assumption of non-coherent sound sources. Therefore, it is challenging to avoid algorithmic contamination, leading to biased results in the high-frequency band of the real test results.

3 Results and discussion

3.1 Aerodynamic excitation characteristics

The pulsating flow field is analyzed in terms of velocity, vortex volume, and vortex structure capture. The Q-criterion is a physical quantity commonly used to describe vortex motion. It is mathematically defined as the magnitude of the curl of the velocity of the air stream, as follows [18]:

$$ Q = \frac{1}{2}||{{\varvec\varOmega}||^{2} - ||{\varvec{S}}||^{2} }, $$

(5)

where ||·|| denotes the second norm of the tensor; ${\varvec{\varOmega}}$ is the vortex tensor, composed of the anti-symmetric part of the velocity gradient tensor; and ${\varvec{S}}$ is the strain rate tensor, composed of the symmetric part of the velocity tensor.

Figure 12 illustrates the velocity distributions for the y = 0 m cross section in both the 1/1 scale model and the 1/8 scale model.

Fig. 12

Velocity distribution in y = 0 m section around pantograph: a 1/1 scale model; b 1/8 scale model

Figure 13 shows the transiently computed time-averaged vortex distribution clouds for the y = 0 m section of the pantograph region. The time-averaged sampling duration is 0.05 s.

Fig. 13

Vortex distribution clouds in y = 0 m cross section around pantograph: a 1/1 scale model; b 1/8 scale model

In the lower region of the pantograph in both models, flow conditions of the bottom components change with the size of the pantograph cavity: The upstream of the pantograph cavity is divided by the shear layer, forming a “dead water zone” in the bottom region, which greatly reduces the flow velocity in front of the base and insulators. In the 1/8 model, the average flow velocity in front of the insulator is about 7.8 m/s, which is about 90% lower than that of the inlet flow directly impacting the insulator of the pantograph, thus forming a low velocity and low vortex zone in front of the bottom structure; the upper end of the insulator and the base frame is subjected to the shear layer, the action of the airflow from above, and the return flow from the back wall of the cavity, while the downstream base frame of the pantograph cavity, insulator and other structures behind the formed lower velocity zone of strong vortex appear to have no obvious features.

The velocity distributions in the y = 0 m cross section of the 1/1 scale model and the 1/8 scale model are illustrated in Fig. 14.

Fig. 14

Distribution of instantaneous vortex structure identified by Q isosurfaces: a 1/1 scale model (Q = 2.5 × 10⁵ s^-2); b 1/8 scale model (Q = 2.0 × 10⁵ s^-2)

The distribution of spatial vortex structures in Fig. 14 shows that the flow field characteristics in the pantograph region of the two scales are similar. The vortex structure formed by the incoming flow passing through the pantograph can be divided into three parts.

• (1) The development of the vortex structure in the upper region: Flow separation of the fluid occurs at the panhead. This separation creates a quasi-two-dimensional spreading vortex shedding in the wake region, which is distributed axially along the panhead. Over time, this shedding vortex structure gradually develops into an ordered pattern as it propagates downstream.

• (2) The development of the vortex structure in the intermediate region: The fluid forms a vortex packet in the arm knuckle. Part of the vortex packet acts along arm knuckle and separates after the front; it finally ends at the upper arm frame and lower arm, and part of the airflow is separated in from of crescent-shaped separation bubbles, interacting with the slanting arm structure such as the upper arm frame and lower arm to shed vortices, which are heavily doped and form a large number of irregularly shaped vortex structures. Subsequently, the vortex structures formed in the intermediate region are divided into three parts: some of which act on the panhead structure and fuse with the vortex structures generated by the beam in the panhead region, and some of which move downward and fuse with the large-scale vortex structures in the cavity, while the main vortex structures still move downstream along the airflow.

• (3) The development of the vortex structure in the bottom region: A low-speed small-scale vortex structure is formed inside the pantograph cavity. Pie-shaped vortex packages are formed in the middle of the leading edge of the cavity, and strip vortices are generated in the opening of the cavity in front of the pantograph. These strip vortices are broken and regenerated by the push/pull shafts, insulators, and base. Additionally, a series of larger-scale horseshoe vortices are formed at the rear of the cavity and develop downstream as well as to the two sides. The first large-scale horseshoe vortex, which is close to the panhead, affects the downstream development of the shedding vortex at the panhead to some extent.

3.2 Principal scale analysis of flow field structures

According to the geometrical characteristics of the new pantograph region with the above-mentioned basic features of flow field spatial distribution, it can be observed that the vortex structure of the new pantograph region exhibits two distinct scale features. These features are located in the upper part (known as the panhead region) and the lower part (known as the concave cavity region) of the pantograph region.

3.2.1 Panhead region

In the upper region, the typical winding vortex detachment phenomenon at the rear of the panhead region is primarily generated by the panhead component. The downstream progression of its vortex system is largely suppressed by vortices induced from the structure below the pantograph. Figure 15 demonstrates the distribution cloud diagram of the Q value for the y = 0 m cross section at different moments. The white circles refer to the vortex structures that have formed and are gradually moving downstream, and the white boxes refer to the vortex structures that are forming as the flow passes over the panhead. It can be observed that a typical Kamen vortex street phenomenon is formed behind the rectangular column of the panhead in the 1/1 scale model. At this point, it can be considered that one vortex shedding cycle has been completed. From this, the vortex shedding frequency f_1/1 of the panhead flow can be deduced to be in the range of 303–333 Hz according to its period T_1/1 range from 30∆t₁ to 33∆t₁, and then, the dimensionless vortex shedding frequency St_{1/1, panhead} is found to be in the range of 0.168–0.184 under the 1/1 model. The Strouhal number is calculated as follows [19]:

$$ {{St}} = \frac{fL}{{u_{\infty } }}, $$

(6)

where $f$ is the frequency, $L$ is the geometric characteristic length, and $u_{\infty }$ is the inflow velocity.

Fig. 15

Cloud diagram of the distributi on of Q values in the longitudinal plane of symmetry in the region of the panhead at different moments during one cycle of the 1/1 scale model ( ∆t_1/1 = 0.0001 s, Re = 4.29 × 10⁵): a t₁ = t₀; b t₂ = t₀ + 3∆t; c t₃ = t₀ + 6∆t; d t₄ = t₀ + 9∆t; e t₅ = t₀ + 12∆t; f t₆ = t₀ + 15∆t; g t₇ = t₀ + 18∆t; h t₈ = t₀ + 21∆t; i t₉ = t₀ + 24∆t; j t₁₀ = t₀ + 27∆t; k t₁₁ = t₀ + 30∆t; l t₁₂ = t₀ + 33∆t

Figure 16 gives the Q value distribution cloud diagram of y = 0 m cross section at different moments under the working condition of 1/8 scaled model; it can be found that after one vortex shedding cycle, the vortex structure distribution behind the panhead is formed at t₉–t₁₀ moments which is very similar to that at t₁ moment, and its vortex shedding cycle T_1/8 ranges from 9∆t₂ to 10∆t₂. We can deduce that its vortex shedding frequency f_1/8 of the panhead flow is in the range of 2000–2222 Hz, and its St is in the range of 0.129–0.145 from Eq. (6).

Fig. 16

Cloud view of the distribution of Q values in the longitudinal plane of symmetry in the region of the panhead at different moments of the 1/8 scale model (∆t_1/8 = 0.00005 s, Re = 4.2 × 10⁴): a t₁ = t₀; b t₂ = t₀ + 3∆t; c t₃ = t₀ + 6∆t; d t₄ = t₀ + 9∆t; e t₅ = t₀ + 12∆t; f t₆ = t₀ + 15∆t; g t₇ = t₀ + 18∆t; h t₈ = t₀ + 21∆t; i t₉ = t₀ + 24∆t; j t₁₀ = t₀ + 27∆t; k t₁₁ = t₀ + 30∆t; l t₁₂ = t₀ + 33∆t

The layout and naming of the speed signal acquisition point array are shown in Fig. 17. The power spectral analysis of time-domain velocity signals at 25 measurement points from 1/1 model simulation results is shown in Fig. 18, where the data are present in categories by different Z-values. The horizontal coordinate represents the dimensionless frequency St (with the transverse diameter of the panhead as the characteristic length), while the vertical coordinate represents the power spectral density Eu. In the analyzed regional space, the wake caused by the panhead structure is reduced from bottom to top due to the interference from other structures. Combined with the symmetry of the spectra of each point, it can be found that from Z = 2 (Fig. 17b), the spectra of each point with lateral symmetry in spatial location have better symmetry similarity. This suggests that, for the simulation calculations of the 1/1 scale model, the disturbance of the flow field in the region above Z = 2 is primarily caused by the vortex shedding resulting from the panhead. Based on this analysis, the dimensionless vortex shedding frequency (St_1/1) of the panhead can be obtained as 0.17 under the 1/1 condition. Additionally, the spectra of the rear measurement point in the panhead (Fig. 18c) show two peak frequencies, which correspond to the onefold and twofold vortex shedding frequencies, respectively.

Fig. 17

Schematic diagram of velocity signal acquisition array points: a location of panhead sites; b nomenclature of sites

Fig. 18

Velocity power spectra of 1/1 scale model (Y is taken as 1, 2, 3, 4, 5, respectively): a Point-1Y; b Point-2Y; c Point-3Y; d Point-4Y; e Point-5Y

Figure 19 shows the velocity power spectra of the points directly behind the panhead in the 1/8 model. It is evident that there is a prominent peak frequency at the center measurement point behind the panhead, St_Peak-33 = 0.26, and it can be inferred that the vortex shedding frequency caused by the panhead in this case is St_{1/8, panhead} = 0.13.

Fig. 19

Velocity power spectra of Point-3Y in the 1/8 scale model

The transverse and longitudinal side lengths of the rectangular panhead are 54 and 24 mm, respectively, and their ratio is B/H = 2.25. When the aspect ratio of the rectangular cross section is greater than 2, the vortex shedding pattern of the rectangular winding varies across different Reynolds number ranges, resulting in a discontinuous change in its Strouhal number (St). The flow pattern that turbulence detaches downstream at the leading edge position and reattaches at the following edge is referred to as Mode I. Its vortex detachment frequency, expressed as a dimensionless value, ranges from approximately 0.16 to 0.17 across a wide range of Reynolds numbers. The flow pattern that turbulence separates directly from the leading edge of the rectangle is called Mode II. This mode corresponds to a vortex shedding dimensionless frequency, St, ranging from approximately 0.13 to 0.14 across a wide range of Reynolds numbers. Thus, Modes I and II coincide with the vortex shedding frequencies at the panhead of the 1/1 and 1/8 scale models, respectively, and also confirm the typical differences in the flow separation patterns at the panheads of the two models.

3.3 Concave cavity region

Based on the results of identifying the equivalent surface vortex structure by the Q-criterion, it can be observed that the lower part of the pantograph region exhibits a distinct periodic vortex detachment structure.

Figure 20 provides a schematic representation of the distribution of the 3D vortex structure identified by the Q criterion at different time intervals in the operating condition of the 1/1 scale model. The time interval is 30∆t = 0.003 s.

Fig. 20

Instantaneous distribution of vortex structures identified by Q-equivalent surfaces in the pantograph region of the 1/1 scale model (Q = 2 × 10⁵ s^-2): a t₁ = t₀; b t₂ = t₀ + 30∆t; c t₃ = t₀ + 60∆t d t₄ = t₀ + 90∆t; e t₃ = t₀ + 120∆t; f t₄ = t₀ + 150∆t

According to the development history of the vortex structure demonstrated by the spatial distribution of the vortex structure at different moments in Fig. 20, it can be observed that there is a development cycle of the large-scale horseshoe vortex in the lower part of the pantograph. For the sake of clarity, as per their generation times the vortexes will be referred to as horseshoe vortexes 01 and 02, respectively. They will be labeled with an ellipse and a rectangle, as shown at the top of the figure for number correspondence. From the generation of the horseshoe vortex 02 in Fig. 20d–f, it can be observed that the intermediate large-scale horseshoe vortex is formed behind the opening of the pantograph cavity. The vortex structure 02 shown in Fig. 20 f aligns with the position and developmental state of the vortex structure 01 shown in Fig. 20 a, and the middle large-scale vortex structure corresponds to the previously shed vortex structure with the same developmental history after 120∆t–150∆t. That is, the development period of the bottom horseshoe vortex structure is in the range of 120∆t–150∆t, i.e., 0.012–0.015 s, which corresponds to the main shedding frequency of lower vortexes in the 1/1 scale model, f_{1/1, cavity}, in the range of 66–83 Hz.

The distribution of 3D vortex structures at different moments of the 1/8 scale model is shown in Fig. 21 with a time interval of 10∆t = 10 × 0.00005 s = 0.0005 s.

Fig. 21

Instantaneous distribution of vortex structures identified by Q-equivalent surfaces in the pantograph region of the 1/8 scale model (Q = 2 × 10⁵ s^-2): a t₁ = t₀; b t₂ = t₀ + 10∆t; c t₃ = t₀ + 20∆t; d t₄ = t₀ + 30∆t; e t₅ = t₀ + 40∆t; f t₆ = t₀ + 50∆t; g t₇ = t₀ + 60∆t; h t₈ = t₀ + 70∆t

The location and developmental status of the vortex 01 structure shown in Fig. 21a and g are essentially overlapping. In other words, the intermediate large-scale vortex structure aligns with the last shedding vortex structure of the same developmental history after 50∆t–60∆t. The development period of the bottom horseshoe vortex structure is in the range of 50∆t–60∆t, i.e., 0.0025–0.003 s, which corresponds to the main frequency f_{1/8, cavity} at 333–400 Hz for the lower vortex shedding of the 1/8 scale model. In both the 1/1 and 1/8 scale models, the large-scale shedding cycle in the lower part of the pantograph region is essentially the same as the development cycle of the spreading strip vortex at the opening of the forward cavity. Combined with the previous analysis of the fundamental characteristics of the vortex structure, it can be inferred that the shedding frequency of the rear large-scale horseshoe vortex is synchronized with that of the spreading strip vortex at the front of the pantograph cavity.

Figure 22 gives a cloud plot of the distribution of Q values in the cavity region of the pantograph at different moments of the 1/8 model y = 0 m section, showing a cycle t₁–t₁₂ of the shedding of the strip vortex in the pantograph cavity: a pie-shaped vortex packet is formed in the middle of the cavity leading edge, and the strip vortex 01, which is generated in the opening of the cavity in the front of the pantograph, gradually moves downstream and arrives at the position of the previous shedding of the strip vortex 02 after 55∆t, when the next strip vortex structure has been formed and arrives at the same location as the strip vortex 01 at the moment t₁. It can be determined that the period T_1/8,cavity is about 55∆t, and the vortex shedding frequency is about 363 Hz, which is basically the same as the development period of the large-scale horseshoe vortex structure behind the bottom.

Fig. 22

Cloud diagram of Q value distribution in the longitudinal symmetry plane in the region of the pantograph cavity at different moments of the 1/8 scale model (∆t = 0.00005 s, Re = 4.2 × 10⁴): a t₁ = t₀; b t₂ = t₀ + 5∆t; c t₃ = t₀ + 10∆t; d t₄ = t₀ + 15∆t; e t₅ = t₀ + 20∆t; f t₆ = t₀ + 25∆t; g t₇ = t₀ + 30∆t; h t₈ = t₀ + 35∆t; i t₉ = t₀ + 40∆t; j t₁₀ = t₀ + 45∆t; k t₁₁ = t₀ + 50∆t; l t₁₂ = t₀ + 55∆t

According to Sect. 1, it is known that the pantograph cavity is an open cavity. Part of the shear flow, which is separated from the leading edge of the cavity, is obstructed at the bottom frame of the pantograph and does not reach the following edge of the pantograph cavity. This obstruction, in turn, affects the period of self-sustained oscillation in the cavity. The distances of the cavity leading edge from the bottom frame and the cavity following edge of the 1/1 model are 1.94 and 3.38 m, respectively. The full-size distance of the cavity depth is 0.46 m. The power oscillation frequency and resonance oscillation frequency are calculated according to Eqs. (7) and (8) and shown in Tables 3 and 4. Combined with the primary frequency range of vortex shedding depicted in Fig. 21, it is evident that the third-order dynamic oscillation frequency of the leading edge and bottom frame of the 1/1 scale model cavity is approximately 67.6 Hz. Additionally, its first-order resonance oscillation frequency is around 63.4 Hz. These frequencies are in close proximity to each other and have the potential to induce the phenomenon of cavity resonance oscillation. The fifth-order and sixth-order dynamic oscillation frequencies of the leading and following edges of the 1/1 model cavity are approximately 66.9 and 81.1 Hz, respectively. These frequencies are significantly different from the resonance oscillation frequency and can cause the occurrence of cavity dynamic oscillation. Combined with the primary frequency range of vortex shedding depicted in Fig. 21, the cavity leading edge and bottom frame of the 1/8 model exhibit a second-order power oscillation frequency of approximately 343.9 Hz, and a first-order resonance oscillation frequency of about 376.1 Hz. These frequencies are in close proximity to each other and can induce cavity resonance modes.

Table 3 Frequency of power oscillations and resonance oscillations inside the pantograph cavity (cavity leading edge and bottom frame)

Table 4 Frequency of power oscillations and resonance oscillations within the pantograph cavity (cavity leading edge and cavity following edge)

In summary, the flow field in the concave cavity region of the 1/1 scale model consists of two main types of scales. The first type is the airflow resonance oscillation formed by the cavity leading edge and the bottom frame. The second type is the airflow dynamics oscillation mode formed by the cavity leading edge and the following edge. The main scale of the flow field in the concave cavity region of the 1/8 scale model is the resonance oscillation mode formed by the cavity and the bottom frame.

The semi-empirical formula for the modal frequency (f_n) of the cavity hydrodynamic oscillations is as follows [20]:

$$ f_{n} = \frac{n - 0.25}{{M + 1/0.57}}\cdot\frac{{U_{0} }}{L}, $$

(7)

where $U_{0}$ is the incoming flow velocity, $L$ is the length of the cavity along the flow direction, $M$ is the Mach number, and $n$ is the model order.

The semi-empirical formula for the resonance frequency ($f_{m}$) is as follows:

$$ f_{m} = \frac{{\left( {2m - 1} \right)C}}{{4H\left[ {1 + 0.65\left( {L/H} \right)^{0.75} } \right]}}, $$

(8)

where $C$ is the ambient sound velocity, ${ }H$ is the depth of the cavity, and $m$ is the resonance frequency order.

3.4 Sound source characterization

In this paper, the aerodynamic noise source analysis in the pantograph region is conducted using the transient calculation results from the data in the last 0.4 s. Figure 23 shows the distribution of the root mean square value of the surface pulsation pressure ($p_{{{\text{rms}}}}^{\prime}$) in this region.

According to Fig. 23, the intensity distribution patterns of the aerodynamic noise source in the two scales of the PCCS are significantly different. In the 1/8 scale model, there is a larger region of sound source at the following edge of the cavity and the upper frame and lower arm surfaces, which is significantly stronger than that of the 1/1 model. To quantify the contribution of each component to the aerodynamic noise source, Fig. 24 shows its equivalent sound source energy as a percentage of their respective total equivalent sound source energy in a histogram; the calculation process is described in the literature [8]. According to this figure, the energy share of the sound source is similar for most components of the two models, including the middle region structure of the bottom frame, the upper arm frame, and the lower arm. However, there is a significant difference in the distribution of sound source energy between the panhead, supporting insulators, and the pantograph cavity in the two models. This difference is particularly noticeable in the panhead structure, where the sound source energy distribution for the 1/1 scale model and the 1/8 scale model is 11.3% and 0.7%, respectively.

Fig. 23

Cloud view of $p_{{{\text{rms}}}}^{\prime}$ distribution on the surface of the PCCS: a 1/1 scale model; b 1/8 scale model

Fig. 24

Sound source energy shares of PCCS components in the 1/1 and 1/8 scale models

3.5 Radiated noise characteristics

Figure 25 shows the sound pressure level of the radiated noise from each component at the far-field radiation point. Compared to the 1/1 scale model, the sound pressure level in the panhead region of the 1/8 scale model is significantly reduced, with a reduction value ranging from 4 to 12 dB. Specifically, the panhead structure experiences an 11 dB reduction. On the other hand, the sound pressure level of the central armature region, as well as most of the components in the bottom region, is elevated by approximately 5 dB.

Fig. 25

Sound pressure level distribution of radiated noise by PCCS components in the 1/1 and 1/8 scale models

Figure 26 shows the contribution of radiated noise from each component. The main components contributing to the radiated acoustic energy of the 1/1 scale model are the panhead, panhead support, panhead knuckle, and pantograph cavity, respectively. The main components of the 1/8 model that radiate acoustic energy are the balance rod, the upper arm frame, the lower arm, and the pantograph cavity. The distribution of the radiated sound intensity varies considerably between the two scale models, particularly for the panhead and pantograph cavity components. The two components are the uppermost and lowermost structures in the pantograph region, respectively.

Fig. 26

Distribution of the percentage of radiated noise of each component in the pantograph region for the 1/1 and 1/8 scale models

Table 5 provides the ratio of equivalent sound source energy to radiated sound energy in various regions of the pantograph under the two different working conditions. From the table, it can be seen that the contribution of the lower and middle parts of the 1/1 and 1/8 models to the radiated noise in the region is 27.3% and 87.2%, respectively, and it can be found that the main acoustic energy components of the 1/8 model are concentrated in the lower and middle parts of the region. The 1/8 scale model exhibits a relatively high radiation efficiency of the sound source in the middle and lower regions. Its ratio of radiated sound energy is significantly larger than that of the 1/1 scale model. However, the ratio of the radiated sound energy in the panhead region of the 1/8 scale model is much lower that of the 1/1 scale model. The smaller the model scale, the larger the bottom perturbation. This causes the sound source energy to expand to both sides, resulting in an overall increase in the radiation efficiency of the bottom.

Table 5 Percentage of source energy and radiated sound energy statistics by block region (%)

The equal bandwidth spectra of radiated noise for the key components and the entire system at the two scales are shown in Figs. 27 and 28, respectively.

Fig. 27

Equal bandwidth spectra of the radiated noise of main components in subregions of the pantograph of the 1/1 scale model: a panhead region; b middle region; c bottom region; d pantograph region as a whole

Fig. 28

Equal bandwidth spectra of the radiated noise of main components in subregions of the pantograph of the 1/8 scale model: a panhead region; b middle region; c bottom region; d pantograph region as a whole

Comparing Figs. 27 and 28, it is evident that the radiated sound spectra of most components within each subregion exhibit some similarity. However, there are noticeable variations in the radiated sound spectra of components across different regions. Specifically, the lower structure of the pantograph produces more low-frequency sound, making it the primary source of low-frequency noise. Conversely, the upper region predominantly contributes to the higher frequency range, thus being the main source of high-frequency noise. The spectra of each component of the two scales exhibit significant differences, and there is no consistent scale similarity law. Therefore, it is necessary to perform spectral shape scale correction.

For the 1/1 scale model, its total radiated noise in the 40–100 Hz range is significantly larger than in other frequency domains, and it mainly originates from the bottom of the cavity. Combined with the main flow field structure in this region, it is reasonable to believe that it originates from the resonance oscillations of the airflow caused by the cavity guiding edge and the bottom frame, as well as the dynamic oscillations of the airflow caused by the cavity guiding edge and the following edge. Meanwhile, the 1/1 scale model has a total radiated noise spectrum with peak sound pressure levels at f₁ = 307 Hz and f₂ = 571 Hz, respectively. The f₁ is consistent with the peak vortex shedding frequency of the panhead. The source of the f₂ peak frequency is relatively complex. Peak frequencies close to them appear in the spectra of most components, but it can be noted that their frequencies are different yet closer. Therefore, it is more likely that f₂ is a result of the superimposed effect of the components.

For the 1/8 scale model, the peaks of the total radiated noise spectra are f₁ = 280 Hz, f₂ = 1970 Hz, and f₃ = 3820 Hz, respectively. The frequency f₁ is similar to the oscillation frequency of the shear layer caused by the pantograph cavity and the bottom frame. The frequency f₂ mainly comes from the frequency of vortex shedding from the panhead square column winding. The source of the peak frequency f₃ is complex and not identical to the source of f₂ in the 1/1 model. The spectra of the panhead region in Fig. 28a are very similar to those of the middle region in Fig. 28b, showing similar peak frequencies in the middle and high frequencies. It is hypothesized that the peak is a result of the coupling and superposition of the spectra of the middle and upper components.

4 Conclusion

This paper comprehensively compares the aerodynamic acoustic performance of the PCCS at both 1/1 and 1/8 scales. It reveals that the two aerodynamic noise spectra do not follow a uniform scale similarity law. The study also identifies their respective aerodynamic sounding mechanisms. The main conclusions are as follows:

• (1) The 1/1 scale model fluid is reattached to the cylinder after the separation from the panhead leading edge, while the 1/8 scale model fluid is directly separated at the panhead leading edge. The vorticity frequency between the two will also vary. The Reynolds number around the panhead of the 1/1 scale model is approximately 4.3×10⁵, and the frequency at which vortex shedding occurs is 303 Hz, corresponding to a St of 0.17. The Reynolds number around the 1/8 panhead is approximately 5.4×10⁴, and the frequency of vortex shedding is 2000 Hz, which corresponds to a St of 0.13.

• (2) In the 1/1 scale model, there are two main flow modes in the cavity region, including the airflow resonance oscillation formed by the leading edge of the cavity and the bottom frame, and the airflow dynamic oscillation mode formed by the leading edge of the cavity and the following edge. The primary flow pattern in the 1/8 scale model cavity region is the resonant oscillation mode that occurs between the cavity and the bottom frame.

• (3) The ratio of the source energy in the panhead region of the 1/1 scale model is significantly higher than that in the 1/8 scale model, while the source energy in the cavity region is significantly weaker than that in the 1/8 scale model.

• (4) The low-frequency region of the total radiated noise of the 1/1 model mainly originates from the concave cavity region, which is generated by the two mechanisms: airflow resonance oscillations formed by the cavity leading edge and the bottom shelf , the airflow power oscillations formed by the cavity leading edge and the following edge. Its high-frequency region is originated from the panhead region, and the peak noise at 307 Hz is originated from the panhead vortex shedding.

• (5) The low-frequency region of the total radiation noise of the 1/8 scale model mainly originates from the cavity region. The peak noise at 280 Hz is closely associated with the airflow resonance oscillation created by the leading edge of the cavity and the bottom frame. The high-frequency region originates from the panhead region, while the 1970 Hz peak noise is caused by the panhead vortex.