Flow and sound fields of scaled high-speed trains with different coach numbers running in long tunnel

Abstract

Segregated incompressible large eddy simulation and acoustic perturbation equations were used to obtain the flow field and sound field of 1:25 scale trains with three, six and eight coaches in a long tunnel, and the aerodynamic results were verified by wind tunnel test with the same scale two-coach train model. Time-averaged drag coefficients of the head coach of three trains are similar, but at the tail coach of the multi-group trains it is much larger than that of the three-coach train. The eight-coach train presents the largest increment from the head coach to the tail coach in the standard deviation (STD) of aerodynamic force coefficients: 0.0110 for drag coefficient (C_d), 0.0198 for lift coefficient (C_l) and 0.0371 for side coefficient (C_s). Total sound pressure level at the bottom of multi-group trains presents a significant streamwise increase, which is different from the three-coach train. Tunnel walls affect the acoustic distribution at the bottom, only after the coach number reaches a certain value, and the streamwise increase in the sound pressure fluctuation of multi-group trains is strengthened by coach number. Fourier transform of the turbulent and sound pressures presents that coach number has little influence on the peak frequencies, but increases the sound pressure level values at the tail bogie cavities. Furthermore, different from the turbulent pressure, the first two sound pressure proper orthogonal decomposition (POD) modes in the bogie cavities contain 90% of the total energy, and the spatial distributions indicate that the acoustic distributions in the head and tail bogies are not related to coach number.

1 Introduction

Nowadays, high-speed trains are playing an increasingly important role in long-distance passenger transport. With the development of technology, the operating speed of trains keeps increasing, generally reaching more than 300 km/h. It is accompanied with larger aerodynamic load and aerodynamic noise. When the trains run in tunnels, the average values and fluctuations of aerodynamic forces will increase significantly due to the large increase of the blockage ratio [1]. This aerodynamic effect will cause many problems in high-speed trains, including increased energy consumption and aural discomfort. Therefore, it is of great significance to study the aerodynamic characteristics of trains in tunnel to improve the running efficiency and passenger comfort.

Aerodynamic characteristics of high-speed trains can be obtained by real train tests and moving model tests. It is generally considered that the data obtained from the real train tests are the most realistic because it is completely consistent with the actual working conditions of high-speed railways. Aerodynamic effects in tunnel were investigated by a series of field measurements near the portal and the shaft or the adit of the tunnel during normal operation [2]. It is found that the pass-by of a train inside the tunnel induces an immediate local pressure drop due to aerodynamic drag. The fluctuating pressure of Korean high-speed trains running on Seoul-Busan high-speed line was measured and estimated [3], revealing that the maximum pressure change has definite relation with the train speed and the train shape. The moving model tests are carried out as well due to their low cost, high repeatability and minor random error. A new test method that can precisely measure the aerodynamic drag coefficient of a high-speed train passing through a tunnel was adopted on the moving model device at Central South University in China [4]. It has indicated that the difference between moving model test and numerical simulation is less than 7%. Li et al. [5] used the double-line tunnel and 1:30 eight-coach train model to explore the pressure effect on the platform screen door induced by the high-speed trains. The results reveal that the increase of train speed leads directly to the increase of pressure variation amplitude on the surfaces of train and platform screen door. Xie et al. [6] conducted moving model tests to collect the interior pressure transients under a series of speeds. In terms of the interior comfort in tunnel, the increase of the train speed induces directly aural discomfort, and the aural sensations are better for passengers in the tail compared with in the head.

A lot of researches also have been carried out by conducting numerical simulations with static mesh method because of its convenience, flexibility and strong ability in data processing. The flow field and aerodynamic noise sources obtained by the static mesh method are barely different from the dynamic mesh method, and the difference in the sound pressure level was less than 2 dB(A) [7]. Diedrichs et al. [8] analyzed the unsteady pressure load on the train body in tunnel by full-scale two-coach high-speed trains and concluded that, compared to the ICE2 model, the Shinkansen model induced about twice the lateral unsteady load, which led to worse train body vibration. Li et al. [9] found that vortices mainly distributed in the bogie and inter-coach area, and the negative peaks of the aerodynamic lift decreased with the increase of the tunnel length. According to Ref. [10], the upside of upstream coach had larger sound source energy than that of the downstream coach, but the downside of coach was on the contrary.

In order to extract dominant flow structures in turbulence more effectively, modal decomposition methods of flow field including cluster-based reduced-order model (CROM), proper orthogonal decomposition (POD) and spectral proper orthogonal decomposition (SPOD) are developed. Östh et al. [11] analyzed the wake of the high-speed train model obtained from large eddy simulation (LES) by CROM and extracted two main flow structures, namely longitudinal vortices and vortex shedding, which were associated with states of the aerodynamic drag. Ferrari et al. [12] presented a novel surface visualization to convey the spatiotemporal changes undergone by clustered vortices in the wake of high-speed trains. Through dimensional reduction of 3D volumetric vortices into 1D ridges and physics-based feature tracking, the behavior of vortices in the wake was clearly displayed on 3D surfaces. The spectral estimation parameters including block number, frequency resolution and cutoff frequency were investigated. As the number of blocks, cutoff frequency and frequency axis resolution increased, the SPOD results for the train wake became better [13].

Aerodynamic characteristics and sound source generated by high-speed trains in tunnel have been studied by experiments and simulations, but the effect of coach numbers on aerodynamics and acoustics has not been clearly understood. Therefore, the static method was used in this paper, and the flow field, aerodynamic forces and near-field sound of scaled high-speed trains with different coach numbers in long tunnel were obtained by LES and acoustic perturbation equations (APEs). All numerical simulations in this work were conducted with Star-CCM + . The current static methods contribute to computational stability and reduce computational time. It should be pointed out that when a train runs in a long tunnel, the pressure waves at the entrance and exit of the tunnel could have a small impact on the aerodynamics and acoustics of the train. The structure of this paper is presented as follows: Methodology is introduced in Sect. 2, and the accuracy of the simulation is verified by the same scale model experiment in Sect. 3; the flow field and acoustic characteristics of trains with different coach numbers are analyzed in Sect. 4; the conclusions are drawn in Sect. 5.

2 Methodology

2.1 High-speed trains models

At present, high-speed trains with three coaches running in tunnel are mostly used to simulate the characteristics of flow field and aerodynamic noise, which include the head, the tail, and the middle, etc., as Fig. 1 shows. However, due to three-coach mismatch, the stowed and raised pantographs are located separately. In order to clarify the influence of the coach number on the aerodynamic forces and near-field noise of the train, eight-coach model is constructed, where the stowed and raised pantographs are located on the 3rd and 6th coach, respectively. Six-coach model has the stowed and raised pantographs located on the 2nd and 5th coach, respectively. Even though six-coach model does not exist in reality, it can be viewed as an intermediate case between the commonly used three-coach model and the more realistic eight-coach model to investigate the influence of coach number on the flow field and aerodynamic noise characteristics in the long tunnel. Given the limited computing resources and time, a 1:25 scale train model is adopted with the width W and the height H of 133 and 153 mm, respectively. The distance between the head of the raised and stowed pantographs from the roof is 73 and 21mm, respectively. The total lengths of the three, six and eight coaches are 3232, 6448 and 8592 mm, respectively. For the three train models, they all have an entrance wind speed of 97.2 m/s, of which the Reynolds number is 9.4 × 10⁵ corresponding to the train height.

Fig. 1

1:25 scale high-speed train models with different coaches

The same standard cross section of single tunnel in China [14] is adopted, and it is of excellent arc. After scaling down, its radius and the circle center away from the ground are 235 and 65mm, respectively. The train is on the Chinese standard track, and the distance between the head nose tip and the tunnel entrance is about 10H, and that between the tail nose tip and the tunnel exit is about 25H.

The curved segment of the head, tail, and pantograph is divided into triangular mesh with the size of 0.8 mm. The surface mesh size of the bogie and its cavity is 1.6 mm, and the straight section and inter-coach gap are 4.0 mm.

The track has a surface mesh size of 1 mm, the ground plane 6 mm, and the tunnel wall, the inlet and outlet of the computational domain 8 mm. The grid distribution of train with different coach numbers remains the same. In order to solve the flow and noise better, the refined zone with the size of 2.0 mm is set in the key regions of the flow field including the first and last bogies and the pantographs. In addition, the whole train is wrapped in a refined zone with a volume grid of 4 mm, as Fig. 2 shows. After the generation of the volume mesh, the cross-sectional diagrams of whole trains are shown in Fig. 3. The same prism layer mesh is used for each case, that is, the Y + corresponding to the first prism layer of the train surface is about 1, the growth rate is 1.2, and the number of layers is 18. Due to the moving wall of the ground and the tunnel wall, the relative near-wall velocity becomes small and the number of layers is 9; thus the requirement of Y + can be satisfied. With the basic volume grid size of 8.0 mm and the trimmer mesh generation method, the numbers of volume mesh are 35 million, 54 million and 65 million for the train models with three, six and eight coaches, respectively.

Fig. 2

Refined zone distribution: a volume grid of 2 mm; b volume grid of 4 mm

Fig. 3

Volume mesh distribution for trains with 3, 6 and 8 coaches

2.2 Numerical model

Segregated incompressible LES is used in unsteady flow calculation. The large-scale eddies are solved directly, and the small-scale eddies are processed by sub-grid model of wall-adapting local eddy-viscosity (WALE) [15]. The advantage of this model is that it does not require any form of near-wall damping, but automatically provides accurate scaling at the wall. The equations of LES are obtained by a spatial filtering. Each solution variable $\varphi$ is decomposed into a filtered value $\tilde{\varphi }$ and a sub-grid value $\varphi^{\prime}$ as Eq. (1) shows, where $\varphi$ represents velocity components, pressure and species concentration, etc. Eq. (2) is the filtered mass equation, Eq. (3), is the filtered momentum equation, and Eqs. (4) and (5) are obtain the turbulent stress tensor.

$$\varphi = \tilde{\varphi } + \varphi^{\prime},$$

(1)

$$\rho \left( {\nabla \cdot \tilde{v}} \right) = 0,$$

(2)

$$\rho \frac{{\partial \tilde{v}}}{\partial t} + \rho \left( {\tilde{v} \cdot \nabla } \right)\tilde{v} = - \nabla \tilde{p}I + \nabla \cdot \left( {T + {T}_{{{\text{SGS}}}} } \right) + f_\text{b} ,$$

(3)

$$T_{{{\text{SGS}}}} = 2\mu_{{\text{t}}} S - \frac{2}{3}\left( {\mu_{t} \nabla \cdot \tilde{v} + \rho k} \right)I,$$

(4)

$$S = \frac{1}{2}\left( {\nabla \overline{v} + \nabla \overline{v}^{T} } \right),$$

(5)

where $\rho$ is density and the value is constant; $\tilde{v}$ and $\tilde{p}$ are the filtered velocity and pressure, respectively; $I$ is the identity tensor; $T$ is the stress tensor; $T_{{{\text{SGS}}}}$ is the sub-grid scale stress tensor; $f_{{\text{b}}}$ is resultant force of the volume force (including gravity and centrifugal force, etc.); $S$ is the mean strain rate tensor; $k$ is the sub-grid scale turbulent kinetic energy; $\mu_{{\text{t}}}$ is the sub-grid scale turbulent viscosity and described by WALE sub-grid model; $\overline{v}$ is the time-averaged velocity.

To solve the near-field sound pressure, the APEs developed by Ewert and Schroder is used [16]. APEs, the physical damping term can effectively remove the numerical spurious waves generated from mesh-coarsening transitions. When sound waves propagate in a viscous medium, the equation is

$$\frac{1}{{c^{2} }}\frac{{\partial^{2} p^{{\text{a}}} }}{{\partial t^{2} }} + \frac{{2\overline{v}}}{{c^{2} }} \cdot \nabla \frac{{\partial p^{{\text{a}}} }}{\partial t} + \frac{{\left( {\overline{v} \cdot \nabla } \right)}}{{c^{2} }}\left( {\nabla \cdot \overline{v}p^{{\text{a}}} } \right) - \nabla^{2} \left( {p^{{\text{a}}} + \tau \frac{{\partial p^{{\text{a}}} }}{\partial t}} \right) = - \left( {\frac{1}{{c^{2} }}\frac{{\partial^{2} p^{\prime}}}{{\partial t^{2} }} + \frac{{2\left( {\overline{v} \cdot \nabla } \right)}}{{c^{2} }}\frac{{\partial p^{\prime}}}{\partial t} + \frac{{\left( {\overline{v} \cdot \nabla } \right)}}{{c^{2} }}\left( {\nabla \cdot \overline{v}p^{\prime}} \right)} \right),$$

(6)

$$\tau = \chi \frac{\Delta t}{{\uppi \lambda }},$$

(7)

where c is the speed of sound; p^a is the acoustic pressure; $p^{\prime}$ is the hydrodynamic pressure; $\chi$ is the damping coefficient; ${\Delta }t$ is the time step; $\lambda = c{\Delta }t/{\Delta }x$ is the local Courant-Friedrichs-Lewy (CFL) number, where ${\Delta }x$ is the grid size.

2.3 Computational setup

The inlet of the computational domain is set as velocity inlet with the value of 97.2 m/s. The pressure reference point is set at the center of the inlet. The outlet is set as the outflow with the split ratio of 1.0. The train is stationary, and the ground and tunnel wall move at 97.2 m/s.

The $k{-}\varOmega$ SST turbulence model [17] is used for steady state, and its convergence is observed after 5000 iterations. The initial time step is 0.0005 s, and the inner iteration number is 10. After 1000 timesteps, the time step is changed to 0.00005 s, and inner iteration number is 8 during a time step. Thereafter, the averaged velocity and pressure are monitored, and the sound source zone and damping zone are set, as Fig. 4 shows. The numerical representation of the legend in Fig. 4 represents noise source weighting coefficient, which defines the space within which noise is generated and is a dimensionless quantity with the values from 0 to 1. The APE solver starts at 0.57505 s and the data sampling begins when the flow and sound fields reach unsteady stability at 0.6 s. During the 4000 steps of data sampling, more than two times the flow-through time, turbulent pressure and near-field sound pressure are obtained from the LES and APE solvers, respectively. Meanwhile, the time discretization of the implicit unsteady solver is set to the 2nd order, and the spatial discretization is set to the 2nd-order bounded differencing.

Fig. 4

Noise zone

3 Numerical comparison and test validation

Due to the limited experimental condition, a 1:25 scale high-speed train with two coaches is processed to perform pressure measurements in a closed-jet wind tunnel. Meanwhile, the related simulation is constructed, in which the geometric features and dimensions of the train model are as consistent as possible with those of the experiment. For the specific _tsettings of experiment and simulation, please refer to Ref. [18].

3.1 Grid independence

In order to ensure grid independence, three different mesh strategies are set up and shown in Table 1. In the simulation, the free flow velocity is uniformly set at 20.2 m/s, and the same LES solver is used to ensure that the comparison is not affected by other factors. And the volume mesh diagrams of three different mesh strategies are shown in Fig. 5.

Table 1 Three different mesh strategies

Fig. 5

Grid distribution on the mid-sections: a coarse mesh; b mid-mesh; c fine mesh

Thirteen and seven measuring points are arranged on the top and side of the train, as Fig. 6 shows. Results of the time-averaged pressure coefficient C_p from the numerical simulations and experiment are shown in Fig. 7. Pressure coefficient C_p is defined as

$$C_{{\text{p}}} = \frac{{p - p_{\infty } }}{{p_{0} - p_{\infty } }},$$

(8)

Fig. 6

Position of the measuring points

Fig. 7

Comparison of the pressure coefficient of measuring points for grid independence: a top; b side

where p is the local pressure; $p_{\infty }$ is the reference pressure; $p_{0}$ is the total pressure at the reference point.

In Fig. 7, it can be found that the pressure coefficients obtained from simulation under different meshes of each measuring point on the top of the first coach are in good agreement with the experimental data. And their differences are all within 0.1, indicating that there is no large error in the experiment and simulation. The overall comparison between the simulation data and the experiment data shows that the gap between them gradually increases from T1, and the difference between simulation and experimental data at points T10, T11, T12 and T13 remains basically unchanged. The difference at T10 can be attributed to the assembling of 3D printed components, as the interface of the upper part and lower part is attached and smoothed with the metal and black friction tapes, leading to surface unevenness. The simulated pressure coefficients of the side measurement points are generally about 0.15 larger than the experimental data. A possible reason is the three threaded steel rods in the experiment to suspend the train in the air, through which the data lines from the pressure scanning valve inside the train are led out, as Fig. 8 shows. The absolute symmetry is difficult to achieve, when positioning the suspended train model.

Fig. 8

1:25 scale high-speed train model in wind tunnel

The pressure coefficients of the corresponding top measurement points obtained from simulation under different meshes are generally small, and their differences become slightly larger for the side measurement points. Overall, the results of coarse mesh with the base size of 12 mm are slightly worse than those of medium and fine meshes, and the medium mesh with the base size of 8 mm has enough accuracy to obtain the pressure of the top and side measuring points of the train body.

3.2 Discretization scheme

A comparative validation was conducted on different discretization schemes for the convective flux settings. Bounded-central discretization scheme applies a boundedness criterion that makes the scheme more robust than central-differencing alone, which is recommended for the LES of complex turbulent flows. The MUSCL (monotonic upstream scheme for conservation laws) 3rd-order/CD (central-differencing) discretization scheme is a blend between a MUSCL 3rd-order upwind scheme and the 3rd-order central-differencing reconstruction scheme. It provides reduced dissipation when compared with the second-order scheme. At the same time, it is robust and capable of simulating steady and unsteady flows from incompressible to high-speed compressible regimes.

From Fig. 9, it can be observed that the pressure coefficients obtained from both schemes are essentially consistent. Meanwhile, power spectral density (PSD) is calculated using MATLAB. The sampling frequency and time are 2500 Hz and 2s, respectively. Hanning window is used and the window length is 1024 where the overlap ratio is 0.5. The pressure PSD plots at T4 and S2, as Fig. 10 shows, reveal that bounded-central scheme and MUSCL-3rd-order/CD scheme exhibit a similar behavior in the frequency band above 200 Hz. However, in the low-frequency range, the former exhibits a peak around 30 Hz that closely matches the experimental results, which can be selected in the subsequent simulations.

Fig. 9

Comparison of the pressure coefficient of measuring points with discretization schemes: a top; b side

Fig. 10

Comparison of the pressure PSD with discretization schemes: a T4; b S2

3.3 Train velocity

In order to investigate the changes in the averaged pressure coefficient and PSD of measurement points on the body surface under different train velocity, 97.2 m/s for subsequent attention and the current 20.2 m/s are selected. As Fig. 11 shows, the pressure coefficients obtained from both simulations are essentially consistent, indicating an accurate fit for the case of 97.2 m/s. In the pressure PSD plots at T4 and S2 illustrated in Fig. 12, the simulation at 97.2m/s exhibits a difference of approximately 20 dB/Hz compared to that of 20.2 m/s in the frequency range of 30–600 Hz, where the difference remains stable. The sampling time and processing parameter settings for PSD are consistent with the PSD of pressure with different discretization schemes in Sect. 3.2. This observation is consistent with the enhanced fluctuation of airflow on the vehicle surface due to increased velocity. And in the PSD at T4, a peak is located around 70 Hz, which only occurs at 97.2 m/s. This can be attributed to the generation of new fluctuations in the space between the roof of the train and the tunnel by higher-speed airflow.

Fig. 11

Comparison of the pressure coefficient of measuring points for different velocities: a top; b side

Fig. 12

Comparison of the pressure coefficient PSD for different velocities: a T4; b S2

4 Numerical analysis of trains with different coach numbers

4.1 Time-averaged velocity

The dimensionless streamwise velocity contour of the middle section under different coaches is shown in Fig. 13, where V is the velocity of the train, which is 350 km/h and u is the velocity of the airflow. Due to the increased blockage ratio within the tunnel, the interaction of the train with the tunnel results in an approximately 25% increase in the maximum velocity. By comparison, it can be found that the velocity gradient change along the outer curved segment of the head is almost the same, while the velocity distribution at the top of the tail is different due to the flow field disturbance caused by the pantograph. The wake induced by the stowed pantograph in the six-coach and the eight-coach trains is basically the same, which mainly covers the space over the next two coaches. The distribution of wake velocity induced by the raised pantograph is basically the same in three-coach and six-coach trains. There are two coaches behind the raised pantograph in eight-coach train, whose roof has larger velocity recovery area than the cases of three-coach and six-coach trains. The velocity gradient is larger in the transition area from the upper part of the tail to leave the train.

Fig. 13

Time-averaged dimensionless streamwise velocity distribution on train mid-section: a eight coaches; b six coaches; c three coaches

Figure 14 shows the distribution of time-averaged pressure coefficients of different train models. It is illustrated that the pressure distributions on the head bottom of three train models are basically the same, but there are evident differences on the underbody. The increase in the number of coaches leads to a significant increase in the negative pressure on the rear-coach underbody, and the pressure transition area at the nose end of the tail also extends. For the six-coach and eight-coach trains, the pressure recovery area of the tail nose affected by the wake is significantly reduced. Meanwhile, the negative pressure on the underbody of the six-coach and eight-coach trains gradually increases along the flow direction, while it cannot reflect this characteristic due to its small coach number of three-coach train.

Fig. 14

Time-averaged pressure coefficient of different coaches: a eight coaches; b six coaches; c three coaches

4.2 Aerodynamic force coefficients

The three-direction aerodynamic coefficients of the train with different numbers of coaches are shown in Table 2. Drag coefficient $C_{{\text{d}}}$, lift coefficient $C_{{\text{l}}}$ and side force coefficient $C_{{\text{s}}}$ are defined as

$$C_{{\text{d}}} = \frac{{F_{x} }}{{\frac{1}{2}\rho A_{{{\text{tr}}}} V^{2} }},$$

(9)

$$C_{\text l} = \frac{{F_{z} }}{{\frac{1}{2}\rho A_{{{\text{tr}}}} V^{2} }},$$

(10)

$$C_{{\text{s}}} = \frac{{F_{y} }}{{\frac{1}{2}\rho A_{{{\text{tr}}}} V^{2} }},$$

(11)

Table 2 Mean value of aerodynamic force coefficients of each coach

where ρ is the air density and the value is 1.184 kg/m³; $A_{{{\text{tr}}}}$ is the frontal area of train model and the value is 0.0197 m²; $F_{x}$, $F_{y}$ and $F_{z}$ represent the aerodynamic forces acting on the train in the x, y and z directions, which are known as drag force, lift force and side force.

Table 2 shows the mean values of aerodynamic force coefficients of each coach. With the increase in coach number, the differences in C_d and C_l are not significant for the head and middle coaches, but evident at the tail coach. The tail-coach C_d of the six-coach train and eight-coach train shows a noticeable increase compared to the three-coach train, but relatively small differences observed between the multi-group trains. Except for the head and tail coaches, the middle coach with the highest value of C_d for the six-coach and eight-coach train is the 5th and 6th coach, both of which have raised pantographs, indicating a significant aerodynamic drag induced by the pantograph. Additionally, C_d of the three-coach train shows a significant decrease in the downstream direction, which is different from multi-group trains. In terms of C_l, where a positive value represents an upward direction perpendicular to the ground, the mean values of C_l are downward for the head coach and the maximum lift occurs on the tail coach for all three types of train sets, reflecting the effect of the wake. In addition, the mean values of C_s for each coach are much smaller compared to C_l and C_d, attributed to the lateral symmetry of the model, where the perturbations cancel out in the mean values.

Comparatively, as Table 3 shows, the distribution of STD (standard deviation) for individual coaches is consistent across the three cases, with minimal differences in the values of the head coaches. The STD of C_d, C_l and C_s values of individual coach increases in the streamwise direction. As the coach number increases, the overall STD of the entire train also rises. Compared to C_d and C_l, STD of C_s exhibits a larger increase from the head to the tail, reaching 0.0371, which is related to the hunting motion at the bottom of the train [19].

Table 3 STD of aerodynamic force coefficients of each coach

For PSD of aerodynamic force coefficients, the sampling frequency and time are 20,000 Hz and 0.2 s, respectively. Hanning window is used and the window length is 1024 where the overlap ratio is 0.5. Figure 15 shows the results of trains with different numbers of coaches. In general, aerodynamic coefficient fluctuation energy of the eight-coach train is slightly higher, the six-coach train is middle and the three-coach one is slightly lower. The aerodynamic coefficients of different coaches have the most obvious difference in the low-frequency band, and their peak frequencies are not completely the same, especially in terms of C_d and C_l. Peaks at 140 Hz occur in only eight-coach and six-coach trains, which indicates that the number of coaches has an influence on lateral force fluctuations. In the frequency band above 500 Hz, the spectra are disordered and they are almost the same.

Fig. 15

PSD of aerodynamic force coefficients: a C_d; b C_l; c C_s

4.3 Turbulent pressure

4.3.1 Total turbulent pressure level

When the train runs, the surface pressure can be segregated into turbulent pressure and sound pressure. The total turbulent pressure levels (TTPLs) on the surface of the three-coach, six-coach and eight-coach trains are shown in Fig. 16, where Fft represents the fast Fourier transform. It can be found that the train body generally presents a state of weak in the front and strong in the back. TTPL is kept at about 125 dB except for the wiper groove, while the TTPL in the pantograph groove and tail near nose tip reaches 145 dB due to the serious separation of airflow. The TTPL of tail is bigger than that of head, and its value is basically higher than 130 dB. This is because the boundary layer gradually develops along the body, and when irregular surfaces such as pantographs and inter-coach gaps are encountered, the flow separation is exacerbated and the turbulent kinetic energy rapidly increases. The TTPL of train bottom is larger and mainly distributed near the rear surface of the bogie cavity and the transition section between the bottom and the side of the train due to the airflow interaction between them.

Fig. 16

TTPL distribution on the train surface (dB): a eight coaches; b six coaches; c three coaches

Unlike three-coach train, stowed pantograph is mounted on six-coach and eight-coach trains. However, the stowed pantographs have a limited effect on the TTPL on the train roofs, where only a smaller range of wake is found. The difference is that the raised pantograph not only greatly improves the TTPL of the area where it is located, but also has a greater impact on the TTPL on the subsequent coaches. The TTPL on the train side and bottom does not change with the different numbers of coaches. It still shows the distribution law of the former weak and the latter strong. This is different from the TTPL distribution of the train running in open air [20], which is caused by the unique flow field formed by trains running in the limited space.

In order to quantitatively compare the difference of turbulent pressure among the coaches of the trains, Table 4 shows the surface-averaged root-mean-square (RMS) of turbulent pressure, which is defined by Eq. (12). It is found that their values are in the range of 132–139 dB and the distribution exhibits the characteristic of small magnitudes near the head and large magnitudes near the tail. Compared with the three-coach train, the head of the eight-coach train is 0.1 dB smaller, and the coach of raised pantograph and the tail is 3.1 and 1.4 dB larger, respectively. Compared with the six-coach train, the head and the tail of the eight-coach train are increased by 0.2 and 0.6 dB, respectively. The coaches of stowed and raised pantographs are increased by 2.7 and 0.2 dB, respectively.

Table 4 Area-averaged root-mean-square turbulent pressure of different coaches (unit: dB)

The aforementioned analysis indicates that, in the study of surface turbulent pressure, the adoption of a three-coach train results in an underestimation of TTPL on all coaches except for the head. For the six-coach train and eight-coach train, TTPL on the head and tail as well as coaches with pantograph does not exhibit significant differences, but TTPL on other coach surfaces is smaller for six-coach train than the eight-coach train. Regardless of whether a commonly used three-coach or six-coach train is adopted, there would still be a discrepancy when compared with the longer eight-coach formation. As the number of coaches increases, the predicted results move closer to the actual values.

$${\text{TTPL}} = 20\lg \frac{{\mathop \sum \nolimits_{i = 1}^{n} \sqrt {\frac{1}{{t_{2} - t_{1} }}\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} p_{i}^{2} \left( {{\varvec x},t} \right){\text{d}}t} A_{i} }}{{2 \times 10^{ - 5} \mathop \sum \nolimits_{i = 1}^{n} A_{i} }},$$

(12)

where TTPL is the surface-averaged RMS of turbulent pressure; t₁ and t₂ are the start and end times of sampling; $p\left( {{\varvec x},t} \right)$ is the turbulent pressure at the element on the train surface presented by vector ${\varvec x}$, which is the coordinates of the center of element i on the surface of the train; A_i is the area of element i.

4.3.2 Frequency spectra of turbulent pressure

The turbulent pressure PSD at measuring points B1, B2 and P1 is shown in Fig. 17, where B1 and B2 are located in the center of the first and last bogie cavity roofs, and P1 is located in the center of the raised pantograph bilge. And the sampling time and processing parameter settings for PSD are consistent with the PSD of aerodynamic force coefficients in Sect. 4.2. As Fig. 18 shows, for each train model, as the frequency increases, the spectra of B1 and B2 show a trend of first increasing and then decreasing. For B1, the frequency band with larger energy is concentrated on 500–3000 Hz, while lower frequency band of 150–2000 Hz is found at B2. Their magnitude exceeds 115 and 105 dB, respectively. The spectrum of B1 of the eight-coach train at the frequency below 1000 Hz are larger than that of the three-coach and six-coach trains. However, the spectra of B2 corresponding to different trains are not very different. For P1, as the frequency increases, the PSD also increases first and then decreases. The frequency band with high energy is concentrated in 100–1500 Hz, and its magnitude exceeds 100 dB. Overall, the number of coaches does not change the tendency of PSD; however, that of the eight-coach train shows small peaks at 270 and 500 Hz.

Fig. 17

Position of monitoring probes

Fig. 18

Turbulent pressure PSD at key points: a B1; b B2; c P1

4.3.3 Reduced-order analysis of turbulent pressure in head and tail bogie cavities

POD [21] has been widely used in the analysis of the flow field around high-speed trains in recent years. In this paper, spatial measuring points are evenly arranged in the first bogie of the head and the second bogie of the tail. For the first bogie, the spatial scope is x = 140–326 mm, y =  − 65–65 mm, z = 0–54 mm, where the lateral, longitudinal and vertical spacings are all 2 mm. There are 173,712 points, as Fig. 19 shows. The arrangement of measuring points in the last bogie is the same as the first bogie.

Fig. 19

The measuring points in the first bogie of the head train

POD analysis is performed on the turbulent pressure in the first and last bogie cavity spaces, and the modal energy proportion is shown in Fig. 20. By comparison, it can be found that except for the 1st-order and 2nd-order modes of the tail, the energy proportions of all modes in different trains are basically the same. For the first bogie located in the head, the proportion of energy in each order mode decreases slowly. The 1st-order mode accounts for about 12% and the 2nd-order mode 10%. It decreases to below 5% after the 8th order. In contrast, the energy proportion of the second bogie of the tail decreases rapidly at first and then goes down slowly. The proportion of the 1st-order and the 2nd-order modes of the three coach is 24.1% and 23.2%, respectively. After the 4th-order mode, the energy proportion decreases to less than 5%. In addition, the 1st-order modes of the six-coach train and eight-coach train are both around 27.6%, and the 2nd-order mode is both almost around 21.2%.

Fig. 20

Energy percentage of turbulent pressure in bogie cavities: a head bogie; b tail bogie

Due to the biggest energy proportion, Figs. 21 and 22 only show the isosurfaces of the 1st-order mode in the bogie cavities of the first bogie of head and the last bogie of tail, where T_p represents the 1st-order POD mode of the normalized turbulent pressure and is dimensionless. It can be seen that the POD mode of them essentially makes no difference. In the first bogie the alternate vortices expand along the flow direction, caused by the flow separation at the trailing edge of the cowcatcher. The vortex near the cowcatcher is relatively smaller and characterized by the alternatively positive and negative variations, while the vortex further away from the cowcatcher is larger. The vortices are caused by the instability of the shear layer as the airflow passes through the cowcatcher. The last bogie has no obvious characteristics along the flow direction, but the positive and negative regions distribute laterally, indicating that the phase of turbulent pressure fluctuation on both sides is opposite. This is mainly due to the alternating shedding of vortices at the left and right edges of the bogie caused by the lateral swaying of the wake flow.

Fig. 21

The 1st modal contour of head bogie: a three coaches; b six coaches; c eight coaches

Fig. 22

The 1st modal contour of tail bogie: a three coaches; b six coaches; c eight coaches

4.4 Sound pressure

4.4.1 Total sound pressure level

Figure 23 shows the distribution of the total sound pressure level (TSPL) on the surface of each train coach. Compared with TTPL, the gradient of TSPL on the surfaces of different trains becomes smaller, especially in large separation areas such as pantograph and bogie, etc. TSPL on the train roof is in the range of 120–130 dB, and the largest value appears on the pantograph. TSPL at the bottom is in the range of 135–145 dB and that in the bogie area is larger than other areas. For six-coach and eight-coach trains, the phenomenon of the former weak and the latter strong exists, and the more the number of coaches is, the more obvious the change of TSPL is. However, the TSPL in the first bogie region of the three-coach train is the largest and the difference of TSPL between the head and tail is small. The reason for them may be more turbulent bottom airflow due to the increased coach number and the reverberation effect within the tunnel, which has a greater impact on downstream coaches, especially on the underbody of the trains.

Fig. 23

TSPL distribution on the train surfaces (unit: dB): a eight coaches; b six coaches; c three coaches

The surface-averaged RMS of the sound pressure of each coach is statistically analyzed, and it is found from Table 5 that the commonly used three-coach train model has a tendency towards higher mean sound pressure levels at the head coach and lower levels at the downstream coaches compared to longer train models. This is negative to accurate predictions of sound pressure distribution for real trains composed of multiple coaches. For six-coach and eight-coach trains, the surface-averaged RMS of the sound pressure presents same magnitude in head, coaches with stowed and raised pantographs, and a small difference of only 0.7 dB in tail. However, compared with the three-coach train, the surface-averaged RMS of the sound pressure at the head of the eight-coach train is 1.4 dB smaller and the tail is 4.1 dB larger. Therefore, in order to truly present the sound field of the train in the tunnel space, the real marshalling should be used instead of the currently widely used three-coach train model.

Table 5 Surface-averaged root-mean-square sound pressure of different coaches (unit: dB)

4.4.2 Frequency spectra of sound pressure at key points

The sound pressure PSD at B1, B2 and P1 is shown in Fig. 24. Due to the same simulation case, the PSD sampling time and other settings are identical to the turbulent pressure PSD at the key points. For B1 and B2, spectral tendencies are relatively similar, and the peaks of 800 and 2600 Hz exist in all three types of trains. The peak energy of three-coach train is bigger than that of six-coach and eight-coach trains for B1, while it is opposite for B2. The values at 800 and 2600 Hz are 10 dB larger than that at other frequencies. For P1, the peaks at 800 and 1400 Hz exist in all three types of coaches. 800 Hz is the cavity mode of the cross section of the tunnel space, and 2600 Hz is the 1st-order vertical cavity mode of the bogie, and 1400 Hz is the vertical 1st-order frequency between the pantograph bottom tank and the top of the tunnel [18]. Obviously, this is the inherent characteristic train-tunnel system. The peak frequencies are not affected by the coach number, but the energy can be slightly changed.

Fig. 24

Sound pressure PSD at key points: a B1; b B2; c P1

4.4.3 Reduced-order analysis of sound pressure in head and tail bogie cavities

POD analysis is also conducted on the sound pressure in bogie cavity of trains with different coach numbers, and the modal energy proportion is shown in Fig. 25. Both in the first bogie of the head and the second bogie of the tail, the energy ratio drops quickly to a very low level. For three-coach train, the proportion of the 1st-order mode of the head bogie is 71.4%, and that of the 2nd-order mode is 19%; for six-coach train, it is 59.7% for the 1st-order mode and 27.6% for the 2nd-order mode; for eight-coach train, it is 67.3% for the 1st-order mode and 20.6% for the 2nd-order mode. The energy proportion of sound pressure in the tail bogie is more concentrated on the 1st-order mode. For three-coach train, the 1st-order mode accounts for 85.3% and 10.9% of the 2nd-order mode; for six-coach train, the 1st-order mode accounts for 89.3% and the 2nd-order mode accounts for 8.1%; for eight-coach train, the 1st-order mode accounts for 91.3% and the 2nd-order mode accounts for 4.9%; the energy proportion of other modes are all below 1.5%.

Fig. 25

POD energy percentage of sound pressure in bogie cavities: a head bogie; b tail bogie

The PSD of 1st- and 2nd-order mode coefficients of the head bogie and tail bogie of trains with different coach numbers is shown in Fig. 26. In order to better demonstrate the spectral characteristics, the window length is changed to 800, and other parameters are same as that in Sect. 4.2. In general, energy of the 1st-order mode is larger than that of the 2nd-order mode, especially below 1000 Hz. For the head bogie, there are 2 distinct peaks for the 1st-order mode coefficient, whereby the spectra of three train models demonstrate an 800 Hz peak. This phenomenon can also be found in the tail bogie. It is obvious that the energy of the 1st-order mode coefficient around peak frequency of the head bogie of three-coach train head is larger than that of the six-coach and eight-coach trains; however it is opposite for the tail bogie. The 2nd-order mode coefficient has no obvious frequency peak. For the head bogie, the energy is concentrated at 800 ~ 1300Hz, and for the tail bogie, the spectral characteristics of the 2nd-order mode coefficient are nearly the same with those of the head bogie.

Fig. 26

PSD of the 1st- and 2nd-order mode coefficient: a head bogie; b tail bogie

Figure 27 shows the isosurface map of A_p in the cavity space of the bogie, where A_p represents the 1st- and 2nd-order POD modes of the normalized sound pressure and is dimensionless. The high absolute values represent strong fluctuation of sound pressure for the corresponding mode at those locations. By observing Fig. 27a and b, it can be found that the spatial distribution of the 1st-order mode in bogie cavities of the head and tail is basically the same, showing the characteristics of transition from the central high-intensity area to the four-corner low-intensity area gradually. Combined with the fact that the 1st-order mode is mainly concentrated at 800 Hz in Fig. 26, the 1st-order mode of sound pressure largely reflects the resonance of the overall tunnel noise in the bogie cavity, and the energy also occupies the main part of the noise in the bogie cavity. By observing Fig. 27c and d, it can be found that the 2nd-order mode of the tail is laterally symmetrical, and high-intensity areas located in the both ends of the streamwise direction in the bogie cavity with the trend to be neutralized in the middle section. According to the acoustic characteristics, the 2nd-order mode distribution of sound pressure in the bogie cavity shows the oscillation in both ends and gradual attenuation near the middle plane. Combined with the 2nd-order mode spectrum in Fig. 26, the 2nd-order sound pressure mode mainly reflects the noise caused by the incoming flow which flows through the complex structure of the bogie after entering the bogie cavity from coach bottom.

Fig. 27

The 1st- and 2nd-order modal contour of sound pressure: a 1st-order mode of the head bogie; b 1st-order mode of the tail bogie; c 2nd-order mode of the head bogie; d 2nd-order mode of the tail bogie

5 Conclusion

Numerical simulation of the 1:25 scale two-coach high-speed trains for different mesh sizes, discretization scheme and train velocity was carried out. The averaged and transient pressure of the key measuring points on the train surface obtained by simulation were consistent with the test results, which showed that the WLES with mid-mesh and bounded central scheme can accurately obtain the unsteady flow fields of the train in long tunnel.

The unsteady flow fields of high-speed trains in long tunnel with the same scale and different coach numbers were carried out. As the number of coaches increases, the average and standard deviation of aerodynamic drag and lift also increase, which suggests that a longer train leads to greater disturbance and energy consumption in a limited space. The aerodynamic drag coefficients of eight-coach and six-coach trains are large at the head coach, similar to that of the three-coach train. In the middle coaches, the mean C_d value of the coach with the raised pantograph is larger than other coaches. Additionally, unlike the short-group trains, the mean C_d values of the tail coaches of the multi-group trains are much larger than the three-coach train. The mean C_d value of the tail coach of eight-coach train is larger than that of the six-coach train, indicating that the aerodynamic drag force of train tail increases with the coach number. TTPL of middle and tail coaches for eight-coach train is larger than the commonly used three-coach train. POD analysis of turbulent pressure inside the bogie cavities reveals that the proportion of energy in each order mode decreases slowly, and the first order of the head and tail bogies accounts for 12% and 25%, respectively.

The near-field sound pressure is obtained. In the commonly used three-coach model, the overall sound pressure at the tail coach is lower than that at the head coach. However, for the multi-group trains models, despite the fact that the RMS sound pressure at the head coach of eight-coach train is 1.4 dB lower than that of the three-coach train, the sound pressure gradually increases in the streamwise direction, surpassing that of the three-coach train, and 0.7 dB higher than that of the six-coach train at the tail coach. Tunnel walls effect on the acoustic distribution at the train bottom, only after the coach number reaches certain enough value, and then the sound pressure fluctuation presents a streamwise increase. This streamwise increase of multi-group trains can be strengthened by the coach number. We suggest that under long tunnel conditions, multi-group trains such as eight-coach train model are better to reflect the acoustic characteristics of high-speed trains in practical situations. The sound pressure PSD at key points of the three-coach model shows a low–high–low feature, which is different from that of eight-coach train. The peak values of the measurement points in the tail bogie cavity increase as the number of coaches increases. The first mode energy percentages at the head and tail bogies are as high as 65% and 85%, respectively, and the proportion of them increases with the number of coaches. The spatial distribution of the 1st-order mode in bogie cavities presents the characteristics of transition from the central high-intensity area to the four-corner low-intensity area, while the 2nd-order mode is laterally symmetrically distributed, and high-intensity areas are located in the both ends in the flow direction in the bogie cavities with the trend to be neutralized in the middle section.

It is worth pointing out that the current research without considering the possible impact of pressure waves generated by trains entering and exiting the tunnel on the acoustic results. The scaling effect is also a significant factor that changes the acoustic characteristics inside the tunnel, which will be the key content worth studying in the next stage.