Dynamic analysis of the flow fields around single- and double-unit trains

Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 136–150

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Dynamic analysis of the flow fields around single- and double-unit trains Xiao-Bai Li a, b, c, Guang Chen a, b, c, Zhe Wang a, b, c, Xiao-Hui Xiong a, b, c, *, Xi-Feng Liang a, b, c, Jing Yin d a

Key Laboratory of Traffic Safety on Track (Central South University), Ministry of Education, Changsha, 410075, China Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Changsha, 410075, China c National & Local Joint Engineering Research Center of Safety Technology for Rail Vehicle, Changsha, 410075, China d State Key Laboratory for Track Technology of High-Speed Railway, Beijing, 100081, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Formation of trains Aerodynamic force Slipstream Wake flow POD analysis

With the increasing need for passenger transportation, the double-unit formation of high-speed trains (HSTs) has been widely employed. However, the dynamic characteristics of the flow around double-unit HSTs have rarely been studied. In this study, the Improved Delayed Detached Eddy Simulation (IDDES) method is employed to analyse the differences in flow fields around single- and double-unit trains. The numerical case is validated through wind tunnel experiments, mesh independence tests and IDDES assessments. The results show that the existence of the double region concentrates the Strouhal number of the aerodynamic force of the wake at 0.15. For the flow downstream of the double region, the slipstream velocity detected until the far-wake region is higher than that in the single-unit case. The standard deviation of the velocity, which reflects the intensity of the turbulence, is also higher for the double-unit case. For the near wake, the vortex dominate region is dominated by flow structure shedding from the snow plough thus similar for two cases; for the downwash dominate region, the double-unit wake exhibits more turbulent flow structures and lower dominant frequency for velocity. The proper orthogonal decomposition results also indicate more turbulent wakes within the downwash dominate region in the double-unit case.

1. Introduction The flow around high-speed trains (HSTs) is characterized by a relatively high Reynolds number, making it highly turbulent, with large numbers of vortex structures on both spatial and temporal scales. As the running speed of HSTs has increased, the influence of train-related transient pressure or wind gusts on trackside facilities or people may increase, and the aerodynamic force acting on the train could intensify and affect travel safety. Therefore, a deeper understanding of the dynamic behaviour of the flow field is needed, and corresponding research should be conducted to reduce the negative influence caused by the trainside flow field. Baker et al. (2001) studied the slipstream and wake flow conditions of a four-coach train model at a 1/25th scale via a moving-model rig test that included the side wind effect. According to the results, the slipstream around the train can be divided into five different regions: an upstream region, a nose region, a boundary layer region, a near-wake region and a far-wake region. These regions have been set as references for the lateral

investigation on train aerodynamics, especially for the boundary layer on the train surface and the wake characteristics. In the full-scale test conducted by Baker et al. (2014a; 2014b), slipstreams associated with different types of trains were investigated. The results revealed differences among the slipstreams induced by trains with different head shapes and formations. These studies described the fundamental laws of the flow field caused by trains of different types or running conditions through line side velocity probes; however, a more detailed description of the dynamic behaviour of the flow field around trains under different situations still needs to be attained for deeper understanding. For this purpose, plenty of research has been carried out with both numerical and experimental approaches. Bell et al. (2014) carried out a wind tunnel experiment through velocity flow mapping using wake and streamwise measurements with dynamic pressure probes. Through this research, the largest slipstream velocities in the near-wake zone were explained, and the transient nature of the wake was explored. In Bell et al. (2016), the three-dimensional dynamics of a pair of counter rotating streamwise vortices were presented, and the sinusoidal,

* Corresponding author. Key Laboratory of Traffic Safety on Track (Central South University), Ministry of Education, Changsha, 410075, China. E-mail address: [email protected] (X.-H. Xiong). https://doi.org/10.1016/j.jweia.2019.02.015 Received 21 September 2018; Received in revised form 4 December 2018; Accepted 20 February 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 136–150

numerical accuracy and computational costs and has been widely used in studies of train aerodynamics as in Flynn et al. (2014), Wang et al. (2017a; 2017b; 2018) and Xia et al. (2017) and is thus adopted in this paper. In this paper, the Improved Delayed Detached Eddy Simulation (IDDES) method, which can be effectively applied to investigate the unsteady flow around trains, is utilized to identify the differences between the flow fields around single- and double-unit HSTs. The analysis emphasizes the instantaneous behaviour of the flow field. The paper is organized as follows. In Section 2, the numerical method is introduced and validated; Section 3 presents the results in terms of both the timeaveraged and instantaneous flow fields, especially for the wake flow; and Section 4 presents the conclusion of the work.

antisymmetric motion of the counter-rotating streamwise vortices in the wake is observed. Moreover, to investigate the geometric shape of the train on the wake structure, the influence of the roof angle of a train tail on the characteristics of the slipstream and unsteady wake structure was evaluated by Bell et al. (2017) via wind tunnel tests, and multiple flow field measurement approaches were undertaken to identify the impact of the roof angle on the wake vortex pair. Additionally, numerical research, with its post-processing advantages, has been widely applied to determine the features of the flow fields around trains, especially for wakes. Muld et al. (2012a) used proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) to extract the dominant flow structures of a simulated flow in the wake of a high-speed train; the relationships between the two methods and the corresponding requirements for train applications have been discussed. In Muld et al. (2014), trains of different lengths were compared in terms of boundary development, and wake structures were compared in terms of POD and DMD modes. To further analyse the dynamic behaviour of the wake of € the train tail, Osth et al. (2015) used the Large Eddy Simulation (LES) method to obtain the flow field around an HST and conducted a cluster-based reduced-order analysis to extract the dynamic physical mechanisms of the flow field; these extracted flow structures are associated with the longitudinal vortices and the fluctuation of aerodynamic drag. More recently, Wang et al. (2018) explored the effect of bogies on the features of the near-wake flow, and the generation of the strong spanwise oscillation of the wake was discussed. Through the studies mentioned above, the dynamic features of flow structures around trains, especially those related to wakes, have been studied in terms of head shape, train length and bogie. However, as discovered by Baker et al. (2001), the double-unit flow region may influence the operating safety of trains, trackside facilities and pedestrians because the discontinuity region can cause separation, leading to a peak in the slipstream velocity, and the area of influence may reach the wake of the tail. However, as the need for passenger transportation has increased in recent years, the required capacities of high-speed railways have expanded. Increasing the transportation frequency is a poor solution due to the inevitable increase in congestion and the increased risk of accidents (Guo et al., 2018). Thus, double-unit passenger trains are an effective option for meeting the requirements of railway transportation; therefore, the differences in the flow fields caused by trains of different unit formations should be understood in detail. In terms of existing papers that consider double-unit trains, Niu et al. (2017) analysed the influence of the coupling region on aerodynamic performance for various running conditions; however, the slipstream and flow field were not the focus of this work. Guo et al. (2018) used the Detached Eddy Simulation (DES) method to compare the slipstream in terms of different trackside measurement points, but the paper primarily focused on the average flow field rather than the instantaneous or dynamic behaviour of the flow field. Thus, a detailed analysis of the dynamic behaviour around singleand double-unit trains should be performed for comparative purposes. As most experimental methods do not provide complete flow field information but rather information on a few measurement points or a plane, direct observations of the mechanisms associated with the differences between the flow fields induced by single- and double-unit trains can be difficult to obtain. Because of these complexities, numerical simulation is ideal for investigating flow field dynamics. With developments in computational power and the need for higher accuracy in numerical predictions, scale-resolving simulations, including DES and LES, have been widely applied in train aerodynamics. However, obtaining the highly turbulent flow structure around a train in a relatively highReynolds-number field based on LES would require considerable € computational resources, as noted by Hemida et al. (2014), Osth et al. (2015) and Wang et al. (2017a). Additionally, LES is highly dependent on grid size; a coarse mesh may not capture the main flow field information and thus produce an incorrect result. Thus, the application of LES for an HST in a high-Reynolds-number field is limited. DES, which is a hybrid method of URANS and LES, has achieved a good balance between

2. Numerical method 2.1. Geometric model Fig. 1 shows the geometric model of the high-speed train studied in this paper. The model is a simplified version of CRH2 and consists of 6 cars, including the head, tail and four intermediate cars (labelled M1, M2, M3 and M4, as shown in Fig. 1(a) and (b)), which is the same formation that was studied by Guo et al. (2018). Because the model in this study is at a scale of 1:8, the reference height H, which is defined as the distance from the top surface of the train to the ground, is 0.5 m, including the distance from the bottom of the wheels to the ground, which is equal to the height of the rail, i.e., 0.235 m according to the TSI standard at full scale. In reality, the two trains are different lengths, and a discontinuity region exists; in this study, the lengths of the two trains are equal to eliminate differences in the flow field due to the length of the trains. Thus, the total length of the train in each case is 37.5H. The original shape of the streamline area at the train head is preserved. Moreover, the bogies and wind shields are simplified to improve the quality of the grid. Although it differs from the real full-scale train running on the line, a simplified model with a high-quality grid will provide an adequate understanding of the flow mechanisms induced by the train geometry beyond those linked to the complicated underbody structure like bogies etc. 2.2. Computational domain and boundary conditions A sketch of the dimensions of the computational domain is shown in Fig. 2. The train is placed 10H from the velocity inlet, where a constant velocity with a low turbulence intensity of 1% is defined (Wang et al., 2017a; 2017b; 2018) to simulate the relative motion between the running train and the air. The corresponding Reynolds number based on the upwind velocity and the reference height of the train is 1.67
106. The side face and upper face of the domain are set to 10H from the model area, and a symmetry plane is assigned at the boundary. For the

Fig. 1. Geometry of the computational model. (a) 3 þ 3 double-unit train. (b) Single-unit train with 6 cars. (c) Streamline zone of the computational geometry. (d) Corresponding streamline zone of the full-scale CRH2. 137

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Fig. 2. Sketch of the computational domain. (a) Top view. (b) Front view.

trimmer mesh, the grid size is doubled at the interfaces between refinement zones. A prism layer with 18 extruding cells based on a growth rate of 1.2 is attached to the surface of the train body to resolve the near-wall flow structure. The average y þ value of the first cell attached to the train surface is approximately 0.9, which meets the recommendation of using a value less than 1.5 proposed in the STAR-CCM þ user guide (SIEMENS, 2017). To effectively blend the URANS area and the LES area at an appropriate position, wall-parallel spacing should be implemented for the near-wall grid at a reasonable value. In this paper, to assess the grid dependence on the results, two sets of grids that only differ in their wall-parallel spacing are generated for the double-unit case. The average xþ and z þ values for the train surface are 600 and 400 for the coarse and fine meshes, corresponding to the size of surface mesh 0.012H and 0.008H, respectively, for a total of 36 million (Fig. 3) and 58 million grids, which are both at the same level as those in Guo et al. (2018), who studied a 1/8th-scale HST with 6 cars as the geometric model. The results of the numerical validation study are presented in Section 2.5.

leeward-side boundary, a constant zero-pressure outlet is utilized, and the boundary is placed 30H from the tail of the train to ensure that the wake of the flow is fully developed. These lengths are sufficient for neglecting the influence of far-field boundaries in numerous numerical € studies on vehicle bluff body flows, according to Osth et al. (2015). The origin of the coordinate system is placed at the nose point of the tail at ground height. The positive x direction is the flow direction, the lateral direction is defined as the y direction, and z represents the height above the ground, as displayed in Fig. 2. 2.3. Meshing strategy As noted above, the quality of IDDES is highly dependent on the mesh resolution because the blending process between URANS and LES is largely influenced by the cell size and the capability of the solver that determines the LES area to resolve small-scale flow structures based on the cell size. In this study, to achieve high precision, the domain is discretized based on an unstructured trimmer grid in STARCCM þ developed by Siemens. The mesh is constructed for local refinement around the train, where the flow field exhibits considerable separation and reattachment, and at the boundary layer, where substantial turbulence exists. In addition, the tail wake area, as a main focus of this study, is refined so that the IDDES solver can capture turbulence structure details. The detailed refinement zone is shown in Fig. 3(a) and (b). The size of the near-wake zone, which is included in the red wireframe, is the finest level, which is the same as that of the train surface. The detailed topology of the wake grids can be visualized in Fig. 3(c) and (d). For the

2.4. Numerical model and brief description of the solver As in DES, if the size of the wall-parallel spacing is too small, the LES length scale is less than the RANS length scale, and the near-wall flow is solved based on LES, which is often incorrect because the mesh is too coarse to capture the near-wall turbulence; the SGS model is often too small to replace the unsolved turbulence structure. This problem was termed ‘modelled-stress depletion’ (MSD), which in some situations can lead to early separation, or ‘grid-induced separation’ (GIS) (Ashton, 2017). The IDDES formulation (based on SST k-omega), as a hybrid method combining DDES and WMLES, has advantages for overcoming MSD and GIS, as noted above, and for reducing the limitation of the Reynolds number for near-wall flow. In IDDES, an alternative and more physically justified definition of the subgrid length scale, which does not require different SGS model constants for wall-bounded and free turbulent flows, is proposed: Δ ¼ minfmax½Cw dw ;  Cw hmax ;  hmax g

(1)

where Δ is the subgrid length scale, Cw is an empirical constant that is set to 0.15, dw is the distance to the wall, hmax is the local maximum grid spacing, and hwn is the grid step in the wall-normal direction. With this formulation, the wall-distance and grid spacing are considered, and a linear increase between the limitation of the grid step in the wall-normal direction and the local maximum grid spacing is realized. The DDES branch of IDDES becomes active only when the inflow conditions do not have any turbulent content and, in particular, when a grid of the ‘boundary layer type’ precludes the resolution of the dominant eddies. The WMLES branch is active only when the inflow conditions used in the simulation are unsteady and impose turbulence and the grid is fine enough to resolve boundary layer dominant eddies. A full description of the utilized IDDES formulation is given by Shur et al. (2008). This study employs a segregated flow solver, and the employed pressure-velocity coupling method is SIMPLE. The convective term is discretized based on a hybrid scheme (Travin et al., 2000) that switches between a bounded central-differencing scheme and a second-order upwind scheme (BCDS). The area that utilizes the bounded

Fig. 3. Computational grid around the tail of the train. (a) Top view of refinement zone. (b) Side view of refinement zone (c) Side view of computational mesh. (d) Top view of computational mesh. (e) Prism layer. 138

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central-differencing scheme is 15% of the entire domain. For the turbulence term, the second-order upwind scheme is utilized. A second-order implicit scheme is adopted for the temporal term in the transition simulation. The discretized time step is 0.025t* (where t* is the non-dimensional time equal to H/U∞) for the coarse grid and 0.015t* for the fine grid. These values are sufficient for IDDES in the flow field around trains according to Wang et al. (2017a). In fact, the time step is chosen so that the convective Courant number is less than 1 for most cells and the mean convective Courant number for the wake domain, which is the most important research focus and has the finest grid, is 0.6. The Courant number only exceeds 1 in very small parts of the grids, and several studies (Flynn et al., 2014, 2016; Wang et al., 2017a) have shown that this minor infringement on the Courant number requirement is unlikely to have a significant effect on the flow. The simulation is initialized based on a converged-state RANS simulation. The aerodynamic force coefficient is monitored, and if the variance decreases and the time history curve converges, the IDDES result reaches a stable state. In this case, the data are considered reliable. Thus, the passage time of the IDDES simulation is 62.5t*, and the following 162.5t* of the simulation, which corresponds to 10.4 s at full scale, is adopted. These times and the total time of data collection are similar to those reported by Wang et al. (2017a) and Xia et al. (2017). Fig. 5. Comparison of the velocity profiles for coarse and fine meshes: (a) timeaveraged slipstream at platform height; (b) time-averaged streamwise velocity at the streamline zone of the tail.

2.5. Numerical validation This section describes the validation of the simulation using data from existing research. Additionally, a mesh independence study and an assessment of the IDDES simulation are presented. Fig. 4 compares the time-averaged pressure coefficient distribution along the centreline of the train extracted from the fine and coarse mesh simulations with wind tunnel data from Zhang et al. (2018), who utilized a 1/8th-scale CRH2 model with 3 cars. The figure shows that the pressure coefficients obtained from the simulations of coarse and fine meshes exhibit good agreement, except for deviations at the inter-car gap, where full separation exists, and the discontinuous region, where the flow structures are complicated. Compared to the experimental data, the pressure distribution along the centre line of the train head is well captured by the numerical simulation, but discrepancies are present for the tail. Notably, both the fine and coarse grids failed to predict the lowest pressure obtained in the experimental study of Zhang et al. (2018). Moreover, because the experimental model used by Zhang et al. (2018) only consisted of 3 cars, and the growth of the boundary layer might differ from that simulated in this paper, the ground configuration and double-unit regions will also display different wake flow fields. Generally, the numerical simulations reflect the general flow field around the train body. Fig. 5 provides a comparison of the velocity fields around the train obtained from the fine and coarse meshes. In Fig. 5(a), the slipstream is recorded and compared. The slipstream is defined as the horizontal velocity measured at two fixed probes on the ground located 3.0 m from the

centre of the train (COT) and 0.2 m and 1.4 m from the top of the rail (TOR) at full scale. These two positions are defined as the positions of the trackside and platform. Equation (2) presents the definition of the slipstream: USlipstream ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2GF þ V 2GF

(2)

where UGF and VGF are the streamwise and transverse components, respectively, of the normalized velocity based on a ground-fixed coordinate system. These components can be calculated as shown in Equation (3): UGF ¼

U∞  UTF VTF ; VGF ¼ U∞ U∞

(3)

where UTF and VTF are the streamwise and transverse components, respectively, of the velocity based on a train-fixed coordinate system. These values can be directly obtained in the simulation. Fig. 5(a) gives the time-averaged slipstream measured at platform height. The comparison shows that the profiles of the slipstream are all similar to the basic trend observed by Baker et al. (2014a). The slipstream results obtained using the coarse and fine grids exhibit good agreement, except for the minor deviations in the near-wake region.

Fig. 4. Pressure coefficient distribution compared with that of Zhang et al. (2018). 139

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In addition, in Fig. 5(b), the time-averaged streamwise velocity at the streamline zone of the tail is presented. The flow at the tail surface exhibits little separation, and no evident shear layer can be detected. In this case, the long streamline zone of the train body and the slope of the tail are not sufficiently steep to induce full separation. For the two sets of grids, the streamwise velocity profiles at the streamline zone of the tail are almost identical. This result suggests that the coarse mesh is sufficient for predicting the appropriate flow topology in the wake area. In Fig. 6, the contours of the percentage of resolved turbulent kinetic energy for the coarse mesh are presented. The field is in an instantaneous state. The resolved turbulent kinetic energy is obtained using Equation (4):  0 0 0 0 0 0 TKEresolved ¼ 0:5  u u þ v v þ w w

Table 1 Aerodynamic drag force. Parts

Cx Ave Double

Single

Double

Single

Double

Head M1 M2 M3 M4 Tail

0.1765 0.0955 0.0917 0.0783 0.0864 0.1266

0.1715 0.1004 0.0464 0.1709 0.0879 0.1271

0.2020 0.1237 0.1164 0.1140 0.1199 0.1564

0.1964 0.1306 0.0752 0.2013 0.1272 0.1625

0.0091 0.0093 0.0087 0.0102 0.0107 0.0092

0.0089 0.0096 0.0092 0.0096 0.0106 0.0100

C x ¼ Fx

  0:5ρU 2∞ A

(4)

where u’, v’ and w’ denote the velocity fluctuations in three different directions. Because the IDDES cannot resolve all turbulent flow due to the restrictions of the grid size, theunresolved turbulence are modelled with the SGS model in LES. Thus, the percentage of resolved turbulent kinetic energy relative to the total turbulent kinetic energy provides an indicator of the quality of the IDDES simulation (Davidson, 2009). As shown by the contours, most of the area around the train wake displays percentages greater than 80%, which is acceptable for resolving minor turbulence structures. In the zones near the train body, the flow is more likely to be modelled by URANS because the grid scale may be too large for LES to identify the boundary layer turbulence and produce a relatively high SGS viscosity to resolve the flow structures. This approach is ideal because the URANS model can provide an acceptable representation of boundary flow and produce a relatively high turbulent viscosity. Based on the validation above, we can conclude that the coarse grid simulation of 3-dimensional turbulent flow around the train containing 6 cars is of acceptable accuracy compared to the fine grid simulation. In fact, the grid number is acceptable for the slipstream and wake analyses compared to those used by Wang et al. (2017a; 2017b) and Xia et al. (2017), who focused on resolving the flow structures developed by the complicated bogies and ground features that occupied a large amount of the grid number. However, these factors are not the focus of the present study. Thus, the coarse mesh results are adopted for further analysis.

σ

Max

Single

(5)

where Fx is the aerodynamic drag force; ρ is the density of the air, which is 1.225 kg/m3 in the simulation; A is the reference area of the train, which is equal to 0.175 m2 at the model scale; and U∞ is equal to 60 m/s. Comparing the two cases with single- and double-unit trains, differences exist in the average force coefficient values at the head, tail, and first and fourth intermediate cars, but the largest difference, which was still less than 5%, was observed at the first intermediate cars. For the second and third intermediate cars, the aerodynamic force greatly differed because the train had different shapes based on the single- and double-unit formations. As the back slant for M2 in the double-unit case is not steep enough to induce a full separation, which can be inferred from Fig. 5, the downwash will directly crash into the surface of M3 to form a highpressure zone. This will cause a significant decrease in aerodynamic drag for M2 and increase in aerodynamic drag for M3 in the double-unit situation compared to single-unit one. Notably, the aerodynamic drag force for M3 is nearly the same as the head in the double-unit case. This phenomenon is caused by several factors including the above one with the high pressure zone. Moreover, for the head train, a local acceleration of the flow will be formed at the snow plough; thus, a low pressure zone will be imposed at the first bogie set, which will contribute to a large portion of the aerodynamic drag of the head train. While for M3 in double-unit case, as the underbody flow has been decelerated by several bogies, the low-pressure zone will not be observed in the bogie set of M3. These factors lead to a similar aerodynamic drag for the head and M3 in double-unit case. In general, the discontinuous area increases the total aerodynamic force on the train in terms of both the average and maximum values. For the standard deviation, a maximum difference of approximately 8% was observed at the tail train; therefore, a double-unit region can likely disturb the flow, increase the turbulence intensity and result in force fluctuations at the tail. Because the downstream area is more influenced by the discontinuous region than the upstream area in terms of force fluctuations, more temporal details of the aerodynamic force at the tail should be obtained. Fig. 7 shows the power spectral density of the time history value of the aerodynamic drag force coefficient at the tail of the train in both singleand double-unit scenarios based on the frequency domain. The frequency was transformed to the corresponding Strouhal number (St) based on the reference height and the oncoming wind speed, which ranged from 0.1 to 10, because the harmonics with periods longer than 1/10th or shorter than 10 times the integration time are not correctly simulated by the numerical computation (García et al., 2015). The power spectral distribution shows that the density value decreases for St greater than 1, which is expected at higher frequencies because the small turbulent eddies that form have shorter wavelengths and thus have a more rapid loss of coherence than the larger eddies (Bouferrouk, 2013). At the tail of the train, the coupling region significantly increases the power spectral density at a St value of 0.15, which is likely a consequence of the flow interaction with the coupling region. Specifically, the flow downstream is dominated by the periodic turbulent flow; thus, the aerodynamic force at the tail displays a higher peak in the frequency domain, and in the single-unit case, the spectrum is more dispersed.

3. Numerical results This section is organized as follows. First, the aerodynamic force is analysed to determine average value, frequency information and probability distribution. Then, the slipstream at the side probes around the trains is presented for both the average flow field and an instantaneous gust. In addition, the flow structures around the train, such as the boundary layer and wake region, are analysed.

3.1. Aerodynamic force Table 1 shows the aerodynamic drag force coefficients obtained in the simulations of both single- and double-unit trains. The average values together with the maxima and standard deviations are presented. The definition of Cx is as follows:

Fig. 6. Contours of the percentage of the resolved TKE. 140

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will be more concentrated at a certain frequency, which is a St value of 0.15 in this study. The flow around the train will be discussed in the following section in further detail. 3.2. Slipstream 3.2.1. Time-averaged slipstream The slipstream is caused by the movement of the running train, which forces the air to flow mainly due to its viscous characteristic. Additionally, as part of the HST acceptance procedure, the slipstream caused by trains has been studied in several existing papers (Flynn et al., 2014, 2016; Wang et al., 2017a; 2017b; Xia et al., 2017). In this section, to provide a basic understanding of the distribution characteristic of the slipstream along the train, the time-averaged flow field is adopted to extract the slipstream velocity at the trackside and at platform height in both cases. Additionally, the standard deviations of the values at the corresponding positions noted above are plotted. The slipstream is discussed in terms of the five different regions described by Baker et al. (2001): the upstream region, nose region, boundary layer region, near-wake region and far-wake region. In terms of both the time-averaged slipstream and the standard deviation values, the values before the doubled region exhibit good agreement for different cases at the same measurement position. The probe at the trackside position provides a greater value for both the timeaveraged slipstream and standard deviation in this area. Additionally, increases in both values along the train were observed, and they were likely caused by the growth of the boundary layer at the surface of the train, where considerable turbulent flow is present. When the flow reaches the doubled region, differences occur. Increases in both the timeaveraged and standard deviation values were detected for the doubleunit case. The slipstream measured within the boundary layer region seems to accelerate, and the standard deviation value, which indicates the turbulence intensity, also increases. The influence of the doubled region continues downstream to the near-wake region, and the timeaveraged slipstream caused by a double-unit train is greater than that caused by a single-unit train in this area, but the differences still decrease as the flow moves further downstream to the wake. The difference for the STD value decreases for the trackside probe in the near-wake region, and the two curves in Fig. 9(b) largely coincide, likely due to the flow structure of the tail. The tail longitudinal vortex contains considerable turbulence intensity and is always located close to ground, according to € previous studies (Osth et al., 2015; Wang et al., 2018); thus, the differences in turbulence conditions induced by the doubled region seem to be recovered in the wake. At the platform height, a difference still exists, which indicates that the platform height of the near-wake region was still extensively influenced by the upstream turbulence induced by the double region, and the turbulence structure in this area was different for the single- and double-unit cases. In the far-wake region, the differences appear to be smaller, which suggests that the influence gradually dissipates.

Fig. 7. Comparison of the power spectral density of the aerodynamic drag force coefficient at the tail of the train for single- and double-unit formations.

The maximum force can be used as an assessment criterion for the operating safety of the train, but it may greatly depend on the monitoring period. Thus, the maximum value noted above considers only the extreme situation of approximately 10.4 s at full scale, and it is more informative if the corresponding probability distribution function can be given. In this paper, the Gumbel distribution, which was also utilized by García et al. (2015), is adopted for the aerodynamic forces at the tail and head of the train in both the single- and double-unit cases. The entire time period is adopted in the analysis. Equation (6) shows the distribution function:    b   b ¼ exp  exp C  μ P Cx < C β

(6)

pffiffiffi b is the argument of the probability density function, with πβ= 6 where C being the standard deviation and μ being the mode. μ and β can also reflect the average value and the fluctuation degree of the aerodynamic force, respectively, and the values are given in Table 2 for each case. These values display the same trend as the average and standard deviation values presented in Table 1. Fig. 8 shows the corresponding plot of the numerical results and the target Gumbel distribution. A good fit was achieved between the numerical and analytical results. Equation (7) is the estimation of the return period of an extreme load: tr ¼

1  t b c 1  P Cx < C

(7)

where tc is the time period adopted for the probability analysis, which is 10.4 s at full scale in this study. In conclusion, the existence of the discontinuous region has little effect on the average value of the aerodynamic force coefficient for different unit formations. However, a notable effect was observed for the second and third intermediate cars, which were largely different in shape. The influence on the coupling region lies mostly in the standard deviation and the frequency domain, as the flow downstream may contain a higher turbulence level and more periodic motion. Therefore, the spectrum of the aerodynamic force of the tail of the double-unit case

3.2.2. Gust analysis Gust assessment is part of the HST acceptance procedure, and this study is implemented following the CEN European Standard (2013). In the simulation, moving measurement probes were arranged on both sides of the COT at a distance of 3.0 m at full scale, and the vertical distances from the TOR were 0.2 m and 1.4 m at full scale. These distances are defined as the heights of the trackside and the platform, respectively, and meet the requirements of the standard. The first set of 4 probes was placed at the inlet of the computational domain and released at a speed equal to the upstream inlet velocity after the flow field reached a steady state. The measurement consisted of 12 sets of measurement probes, which yielded 24 total samples at the trackside and platform heights and met the requirement of at least 20 individual runs proposed by TSI (TSI HSRST, 2008). The distance between two independent measurements was 24 m at full scale, which satisfied the requirements of the standard.

Table 2 Parameters of the probability distribution. Parts

Head Tail

μ

β

Single

Double

Single

Double

0.1719 0.1220

0.1675 0.1226

0.0071 0.0072

0.0070 0.0078

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Fig. 8. Probability distributions of the force coefficients for (a) the head of the train in the single-unit case; (b) the tail of the train in the single-unit case; (c) the head in the double-unit case; and (d) the tail in the double-unit case.

Fig. 9. Slipstreams for the (a) time-averaged field and (b) standard deviation value.

actually can be explained by the instantaneous vortex distribution around the train. According to the comparison between the single- and double-unit cases, it is clear that the existence of the coupled region greatly disturbs the downstream flow, and the longitudinal vortex structure with low momentum caused by the discontinuous region induces more peak values in individual tests for USlipstream just behind the double-unit region, especially at the trackside height. In the single-unit case, peak values of USlipstream typically occur in the near-wake region based on the instantaneous measurements. Without the double-unit region, flow around trains within the boundary layer region will be mostly affected by bogies and inter-car gaps at lower heights, which can be identified by the comparison between Fig. 10(c) and (d), as more peaks are detected within the boundary layer region in Fig. 10(c). The ensemble-averaged curve also shows an increase in the slipstream velocity just behind the coupled region. The same trend was previously observed by Flynn et al. (2014, 2016), Wang et al. (2017a; 2017b) and

The slipstream velocity was recorded during each individual run with a moving probe. The USlipstream values measured at different positions and in different cases are presented in Fig. 10 with the blue line, along with the ensembleaveraged curve denotes as the red line. The peak value for each run is also plotted as a black dot, and the peak value for the ensemble-averaged curve is plotted as a brown dot. Additionally, in accordance with the requirement of the standard, the equivalent of a 1-s moving average filter, which is 0.125 s at the model scale, is applied to each dataset and presented as a blue curve in Fig. 11. Fig. 11 also shows the ensembleaveraged curves and the corresponding peak values. Similar to the full-scale testing (Baker et al., 2014a; 2014b) and moving-model experiments (Bell et al., 2015), gust analysis shows a large run-to-run variance. The phase for the periodic vortex motion and the turbulence structures is different in each run. The instantaneous measurement of the slipstream which presented in the picture as blue line

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Fig. 10. Gust measurements without a 1-s MA for the (a) trackside height of a double-unit train, (b) platform height of a double-unit train, (c) trackside height of a single-unit train and (d) platform height of a single-unit train.

Fig. 11. Gust measurements with a 1-s MA for the (a) trackside height of a double-unit train, (b) platform height of a double-unit train, (c) trackside height of a singleunit train and (d) platform height of a single-unit train.

The TSI value (also known as the maximum slipstream value) is calculated as the mean of the peak velocity plus two standard deviations. The TSI value reflects the maximum slipstream velocity that is likely to occur at the measurement locations based on a 95% confidence interval, and this assessment is integrated into the HST acceptance procedure. In Table 3, the TSI values are presented for data both with and without the 1-s moving average. The results display the same trend as that observed in the previous analysis, in that the TSI value is considerably larger in the double region, especially at the trackside probe both with and without the 1-s MA.

Xia et al. (2017). The phenomena described above can also be observed in Fig. 11 for the application of the moving average. In Fig. 12, the slipstream extracted from the averaged field is compared to the ensemble-averaged result based on the raw data. It is important to note that the ensemble-averaged curve can reflect certain aspects of the average flow field but is also considerably influenced by the samples because the flow field is highly turbulent. This effect can be observed in Fig. 11(c), as an unexpected peak can be observed at the second intermediate car, with an USlipstream value of 0.42. This peak caused the ensemble-averaged curve at this point to increase in slope, which was not expected, and Fig. 12(b) further illustrates this discrepancy. Alternatively, if the number of samples is sufficiently large, the ensemble-averaged curve will more closely resemble the curve of the time-averaged flow field.

3.3. Flow structure analysis The above findings indicate that differences exist between the single143

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Fig. 12. Comparison of the slipstreams obtained from the average field and ensemble-averaged field for the (a) double-unit case and (b) single-unit case. Table 3 Gust assessment. Without a 1-s MA

Trackside Platform

Double unit Single unit Double unit Single unit

With a 1-s MA

UP

σ

UP þ 2σ

UP

σ

UP þ 2σ

0.3862 0.3030 0.2300 0.1912

0.0938 0.0601 0.0729 0.0574

0.5738 0.4232 0.3759 0.3061

0.1966 0.1657 0.1084 0.0979

0.0329 0.0267 0.0251 0.0232

0.2623 0.2192 0.1586 0.1442

and double-unit cases. With the existence of the discontinuous region, the flow downstream will differ in terms of the turbulent flow structure. In this section, to identify the differences between these two cases, the flow structures downstream, including the time-averaged boundary layer and the wake structure, are compared based on the time-averaged field and frequency distribution. Additionally, a POD analysis is conducted to obtain the energy distribution of the flow pattern of the wake for comparison. 3.3.1. Boundary layer In this section, the distribution of the boundary layer at the upper surface of the train is plotted. Because the largest difference in the flow fields of the single- and double-unit trains exists within the downstream area of the double-unit region, as discussed in Section 3.2, only the boundary layer of the last three train cars is considered. Several parameters determine the thickness of the boundary layer: the boundary layer thickness (δ), the displacement thickness (δ*) and the momentum thickness (θ). In this paper, the momentum thickness is adopted to determine the thickness of the boundary layer. Equation (8) gives the definition of the momentum thickness for the train. Z θ¼

0

δ

UGF ð1  UGF Þdz

Fig. 13. Momentum thickness of the upper surface of the trains.

(8)

2014). As the flow moves further downstream to the streamlined zone, the momentum thickness decreases, which suggests that acceleration occurs close to the tail. Because differences exist in the momentum thickness of the boundary layer and the turbulence intensity, the flow structure of the wake zone is influenced. This influence is investigated in the next section.

Munson et al. (2002) proposed an empirical formula for the momentum thickness as a function of the downstream position x for a flat plate, as shown in Equation (9).  1=5 ν θðxÞ ¼ 0:0360 x4=5 U∞

(9) 3.3.2. Description of the wake Fig. 14 illustrates the flow structure of the wake zone for the average flow field. In Fig. 14(a) and (b), the streamline projected at the surface of the tail of the train is based on the mean wall shear stress. The scalar variable used to colour the train surface and the plane is the pressure coefficient Cp (Cp ¼ P/0.5ρU2∞), and the vortex core (labelled VC in the picture) is plotted based on the average velocity field. In Fig. 14(c) and (d), the iso-surface of the Q criterion (Q ¼ 1) based on the time-averaged flow field is shown, coloured by the x-vorticity (ωx). This procedure has also been carried out by Wang et al. (2018) for illustrating the topology of the wake flow structure. Through the picture, acceleration occurs at the end of the streamlined

In this study, the momentum thicknesses in the single- and doubleunit cases and the theoretical values are plotted in Fig. 13. Notably, the windshield influences the boundary layer distribution, and the momentum thickness undergoes a sudden change in this area. In the double region, the flow will experience separation and reattachment near the windshield; thus, the momentum thickness in the double-unit case will initially decrease and then increase again after the double region. This condition makes the momentum thickness downstream lower than that in the single-unit case. The evolution of the boundary layer in the singleunit case is similar to the flow scenario involving a flat plate, and the plot of the single-unit case in Fig. 13 shows good agreement with the theoretical case. This result also validates the simulation results (Muld et al., 144

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Fig. 14. Flow structure of the wake zone. (a) Streamline of double-unit train. (b) Streamline of single-unit train. (c) Flow topology of double-unit train. (d) Flow topology of single-unit train.

3.3.3. Frequency analysis of the wake In this section, the frequency characteristics of the wake are presented. Two probes were placed in the near-wake region, and the time history of the velocity was recorded. The two points were placed at the trackside and platform heights (i.e., 1H from the tail of the train and 0.5H from the COT, respectively), which are located in the vortex-dominated region and the downwash-dominated region, respectively. The frequency was analysed in terms of the VTF values obtained by the corresponding probes in each simulation case. Fig. 15 shows the frequency analysis results, with the frequency translated into the Strouhal number. In addition, to improve the comparison with the frequency information, the power spectral density was normalized by the corresponding maximum value. For the trackside probe, the dominant frequency occurred at a Strouhal number of nearly 0.12 for both the single- and double-unit cases. For the vortex-dominate region of the near-wake, the difference for the dominate flow structure for two cases has been eliminated, as the upstream underbody flow will go through several bogies and snow ploughs and interact with the ground. This process will introduce turbulence production that is even more severe than that of the doubled region. Thus, the dominate wake structure of trackside area V1 might be identical for both cases, and the flow of vortex dominate region was mostly influenced by the tail structures rather than the upstream turbulence, as illustrated in Section 3.2.1 and Section 3.3.2; thus, the dominant frequency was similar in the trackside area for both single- and doubleunit cases. Moreover, for the higher frequency, differences still exist due to the upstream turbulence condition of the boundary layer region. For the platform probe, which is located in the downwash dominate region, differences occur in the dominant frequency of different cases. The double-unit case has a dominant St of 0.11, and the St is 0.14 for the single-unit case. This result is observed because the flow within the boundary region is greatly influenced by the upstream condition before the flow approaches the wake, as found in Section 3.2.1 and illustrated in Section 3.3.2. For differences in the dominant frequency of the vortex shedding, as introduced in Section 3.3.1, the boundary layer thickness in the double-unit case is small, which would result in rapid vortex shedding according to Muld et al. (2014). However, the UTF value in the boundary layer region is lower for the double-unit case than for the single-unit case, as discussed in Section 3.2.1. Therefore, vortex shedding

zone, and a negative pressure (NP) can be seen at this location with no obvious separation. Then, the flow moves downstream, and a positive pressure occurs when the turning point on the surface is reached because the flow decelerates. After the turning point, the surface flow separates into three branches, two of which are separated by the nose shape towards both sides of the train instead of downward. The bending surface between the side and the nose area causes the interaction of the surface flow originally attached on the side and top surface, which leads to flow separation and can be observed with the separation line (labelled SL in the figure). The third branch continues travelling downstream along the central line of the train nose and does not exhibit clear separation, as illustrated in Fig. 5(b). The separated flows mentioned above interact and form a pair of longitudinal counter-rotating vortices, which is labelled as V1 in Fig. 14(c) and (d), and travel along the surface of the train. However, a pair of secondary vortices (denoted as vortex pair V2) is formed at the snow plough due to the underbody flow. As V1 and V2 propagate downstream, the two vortex pairs meet at approximately 0.6H downstream of the tail and then merge into one vortex structure (denoted as vortex pair V3). With the slipstream projected in the plane behind the tail, the downstream process of vortex pair movement can be visualized. Comparing the two cases reveals that the single-unit case has a more distinct separation line than the double-unit case in terms of the flow from the top surface, which is caused by the coupled region in the doubleunit case. In this scenario, the boundary layer downstream becomes more turbulent and thus slows separation. For the lower part of the wake, the two cases exhibit little difference, as the dominant vortex in this area mainly arises from the structure of the snow plough, and the differences observed upstream of the wake gradually disappear, as discussed in Section 3.2.1. With the wake topology mentioned above, the near-wake region can be roughly divided into two regions: the vortex-dominate region, which is close to the ground where the longitudinal vortex plays the most important role, and the influence on the turbulence intensity in this area may be more noticeable by the tail underbody structures such as snow plough, which can be identified in Section 3.2; and the downwash dominate region, which is above the vortex core for certain distance, where the upstream turbulence of the boundary layer region of the train surface may have distinguished influence, while the influence from the tail longitudinal vortex may decrease with increasing height. 145

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Fig. 15. Normalized PSD of UGF for (a) probes at trackside height and (b) probes at platform height.



εOrth ¼ σ i;1 :σ i;2

will slow down, and this phenomenon will be associated with a smaller St compared to the single-unit case, as displayed in Fig. 15(b). Additionally, as seen in Fig. 15(b), more peaks occur in the high-frequency region in the double-unit case because the boundary layer downstream of the doubled region contains considerable turbulence, which would result in more small-scale turbulent flow structures with a high oscillating frequency in the downwash dominate region of the near-wake for the double-unit case. Moreover, the dominant frequency obtained displays good agreement with those previously reported by Baker (2010) and Xia et al. (2017), which again validates the research.

εL2

(10)

where u, v and w are the combinations of UTF, VTF and WTF, respectively, at each corresponding grid point obtained from the instantaneous flow field. According to the POD method, the flow field is then decomposed into a basis function and a set of mode coefficients as follows: ut ¼

N X

ai ðtÞσ i

(12)

where σ i,1 and σ i,2 denote the basis function of the ith POD mode calculated, considering the time steps between snapshots or the total time. Notably, because the signs of the modes are actually meaningless, when comparing the modes obtained based on different parameters, attention should be paid to whether the signs of the modes differ. In this study, the first 7 POD modes are compared in terms of the convergence for both the time step between snapshots and the total time, and all the results are compared with the modes obtained for a time step between snapshots of 0.125t* and total time of 160t*. The results plotted in Fig. 16 and Fig. 17 show that the first 7 modes converged for a time step between snapshots of 0.5t* in terms of εOrth. For the total time, the first 5 modes converged when the total time reached 140t*, and the first 7 modes converged at a total time of 150t* in terms of εOrth. For the L2 norm criterion, a time step less than 0.25t* is adequate for all the modes considered above, and modes 6 and 7 display slight discrepancies at total times of longer than 150t* that are still within an acceptable stage. Thus, the parameters chosen for the POD analysis provide acceptable results for the lateral analysis. Fig. 18 shows the basis function of the first 4 POD modes for the V component and the associated frequency information, which is obtained through the corresponding mode coefficients. The scalar colour bars in each mode are different, but all are centred at 0. The 1st mode represents the time-averaged flow field in the horizontal plane, which is similar for the two cases and exhibits a symmetrical flow pattern about the COT. Notably, the flows move away from each other when moving downstream because of the counter rotating motion of the field. The 2nd mode denotes the periodic vibration of the dominant vortex core in the þy and –y directions in the horizontal plane, and the structure is approximately 2.0H in the streamwise direction. The two branches of the structures move away from each other. These structures are similar in shape and size for the double- and single-unit cases. Notably, the sign of the 2nd POD basis function is completely opposite in different cases due to phase differences of the corresponding mode coefficients. For the 3rd mode, there is a turning point for the flow nearly 1.1H downstream of the tail nose, and the wake separates into two regions, which denotes a direction reversal of the flow pattern. The structure upstream of the turning point also has two branches that move away from each other. Unfortunately, the vertical plane used to conduct the POD analysis did not include the entire flow structure downstream of the turning point; nevertheless, we can conclude that the downstream structure is of a similar size to the structure in mode 2. The 4th mode contains several minor structures, and the length in the streamwise direction is approximately 1.2H. The flow changes direction at several turning points when travelling downstream, which indicates that the

3.3.4. Proper orthogonal decomposition (POD) of the wake POD analysis is used to extract the flow pattern based on the energy distribution. In this study, the velocity field of the downwash dominate region (0.25H from the TOR) was recorded as a series of snapshots. The time step between snapshots was 0.125t*, which, according to Muld et al. (2012a), was sufficient for capturing the dynamic flow modes. The total time required to conduct the research was 160t*. The snapshots obtained previously were arranged into discrete flow field matrixes U, V and W, which can be defined as follows: U ¼ ½u1 u2 … uN  V ¼ ½v1 v2 … vN  W ¼ ½w1 w2 … wN 



2 ¼ jσi;1  σi;2 j2

(11)

i¼1

where σ i is the basis function, ai(t) is the mode coefficient, and N is the number of finite modes. This equation indicates that the velocity field at a fixed time can be reconstructed by all POD modes and their corresponding coefficients at that time step. A full description of the method used in this paper was provided by Chen et al. (2012). The first step in the POD study involved convergence analysis. As demonstrated in a previous study (Muld et al., 2012a; 2012b), the time step between snapshots and the total analysis time influence the calculation of the POD mode coefficients; therefore, the results obtained based on specific parameters should be evaluated to ensure that they are within a reasonable range. There are two ways to compare POD modes: assess the orthogonality between modes, εOrth, or assess the L2-norm of the difference between modes, εL2.

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Fig. 16. Convergence curves for different time steps between snapshots in terms of (a) εOrth and (b) εL2.

Fig. 17. Convergence curves for different total times in terms of (a) εOrth and (b) εL2.

Fig. 20, the cumulative energy fraction of the first 50 modes is plotted for the single- and double-unit cases. As mode 1 is actually the average flow field, the corresponding energy contribution is not considered. The cumulative energy fractions of the modes that reflect the fluctuations in the components of the flow field are presented in Fig. 18. The cumulative energy percentage is higher in the single-unit case than in the double-unit case, which indicates that the energy was more concentrated in the motion of the large-scale vortex in the single-unit case and that the energy in the double-unit case is distributed in small-scale turbulent flow structures. This phenomenon corresponds to the earlier findings discussed in this paper. Notably, the height of the surface used in the POD study was 0.25H from the TOR, which is close to the platform height, where more turbulence was observed in the double-unit case.

length of the turbulence structure of the POD mode decreases. The 2nd and 3rd modes are more likely to undergo phase shifting at the same frequency. Moreover, the dominant frequency for the 2nd and 3rd modes in the single- and double-unit cases occurs at St values of 0.14 and 0.11, respectively. This result is in good accordance with the frequency findings of the wake probes discussed in Section 3.3.3. For the 4th POD mode, more small flow structures with high dominant frequency were found at St values of 0.21 and 0.19 for the single- and double-unit cases, respectively. Fig. 19 shows the reconstructed fields based on the magnitude of the velocity. The time t ¼ 160t* was chosen to reconstruct the instantaneous flow field, and the first 5, 50 and 500 modes are selected to reconstruct the flow field. The flow field reconstructed using the first 5 modes reflects the average flow field for a phase of the dominant vortex of the wake flow, and a spanwise wake oscillation can be detected. For the flow based on the first 50 modes, a large-scale instantaneous vortex can be observed moving downstream. These vortices contain a large amount of the energy of the flow field. Based on 500 modes, more small-scale structures that dissipate gradually are captured. A snapshot of the instantaneous flow field cannot be used to compare the single- and double-unit cases but can illustrate how the modes influence the corresponding flow field. To qualitatively compare the modes calculated from the different train formations, the energy of each mode was obtained based on the 2norm of the mode coefficients for the corresponding POD modes. In

4. Conclusion This study employ the IDDES method to compare the dynamic behaviours of the flows around single- and double-unit trains. A validation process is implemented through an existing wind tunnel test, grid independence study and the IDDES assessment. The aerodynamic forces, slipstreams around the trains and flow structures around the trains, especially the wake zone, are analysed. This study focuses on not only the time-averaged flow field but also the instantaneous and dynamic behaviours of different cases for comparison. The study highlights the

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Fig. 18. The first 4 POD modes for single- and double-unit cases together with the frequency information obtained through the corresponding mode coefficients.

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Fig. 19. Reconstructed field according to the first n POD modes.

probes at platform height for the corresponding cases. The energy fraction results also indicate that the double-unit case produces more turbulent wake at platform height. Acknowledgements The authors acknowledge the computing resources provided by the High-Speed Train Research Centre of Central South University, China. This work was supported by the National Key R & D Program of China (Grant No. 2016YFB1200506-03), the State Key Laboratory for Track Technology of High-speed Railways (Grant Nos. 2017YJ164 and 2017G004-A), the China Academy of Railway Sciences, and the Graduate Student Independent Innovation Project of Central South University (Grant Nos. 2018zzts504 and 2018zzts508).

Fig. 20. Energy fractions for the POD modes in the single- and doubleunit cases.

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(1) The influence of the coupling region on the aerodynamic force is most strongly related to the standard deviation and the frequency domain, as the periodic oscillating flow downstream is related to the dominant frequency of the aerodynamic force of the wake. The dominant St value is 0.15 in this study. (2) For the slipstream around the trains, the double region greatly increases the slipstream velocity in downstream regions until the far-wake zone is reached. Additionally, the standard deviation of the slipstream, which indicates that the turbulence intensity increases, and the variations in the slipstream are larger in the downstream boundary layer region for the double-unit case. In the near-wake region, the vortex dominate region was more dominated by the flow separated from the underbody structure of the tail like the snow plough; thus, the upstream influence on velocity variations is limited. Moreover, the flow fields for the two cases are much more similar in this region. In the downwash dominate region, the double-unit case displays a more pronounced fluctuation phenomenon and large velocity variations. (3) The frequency of the wake flow at a fixed point displays the same trend as that observed in the lower region, i.e., the St values are similar for the single- and double-unit cases. Based on the measurements of the platform height probes, differences exist in the flow field characteristics, with a lower dominant frequency for the double-unit case. This lower frequency is not directly related to the thickness of the boundary layer but rather to the low UTF value within the boundary layer at the train surface. (4) The POD analysis results support the finding that more turbulence structures exist at the downwash dominate region in the wake zone in the double-unit case. The 2nd and 3rd modes, which display the most energetic flow structures, have the same dominant frequency with the frequency content of the velocity at fixed 149

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Guo, Z.J., Liu, T.H., Chen, Z.W., Xie, T.Z., Jiang, Z.H., 2018. Comparative numerical analysis of the slipstream caused by single and double unit trains. J. Wind Eng. Ind. Aerod. 172, 395–408. Hemida, H., Baker, C.J., Gao, G., 2014. The calculation of train slipstreams using largeeddy simulation. Proc. Inst. Mech. Eng. - Part F J. Rail Rapid Transit 228 (1), 25–36. Muld, T.W., Efraimsson, G., Hennigson, D.S., 2012a. Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition. Comput. Fluids 57, 87–97. Muld, T.W., Efraimsson, G., Henningson, D.S., 2012b. Mode decomposition on surfacemounted cube. Flow, Turbul. Combust. 88 (3), 279–310. Muld, T.W., Efraimsson, G., Henningson, D.S., 2014. Wake characteristics of high-speed trains with different lengths. Proc. Inst. Mech. Eng. - Part F J. Rail Rapid Transit. 228 (4), 333–342. Munson, B.R., Young, D.F., Okiishi, T.H., 2002. Fundamentals of Fluid Mechanics, fourth ed. John Wiley & Sons, USA. Niu, J.Q., Zhou, D., Liu, T.H., Liang, X.F., 2017. Numerical simulation of aerodynamic performance of a couple multiple units high-speed train. Veh. Syst. Dyn. 55 (5), 681–703. € Osth, J., Kaiser, E., Krajnovic, S., Noack, B.R., 2015. Cluster-based reduced-order modelling of the flow in the wake of a high speed train. J. Wind Eng. Ind. Aerod. 145, 327–338. Shur, M.L., Spalart, P.R., Strelets, M.K., Travin, A.K., 2008. A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities. Int. J. Heat Fluid Flow 29 (6), 1638–1649.

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