Impact of the gap distance between two adjacent external windshields of a high-speed train on surrounding flow characteristics: an IDDES study

ABSTRACT The external windshield is a key component in high-speed trains for drag reduction. However, the installation gap between external windshields, which is intended for improving curve-passing ability, could lead to strong aerodynamic instability in train-end area and therefore significant operating risk for train. This study aims to advance this field based on detailed numerical simulation, investigating the related flow features in this area, particularly from a time-dependent perspective. Flow field around external windshields with different installation gaps (10, 20 and 30 mm) is resolved based on Improved Delayed Detached Eddy Simulation (IDDES). The aerodynamic forces on different windshield components, turbulent flow fields around the train-end area, and corresponding low-dimensional modes are analyzed. The results show that the difference in the external windshield gap can significantly affect the related aerodynamic forces, with the 20 mm case having the highest time-averaged values. The aerodynamic spectrum shows primary, secondary and tertiary peak frequencies with a relatively high power spectral density (PSD). Decreasing the gap distance generally increases the overall PSD in the Strouhal number ranges considered, particularly for the secondary and tertiary peaks. The load forms and their energy distributions under different external windshield gaps are obtained through the proper orthogonal decomposition analysis.


Introduction
The high-speed train, as a popular mode of intercity transportation, has been rapidly developed owing to the advantages of high efficiency, comfort, safety, and environmental protection. High-speed trains are being operated at speeds of up to 350 km/h, such as the Chinese Standard EMU. This has worsened the aerodynamic problems; for example, a significant increase in aerodynamic drag not only decreases railway transportation efficiency, increases energy consumption, and leads to wastage of resources, but also produces noise, affecting the surrounding environment along the railway. With increasing operating speeds, the aerodynamic behavior of high-speed trains has attracted significant attention, primarily because it accounts for approximately 80% of the total drag (Tian, 2019). Hence, it is necessary to advance the optimization of high-speed trains in terms of aerodynamic characteristics.
Much effort has been made to improve the aerodynamic performance of high-speed trains; for example, strong flow separation occurs in the bogie region, forming a non-negligible resistance and making this region one of the objects that requires smoothening . Yang et al. (2010) compared and analyzed the effect of bogie area with and without firings on the aerodynamic force of the train. The results showed that the aerodynamic drag of the bogie area at one end of the head car was the highest, the aerodynamic resistance of the train could be significantly reduced after each bogie was equipped with firings, and the reduction in the aerodynamic drag after the head bogie was equipped with firings accounted for approximately 50% of the total reduction. Zhang et al. (2018) studied the cut out angle of the partition wall in the bogie area and compared the influence of different cut out angle partition wall arrangements on the aerodynamic performance of the train. The research results showed that the cut out angle of the partition wall has a greater influence on the surrounding flow field and that the resulting aerodynamic resistance varies significantly, in which the solution with the best drag reduction effect reduces the aerodynamic resistance of the entire train in three groups by 2.92%. Kong and Yang (2017) analyzed the influence of different sinking heights and contours of the pantograph installation platform on the aerodynamic resistance of a high-speed train. After the pantograph was appropriately sunk, the aerodynamic resistance of the pantograph decreased. With the sinking height reaching 450 mm, the aerodynamic resistance of the pantograph decreased by 52.92%, and the resistance of the entire vehicle decreased by 6.19%. In addition, a pantograph deflector structure was introduced. By adding a deflector of a suitable shape, the aerodynamic drag in the pantograph area can be considerably reduced, and the performance of the pantograph system can be improved. An external windshield is installed in high-speed trains to realize train body smoothing and thus reduce the drag and noise. Peters (1982) conducted a wind tunnel test and found that the aerodynamic resistance generated by the connected parts of the vehicle (i.e. the windshield) has a 4% contribution to the overall vehicle resistance. Niu et al. (2019) found that the external windshield acts as a plate, slowing down the strong flow separation, thus improving the flow field around the connected parts of the vehicle and reducing the aerodynamic drag under normal operations. Suzuki et al. (2009) studied the effect of the outer windshield on the unsteady aerodynamic force of a train in a tunnel through on-track tests and a wind tunnel test. The outer windshield was found to be an effective countermeasure against pressure fluctuations, which also reduces the aerodynamic resistance of the train. These studies indicate that smoothing the train body by installing an external windshield can serve as an additional process to reduce the aerodynamic drag of high-speed trains.
In particular, the reduction in the aerodynamic drag could be essentially relevant to the parameters of the windshield. Different windshield structures or installation forms could significantly influence the aerodynamic performance of the train-end area. In particular, the aerodynamic instability introduced by the geometric features of external windshields may worsen. Thus, the influences of the various parameters of the external windshield have been studied. Yang et al. (2012) found that the aerodynamic drag of a closed train with upper and lower external windshields is relatively low and that there is no excessive increase in the aerodynamic lift of the tail car. Moreover, they suggested that considering the aerodynamic performance and easy implementation, a scheme with upper, lower and both-side-enclosed windshields is more suitable than other windshield schemes. Liang and Shu (2003) conducted numerical simulation calculations on the air resistance of small windshields, large windshields, and fully enclosed windshields under six different train running speeds and analyzed the velocity field on the various windshields and the surface pressure distribution on the end wall of the car body. Huang et al. (2012) studied the distribution of the aerodynamic resistance in each compartment of a high-speed train test model under a 2-6 car formation scheme and the influence of two types of windshields with different structural shapes on the aerodynamic resistance of each compartment in the three-car-formation train model. Aiming at three different design forms of the external windshield, namely slotted external windshield, seamless external windshield, and bottom-removed external windshield, the flow field distribution around the external windshield and the aerodynamic load on the external windshield were simulated when the train was running on an open line (Gai et al., 2019). Miao et al. (2021) simulated the flow field around freight high-speed trains under strong cross winds to explore the effect of four windshield solutions on the aerodynamic characteristic of freight high-speed trains, providing insights into the flow velocity, pressure, and surface pressure distributions in the region of a fully closed windshield and helping to reduce the aerodynamic drag and lateral force acting on the entire train during crosswinds; if an opening in the windshield is required for easy access, the opening is recommended at the bottom of the windshield. Niu et al. (2019) studied the influence of the external windshield structure on the aerodynamic performance of a crosswind high-speed train and found that the antioverturning ability of the train with a fully enclosed windshield is relatively optimal and that half and smooth windshields have an evident effect on reducing the train resistance in a wind environment.
Currently, many countries widely use a split external windshield, which has a certain gap between the front and rear parts of the external windshield, to improve the ability of the train when passing curves. However, the external windshield gap significantly influences the aerodynamic performance, with different countries having different standards for this installation gap. Li X et al. (2019) numerically simulated the effects of ten different external windshield gaps on the aerodynamic force, and the data agreed well with the experimental results. An outer windshield gap of 10 mm was recommended for wind tunnel tests. Kukreja (2016) used a 2D model of an Indian train operating at 20 m/s to study the flow field and aerodynamic loss in the external windshield gap. The results showed an evident vortex in the external windshield gap, and the aerodynamic loss could reduced by 5-6% using the filler. Kwon et al. (2001) conducted wind tunnel tests to analyze the shapes of inter-cars. Two external windshield gaps (25 mm after scaled and no gap) were chosen to test the effect on the aerodynamic drag; the inter-car gap with 25 mm increased the drag by 1.3%. Xia et al. (2020) tested four types of gap spacings and concluded that the gap spacing significantly reduces the aerodynamic drag of the head car and increases the aerodynamic drag of the tail car, while the effect on the intermediate car is not significant. Tang et al. (2019) studied the influence of installation gap spacing of the external windshield on the aerodynamic force and time-averaged flow field around the external windshield and its relationship with the deformation. From the above studies, it can be concluded that the external windshield gap has an evident effect on the aerodynamic drag of a train. A proper design of the external windshield gap can help significantly improve the aerodynamic performance.
Most existing studies focused on the influence of the external windshield gap on the time-averaged flow field around the entire train, while studies on the flow structures around the external windshield are lacking, particularly the influence of unsteady aerodynamic characteristics. Due to the significant difference between the scale of the train and the external windshield, their simulation, particularly the unsteady characteristics of the external windshield area, will be more difficult and challenging. The time-dependent flow features could deteriorate the mechanical properties of windshield structures. Notably, when the oscillating frequency of the flow field coincides with the natural frequency of windshield structures, the induced abnormal vibration would certainly affect the operational safety of trains. Therefore, it is necessary to pay more attention on this subject to thoroughly understand the flow behavior of the flow field within this area, including the effect of the installation gap on the phenomenon of interest. Based on the research results, the time-dependent aerodynamic loading on the train end can be better understood, while the relevant conclusions drawn can serve as a basis for potential fluid-structure interaction studies and optimization works based on improving computational resources. These considerations motivated our research.
Three gap conditions (10, 20, and 30 mm) on a 1/20th-scale model are studied, with the help of computational fluid dynamics (CFD) analysis, IDDES, which can provide accurate prediction for engineering research (Farzaneh-Gord et al., 2019;Ghalandari et al., 2019;Mou et al., 2017;Salih et al., 2019). We compare and analyze the flow field characteristics for each case, mainly including the aerodynamic force, flow field and proper orthogonal decomposition (POD) analysis, to determine the impact of the external windshield gap on the flow field. The results obtained from the numerical simulation are reported and discussed. The rest of this paper is organized as follows: Section 2 presents the numerical simulations and mesh independence test. In Section 3, a comparison and analysis of the aerodynamic force, flow field, and POD analysis for each considered case are presented in detail. Finally, the main conclusions are drawn in Section 4.

Model geometry
This study uses a 1:20th-scale high-speed train as the numerical computational model, as shown in Figure 1. The width (W) and height (H) of the train in full scale are 3.36 and 4.05 m, respectively. The height H (0.2025 m) is defined as the characteristic dimension in this study. The train model comprises three cars, including bogies and windshields, as shown in Figure 1(a). The total length of the train is 20H. Because of the negligible influence of the pantograph, window, door, and bogie on the flow field calculation results in the region concerned in this study, these parts are simplified. An enclosed external windshield with different gaps is used for the numerical simulation. Figure 1(b) shows the external windshield model. Figure 1(c) shows the three gaps simulated in this study. Figure 1(d) shows the real configuration of the external windshield.   boundary on the structure of the flow field in the concerned region and ensure the full development of the flow field in the region, the length, width, and height are set to 60.5H, 20H, and 10H, respectively. The distance from the domain entrance to the tip of the nose is 10H, and the length of the wake area is 30H. The train is 0.05H above the ground. The side and top of the computational domain are given symmetric boundary conditions. A uniform velocity profile U ref = 97.22 m/s is set at the inlet. The given pressure in the pressure outlet is P ∞ = 0 Pa. The sliding wall boundary conditions are applied to the ground. To simulate the relative motion between the train and the ground, the speed magnitude and direction of the sliding ground are made consistent with the front entrance boundary. The Reynolds number (Re) is 1.23×10 6 based on the height H and speed U train of the train.

Mesh strategy
Mesh generation in the current study is based on the Trimmer technique in STAR-CCM+ 14.02. Hexahedral cell volumes are used to discretize the computational domain, and twelve prismatic layers are designed to more accurately capture and simulate the development of the boundary layer along the train surface. The elongation of the prismatic layers is set to 1.2 so that the outermost layer of the prismatic layer can achieve better transition between prism layer and hexahedral grid. In addition, an important point in the mesh generation process is to set several appropriate local refinement zones to in order to better resolve the flow field with less computational cost and acceptable numerical accuracy. In particular, the scales of the geometric features vary in our simulation, for which the transition between the refinement zones should be appropriately arranged to avoid any dissipation in the numerical accuracy, the same as the grid processing technology in Salih et al. (2019). Figure 3 presents the mesh around the train body and the mesh of the external windshield. In this study, three sets of grids (namely coarse, medium, and fine grids) are generated to conduct a mesh independence test to determine the appropriate mesh resolution while ensuring both efficiency and accuracy. Regarding the prismatic layer and encryption topology, the three groups of grids are the same, differing only in terms of the spatial resolution to ensure the uniqueness of the variables. Table 1 presents the detailed parameters of the three groups of grids expressed in viscous units; the stream-wise and span-wise dimensions are obtained from the train surface. Note that the identical wall-normal distance in all the three sets of grids corresponds to the viscous unit n+ equaling 0.6, meeting the requirement that this value should be less than 1.0 as mentioned in CD-Adapco (2014) and finally adequately capturing the velocity gradient development within the boundary layer region.

Numerical method and solver setting
The principle of scale-resolved simulations, such as the IDDES method used in our current simulation, is to simulate the boundary layer near the wall using Reynolds-averaged Navier-Stokes simulations (RANS) and to capture large-scale turbulence away from the wall using large eddy simulations (LES). The IDDES method is a hybrid approach combining DDES and wall-modeled large eddy simulation (WMLES) with an improved delayed shielding function to achieve higher accuracy in the RANS-LES mixing region. The DDES and WMLES complement each other in terms of turbulent content and dominant eddies simulation, improving the numerical simulation accuracy of IDDES. Based on the IDDES method, problems, such as 'modeled stress depletion' and 'grid-induced separation,' can be effectively overcome. Shur et al. (2008) describes the IDDES turbulence model in more detail. The IDDES technique has been regularly applied to engineering applications concerning train aerodynamics, such as simulating the aerodynamic drag, slipstream, and wake flow (Wang et al., , 2020a(Wang et al., , 2020b. In this study, the IDDES based on the shear-stress transport k-ω is applied to predict the flow structure around the external windshield with various installation gaps. The current study employs the segregated flow solver, and the SIMPLE algorithm is used for the pressure-velocity coupling process. The convective terms are discretized based on bounded centraldifferencing (BCD) scheme which hybrids with 5% second-order upwind (SOU) scheme, and the diffusive and turbulent terms are discretized in SOU format (Li X et al., 2021). The time marching procedure is in the implicit second-order three-time level scheme, with the discretized time-step size chosen based on the principle of keeping the Courant-Friedrichs-Lewy (CFL) number below 1 in most of the computational grids. Thus, for the three different sets of grids, the time steps are respectively set as 0.0025, 0.0020 and 0.0015 t * (t * is the reference time defined by H/U ∞ ) for the coarse, medium, and fine grids. The simulations are initially performed for 50 t * to achieve a statistically stationary state and then run for 450 t * for field data collection, as shown in Figure 4.

Numerical validation
To verify the accuracy of the numerical method adopted in this study and ensure the reliability of the simulation results, a mesh independence test and full-scale test (Tang et al., 2019) are used to verify the numerical simulation results. For investigating the mesh independence of the numerical predictions, the numerical results obtained from the coarse, medium, and fine meshes are analyzed. Figure 5 shows the positions of the line probe of the pressure and velocity, where X indicates the forward direction of the upstream. Figure 6 compares the time-averaged pressure and time-averaged velocity distribution at the line probe between the three computational meshes. The span-wise distance of the line probe of the pressure from the center line of the train is 0.415H, and the line probe of the velocity is located at the center of the gap. Besides, the height of the two line probes is 0.434H. The two full-scale sensor points are placed at −0.03H and 0.03H away from the center of the gap. By comparing the time-averaged pressure along the line probe in Figure 6(a), the result of the medium mesh presents an evident consistency with that of the fine mesh, while the coarse grid fails to provide an accurate numerical simulation result, particularly for the prediction of the two negative peak values. The enlarged views show that the results of fine mesh and medium mesh both have a good agreement with those of full-scale test, while the results of coarse mesh present relatively large deviation, especially the result of the second sensor point. The deviations of the two test points from fine mesh are 2% and 2% respectively. The deviations of the two test points from medium mesh are 4% and 7% respectively. As shown in Figure 6(b), the result of velocity obtained from coarse mesh fails to accurately predict the positive peak value in the gap, mainly because coarse mesh difficultly captures the small scale flow field. In conclusion, the mesh sensitivity test indicates that the results of fine mesh are closest to that of full-scale test, but the resolution of the medium mesh is sufficient enough for resolution in the present work. Therefore, considering a balance between the numerical cost and precision, all simulation in the following text are conducted using the medium mesh.

Results and analysis
This section presents the results of the gap effects on the external windshield aerodynamics. First, the averaged values, maximum values, and standard deviations of the aerodynamic force on the different parts of the external windshield are chosen for comparison and discussion. Second, the related flow field information, including the mean pressure and velocity, and frequency analysis are presented. Finally, the POD is used to extract the low-dimensional behavior of the coherent structures, to further investigate the time-dependent features of the aerodynamic loads acting on the windshield surfaces.

Time-averaged forces
The external windshield between the head and middle cars is chosen for the subsequent analysis. Because of the change in the curvature of the external windshield structure, the pressure on it changes with the spatial position. This study divides the external windshield into four parts, namely two left parts (L-1, L-2) and two right parts (R-1, R-2), to quantitatively measure the overall force of the external windshield, as shown in Figure 7.     When a high-speed train runs at a speed of 350 km/h, based on the full-scale test, an abnormal vibration of the left and right parts first occurs and appears to be most violent (Niu et al., 2019). Thus, in the analysis, only the lateral forces acting on the left and right parts are considered. The aerodynamic force listed here is under the condition of the train at a scale of 1:20. On the premise of ensuring that the Reynolds number is independent, the results can be scaled to obtain the aerodynamic force of the windshield at the actual scale. Table 2 shows the aerodynamic lateral force on left and right parts of the external windshield obtained in the simulations under the three gap conditions, 10, 20, and 30 mm, including the average values, the maximum values, and standard deviations. The results show that the average values of external windshield 1, containing L-1 and R-1, are nearly five times that of external windshield 2, containing L-2 and R-2. This means that the vibration and deformation of external windshield 1 will be more serious. Evidently, the different gaps affect the average values of the aerodynamic lateral force. The average values under a gap size of 20 mm are the highest among the three gap conditions. As for the average values of L-1and R-1 (0.323 N and −0.327 N), the minimum is when the gap size is 10 mm. In terms of L-2 and R-2, the maximum is when the gap size is 30 mm, with values of 0.069 N and −0.061 N, respectively. The aerodynamic lateral value of L-1 (0.839 N) is maximum in the 20 mm case, whereas the 30 mm case yields the minimum of the maximum values, 0.410 N, which is observed at L-2.
To facilitate the purpose of comparative analysis, the standard deviation values are also proposed in this study to assess the degree of fluctuation in the aerodynamic lateral force. Comparing the standard deviation values, the degree of fluctuation in windshield 2 is stronger than that in windshield 1, increasing by 35% in terms of the numerical terms. Generally, the fluctuation degree of the windshield at different positions decreases with the increase in the windshield gap. The standard deviation value of L-2 is 0.107 N when the gap is 10 mm, which is the highest, while the standard deviation of R-1 is 0.051 N in the 20 mm case, which is the lowest.

Aerodynamic spectrum of aerodynamic forces
In addition to the time-averaged forces and peak value acting on the external windshield, the external windshield vibrates under external excitation, particularly when the unacceptable vibration frequency coincides with the natural frequency of the structure, thereby inducing strong elastic vibrations. If this is the case, the external windshield structure will eventually get damaged. In this study, the total time used for spectrum estimation is 450 t * , from 50 to 500 t * . The power spectral density adopts the Pwelch method in MATLAB, by averaging over 17 sample series. The period of each sample series is 50 t * containing 5000 time steps, with an overlap of 50%, using a Hamming window on each sample series, thus effectively improving the reliability of spectrum estimation. Figure 8 shows the power spectral density of the time history value of the aerodynamic drag force coefficient at the left external windshield 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 0.01-100, meeting the requirement of the range (0.1-10) . The spectrum of the right external windshield is omitted owing to left-right symmetry. Equation (1) gives the definition of the Strouhal number.
Here, f is the frequency, and v and H respectively denote the speed of the train, 97.22 m/s, and the height of the train, which is 0.2025 m after scaling. Based on the data presented in Figure 8, for most of the curves, three peak frequencies with relatively higher PSD can be identified in St ranges of 0.4-0.6, 1-2 and 20-30, referred to as the primary, secondary, and tertiary peaks in the following analysis, respectively. Moreover, we can primarily conclude, from Figure 8, that decreasing the gap distance generally increases the overall PSD in all St ranges for both windshields 1 and 2, particularly for the secondary and tertiary peaks. This generally implies that increasing fluctuation intensity induced by decreasing gap distance covers a wide turbulence scale range. For external windshield 1, the 10 mm case has not been detected with a clear primary peak, but a rather high PSD for the secondary peak. Moreover, a clear boundary between the primary and secondary peaks can be observed for the 30 mm case, but is nonexistent in the 20 mm case. This generally implies that, with decreasing gap distance, the interference between the flow patterns in the primary and secondary peaks gradually strengthens. The strengthening interference can be attributed to the significant growth in the turbulence kinetic energy for flow pattern, which corresponds to the secondary peak, with decreasing gap width. However, the St of the secondary peak seems to be less influenced by the increasing gap width. For external windshield 2, the flow pattern corresponding to the secondary peak seems to have less impact on the force spectrum. The 20 and 30 mm cases exhibit relatively higher PSD in the primary peak range, while only the 10 mm case is observed with a clear secondary peak. For both external windshields 1 and 2, the developments in the tertiary peak with decreasing gap width are almost identical. A monotone increasing St, as well as a monotone decreasing PSD, can be observed for the tertiary peak when the gap width increases, which corresponds to reduced flow propagating period and weak fluctuating intensity. In fact, due to the multiscale feature within the train-end area, the unsteady behavior of the aerodynamic forces can be characterized by various flow phenomenon, for instance the flow separation on windshield 1 and reattachment on windshield 2, the bounded turbulent flow inside the gap, and the largescale low-momentum vortices inside the cavity. Therefore, in our following content, the flow field is further investigated to illustrate the cause of the aerodynamic loads.

Time-averaged flow field
To study the flow field at the external windshields in detail, different spatial sections of the external windshields are taken for analysis. Figure 9 shows the horizontal sections 1-4 and vertical sections 5-7. The heights of Sections 1-4 are 0.11H, 0.20H, 0.49H and 0.89H, respectively, and Sections 5-7 are located 0.0005H, 0.001H and 0.0015H away from end of the head train, respectively. To facilitate the comparison in the subsequent parts of the article, the pressure and velocity are made nondimensional. The pressure and velocity can be defined as follows: Here, C p and C v are the pressure coefficient and velocity coefficient, respectively. The air density, ρ, is equal to 1.225 kg/m 3 . The coming flow velocity U ∞ , which means the speed of the train, is 97.22 m/s. p is defined as the local mean static pressure, and p ∞ is the reference pressure, which is 0 Pa. Owing to the symmetrical sections, only the left halves of the sections are taken for analysis. Figures 10 and 11 compare the pressure contours at Sections 1-7 under three different gap conditions, which are reasons for the vibration of the external windshields. The number in the upper-left corner of the section represents the serial number of the section. As shown in Figure 10, the pressure on the outside of the external windshield is a low negative pressure comprehensively, and the pressure distribution on the outside of the external windshield is different from that at the vehicle height section and external windshield gap. Section 1 is close to the bottom windshield, where the pressure is lower than that in the other areas because of the flow acceleration due to the clearance at the bottom of the train. With the increase in the external windshield gap, a lower negative pressure area is formed outside external windshield 2. The negative pressure in the external environment of the external windshield gradually decreases with increasing section height, which can also be observed in Figure 11. Because Section 2 is located in the shock absorber cavity of the side windshield, when the fluid flows through the end surface of the middle car, a strong flow separation due to the geometric obtuse angle occurs, forming a strong shear flow and a high negative pressure area behind the shear layer. The flow field conditions of Sections 3 and 4 are similar, showing a local negative pressure due to flow acceleration on the outside of external windshield 1. Some of the accelerated high-speed air flows into the windshield through the gap between the two external windshields, and the rest of the high-speed air flows adhering to the surface of external windshield 2 and produces a stagnation effect, forming a local positive pressure area. With the enlargement of the external windshield gap, the ranges of the negative pressure area outside external windshield 1 and positive pressure area outside external windshield 2 increase, which becomes increasingly evident with increasing section height. Similarly, as shown in Figure 11, with the increase in the section height, the negative pressure area outside the upstream external windshield 1 caused by flow acceleration gradually expands, and the strength also gradually increases. The higher the section height, the greater the range and amplitude of the positive pressure in the region between the external windshields 1 and 2, which causes the force on the external windshield 1 to point out from the inside and the force on the external windshield 2 to point in from the outside, eventually leading to opposite forces on the external windshields 1 and 2. This results in a mutual movement trend between the windshields, finally leading to the deformation and even vibration of the external windshield. When the external windshield gap reaches 20 mm, the range and pressure value of the local negative pressure and local positive pressure at Sections 3 and 4 increase. Based on the pressure contour of the 20 mm case shown in Figures 10 and 11, the environmental pressure in the inner cavity of the 20 mm windshield is significantly higher than those under the other two working conditions, resulting in the maximum lateral force on the external windshield surface in the 20 mm case. However, there is no evident difference in the cross-sectional surface pressure contours in the 20 and 30 mm cases. Therefore, in this study, a line probe is established at different cross-sections near the outside of the external windshield to understand the influence of the external windshield gap on the outside pressure distribution of the external windshield in detail.
For the same positions of line probes 1 and 4 and line probes 2 and 3, the positions of the four line probe lines are as shown in Figure 12. The span-wise distances of the four line probes from the center line of the train are 0.375H, 0.425H, 0.425H and 0.375H. Figure 12 shows the pressure distribution curves outside the four sections of the external windshield along the line probe. As shown, the pressure distribution along the four cross-sectional probes varies with the external windshield gap. As shown in Figure 13(a), the pressure increases at the gap of the external windshield and decreases sharply at the surface of external windshield 2, as the air flow is accelerated by the gap, forming a negative pressure area behind the gap. With the increase in the external windshield gap, the pressure difference changes more. The pressure distribution curve in Figure 13(b) shows that the gap size mainly changes the differential pressure, and the larger the gap, the greater the differential pressure. Because Section 2 is located at the opening of the transverse pressure reducer, the strong shear flow effect formed by the airflow passing through the opening has a more evident influence on the pressure distribution than the external windshield gap. Figures 13(c) and (d) show that the pressure distribution trends along the line probe at Sections 3 and 4 are similar, consistent with the flow state shown in the pressure contours ( Figure 10). A part of the airflow accelerates near the gap of external windshield 1, resulting in a pressure drop, and the remaining airflow stagnates on the surface of external windshield 2, forming a positive pressure area. The fluctuation in the pressure is proportional to the gap size of the external windshield. The pressure fluctuations under gap sizes of 20 and 30 mm, which are similar, are evidently greater than that under a gap size of 10 mm. Overall, the pressure fluctuation varies with the different gaps of the external windshield, that is, the larger the gap width of the external windshield, the more evident the pressure fluctuation.
As shown in Figure 14, the air flow separates at the external windshield gap, then enters the cavity at the gap near external windshield 2, and finally converges to form several multiscale vertical vortex structures. Under a gap size of 20 mm, the vertical vortex structure with a small scale is formed at the position of Section 1, but no evident vortex structure is observed under gap sizes of 10 and 30 mm. With the enlargement in the external windshield gap, the airflow in the inner windshield cavity in Section 2 becomes stable gradually, and the vertical vortex structure with a regular shape is formed under the 30 mm gap condition. A single vortex with a regular shape at the time-averaged scale is formed in the inner windshield cavity of Section 3 under the three gap conditions. The airflow states at Section 3 cavity of 10 and 20 mm windshields are similar, including the direction of the airflow into the cavity and the direction of the vortex. However, when the gap between the external windshield reaches 30 mm, the airflow state changes: the airflow enters the cavity from the second side of the external windshield to form a vortex, and the flow direction of the vortex changes from anticlockwise to clockwise. This leads to the acceleration of the airflow and a decrease in the pressure inside external windshield 2, and finally, the lateral force on external windshield 2 significantly  reduces under the 30 mm working condition. At the position of Section 4, only a small-scale vertical spiral flow is observed under a gap size of 20 mm. Combined with that shown in Figure 15, the flow direction vortex structure is mainly formed at the height of Section 4. As shown in Figure 15, due to the evident air separation phenomenon at the external windshield gap, only a small number of irregularly shaped stream-wise vortices can be observed in the area with higher Section 7. A large amount of stream-wise momentum is introduced from the bottom transverse damper, which disturbs the development of the stream-wise vortices in the area with a lower height, making it difficult to observe the steady stream-wise vortices on the average scale, and the external windshield gap has little effect on the formation of the time-averaged flow vortex structure. The airflow enters the inner and external windshield cavities to form periodic vortex shedding, which forms a pulsating airflow, resulting in unstable   stress and vibration of the rear capsule. In conclusion, because the flow pattern in the cavity is complex, there are multiperiod flow structures with different directions and different scales in the cavity, which evidently causes a wide frequency pulsation of the aerodynamic force. To further explore the influence of the external windshield gap on the aerodynamic performance of the windshield in more temporal detail, it is necessary to compare and discuss the unsteady characteristics under the three working conditions.

Frequency analysis
In this section, the spectrum information of the spatial pressure at the monitoring points is compared and analyzed. Three monitoring points are placed at the height of Section 3, which presents an evident vertical vortex structure, as shown in Figure 16. Figure 17 shows the frequency analysis result of the three points, the same as the method of obtaining spectrum information in Section 3.1.2.
As discussed previously, the upstream flow particles undergo massive separation and reattachment on the external surfaces of the windshields, with part of the shear flow propagating into the cavity through the gap, forming a large-scale vertical vortex. The separated flow introduces a large amount of turbulence kinetic energy to the mean flow field, which significantly alters the frequency information of the flow quantity in the train-end area. Similarly, the three peak frequency regimes, mentioned in Section 3.1.2, are used in the current discussion, namely the primary, secondary, and tertiary peaks, which respectively correspond to St ranges of 0.4-0.6, 1-2 and 20-30. As shown in Figure 17, all the curves can be observed with a clear primary peak except for point 2 of the 10 mm case. As this point is located on the external surface of the downstream windshield, the rather high turbulence kinetic energy in a wide scale range induced by the small gap size significantly increases the overall PSD at all the frequency points, leading to the elimination of the primary peak. Moreover, the St ranges for the primary peak in all the considered points are generally identical and seldom influenced by the gap size. This implies that the primary peak could be related to the separation of the upstream turbulent boundary layer at the rear of the upstream windshield and could spread to all the downstream points. The secondary peak can be found only in Figure 17(c) in which the spectra of point 3 for various gap sizes are presented. Similar to that presented in Section 3.1.2, the St ranges for the secondary peaks are also less affected by the gap widths, and an evident interference between the primary and secondary peaks can be identified for the 10 mm case. As the oscillating frequency for the flow pattern related to the secondary peak is much lower than that of the tertiary peak, and unique to the cavity area, the secondary peak could be related to the oscillation of the vertical vortex inside the cavity. While for the high St range, only point 3 can be found with a clear tertiary peak, with the other two points only exhibiting overall higher PSD in all high St points. Thus, the tertiary peak could be the consequence of the bounded shear flow inside the gap. Notably, for wall-bounded flows that propagate in the wall-parallel direction, the dominant frequency should increase with decreasing wall spacing, which is opposite to our observation. The regularity in the tertiary peak is due to the different span-wise velocities under different gap conditions. With increasing gap width, the separated flow would naturally propagate in the span-wise direction, and the flow between the windshields will be characterized by a higher span-wise velocity, which makes the physical quantities to oscillate more rapidly.

Proper orthogonal decomposition (POD) analysis
The POD, as a widely used technique in data dimensionality reduction, flow field analysis, and image processing, is derived from the statistical analysis of vector data. In the POD, the modes are sorted by energy from highest to lowest after decomposing in orthogonal modes. Regardless of the given number of modes, the maximum amount of energy is contained by the design of the POD modes. Fan and Li (2012) and Luo et al. (2014) performed a simple analysis of the multisegment wing flow using the POD method and found that the POD method has a fast and effective prediction function for flow field information. Muld et al. (2012) used the POD method to extract the flow field structure in the near wake region of an ATM train model. They successfully identified a pair of counter-rotating flow vortices, found the dimensionless frequency St = 0.085 of the vortex pair shedding, and discovered the bending motion of the vortex pair during the evolution process by flow field reconstruction In this study, the POD analysis, originally proposed by Sirovich (1987), was adopted and applied to study the HST wake structure based on numerical (Muld et al., 2012) and experimental data (Bell et al., 2016b).
The principle of the POD is to find a set of orthogonal basis functions that best approximates the flow field quantities, expressed as follows: where Q(x, t) is the mean-subtracted flow quantity at a specific instant in time, such as the velocity and pressure; φ j (x) is the derived basis function, and a j (t) is the time coefficient for the mode at the current flow time.
To achieve this, first, the mean-subtracted snapshot data of the flow field should be arranged in a matrix, as expressed in Equation (5). m and n are the total number of grid points in the flow field and the total number of snapshots, respectively.
The temporal correlation matrix can be calculated as: where W is a positive-definite Hermitian tensor of appropriate dimension, which can be replaced by a diagonal matrix containing the volume of each cell on the diagonal. To fulfill the best approximation of Equation (4), the eigenvalue problem related to matrix C should be considered: The basis function φ j (x) should be calculated by projecting Q(x, t) onto the eigenvector β j , with subsequent normalization. The basis functions represent the extracted flow pattern. Only when all the basis functions are added together, the real flow field is observed, according to the definition of Equation (4). A real flow pattern appears prominent only when the flow structure in each flow field is nearly the same and in the same location . The coefficients of each mode are then calculated by projecting the mean-subtracted flow field quantities onto the calculated basis functions. In this study, the POD is conducted based on the static pressure on a plane at the location of Section 1. The time step between snapshots is 0.03 t * (t * = H u ∞ ), which, according to Muld et al. (2012), is sufficient for capturing the dynamic flow modes. The total time required to conduct the research is 450 t * . Based on the aforementioned parameter, the total number of snapshots is 15000. Figure 18 shows a schematic of the first four energetic modes. The scalar color bars in each mode are different, but are all centered at 0. The structures of the first and second modes in the 10 mm case show a symmetric or anti-symmetric distribution in space. The first mode in the 10 mm case indicates that the static pressure varies in phase over the section, meaning that the direction of the load acting on the external windshield is inward or outward simultaneously. The second mode characterizes the reverse pressure pulsation on both sides of the section, and the corresponding loading form is that when one side of the external windshield is squeezed inward, the other side of the external windshield is turned outward. The external windshield is torn and broken due to the longterm bearing of this load form. Compared with the first and second modes in the 10 mm case, the corresponding modes in the 20 and 30 mm cases do not show strict symmetry or anti-symmetry, but show a phenomenon wherein the pulsation energy is concentrated on one side. In the 20 mm case, the first and second modes both indicate out-of-phase pressure pulsation outside the external windshield. Different from the 20 mm case, the first and second modes in the 30 mm case show a pressure variation form similar to that in the 10 mm case. Therefore, the loading forms represented by the first and second modes have a strong coupling behavior, and this inference will be further analyzed in the subsequent discussion on the modal energy distribution. In terms of all the cases, the pressure loading forms of the third and fourth modes are similar, which represent the reverse movement of the external windshields 1 and 2 on the same side. The third mode shows that the two windshields of external windshields 1 or 2 are subjected to an out-of-phase load, that is, when one of them is squeezed inward, the other is turned outward, which is opposite to the movement form of the fourth mode, characterized by pressing inward or turning outward simultaneously.
In addition to the mode contours shown in Figure 18, the cumulative energy percentage distribution of the first 30 modes is shown in Figure 19. As shown in Figure 19, the energy is more concentrated in a few energetic modes. The energy concentration degree of the three cases from large to small is 10 mm, 20 and 30 mm respectively. For example, the total energy proportion of the first four modes for the 10 mm case is 0.721, whereas the corresponding values are 0.719 and 0.685 for the 20 and 30 mm cases, respectively. The energy is more concentrated in the first four modes, which means that the load forms of the first four modes have a major impact on the abnormal vibration and deformation of the windshield and that they dominate the flow field most. The proportions of the first and second-order modal energies are close, and the proportions of the third and fourth-order modal energies are close, indicating that these two groups of load forms have a similar influence degree on the aerodynamic fatigue of the external windshield. It is confirmed, once again, that modes are mutually coupled.

Conclusions and future works
In this study, the aerodynamic characteristics, particularly the flow field, around an external windshield under three different gap cases were investigated using IDDES at Re = 1.23×10 6 . The conclusions are summarized as follows: (1) For the aerodynamic force, the average values of 20 mm were the highest among the three gap conditions. In the aerodynamic spectrum, primary, secondary, and tertiary peak frequencies with a relatively higher PSD could be identified in the St ranges of 0.4-0.6, 1-2 and 20-30, respectively. Decreasing the gap distance generally increased the overall PSD in all St ranges, particularly for the secondary and tertiary peaks, which generally implies that increasing the fluctuating intensity induced by decreasing gap distance covers a wide turbulence scale range.
(2) The environmental pressure in the inner cavity of the 20 mm windshield was significantly higher than those in the other cases, resulting in the maximum lateral force on this external windshield surface. For the pressure outside the windshield, the greater the gap width of the external windshield, the more evident the pressure fluctuation. The upstream flow particles underwent massive separation and reattachment on the external surfaces of the windshields, with part of the shear flow propagating into the cavity through the gap, forming a large-scale vertical vortex. The St numbers for the primary peak at all the considered points were generally identical and seldom influenced by the gap size, which implies that the primary peak could be related to the separation of the upstream turbulent boundary layer at the rear of the upstream windshield and spread to all the downstream points.
(3) The structures of the first and second modes in the 10 mm case showed a symmetric or anti-symmetric distribution in space, and the corresponding modes in the 20 and 30 mm cases showed the phenomenon of concentrated pulsation energy on one side. The pressure loading forms of the third and fourth modes represented the reverse movement of external windshields 1 and 2 on the same side. The load forms of the first four modes had a major impact on the abnormal vibration and deformation of the windshield and dominated the flow field.
In this paper, the effect of installation gap of external windshield on the flow field around the train end is studied in depth, mainly analyzing the aerodynamic forces on different windshield components, turbulent flow field near the end of train and the corresponding low-dimensional modes. However, in the actual operation of the train, the external windshield structure is not rigid body but elastic rubber structure, which will appear vibration and deformation problem, serious will lead to the external windshield tear under the action of aerodynamic load. In this study, the vibration and deformation of the external windshield and the interaction between the flow field and the structure are not considered. Next step, the fluid-structure coupling method can be used to study the coupling effect of the interaction between the external windshield structure of highspeed trains and the surrounding flow field, or the geometric shape characteristic parameters of the external windshield structure could be optimized by numerical analysis method to improve the reliability of the external windshield structure and train operation security.