1 Introduction

The continual accelerating operating speeds as well as the security and reliability of train brakes have become increasingly important. Since drag scales with dynamic pressure, drag losses should scale quadratically with flow velocity. However, the ratio of pressure drag to viscous drag is important; as the train length increases, viscous losses dominate. According to previous studies (Brockie and Baker 1990; Schetz 2001; Raghunathan et al. 2002; Baker 2014; [5, 6]), aerodynamic drag increases rapidly with train speed and constitutes more than 80% of the total drag when the train speed reaches 300 km/h. Japan is a global leader in high-speed magnetic levitation (maglev) train engineering, and the test speed of maglev passenger trains exceeded 600 km/h in 2015 [5]. Germany achieved the manned operation of a maglev test train at a speed of 501 km/h in 2003 [6]. China is currently developing a 600 km/h maglev train. Other high-speed rail vehicles are under development, such as a high-speed evacuated tube train system whose speed may reach 3000 km/h. Braking at high velocities is challenging for an optimal train operation. This study investigates a method to supplement conventional braking through an increase in vehicle aerodynamic drag.

Considerable research has been conducted on the application of aerodynamic braking to rail transit systems. Most aerodynamic brake plates are installed on a vehicle roof (Puharić et al. 2010; Puharić et al. 2011; Puharić et al. 2014; Xi et al. 2011). The number of plates varies, typically one or more (Puharić et al. 2010; Puharić et al. 2011; Puharić et al. 2014; Xi et al. 2011; Yoshimura et al. 2000). If the brake plate is installed on the vehicle roof, then the vehicle surface will be uneven, which will change the flow field near the train surface and deteriorate the train’s aerodynamic performance (Jianyong et al. 2014; Takami and Maekawa 2017). A suitable installation location for the vehicle is necessary. The vehicle profile around the vehicle connection is complex, which causes extreme changes in the flow field (Dai et al. 2018; Niu et al. 2016a). Using an outer windshield to fill vehicle-connecting parts can effectively reduce a train’s aerodynamic drag (Niu et al. 2019; Li et al. 2019; Watkins et al. 1992). Consequently, the outer windshield functions as a plate that improves the flow field around the vehicle-connecting parts and decreases the aerodynamic drag under normal operations; and as a brake plate when the train is braking.

In this study, the aerodynamic characteristics of a train with aerodynamic braking plates were simulated, the effects of aerodynamic braking plates installed at the car-connecting parts on the increase in the braking force were analysed, and the aerodynamic characteristics of a high-speed train with different aerodynamic braking plate configurations were compared. The key aims of this study are as follows: (i) to elucidate the braking effect(s) of an aerodynamic braking plate installed at car-connecting parts, and (ii) to understand the underlying mechanism(s) of an aerodynamic braking plate installed at the car-connecting parts. This knowledge can assist engineers in designing aerodynamic brakes. A numerical simulation including models, methods, computational domains, and grid generation, among other factors, is introduced in Sect. 2. A grid resolution analysis and a comparison with a wind tunnel test are presented in Sect. 3. The results and discussion are presented in Sect. 4. Conclusions are presented in Sect. 5.

2 Computational Fluid Dynamics

2.1 Description of Model Train Geometry

As shown in Fig. 1a, the high-speed train model was composed of two identical streamlined vehicles, and their lengths (L is for the entire train) for the full-scale model were 53 and 26.25 m, respectively. The height of the train model (H is the distance from the top surface of the rail to the ground in the vertical direction) was 3.7 m, which was used as the characteristic length in this study. Figure 1b shows the size of the plates; the total frontal areas of the different configurations of plates in the three cases were the same, and the detailed dimensions of the different configurations of the plates are presented in Table 1. Figure 1c shows four configurations of aerodynamic braking plates, i.e. C-0 (plates are closed), C-1 (large upstream plates), C-2 (large downstream plates), and C-3 (small upstream and downstream plates), installed at the car-connecting part of a China railway high-speed train model. Finally, a 1:8 scaled train model was used in all simulations.

Fig. 1
figure 1

Configuration and size of full-scale models of train and aerodynamic braking plate: a and b show the sizes of the train model and aerodynamic braking plate models, respectively, and c shows the braking plate configuration at the vehicle connection

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Table 1 Key dimensions of different plate configurations for the full-scale model (unit: m)
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2.2 Outline of Turbulence Treatment

The flow field around the aerodynamic braking plate exhibited strong separation and unsteady characteristics. Therefore, an appropriate turbulence model and computing method must be used to obtain accurate results. Detached eddy simulation offers several advantages over large eddy simulation and Reynolds-averaged Navier–Stokes equation methods (Spalart 1997; Spalart et al. 2006). Improved delayed detached eddy simulation (IDDES) combines delayed detached eddy and wall-modelled large eddy simulations, partially resolving log-layer mismatches and grid-induced separation, among others (Spalart et al. 2006; Menter and Kuntz 2004; Travin et al. 2002; Gritskevich et al. 2012). This technique is widely used to simulate the aerodynamic characteristics of trains (Niu et al. 2018; Wang et al. 2019b; Li et al. 2019a; 2019b; Tan et al. 2019; Chen et al. 2019; Guo et al. 2019; Paz et al. 2019). A shear–stress transport (SST) κ–ω turbulence model has been proposed to address flows in boundary layers (Menter 1994; Menter 1996; Menter et al. 2003) and has been successfully applied to simulate the aerodynamic performances of trains ([35]; Yao et al. 2012; Jiqiang et al. 2018; Wang et al. 2018). In this study, IDDES with the SST κ–ω turbulence model was adopted to simulate the flow field around a train with an aerodynamic braking plate; this method has been widely used in simulating train aerodynamics (Niu et al. 2018; Xia et al. 2017; Zhou et al. 2019; He et al. 2018; Jia et al. 2017; Maleki et al. 2017; Bhushan et al. 2013; Huang et al. 2016).

2.3 Numerical Aspects of Simulations

To avoid the interference of the computational domain boundary conditions in the simulation of the flow field around the train, the computational domain dimensions were set according to the requirements of the CEN standards (2010), including at least eight characteristic lengths upstream and 16 characteristic lengths downstream. Detailed information regarding the computational domain dimensions is presented in Fig. 2. Combined with Fig. 2, the computational domain boundary conditions were set as follows: the upper and two side surfaces of the computational domain (green) were defined as the symmetry boundary conditions. The train surface was defined as a no-slip wall, and the lower surface of the computational domain (black) was the slip wall (the flow velocity on the surface was equal to the speed of the surface movement (the train speed)). The front and back surfaces of the computational domain (red and blue) were defined as the velocity inlet and pressure outlet, respectively. The inlet values of the turbulent kinetic energy k and specific dissipation rate were calculated using the equations presented in previous studies (Niu et al. 2018).

Fig. 2
figure 2

Dimensions and boundary conditions of the computational domain

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Meshes were created using the snappyHexMesh gridding tool supplied in OpenFOAM 4.1 (OpenCFD). This tool generated hexahedral-dominated grids and has been used to simulate train aerodynamics (Tschepe et al. 2019; Gao et al. 2019; Chen et al. 2018; Gallagher et al. 2018). As shown in Fig. 3a, the computation domain including the train and wake contained two refinement boxes (41 H × 1.62 H × 1.35 H and 31 H × 1.5 H × 0.2 H). The bogie structure was relatively complex, and a fine grid was adopted in this area, as shown in Fig. 3b. According to previous studies (Guo et al. 2020; Wang et al. 2018, 2019c; Weinman et al. 2018), as shown in Fig. 3b, 10 prism layers were added to capture the train’s boundary layers, and an adaptive grid method in FLUENT [56] was used to realise y + around the scaled train model of less than 1 around the train based on initial steady calculation results.

Fig. 3
figure 3

Grids around the train with a closed outer windshield: a longitudinal section grid around the train, b cross-sectional grid around the train, and c grid on the train surface

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Based on previous studies and a comparison with a wind tunnel test, the numerical calculation parameters were set as follows: a second-order accurate implicit scheme with a dual time step format based on FLUENT’s pressure solver was applied to the calculation, with a physical time-step of 1.0 × 10−5 s; a bounded central differencing was used to discretise the convection terms, and a semi-implicit pressure-linked equations-consistent algorithm was used to assess pressure–velocity coupling. The calculation revealed that the CFL number was less than 1.0 in more than 99% of the cells, and the residuals of each equation were less than 10−4 for each time step.

2.4 Description of Post-Processing Data

For the convenience of comparison and analysis in this study, the aerodynamic drag (Fd), aerodynamic lift force (Fl), pressure (p), and slipstream velocity (u) were nondimensionalised as follows:

$${\text{C}}_{\text{d(l)}} = \frac{{{\text{F}}_{\text{d(l)}} }}{{0.5\rho {\text{U}}_{\text{m}}^{2} {\text{S}}}}$$
(1)
$$\text{C}_{\text{p}} \text{ = }\frac{{\text{p}{ - }\text{p}_{\text{0}} }}{{\text{0}{.5\rho U}_{{\text{tr}}}^{\text{2}} }}$$
(2)
$$\text{C}_{\text{u}} \text{ = }\frac{{\sqrt {\left( {\text{U}_{\text{m}} -\text{u}} \right)^{\text{2}} \text{ + v}^{\text{2}} \text{ + w}^{\text{2}} } }}{{\text{U}_{\text{m}} }}$$
(3)

where ρ, the air density, is 1.225 kg/m3; Um, the velocity of the incoming flow, is 60 m/s; S, the cross-sectional area of the uniform train body, is 0.175 m2 for a 1:8 scaled model. p0 is the reference pressure; in this study, it was 0 Pa. Cd, Cl, Cp, and Cu are the aerodynamic drag coefficient, aerodynamic lift force coefficient, pressure coefficient, and slipstream velocity ratio, respectively.

As demonstrated by the time-history curve of the train’s aerodynamic forces and the flow field variables monitored by measured points around the train, the time-history curves fluctuated near a constant value. Therefore, the results used in the comparison and analysis in this study were based on time-averaged data from nondimensional time tUm/S1/2 of 72 to 287, which was approximately 14 times the time required for the flow to pass the length of the train. The sampling time in this study approached those in references (Wang et al. 2017, 2018, 2019c) and was much longer than those in references (Wang et al. 2018, 2019a, 2019c; Dong et al. 2019; Noguchi et al. 2019).

As shown in Fig. 4, 103 sampling lines were arranged on one side of the train to obtain the flow field variables (including the pressure and velocity distribution) around the train to analyse the effect of the plate on the flow field around the train.

Fig. 4
figure 4

Position of sampling lines around full-scale train model

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3 Validation of Simulation Approach

To ensure the accuracy of the calculation method used in this study, the numerical results were compared with the wind tunnel test data. In the numerical simulation, the models of the train and ballast and the size of the boundary conditions of the computational domain were the same as those for the wind tunnel test. Detailed descriptions of the wind tunnel tests have been presented in previous studies (Niu et al. 2016, 2017). The train models used in the experiment and simulation for verification are presented in Fig. 5, and the difference between the train model in the simulation for verification and the train model in the calculation in this study was the vehicle connection. To determine the reliability of the grid density used in this study, three types of grid densities (coarse, medium, and fine) were used to divide the computational domain, and the grid generation was that described in Sect. 2.3. Three grid densities were selected in the generation resolution to determine the appropriate grid resolution. The grid parameters for the three grid resolutions are presented in detail in Table 2. The initial grid on the model surface is shown in Fig. 6.

Fig. 5
figure 5

Train models used in simulation b for verification and experiment a

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Table 2 Grid parameters for 1:8 scaled model with three grid resolutions
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Fig. 6
figure 6

Grid on train surface with three types of grid densities

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As shown in Table 3, the differences in the mean Cd between the Medium and Fine grids for the head and tail cars in the numerical simulation were relatively small (not exceeding 2.4% and 4.3%, respectively) relative to those between the Coarse and Medium grids. Furthermore, Table 3 shows that the difference in Cd between the numerical simulation (Medium) and experiment was less than 7%, but the difference in Cl between the numerical simulation (Medium) and experiment was reasonably large, which was primarily due to the higher sensitivity of the flow field at the bottom of the train compared with that of the installation of the train model. The model installation location and accuracy between the numerical calculation and test differed.

Table 3 Mean values of Cd and Cl of each car in a train between the wind tunnel test and numerical simulation
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Figure 7 shows that the effect of grid density on the train slipstream was greater at locations closer to the train. Furthermore, Fig. 7 shows that the slipstream velocity distribution law along the train between the numerical simulation and experiment was similar, and that most of the differences in measuring points in sampling lines of Lines-1, -2, and -3 did not exceed 4%.

Fig. 7
figure 7

Comparison of train slipstream (Cu) between experiment and numerical simulation: a, b, and c show the mean Cu along Lines-1, -2, and -3, respectively

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Figure 8 shows that the distribution law of the surface pressure coefficient along the train in the numerical simulation agreed with the experimental data, although relatively large differences occurred in the individual measurement points between the numerical simulation and experiment. Furthermore, Fig. 8 shows that the mesh density primarily affected the distribution and value of pressure around the connection of two adjacent vehicles.

Fig. 8
figure 8

Comparison of mean Cp along the upper surface of the train between the previous experiment and simulations with three types of mesh densities

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Distribution of mean values of Cp and Cu along Line-1 under the train was presented in Fig. 9, which shows that both Cp and Cu under the train with the Medium mesh were closer to those with a Fine mesh relative to that with a Coarse mesh, and it also shows that the fluctuation of Cp and Cu was primarily concentrated in the region around the bogie.

Fig. 9
figure 9

Distribution of mean values of Cp a and Cu b along Line-1 under the train

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To determine the aerodynamic force of the train and the flow field around the train, such as the slipstream and pressure, the medium mesh density is considered appropriate for simulating the aerodynamic forces in this study.

Another comparison between the numerical simulation and wind tunnel test has been conducted in previous studies (Niu et al. 2019; Takami and Maekawa 2017); the grid generation and numerical method used in the previous study were the same as those in this study. The results showed that the maximum difference in the aerodynamic drag of the vehicles was approximately 5%. Therefore, the numerical method used in this study predicted the aerodynamic characteristics of trains reliably.

4 Results and Discussion

4.1 Analysis of Time-Averaged and Unsteady Aerodynamic Forces

As shown in Table 4, opening the aerodynamic braking plates exerted a significant effect on the high-speed train’s aerodynamic forces. Overall, the aerodynamic forces of the vehicles increased significantly. However, the braking plate on each vehicle significantly affected the aerodynamic drag, in particular each vehicle’s aerodynamic drag. The data comparison in Table 4 shows that the opened plate would significantly affect the corresponding vehicle side, and that opening both sides would significantly affect the upstream vehicle. Opening the large downstream plates exerted the greatest effect on the aerodynamic drag of the entire train, followed by opening the large upstream plates. The difference between the first two did not exceed 3%, and opening the small upstream and downstream plates exerted the least effect. Additionally, Table 4 shows that the difference in the vehicle aerodynamic drag among the three cases was significant, demonstrating that the vehicle aerodynamic drag was sensitive to the opening direction of the braking plates installed in the ICG. As shown in Table 4, each vehicle’s aerodynamic lift force increased when the braking plates were opened. Opening the upstream braking plates exerted the greatest effect on the head vehicle’s aerodynamic lift force, whereas opening the downstream braking plates exerted the greatest effect on the tail vehicle’s aerodynamic lift force. Furthermore, Table 4 shows that opening the braking plates exerted a much greater effect on the tail vehicle’s aerodynamic lift force than on the head vehicle’s aerodynamic lift force. This was primarily because the airflow separated from the plate reduced the pressure of the flow field on the top of the tail vehicle, and that the lift force of the vehicle was primarily caused by the pressure difference between the upper and lower parts of the vehicle; therefore, the lift of the tail vehicle increased.

Table 4 Mean values of high-speed train’s aerodynamic force coefficients averaged over 215 convective time units when the braking plate was opened (C-1, C-2, and C-3) or closed (C-0)
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As shown in Table 5, the opening of the braking plate increased the fluctuation in the vehicle aerodynamic forces, and the standard deviation (SD) values were relative to the mean value in the entire sampling interval (tUm/S1/2 of 72 to 287). Opening the upstream braking plate exerted the greatest effect, with a maximum change of up to 1.1 times. Opening the downstream braking plate is second, with a maximum change of approximately 1.0 times. Opening the two small upstream and downstream braking plates exerted the least effect, with a maximum change of less than 30%. Furthermore, Table 5 shows that opening the braking plate exerted a much greater effect on the aerodynamic drag of the head vehicle than on the tail vehicle. However, opening the braking plate exerted a greater effect on the tail vehicle’s aerodynamic lift force than on the head vehicle, except in the case of two small braking plates. Opening the two small upstream and downstream braking plates exerted little effect on the aerodynamic lift force of the vehicles, with a maximum change of approximately 15%. The fluctuation of the aerodynamic force of the train and plate is shown in Fig. 10, and the phenomenon was consistent with the data in Table 5.

Table 5 Standard deviation (SD) values of the high-speed train’s aerodynamic force coefficients over 215 convective time units when the braking plate was opened (C-1, C-2, and C-3) or closed (C-0)
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Fig. 10
figure 10

Nondimensional time histories of Cd and Cl of the head car, tail car, and plate: a, b, and c are the Cd of the head car, tail car, and plate, respectively; d and e are the Cl of the head car and tail car, respectively

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In summary, from the perspective of increasing the train braking force, opening the upstream braking plate was the best option; however, the vehicle vibration will increase significantly. Opening the downstream braking plate improved the braking force of the vehicle by an additional 10% compared with opening the upstream braking plate. The vehicle vibration caused by opening the downstream braking plate was much smaller than that caused by the opening of the upstream braking plate. Therefore, opening the downstream plate was a superior option based on the project’s parameters.

4.2 Flow Structure Around ICG

As shown in Fig. 11, three large vortices (vortex-1, vortex-2, and vortex-3) appeared around the ICG. Vortex-1 was formed when the incoming flow was blocked by the plates; vortex-2 was formed by the backflow of vortex-3 blocked by the plates; and vortex-3 behind the plates formed by the airflow separation at the top of the plates was the largest among the three vortices. In the cases of opening the large upstream and downstream plates, the distances from the core of the largest vortex-3 to the corresponding plate was the same but larger (∆x) than that in the case of opening the two small plates. A comparison of the flow streamlines of C-1, C-2, and C-3 demonstrated that no large vortices formed in the ICG owing to the small downstream plate for C-3. Furthermore, Fig. 11 shows that the velocity of the flow field behind the large upstream plates was much lower than that behind the large downstream plates based on comparing the velocity contours of C-1 and C-2. This was related to the ICG. The range of the low-velocity area caused by opening the two small plates was much smaller than that caused by the large upstream and downstream plates.

Fig. 11
figure 11

Time-averaged flow streamlines around the vehicle-connecting part with three types of aerodynamic braking plates on the symmetry plane rendered for the mean velocity ratio U/Um

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As shown in Fig. 12a, the pressure distribution around the vehicle-connecting parts changed significantly when the plate was opened. The airflow in front of the plate was blocked to form a positive pressure area, whereas that behind the plate decreased because the airflow at the top of the plate separated to form a negative pressure area. Additionally, Fig. 12a shows the clear differences in the effects of varying configurations of the aerodynamic braking plates. The positive pressure range in front of the downstream plate was the largest, primarily due to the gap at the vehicle-connecting part. In other words, the gap increased the plate’s windward area, and the negative pressure ranges behind the large upstream and downstream plates were the same. The blue lines around the vehicle-connecting parts in Fig. 12b indicate that the Δ × 1 of C-1 was close to the Δx2 of C-2, and that both were greater than the Δx3 of C-3. The negative pressure range in the y-direction for C-1 and C-2 was greater (Δy) than that for C-3, demonstrating that the pressure range affected by opening the two small plates was smaller than those of the other two plate configurations, which were close to each other.

Fig. 12
figure 12

Comparison of pressure contour a and isopiestic line b around the high-speed train’s vehicle-connecting part with different aerodynamic braking plate configurations

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Figure 13a shows that the boundary layer profiles over the vehicle-connecting part and tail vehicle changed significantly when the plates were open. The effects of various plate configurations on the boundary layer profile over the vehicles differed significantly. Opening the large upstream and downstream plates exerted the greatest effect, whereas opening the two small plates exerted the least effect. Figure 13b shows the effects of various plate configurations on the boundary layer over the vehicle. For the lower flow field around the train, the plate configurations exerted a slight effect on the profiles of the boundary layer over the train but exerted a significant effect on the scope of the boundary layer. For the upper flow field around the train, both the profile and scope were affected by the plate configuration.

Fig. 13
figure 13

Two-dimensional comparison of boundary layers over the train: a symmetry plane and b cross section

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Figure 14a shows that opening the plate resulted in many detached vortices at the vehicle-connecting part that flowed downstream, thereby affecting the flow field structure around the downstream vehicle. Furthermore, Fig. 14a shows that the range of vortices in the train’s wake with opened plates was longer than that with closed plates, implying that the wake vortices were strengthened by the upstream turbulence airflow enhanced by opening the plate. As shown in the vortices in the wake of Fig. 14, the vortices in the wake of the train with closed plates were larger than those with opened plates because the separation vortex and strong turbulence caused by the opening of the plate separated the large vortices in the wake and accelerated the decay of large vortices in the wake. Opening the plate exerted a more significant effect on the generation, development, and distribution of vortices around the train; however, no clear difference was observed among C-1, C-2, and C-3.

Fig. 14
figure 14

Comparison of instantaneous isosurfaces corresponding to the Q criterion (Q = 100,000) rendered for the mean velocity ratio U/Um: a top view and b three-dimensional view

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As shown in Fig. 15, abrupt changes occurred in the pressure and velocity distributions around the vehicle-connecting parts. The basic characteristics of the pressure and slipstream velocity distributions along the train remained unchanged with respect to the distance from the vehicle. The closer the value was to the vehicle, the larger was the variation amplitude of the pressure and slipstream velocity. Figure 15a shows a significant difference in the slipstream disturbance caused by the opening of the large upstream and downstream plates. As shown in Fig. 15c, the difference in the slipstream velocity around the vehicle-connecting parts between C-1 and C-2 was small far away from the vehicle. The difference in the effects of the large upstream and downstream plates on the surrounding slipstream was primarily reflected in the area near the vehicle-connecting parts, and the changes in the slipstream at other positions were similar. Figures 15b, c show that the maximum pressure value around the vehicle-connecting parts caused by the large upstream and downstream plates were similar and much greater than those caused by the two small plates. The minimum pressure value around the vehicle-connecting parts caused by the large upstream plate was slightly greater than that of the large downstream plate, and both were much greater than that caused by the two small plates. This analysis demonstrated that the pressure field sensitivity to the plate configurations was lower than the velocity.

Fig. 15
figure 15

Comparison of distribution of slipstream velocity ratio and pressure coefficient along sampling lines around the high-speed train for four cases: a and b are along the line at the position of the square mark (Fig. 4); c and d are along the line at the position of the triangle mark (Fig. 4)

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To present the effects of different open-plate configurations on the maximum value distribution of the time-averaged slipstream velocity (Cu-max), the maximum and minimum values of the time-averaged pressure (Cp-max and Cp-min) around the high-speed train were assessed in the x-direction. The ratio of the maximum values of the time-averaged slipstream velocity along the sampling lines around the train in C-i to that in C-0 was denoted Rui-max, whereas the ratios of the maximum and minimum values of the time-averaged pressure along the sampling lines around the train in C-i to that in C-0 were denoted Rpi-max and Rpi-min, respectively. Rui-max, Rpi-max, and Rpi-min were calculated using Eqs. (4–6), respectively.

$$R_{{u}_{i - \hbox{max} }} = \frac{{\frac{{C_{{u}_{i - \hbox{max} }} }}{{C_{u0 - \hbox{max} } }}}}{{\hbox{max} \left( {\frac{{C_{u1 - \hbox{max} } }}{{C_{u0 - \hbox{max} } }},\frac{{C_{u2 - \hbox{max} } }}{{C_{u0 - \hbox{max} } }},\frac{{C_{u3 - \hbox{max} } }}{{C_{u0 - \hbox{max} } }}} \right)}}$$
(4)
$$R_{{p}{i - \hbox{max} }} = \frac{{\left| {\frac{{C_{{p}{i - \hbox{max} }} }}{{C_{p0 - \hbox{max} } }}} \right|}}{{\hbox{max} \left| {\frac{{C_{p1 - \hbox{max} } }}{{C_{p0 - \hbox{max} } }},\frac{{C_{p2 - \hbox{max} } }}{{C_{p0 - \hbox{max} } }},\frac{{C_{p3 - \hbox{max} } }}{{C_{p0 - \hbox{max} } }}} \right|}}$$
(5)
$$R_{{p}{i - \hbox{min} }} = \frac{{\left| {\frac{{C_{{p} {i - \min}} }}{{C_{p0 - \hbox{min} } }}} \right|}}{{\hbox{max} \left| {\frac{{C_{p1 - \hbox{min} } }}{{C_{p0 - \hbox{min} } }},\frac{{C_{p2 - \hbox{min} } }}{{C_{p0 - \hbox{min} } }},\frac{{C_{p3 - \hbox{min} } }}{{C_{p0 - \hbox{min} } }}} \right|}}$$
(6)

As shown in Fig. 16, the maximum slipstream velocity values around the train increased significantly when the plates were open, and the affected area was primarily concentrated in the area where the plates were located. The total Ru-max of all of the sampling lines for C-1, C-2, and C-3 were 2.73, 2.98, and 1.83, respectively, demonstrating that opening the large downstream plate exerted the greatest effect on the surrounding slipstream. This finding supports the aerodynamic forces analysis in Sect. 4.1. Additionally, Fig. 16 shows that the ranges of the affected areas in C-1 and C-2 were larger than that in C-3, indicating that the larger plate exerted a greater effect on the surrounding flow field. The isolines of the maximum slipstream velocity values between C-1 and C-2 differed, demonstrating that the effects of opening the large upstream and downstream plates on the surrounding slipstream differed.

Fig. 16
figure 16

Comparison of distribution of maximum values of time-averaged slipstream velocity along sampling lines around the high-speed train in three cases: a, b, and c show cases of Ru1-max, Ru2-max, and Ru3-max, respectively

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The peak pressure values around the train were caused by opening the plates and appeared primarily at the vehicle-connecting parts, as demonstrated in Fig. 16. Figure 17a shows that the maximum values of pressure distribution around the three plate configurations differed. The range of the highlighted area in Rp2-max was the largest, demonstrating that opening the downstream plate exerted the greatest effect, which was consistent with Fig. 12a. Figure 17b shows that the difference in the minimum pressure distribution value around the train between C-1 and C-2 was small, but the range in the highlighted area was much greater than that in C-3, demonstrating that the effects of opening the large upstream and downstream plates on the minimum pressure value around the train were almost the same and the largest.

Fig. 17
figure 17

Comparison of distribution of maximum a and minimum b values of the time-averaged pressure coefficient along the sampling lines around the high-speed train in three cases along sampling lines at different positions

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As shown in Fig. 18a, a significant pressure increase occurred at the vehicle-connecting part when the plates were open. The laws of pressure variation caused by the three plate configurations were the same, but the variation amplitude differed; that of C-2 was the largest, that of C-1 was slightly smaller, and that of C-3 was the smallest, consistent with the pressure distribution shown in Fig. 12a. Figure 18b shows that opening the plates significantly affected the downstream flow field, resulting in significant pressure fluctuations, and that the effects of the three plate configurations were similar.

Fig. 18
figure 18

Comparison of distribution of time-averaged pressure coefficient and its RMSE (Root Mean Squard Error) along the upper and lower surfaces of the high-speed train in four cases

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As shown in Fig. 19, two symmetrical vortices appeared in the train’s wake without opening the plate that formed under the train tail nose tip. The distance (D1) between the two symmetric vortices increased with the distance from the train. This phenomenon was also observed when the plates were open, but D1 in the same section and different cases changed when the plates were opened with no obvious trend. Furthermore, Fig. 19 shows that two additional symmetrical vortices appeared in the wake when the plates were opened. This pair of vortices formed at the gap between the plates and subgrade, which is clearly shown in Fig. 14. The distance (D2) between these two symmetric vortices increased with the distance from the train. The distances between the two symmetric vortices (D2) caused by opening the large upstream plates (C-1) and large downstream plates (C-2) were almost the same, which were slightly wider than that caused by opening the two small plates (C-3).

Fig. 19
figure 19

Comparison of longitudinal vortex structures in three cases at the vertical planes in the wake region at distances of x = 1.0 H, 2.0 H, and 3.0 H to the tail nose

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5 Conclusions

A comparative analysis of the aerodynamic characteristics of a train with different aerodynamic braking plate configurations yielded the following conclusions:

  1. 1.

    The train’s aerodynamic drag significantly increased the drag by opening the plates installed in the ICG. In particular, the aerodynamic drag of the entire train increased by approximately 1.6 times by large downstream plates.

  2. 2.

    The flow field around the ICG was affected by the opening of the plates. The vortex distribution around the ICG differed among the three plate configurations. Compared with the velocity field, the pressure field distribution was more sensitive to the plate.

  3. 3.

    A pair of vortices at the lower part of both sides of the train formed when the plate was opened, merged into the wake, and exerted little effect on the pair of vortices in the wake formed under the train tail nose. The location and intensity of the vortices in the wake were affected, but the differences were insignificant to the first order.

  4. 4.

    Opening the plates exerted a significant effect on the flow field around the vehicle-connecting part. The airflow turbulence behind the plates intensified when the plates were opened, which reduced the slipstream velocity, in particular those of the large plates.