Numerical study on aerodynamic resistance reduction of high-speed train using vortex generator

Vortex generator (VG) is one of the potential technical tools to reduce the aerodynamic resistance of high-speed trains. The passive control resistance reduction research of a high-speed train is carried out by using VG. Three typical installation locations, including flow separation point, boundary layer mutation and streamline transition location, and several nearby locations were selected to study the effect of VG location on the train aerodynamic characteristics. The results show that the tail car drag is significantly reduced when the VG is arranged at the boundary layer mutation, the tail car’s resistance can be reduced by 15.42%. The tail car reduces resistance by 3.87% when VG is set at the flow separation point. By analyzing the flow field structure, we found that the VG arranged in front of the separation point triggers the flow separation, which destroys the balance between the separation and longitudinal vortex. It effectively reduces the strength of the separation vortex, thereby reducing the tail car’s aerodynamic drag. The research provides a new thought for the aerodynamic resistance reduction of high-speed trains, and is of great significance to break the limitations of traditional aerodynamic drag reduction technology of trains.


Introduction
High-speed trains are an essential means of transportation for people due to the safety, comfort, energy saving and environmental protection (Sun et al., 2020;Tian, 2019).In recent years, Japan, Germany, France and other countries have begun to develop the next generation of high-speed trains, including the Japanese ALFA-X, German Velaro Novo and French TGV M, etc.At present, China has also started the development of CR450 highspeed EMUs with a speed of 400 km/h.
As the train's running speed increases, it will inevi tably lead to a dramatic increase in the train's aerodynamic resistance.The aerodynamic resistance takes up 75% to 85% of the total resistance, when the train is running at a speed of 300 km/h (Li, et al., 2021;Zhou et al., 2021).As train's speed increases to 400 km/h, the aerodynamic resistance will account for more than 90% (Yang et al., 2012;Yu et al., 2021).The energy consumption of train operation is positively related to the cube of speed (Wang, Wang, et al., 2022;Wang, Zhu, et al 2022).Therefore, the resistance reduction is a key research for the next generation of high-speed trains.The traditional train resistance reduction technology mainly includes the head optimization and structural CONTACT Tian Li litian2008@home.swjtu.edu.cnsmoothing design of bogies, pantographs, windshields and etc.However, the above optimization provided by the train is limited.Flow control is a leading field of applied fluid technology, and many related results have been achieved (Verma & Manisankar, 2022;Chung et al., 2021).Because of its simple shape and excellent control effect, VG has become a common passive control technology of turbulent flow.Currently, VGs have been widely used in aerospace, fluid machinery, metallurgy, automobiles, ships and other areas (Gönül et al., 2022;Zhao et al., 2022), which can effectively increase the lift force and reduce resistance, and inhibit flow separation.Aftab and Murthy (2012) used three arrangements for mounting VGs on the ONERA M6.The numerical calculation method is used to investigate the flow over the wing.The results show that the VGs can increase the lift force and reduce resistance on the wing, and the most effective arrangement is the co-rotating clockwise.Gibertini et al. (2015) examined the effect of VGs on helicopter aerodynamic resistance by computational fluid dynamics and wind tunnel tests, and they found VGs can reduce helicopter aerodynamic resistance, and the maximum resistance reduction can reach about 5%.Shiva et al. (2020) modified the existing aircraft wing geometry and added VGs.The research shows that VGs can increase the lift and reduce the resistance of the wing.Said et al. (2021) investigated the effect of the micro VGs on the resistance at different angles of attack of the wing and explored the optimal position.The test results showed that the optimal installation position of the micro VG was 30% of the chord length.Katz and Morey (2008) conducted wind tunnel experiments by installing several VGs on the undersurface of a racing car.Aider et al. (2010) applied the VG to a modified Ahmed model and showed that the resistance could be reduced by 14% and lift could be reduced by 60%.Ali et al. (2013) used the VG to analyze the resistance reduction of the automobile model.They found that the triangular VG with a height equal to the boundary layer thickness has the most apparent resistance reduction effect.Selvaraju and Parammasivam (2019) installed VGs at different linear positions on the rear edge of the roof of a typical SUV and studied their aerodynamic characteristics.The research results show that a reasonable layout of VGs can reduce resistance by 9.04%.Brownlie et al. (2016) researched the resistance reduction efficiency of VGs on cylinders, human limb models and full-scale human bodies through wind tunnel experiments and conducted a series of experimental tests.
The results show that the layout of VGs could reduce the aerodynamic resistance of running clothing by 6.8%.Current research work on aerodynamic drag reduction using VGs is focused on wings, aircraft, car and clothing, and many achievements have been obtained.It is certain that the aerodynamic forces, surface pressure distribution and flow field structure of the train will change significantly when a suitably shaped vortex generator is arranged in a reasonable location.However, unlike airplanes and automobiles, high-speed trains have particular wheel-rail contact patterns and marshalling modes, and the train shape is thin and long (Liang et al., 2022;Li et al., 2022).As a result, the findings of many studies on vortex generators are not fully applicable to high-speed trains, and there are few researches on the train resistance reduction technology based on VGs.The flow separation phenomenon occurs in the tail car, and there are boundary layer abrupt change and move away from the tail car body, forming a pair of counter-rotating tail vortices at the tail car nose, which has a significant impact on the tail car aerodynamic drag.In this study, numerical calculation is adopted to study the influence of VGs on the aerodynamic forces and flow structure.It brings a new thought to reduce the aerodynamic resistance of high-speed trains.
This paper is organized as follows.In Section 2, the numerical model, research methodology and layout position of vortex generators are introduced.The influences of the vortex generator on the aerodynamic forces, surface pressure, and flow field characteristics around the train are discussed in Section 3. Section 4 introduces the conclusions and future researches.

Train model
Figure 1 displays a 1/10th scale ICE2 model used in the numerical study.The train is made up of a head car and a tail car with only a streamlined nose.The length, width and height of the train are 3.552, 0.302, and 0.358 m, respectively.There is a 5 mm gap in the windshield region.The numerical simulation model is consistent with the model used by Li et al. (2019).

Vortex generator model
The microramp VG is a research hotspot in the field of aviation, and it has a wide range of application prospects (Babinsky et al., 2009).The layout of the VGs is divided into two types, co-rotation and counter-rotation.According to the research by Aziz et al. (2021), only corotating VGs can reduce train resistance.Therefore, the co-rotating VG layout is adopted in this study.
Figure 2 shows the shape and layout of a microramp VG.The height H of microramp VG is 15 mm.The projected surface is an equilateral triangle, the base-side length is 1.57H, and the leg length is 1.75H.The classic parameters of the VG mainly refer to the work by Verma and Manisankar (2017).Three microramp VGs were mounted on the top of the tail car, the spacing between vortex generators is 16.5 mm.In order to achieve a better resistance reduction, different parameters of the VG are also studied in this paper.

Computational information
Figure 3 displays the computational domain.The computational domain sizes of different VG schemes are kept the same.The length, width and height of the computational domain are 30, 6, and 4 m, respectively.The train is 10 m far from the inlet, and 16 m from the outlet boundary, which satisfies the requirement by EN 14067.There is a gap between the bottom of train body and ground surface, the distance is about 0.05 m.In the numerical simulation, a static model is used and the moving effect of a high-speed train is performed by a relative motion method.Ground is set as a slip wall to simulate the relative motion with respect to the train and the slip velocity is the inlet velocity.The train and bogies are set as fixed   walls.Inlet is set to velocity inlet, based on the actual running speed of ICE2 train, the velocity is 55.56 m/s.Correspondingly, the Reynolds number is 1.4 × 10 6 , which is greater than the critical Reynolds number 2.5 × 10 5 recommend by EN 14067-6.Therefore, the Reynolds number has few effects on the aerodynamic behaviors of the train (Tschepe et al., 2021).Outlet is set to pressure outlet with a surface pressure of 0 Pa.Sym is set to symmetry wall.The air density is 1.225 kg/m 3 .
In no wind conditions, the rail and track have a few effects on the aerodynamic forces, especially for the aerodynamic drag (Jiang et al., 2021).Due to the most concerned aerodynamic force is the aerodynamic drag in this study, the rail and track models are neglected in this study.Therefore, the wheel/rail contact is also neglected.
The SIMPLE algorithm was chosen to solve the pressure-velocity coupling equation, and all variables are discretized in a second-order format.

Computational method
According to previous studies, the Reynolds-averaged (Navier-Stokes) has been widely used to study the aerodynamic performance of high-speed train.Due to the fact that the most concerned indexes are averaged aerodynamic forces and a higher-efficiency calculation for RANS with comparison to unsteady RANS, RANS model is chosen to study the influence of vortex generator on aerodynamic drag of tail car.Meanwhile, SST k-ω turbulence model is more conducive to solving for the boundary layer flow near the train surface.Therefore, the 3-dimensional, steady and incompressible Navier-stokes equation and SST k-ω two-equation turbulence model are adopted.

Grid generation and grid independence
3 refinement regions are established around the train to simulate the flow around the train.Figure 3 shows three mesh refinement boxes around the train.Fine meshes are divided in the near-wall area to capture the turbulent flow details due to the large velocity gradient.The flow field size gradually increases as the grid moves away from the train to ensure the fineness of the grid.The thickness for the first layer has been set to 0.01 mm, the grid growth ratio is set to 1.2.Each set of grids has 12 boundary layers to ensure that y + is around 1. Figure 4 shows the grid details.
The reliability of numerical results are related with the quantity and quality of grid (Roache, 1997).Therefore, three different grid densities are divided for the ICE2 train, and the size of the refinement region in the three sets of grids is kept the same to test the grid sensitivity of the calculation.Table 1 shows the cells number of grids and the difference in the aerodynamic drag force coefficients and lift force coefficients of the head car and tail car between different grid resolutions.The head car aerodynamic drag force coefficient error is 4.95% between the coarse and fine grids, and the lift force coefficient error is 12.5%.The results of the medium and fine grids match well and the errors are within 1%.It can be observed from Figure 5 that the cross-sectional pressure distribution of the train using the medium and fine meshes are in good agreement.It can be seen from the enlarged image that there is a specific difference between the coarse and fine grids, which means that the coarse grid is insufficient to capture the train aerodynamic performance accurately.Based on the above description, on the premise of ensuring the calculation accuracy and considering the calculation efficiency, the medium grid with a

Numerical validation
The numerical results are compared with Li et al.'s result (Li et al., 2019) and wind tunnel test data (Orellano & Schober, 2006) to validate the numerical calculation method.The experiment was conducted in the open wind tunnel T103 of the Central Aerodynamic Institute TsAGI, using a simple 1:10 scaled ICE2 model, which is the Bombardier standard train geometry (Orellano et al., 2006).The trains studied in this paper are the same as those used in the experiments.The aerodynamic forces and moments of the head car are measured by a sixcomponent strain-gauge balance equipped inside the head car.The gap between the head car and tail car is 5 mm.Therefore, the aerodynamic load acting on the tail car will not be transmitted to the strain-gauge balance.The floor consists of an oval floor with a turntable on which the model is fixed to facilitate changing the yaw angle of the model.Several experiments were carried out in wind tunnels with wind speeds of 30 m/s to 70 m/s, Reynolds numbers of 0.6 ∼ 1.4 × 10 6 and yaw angles of −30°to 60°.In this paper, wind tunnel test results with a train running speed of 60.62 m/s and crosswind speed of 35 m/s are chosen for verification.The numerical calculation and test model' size and layout are consistent.Figure 6 compares the numerical calculation results with the previous ones.It can be seen that the numerical result matches well with the results by Li et al. and experimental data, and there are slight deviations at individual points.Therefore, the numerical method used in this study is reliable.

Position and size of vortex generators
VG is one of the most common passive flow control methods.It generates tip vortices by arranging multiple small devices of the same type (e.g.triangle, trapezoid, microramp, microvane, etc.) on an array of object surfaces, introducing the high-velocity, high-energy mainstream into the low-speed, low-energy near wall boundary layer, and exchange mass, momentum and energy between them.Then the boundary layer in the inverse pressure gradient can continue to flow along the surface after obtaining energy to delay the flow separation and reduce the aerodynamic resistance (Lin, 2002;Chen et al., 2022).Based on the above principle, VGs are only installed on the roof of the tail car.Considering the two oppsite running directions for trains, we suppose that the VGs on the roof of the hear car are drawn back to the train body and VGs on the tail car are stretched out.According to the analysis of the flow around ICE2, the VG arrangement position is preliminarily determined.
(1) According to the design of VG, arranging the VG at the separation of the train flow may have a better aerodynamic resistance reduction effect.Figure 7 shows the streamline and velocity distribution around the head and tail cars.We can see that when the air flows over the streamline nose of the head car, some flow moves upward smoothly along the windows of the train, and vortex is generated around the cowcatcher due to the flow rushes over the cowcatcher.The flow separation occurs when the airflow passes over the cab glass of the tail car, forming a larger flow separation area.To further find the specific flow separation point and facilitate the subsequent determination of the particular position of the VG layout, the velocity vector diagram was extracted from the tail car cross-section, as shown in Figure 8.
In the enlarged picture, we can see the specific position where the boundary layer flow separation.The VG is arranged at this position to control the flow separation of the tail car to achieve the purpose of reducing the aerodynamic resistance of the tail car.(2) Figure 9 illustrates velocity distribution around the train.U is the average speed, and U 0 is the inlet speed.The velocity gradient changes significantly as the airflow passes over the tail car streamline nose.
According to the general definition of the boundary layer: (0 ∼ 0.99)U 0 , the boundary layer thickness δ is the distance from the point on the train surface to 0.99U 0 point along the vertical direction.The tail car boundary layer gradually thins from the windshield.When the airflow flows to the transition position between the non-streamlined and streamlines of the tail car, the airflow accelerates at this position, where the tail car boundary layer is the thinnest.As the airflow continues, the boundary layer abruptly moves away from the train surface.In the figure, the VG arranged at the abrupt change position of the boundary layer may significantly affect the tail car's aerodynamic performance.
(3) According to Aziz et al. (2021), when the VG is arranged in the transition position between the streamlined and non-streamlined tail car, the differential pressure drag of the train can be reduced up to 16.11%.Therefore, this position is also one of the VG layout positions of interest in this paper.According to Aider et al.'s research (2010), it was found that the optimal position of the VG is located upstream of the flow separation line, and the VG arranged downstream of the flow separation can still achieve an aerodynamic resistance reduction.Therefore, based on the three layout positions, four additional layout positions are added to explore the law of the layout position of the VG and the aerodynamic resistance reduction rate and to find the best layout position.Figure 10 is a schematic diagram of the VG layout.According to the order from front to back, the eight places are named P 1 ∼ P 8 in turn.A group of VGs is arranged at every 50 mm interval behind P 6 , and two groups are set in total, namely P 7 and P 8 .A group of VGs, namely P 5 , are arranged at an interval of 50 mm in front of P 6 .A group of VGs is arranged 15 mm in front of P 4 , namely P 3 .A group of VGs is arranged 50 mm in front of P 4 , namely P 1 .

Numerical results
In this section, we firstly, analyzed how the VG position affects the aerodynamic force.Then, the surface pressure distribution, the flow field around the train, and the vorticity diagram are analyzed.Finally, the VG's influence mechanism on the tail car's aerodynamic drag is studied.

Aerodynamic force
The changes of the tail car's aerodynamic drag (F d ) and lift (F l ) with the position of the VG are shown in Figure 11.The tail car aerodynamic drag without VGs is 49.36N, and the lift force is 51.11N.
The first observable result is that the VG significantly affects the tail car's F d .The F d has a minimum value at the P 4 .The second result that can be observed is the effect of the position of the VG on the F d shows a regular variation.The arrangement of VGs at and near the abrupt change of boundary layer (P 4 ) has an aerodynamic drag reduction effect on the tail car, and P 4 has the best aerodynamic drag reduction effect.When the VG is gradually away from P 4 , the aerodynamic drag reduction effect of the tail car is also gradually worse.It can be found that the optimal position of the VG is located above the flow separation point, and this is also the same as the Aider et al. 's research (2010).It should be noted that the F l changes significantly more than the F d .At P 4 , the reduction of F l is the most significant.The F l decreases by 47.0%.At the same time, the changing trend of the F l is consistent with the change of the F d .This discovery is surprising.Reducing the F l would make the tail car run more stably, thus improving the ride comfort of the train (Zhang et al., 2023).
Table 2 show the detailed aerodynamic data of trains, including the head car's F d and tail car's F d , the tail car F d reduction rate, the tail car pressure drag and viscous drag.As you can see from the table that the head car's F d is the same when the VGs are arranged at different positions, indicating that the VG on the tail car does not affect the head car's F d , F l and the surrounding flow field.When VG is placed at the sudden change of the boundary layer (P 4 ), the tail car F d reduction rate is 15.42%.When it is put at the flow separation point (P 6 ), the tail car F d reduction rate is 3.87%.When placed at the streamlined transition position (P 2 ), the VG cannot reduce the tail car's F d .
To understand how the VG affects the tail car's F d , the tail car pressure and viscous drag are extracted.The VG's influence on the tail car's pressure and viscous drag is analyzed in detail.It can be seen from the table that with the change of the arrangement position of the VG, the tail car's pressure drag shows significantly changes, and the change law is consistent with the change law of the tail car's F d .When the VGs are arranged in the four positions of P 1 , P 2 , P 7 , and P 8 , the tail car viscous drag hardly change, and the tail car's F d at these four positions does not change significantly.However, when the VGs are arranged in the four positions of P 3 , P 4 , P 5 , and P 6 , the tail car has a slight variation in viscous drag, compared with the pressure drag, the change of the viscous drag is negligible.Therefore, the arrangement of the VG in the tail car mainly changes the pressure drag of the tail car to achieve aerodynamic drag reduction.In contrast, the change in the viscous drag is negligible.

Pressure distribution
The variation of tail car wall shear stress in the prototype and arrangement of the VG is shown in Figure 12.
It is well known that the viscous drag is composed of the shear force between the train surface and the air.From the figure, we can found that the VG would not affect the viscous drag distribution at the front of the tail car.Comparing P 1 , P 2 , P 7 , P 8 with the prototype, it can be seen that P 1 and P 2 have a inevitable increase in the small viscous drag at the cab glass of the tail car in the prototype due to the elimination of the separation bubble by the VG.The viscous drag distribution of the P 7 and P 8 working conditions is consistent with the prototype.The distribution of the viscous drag of the tail car has a specific change in the working conditions of P 3 ∼ P 6 , and the viscous drag distribution of P 3 , P 4 and P 5 has the most significant changes.In general, the tail car's low viscous drag area has increased, while the high viscous drag area close to the tail car's nose has decreased, leading to a reduction of the tail car's viscosity drag in three cases from P 3 to P 5 .The viscous drag distribution of the P 6 working condition is mainly from the concentrated minimum viscous drag area in the prototype.It becomes two stripshaped low viscous drag areas due to the appearance of the VG.However, the viscous drag remains unchanged.
The P 3 ∼ P 5 working conditions also have two stripshaped low-viscosity drag areas behind the VG similar to the P 6 working condition.The viscous drag in the middle of the two strip-shaped regions is relatively high.The pressure distribution between the tail car with VGs and the prototype is shown in Figure 13.In general, the tail car pressure distribution in four working conditions of P 1 , P 2 , P 7 and P 8 is the same as the prototype.The tail car's negative pressure is significantly reduced when the VGs are arranged at P 3 , P 4 , P 5 and P 6 , especially at the top and sides.These four positions also correspond to the four positions where the tail car's F d and F l are most significantly reduced in Figure 11.The negative pressure at the streamlined transition position of the tail car is significantly reduced at these four positions, explaining the reason for the significant reduction in the tail car's F l in Figure 11(a).Comparing the prototype with the four conditions of P 3 , P 4 , P 5 and P 6 , the tail car's negative pressure is significantly reduced, and the pressure distribution at the cab glass has also changed.The pressure in the upper half of the driver's cab glass is slightly lower than the prototype, which is not good for the aerodynamic drag reduction of the tail car, but the pressure change here is only about 200 Pa.The positive pressure of the lower part of the cab glass has increased, which is beneficial to the aerodynamic drag reduction of the tail car, but the increase is only 100 Pa.Therefore, arranging the VGs at P 3 , P 4 , P 5 and P 6 mainly reduce the negative pressure near the VGs and around the tail car, which directly leads to a reduction in tail car aerodynamic drag.A most significant reduction of negative pressure in the tail car of P 4 , which is directly responsible for a reduction in aerodynamic drag on the tail car.

Flow field characteristics
The velocity distribution and streamline at the tail car cross-section with the prototype of the VGs arranged at different positions are shown in Figure 14    causing specific changes to the velocity distribution and streamlining distribution near the tail car.
When the VGs are arranged at P 1 and P 2 , the VGs will make the separation area of the tail car disappear.When the VG is set at the P 7 and P 8 , the flow separation zone exists only in a small area behind the VG.When the VGs are set at the four positions of the boundary layer mutation point (P 4 ), the flow separation points (P 6 ), P 3 and P 5 , the flow separation of the tail car occurs in advance at the rear of the VG.In addition, a sizeable low-velocity area is formed at the VG's rear, and the flow field's velocity around the tail car is reduced to a certain extent in comparison to the prototype.The separation bubble first increases and then gradually decreases and  approaches the tail surface, when the position of the VG from P 3 to P 6 .The separation bubble of P 4 is maximum.At the same time, the F d is the smallest.The flow field distribution of the tail car at these four positions is similar, but the aerodynamic drag of the tail car at these four conditions is significantly different.The current analysis shows that the larger separation bubble has a certain favorable influence on the tail car's F d , but this is inconclusive.Therefore, we need further analysis how the VG affects the tail car aerodynamic drag.From the streamline diagram, we know that when the VGs are arranged P 1 and P 2 , the separation bubbles near the cab glass disappear.When the VGs are arranged in P 3 ∼ P 6 , large separation bubbles will be formed at the rear of the VGs.When the position of the VG moves backward, the separation bubble also decreases.When the VG is arranged at P 7 and P 8 positions, the size of the separation bubble is significantly reduced.
According to the above description, to reduce aerodynamic drag reduction, the optimal position of the VG is P 4 .However, the arrangement of the VG at this position will not weaken the flow separation phenomenon of the tail car but will cause a more intense flow separation phenomenon.In the study of Munoz-Paniagua and García (2020), it was found that the head shape obtained after optimization by a genetic algorithm will create a Ushaped separation zone at the tail car windshield where a small recirculation bubble would be formed.The original head shape does not have a separation area in the driver's cab glass.Therefore, whether the increase of the separation bubble affects tail car's aerodynamic drag and whether it contributes decisively to the tail car aerodynamic performance requires further analysis.
Figure 16 shows the streamlines projected onto the tail car surface, and the purple box is the tail car's separation area in each condition.According to Östh et al. (2015), they were combined with the tail flow diagram of the prototype in the figure.When the air flows pass over the tail car streamlined transition, the airflow accelerates, the development of flow separation occurs (purple box), and the local pressure decreases.The airflow then forms a separation bubble under the action of the main flow, and the airflow is again attached to the upper area of the rear windshield.The airflow then forms a separation bubble under the action of the main flow, and the airflow is again attached to the upper area of the driver's cab glass.An unstable point (blue box) can be clearly observed, resulting from the reconnected airflow and the divergence of the upward airflow along the train surface.It can be seen from the whole that when the VGs are arranged at positions P 1 and P 2 , there is no separation area of the tail car, so there is no saddle point.According to Figures 14 and  15, a giant separation bubble will be formed at the rear of the VG when the VG is arranged at the four positions (P 3 ∼ P 6 ).This also results in a larger separation area corresponding to the purple box in Figure 16.As the VGs are arranged backward from P 3 to P 6 , the separation area gradually decreases, consistent with the variation law of separation bubbles in Figure 15.
When the VGs are arranged at the four positions (P 3 ∼ P 6 ), the area of the original separation area is significantly increased in both the horizontal and vertical directions, and the flow separation also occurs in advance.This behavior was unexpected.According to the previous application principle of the VG to the airfoil, the flow separation of the airfoil is delayed and the separation area is reduced.Combined with Li et al.'s research (2021), it was found that the tail car's aerodynamic force is significantly influenced by the interaction between the transversal and longitudinal vortex.The orange dot in Figure 16 is a stable node SN 1 , and the definition of SN 1 is consistent with that in Li's study.By comparing the eight conditions, it can be found that SN 1 has changed significantly.The SN 1 of P 1 and P 2 was consistent with the prototype.However, the SN 1 at the four positions (P 3 ∼ P 6 ) has changed significantly compared with the prototype, and the position of SN 1 has moved down to a certain extent.The convergence of streamlines around SN 1 is weakened considerably, and the streamlines around nodes are also considerably reduced.According to the distribution change of the tail car's viscous drag in Figure 12 and the Li's research, it can be shown that the interaction between the separation vortex and the longitudinal vortex has changed significantly.However, comparing the three conditions of P 6 and P 3 ∼ P 5 , the reduction of tail car aerodynamic drag is not obvious when the VG is arranged at the position of P 6 .According to the comparison in Figure 16, the change of the SN 1 node of P 6 is the same as that of P 3 ∼ P 5 , but the separation area is small.The distribution of streamlines around the SN 1 point of P 1 , P 2 , P 7 and P 8 are highly consistent with the prototype.Therefore, how the effect between the separation vortex and the longitudinal vortex affects the tail car's aerodynamic drag needs further analysis.
It is worth investigating the influence mechanism of the VG on the tail car aerodynamic drag.Therefore, a couple of reverse rotating longitudinal vortices near the tail car nose and a separating vortex on the back of the tail car were studied.Onorato et al. (1984) found that in the region where the vortex is located, its contribution to the aerodynamic drag increases significantly, and thus has the concept of 'vortex drag'.Baker (2014) also learned that vortex resistance plays a big part in tail car aerodynamic drag.Therefore, a vertical section positioned 42.5 mm in front of the tail car nose is selected, and the vorticity of this section is extracted, as shown in Figure 17.It can be observed from the figure that a law consistent with the change of the tail car aerodynamic drag is consistent.The tail car aerodynamic drag under P 1 , P 2 , P 7 and P 8 conditions is basically consistent with the prototype, and the vorticity diagram of these four conditions is also basically consistent with the prototype.However, the tail car aerodynamic drag of P 3 ∼ P 6 is significantly reduced compared with the prototype, and the vorticity size and the transverse width of the vortex are significantly reduced in these three conditions.Especially at the position P 4 , where the aerodynamic drag reduction rate of the tail car is the largest, the vorticity is significantly reduced.According to the research of Oh et al. (2018), when the lateral width of the train wake vortex decreases, the tail car aerodynamic drag also decreases accordingly.It is proved that the VG changes the vorticity of the tail car, which in turn changes the tail car aerodynamic drag.
According to the above analysis, it can be concluded that the tail car aerodynamic drag depends on the separation vortex and the longitudinal vortex.Therefore, the balance between separated and longitudinal vortex can be changed by means of the VG to achieve the drag reduction of the tail car.Combined with the research by Aider et al. [16] , a reasonable explanation of tail aerodynamic drag and lift reduction was obtained.The VG located in front of the separation point makes the boundary layer separation advance, thus the equilibrium is broken between the longitudinal vortex and the separation vortex.When the VG is arranged at P 4 , the longitudinal vortex intensity and range of the tail car will be significantly reduced, so as to reduce the tail car aerodynamic drag.Meanwhile, the separation bubble behind the VG is significantly increased, and the aerodynamic drag reduced by the longitudinal vortex is greater than the aerodynamic drag generated by the separation bubble, and the final result is that the tail car's F d and F l are reduced.

Effect of streamline nose's length on aerodynamic drag reduction
The streamline nose of ICE2 has a length of 3.9 m.Whereas most high-speed trains have a longer streamline nose to achieve better aerodynamic performances.Therefore, it is very essential to study the arrangement of vortex generators on trains with different streamline nose's lengths.The original streamlined nose length of the 1/10th scale ICE2 model is 390 mm.On the basis of not changing the shape of the ICE2 train, only the streamline length of the train is changed, adding two train models with streamline nose's length of 460 and 550 mm.According to the above results, the vortex generator was arranged at position P 4 to investigate the effect of the streamline length on the aerodynamic drag reduction effect of the vortex generator.
The numerical results are shown in Table 3.It can be observed that streamline nose's length has a significant effect on the drag reduction effect of the vortex generator.When the streamline nose's length increases from 390 mm to 460 mm, the drag reduction ratio of the tail car decreases from 15.42% to 3.07%; When the streamline nose's length is 550 mm, the tail car drag reduction is further reduced to 0.56%.The aerodynamic resistance of the vortex generator is small, and the influence of streamline nose's length on the resistance of the vortex generator can be ignored.
Figure 18 shows the surface pressure on tail cars with three different lengths.The left column shows the tail car without the vortex generator, the right column shows the tail car with vortex generators.It can seen that the streamline nose's length changes and the surface pressure at the streamline transition at the top of the tail car appears to change significantly.With the increase of streamline nose's length, the change of streamline transition point of tail car tends to be gentle, and the acceleration effect of air flow passing through this point is weakened, resulting in a smaller negative pressure value.The vortex generator is arranged on the top of the tail car with a long streamline nose, which cannot change the pressure distribution of the tail car by blocking the airflow.
Figure 19 shows the streamline distribution on the surface and mid-section of the tail car with different lengths.By comparing the streamline distribution on the surface, it can be found that the vortex generator has a few effects on the flow around the tail car when the train has a long streamline nose as shown in the red box in Figure 19.When the streamline nose's length is increased to 450  on the top of the tail car with long streamline length will not damage the strength of the longitudinal vortex of the tail car, and the drag reduction effect of the tail car will be significantly reduced.

Conclusion
With the ICE2 high-speed train model and the microramp VG, which is the current research hotspot, the aerodynamic drag reduction effect of the vortex generator on the high-speed trains is investigated.For the tail car aerodynamic force, surface pressure distribution, velocity distribution around the train, surface streamline distribution of the tail car and vorticity diagram, the optimal placement of the vortex generator is explored, and the mechanism of the influence of the vortex generator on the flow field around the tail car is revealed.From this study, the following conclusions were drawn: (1) The arrangement of VG has obvious influence on the tail car's aerodynamic drag and lift.When the VGs are arranged at the boundary layer mutation point P 4 , the aerodynamic drag and lift of the tail car are the smallest; As the VGs are positioned progressively further away from P 4 , the tail car's aerodynamic drag and lift gradually increases.If the VGs are not arranged properly, the aerodynamic drag of the tail car will increase.(2) The VG is arranged near the boundary layer mutation point, which can achieve a better aerodynamic resistance reduction.When the VG is arranged at the flow separation point, the tail car resistance can be reduced by 3.87%.The tail car's aerodynamic drag can be reduced by 15.42%, when the VG is arranged at the sudden change of the boundary layer.
(3) The reduction of tail car aerodynamic drag and lift is related to the balance between the separation vortex and the longitudinal vortex.The vortex generator arranged in front of the separation point destroys the original balance of the separation vortex and the longitudinal vortex, effectively reducing the strength of the longitudinal vortex, thereby achieving a drag reduction.(4) The streamline nose's length has a significant influence on the drag reduction effect of the vortex generator.With the increase of streamline nose's length, the change of streamline transition of tail car tends to be gentle.The arrangement of vortex generator here cannot change the surface pressure distribution and rear flow field, and the drag reduction effect is therefore less effective.
This paper studies a new passive drag reduction measure for high-speed trains, and compares the drag reduction effects of different placement positions of VGs.The research shows that the best drag reduction effect is obtained by arranging the VG at the boundary layer mutation of the tail car.In this paper, only the ICE2 train with a simple head car shape and short streamline nose was studied, and the research has limitations.Therefore, for complex shapes of head car, longer streamline noses and different vortex generator shapes are worthy to be studied in the future.

Figure 2 .
Figure 2. The shape and arrangement of VG.

Figure 6 .
Figure 6.Comparison of numerical calculation results with literature simulation results and wind tunnel tests.

Figure 7 .
Figure 7. Cross-section streamline and velocity distribution around the train.

Figure 8 .
Figure 8. Cross-section velocity vector diagram around the train.

Figure 9 .
Figure 9. Cross-section velocity distribution around the train.

Figure 11 .
Figure 11.Influence of VG arrangement position on tail car's F d and F l.
and Figure 15.The VGs are placed at different places of the tail car,

Figure 12 .
Figure 12.Wall shear stress of tail car surface at different positions of VG.

Figure 13 .
Figure 13.Surface pressure on tail car with different positions of VG.

Figure 14 .
Figure 14.Velocity distribution in the tail car's cross-section.

Figure 15 .
Figure 15.Streamline distribution at the cross-section of the tail car.

Figure 16 .
Figure 16.Surface streamline of tail car.

Figure 17 .
Figure 17.Vorticity diagram for different positions of VGs.

Figure 18 .Figure 19 .
Figure 18.Surface pressure on tail cars with different streamlined nose's lengths.

Table 1 .
Computational results obtained using different grids.

Table 2 .
Train aerodynamic force results.

Table 3 .
Aerodynamic drag of trains with different streamline lengths (with/without VGs).