Numerical investigation for the influence of the car underbody on aerodynamic force and flow structure evolution in crosswind

Abstract

To investigate the aerodynamic behavior of underbody structure in crosswind conditions, two numerical simulation models have been developed by using computational fluid dynamics method. First, to validate the accuracy of these models, the wind tunnel experiment of model with simplified flat underbody has been designed. The computationally predicted results of realizable k-e model show consistency with experimental data, and the correlation deviation of aerodynamic force is less than 10%. By using such model, the aerodynamic force and flow field of car under steady crosswind are simulated using underbody structure, and the influence on the aerodynamic characteristic has been analyzed. It can be found that the aerodynamic force increased significantly under different yaw angle. The physical mechanism response has been clearly shown by investigating the flow field around car body by vortices visualization technique. The results of this study can be served as a suggestion for studying the vehicle stability of high-speed under crosswind.

Keywords

Aerodynamic behavior underbody structure crosswind computational fluid dynamics wind tunnel experiment aerodynamic force flow field yaw angle

Introduction

A new trend in the development of high-speed cars is to smaller aerodynamic force to improve safety. In extremely windy environments, because of large lateral forces and yawing moment, the vehicles will carry the risk of rollover and go off its original trajectory.^1,2 Vehicle accidents caused by crosswind extend all over the world. Therefore, the running safety of vehicles in crosswind should be paid more attention, and many researches of vehicles’ safety under crosswind have been done by using computational fluid dynamics method and wind tunnel experiments.

For a long time, aerodynamic drag has always been concerned with aerodynamic research. In recent years, more and more attention has been paid to aerodynamic lift with the rapid development of vehicle aerodynamics, and the effect of the deformation and vibration of the car body on the flow field was considered.^3,4 Most present studies have shown that aerodynamic lift has great influence on the stability of high-speed vehicles.^5,6 Aerodynamic lift is sensitive to airflow distribution around the car underbody. However, most works under crosswind condition had been confined to a simplified model which did not consider the effect of underbody details on aerodynamic force,^7–11 and the influence of underbody structure under no-crosswind is such that it changed flow field, which could be lead to the aerodynamic force changing. For example, Gorre et al.¹² showed that there was a considerable air resistance due to underbody components. The contribution of the underbody geometry on total drag was studied by Huminic and Huminic,¹³ and the results showed that the aerodynamic drag increased obviously. Yan et al.¹⁴ studied the external flow of a simplified passenger car model with two schemes adopted: a smoothed complex underbody and a flat underbody, and it indicates that the flow separation and energy loss of fluid was caused by protuberances on the complex underbody, and what’s more, there was a significant change in the lift.¹⁵ There were also a lot of researches on the airflow at the bottom of the train with underbody structure.^16–18 Zhu and Hu¹⁶ studied the aerodynamic behavior of flow past a simplified high-speed train, and it was found that the flow around the bogie was highly unsteady due to strong flow separations, and flow interactions developed there. C Paz et al.¹⁷ presented an innovative numerical methodology to provide a more accurate analysis of the underbody flow, and lower values of the minimum static pressure were observed regarding the underbody flow. Mulligan et al.¹⁸ presented the measurements of full-scale freight train underbody aerodynamics. The result showed that the mean velocity near the wheels was related to the ambient wind speed.

The influence of underbody structure under no-crosswind is such that it changed flow field, which could be lead to the changing aerodynamic force. It is uncertain that the effects of underbody structure on aerodynamics in crosswind condition, but the aerodynamic force was important for the running safety of vehicles in crosswind.¹⁹ It is thus necessary to study the effect of underbody structure on aerodynamic and flow structure in crosswind. Therefore, a more real car with underbody structure was constructed and the aerodynamic characteristic under crosswind was analyzed. Understanding how force and flow structure on a vehicle in crosswind are influenced by underbody structure will offer the possibility to underbody designing and aerodynamic performance optimization.

Numerical model

Governing equations and discretization

The Mach number is quite small when the car moves at a velocity less than 120km/h, so the external flow field of vehicle can be regarded as three-dimensional (3D) incompressible. A proper combination of theoretical modeling and numerical setup, the steady-state Reynolds Averaged Navier-Stokes (RANS) turbulence closure strategies can lead to surprisingly good quantitative predictions of the aerodynamic force coefficients.²⁰ Therefore, the steady-state RANS was employed in current research.

Continuity Equation:

\frac{\partial u_{i}}{\partial x_{i}} = 0

(1)

Momentum equation:

\frac{\partial (u_{i} u_{j})}{\partial x_{i}} = - \frac{\partial p}{\partial x_{j}} + \frac{\partial}{\partial x_{j}} [μ_{eff} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})]

(2)

In this study, the realizable k-e model was used for the RANS portion. Compared to other turbulence models, the realizable k-e model was more accurate to predict the related aerodynamic parameters in vehicle aerodynamics. Plenty of studies haven indicated that the realizable k-e turbulent model was widely used in automobile airflow calculation because of the advantages of accuracy.^21–26 In this model, a new model dissipation rate equation (based on the mean-square vorticity fluctuation at large turbulent Reynolds number) and a new realizable eddy viscosity formulation were constructed. Therefore, this model is more suitable for a variety of flows, including mixing layers, planar and round jets, rotating homogenous shear flows, boundary layers with adverse pressure gradients and flows with separation induced by the geometry of the domain. It is more accurate to predict the related aerodynamic parameters in vehicle aerodynamics. For a steady incompressible flow, the modeled transport equation for k and e were given as shown below.

Turbulence kinetic energy equation:

\frac{\partial (ρ u_{i} k)}{\partial x_{i}} = \frac{\partial}{\partial x_{i}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{i}}] + G_{k} - ρ ε

(3)

Turbulence dissipation equation:

\frac{\partial (ρ u_{i} ε)}{\partial x_{i}} = \frac{\partial}{\partial x_{i}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{i}}] + ρ C_{1} S ε - ρ C_{2} \frac{ε^{2}}{k + \sqrt{υ ε}}

(4)

μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}, C_{μ} = \frac{1}{A_{0} + A_{S} U^{*} \frac{k}{ε}}

C_{1} = max [0.43, \frac{η}{η + 5}], η = S \frac{k}{ε}

U * \equiv \sqrt{S_{ij} S_{ij} + {\tilde{Ω}}_{ij} {\tilde{Ω}}_{ij}}, {\tilde{Ω}}_{ij} = \frac{1}{2} (\frac{\partial u_{i}}{x_{j}} - \frac{\partial u_{j}}{x_{i}})

A_{S} = \sqrt{6} \cos φ, φ = \frac{1}{3} \cos^{- 1} (\sqrt{6} W), W = \frac{S_{ij} S_{jk} S_{ki}}{{\tilde{S}}^{3}}

\tilde{S} = \sqrt{S_{ij} S_{ij}}, S_{ij} = \frac{1}{2} (\frac{\partial u_{j}}{\partial x_{i}} + \frac{\partial u_{i}}{\partial x_{j}})

σ_{k} \equiv 1.0, σ_{ε} \equiv 1.2, C_{2} \equiv 1.9, A_{0} \equiv 4.0

where k is the turbulence kinetic energy, ε is the dissipation rate of turbulence energy, ρ denotes the density, G_k is the generation caused by the average velocity gradient of turbulent kinetic energy, μ is the dynamic viscosity, μ_t the turbulent viscosity, υ is viscosity coefficient, S_ij is average strain rate, Ω_ij is the mean rate of rotation tensor viewed in a rotating reference frame with the angular velocity, C_μ is the function of the average strain rate and the curl, S is the modulus of the mean rate-of-strain tensor, u_i (i = 1,2,3) is the velocity component, x_i (i = 1,2,3) is cartesian coordinates, C₁ and C₂ is constant, σ_k is turbulent Prandtl number of the turbulence kinetic energy, and σ_ε is turbulent Prandtl number of the dissipation rate of turbulence energy.

Computer-aided design model

As the focus is on aerodynamic behavior of underbody structure in crosswind conditions, two geometric models are built. The floor of model A is smooth. The underbody structure of model B is shown in Figure 1 which is based on simplified model A. The underbody retains the key structure of chassis which includes the floor, frame, suspension, transmission system, exhaust system, fuel tank, and spare tires and so on to ensure that the airflow of underbody should be closer to the real situation of car running.

Figure 1.

Geometric model.

The deformation of tires caused by weight is design as shown in Figure 1. The part of the wheel surface of the cylinder is cut off, and then a small lug with a height of 20 mm is drawn from the wheel cross-section to the ground. The small convex platform not only improves the mesh quality of the wheel and the ground surface, but also compensates for the change of the ground clearance caused by the tire deformation.

Numerical model

The computational domain is a big rectangular enclosure. The model A and model B are positioned in a small rectangle with a width of L × B × H, and the small rectangle is positioned in the big rectangle at a distance of 3L from the front inlet, 3B from the side wall, 7L from the back outlet, and 5H from the upper, H is the height of model A and model B, as shown in Figure 2. The computational domain had the same size as the high-speed test section of the wind tunnel, and the location of the model in the domain is also the same.

Figure 2.

Numerical computational model.

The unstructured meshes with size of 0.001 times length are created on surface of model; however, smaller size elements are used in locations of complex curved and wake regions. There are several blocks in the grid topology to obtain high calculation accuracy and efficiency. The finest block of 10 layers prism elements is generated from the vehicle’s surface, and the initial height is 0.3 mm, with a growth rate of 1.1, which can accurately capture the boundary layer flow on the model surface, keeping the y⁺ value within an acceptable range (30–200) in this study. The second finest block is a cuboid near to the prism elements and the mesh is gradually growth to capture more deformation about turbulent flow around model surface. The coarse mesh is generated in other regions. The total number of mesh is about 10 million generated by using ICEM-CFD, as shown in Figure 2.

In current research, to reflect experimental conditions, the boundary conditions imposed to the domain were as follow: a constant velocity (defined as v) of 30 m/s was applied on inlet with a turbulence intensity of 0.5%. In the experiment, the ground boundary layer was eliminated by the pumping system, in order to eliminate the ground boundary layer in the simulation. The ground surface was treated with a moving wall at the same speed of inlet. A symmetry boundary condition was applied to both the top and side walls. A pressure outlet condition that let the flow to exit to the atmosphere was applied at the outlet. The static boundary condition was used for wheels, which matched the wind tunnel condition. In the actual situations, the wind direction is not always parallel with the driving direction, and the yaw angle is often existent. The working condition of vehicle under crosswind was achieved by rotating the vehicle model. Therefore, in this study, the yaw angle β ranged from 0° to 30° with increment of 3°. In current research, the Reynolds number was 7.8 × 10⁶. The solution algorithm for the simulation was based on the well known SIMPLE algorithm for the iterative solution of the steady RANS equations, and the algorithm was of second-order upwind scheme in spatial discretization. All the simulations were fulfilled by the commercial software package Ansys Fluent.

Computation validation

Wind tunnel test of model A was carried out in the closed-circuit subsonic wind tunnel of Hunan University. The cross-sectional area is 3 × 2.5 m, and the maximum speed is 58 m/s. The average turbulence intensity of upstream is about 0.13%. The model was placed at the center of turntable and the aerodynamic force was measured by a six-component balance, as shown in Figure 3. Repeated measurements of MIRA reference cars were carried out to ensure the accuracy of the experimental data. The MIRA reference cars were a group simplified car shapes which were evolved from work undertaken in the early 1980s when European and North American wind tunnel operators began a series of correlation exercises. With the availability of published experimental data and the advantage of simple surface geometry, the MIRA reference car became a popular test case when CFD emerged as a tool for automobile aerodynamics. The flow field around the model was measured by the particle image velocimetry (PIV) system, which was placed on the top of the test section of wind tunnel. The aerodynamic coefficients variation was not obvious when the Reynolds number surpassed 1.2127 × 10⁶ in current experiments.

Figure 3.

Wind tunnel test.

There is a pair of opposite revolving vortex along the longitudinal direction at the symmetry plane. The upper clockwise vortex is formed by the down-wash shear airflow from the back and the lower anti-clockwise vortex is formed by the up-wash shear airflow from the bottom. The size of clockwise vortex is larger than anti-clockwise vortex, and the position of vortex core is much closer to the body. The upper clockwise vortex was measured by PIV. The comparison between measurement and simulation on velocity counter and vortex is shown in Figure 4. The structure and position of the upper clockwise vortex are similar in the recirculation zone. The height of vortex region is approximately equal, but there is a slight difference in the length of the vortex. It could be found that the vortex of simulation is longer than the measurement. The velocity distribution of simulation is similar to the measurement.

Figure 4.

Comparison the aerodynamic date between experiment and simulation (velocity and streamline distribution).

The comparison of aerodynamic force between experiment and simulation are illustrated in Figure 5. The change trend of the aerodynamic force simulation data is consistent to the experimental data and the calculation results of drag force agree well with the experimental results. But the side force and lift force show a certain difference and the error is less than 10%. One possible explanation for the discrepancy is the error of yaw angle between the present calculation and the experiment.

Figure 5.

Aerodynamic coefficient of measurement and calculation for yaw angle range.

Result and discussion

In vehicle aerodynamic, the aerodynamic characteristic is reflected by the aerodynamic force coefficient. In current research, drag force coefficient is reflected by C_D, lift force coefficient is reflected by C_L, side force coefficient is reflected by C_S, and the coefficients were defined as

C_{D} = \frac{F_{x}}{\frac{1}{2} ρ v^{2} S}, C_{L} = \frac{F_{z}}{\frac{1}{2} ρ v^{2} S}, C_{S} = \frac{F_{y}}{\frac{1}{2} ρ v^{2} S}

The pressure coefficient is reflected by C_P, the coefficient was defined as

C_{P} = \frac{(P - P_{ref})}{\frac{1}{2} ρ v^{2}}

where F_x, F_y, F_z are the aerodynamic drag force, side force, and lift force, respectively; S is the front area; P_ref is the reference pressure; and P is the calculation mean pressure.

Aerodynamic force calculation

In vehicle aerodynamics, aerodynamic force, static pressure distribution, and airflow field are the evaluation outcome of aerodynamic behavior. In this research, the aerodynamic coefficients for yaw angle ranging from 0° to 30° were obtained to compare with the aerodynamic characteristic of model A and model B.

Figure 6 shows the relationship between aerodynamic coefficient with yaw angle and the effect of underbody on aerodynamic force. The results revealed that the aerodynamic force of model A and model B had similar change trend with increasing angle. The side force rose linearly with yaw angle, and lift force increased strongly at high yaw angle. Drag force rose to a maximum at about 18°. It is noted that the trend of aerodynamic force is similar as the works described by M Gohlke et al.²⁷ The calculation data showed that the higher lift increased with the yaw angle increasing. The drag force was the least sensitive to the change of yaw angle, while the side force was the most sensitive when yaw angle ranged between 0° and 15°. However, when the yaw angle ranged between 15° and 30°, the lift force was the most sensitive. When the yaw angle ranged between 0° and 3°, the lift force decreased, but when the yaw angle varied from 3° to 30°, the lift force increased gradually and the increment became bigger and bigger.

Figure 6.

Comparison of aerodynamic force calculation results of model A and model B.

Furthermore, the result also revealed that the effect of underbody structure on drag force and lift force was quite obvious. When the yaw angle varied from 0° to 18°, the lift force increased much more, and it increased less when the yaw angle varied from 18° to 30°. But the increment of drag force and side force caused by underbody is no significant in contrast with yaw angle increasing.

Static pressure calculation

In this section, the results of the static pressure on the body surface is presented. Figure 7 presents the static pressure when yaw angle is 15° with views from different direction. In order to observe clearly, the range of pressure was set ranging from −500 to 400. A stagnation point moved windward side with yaw angle increasing. Therefore, the area of positive pressure on the windward side was growing with increasing yaw angle. The area of negative pressure on the leeward side was also growing, which was stretching from the front to the back, and the side force would increase, as described in Figure 6. After adding the underbody, the static pressure on the back was smaller than the model without underbody and the pressure drag would increase, but the pressure on the bottom was much larger than the model without underbody, especially on the back, and the lift force would increase, as described in Figure 6, but there was no significant difference from the top view and front view.

Figure 7.

Pressure distribution on the body (yaw angle = 15°).

Figure 8 shows that the pressure coefficient distribution of Model A and Model B at the body center-line with yaw angle 15°. The trend of pressure coefficient distribution is consistent. The change of Cp was very small from the top of the front bumper to the top of the rear windshield. However, a larger variation of Cp appeared from the top of the rear windshield to the back of decklid. It is indicated that the effect of underbody structure is obvious on flow structure of body surface. The airflow separation point of the rear windshield and the reattachment point of the decklid were further forward.

Figure 8.

The pressure coefficient distribution at the body center-line (yaw angle = 15°).

Due to the underbody surface of the two models is not consistent, the pressure coefficient contour is used, as shown in Figure 9. Comparing to the front end, the phase change of the rear end was larger, which leaded to a more obviously effect on the flow field around the underbody. The pressure coefficient of model B in the rear of underbody was increased significantly. There was large area of positive pressure in the windward face of rear drive axle, fuel tank, spare tires, and silencer. The underbody structure effects lead to the wake vortex separation at difference positions. The change of the space between the underbody and the floor lead to the change of velocity and direction of the flow around the underbody structure.

Figure 9.

The pressure coefficient distribution at underbody surface (yaw angle = 15°).

Flow structure analysis

In this section, the vortex structure is present. Figure 10 presents a comparison of streamline and vorticity distribution w_y at the model center-line between model A and model B. By comparing the vortex structure of these two models, it was founded that:

The structures of vortex were similar, and there were two counter-rotating structures. The upper clockwise vortex (defined as C) was induced by down-wash shear flowing from the back, and the lower anti-clockwise vortex (defined as D) was induced by up-wash shear flowing from the bottom.

Comparing with model A, the vortex core of model B was much closer to the body. Vortex core represented a zone of low pressure, which was induced to the increasing force on the surface of the body. Moreover, the underbody structure of model B was impacted directly by high-speed airflow and there existed a zone of positive pressure, as described in Figure 7. For these reasons, the drag fore of model B was larger than model A.

The diffusion range of vortex C expanded with rising yaw angle, while the vortex D was close to the surface of the car when yaw angle was less than 18°. When the yaw angle exceeded 18°, the vortex D was dragged quite far from the surface of car. For these reasons, the drag force of model A and model B increased first and then decreased with yaw angle increasing.

The airflow of mode B under the vehicle was complex, which was with very high negative vorticity. When the air of the bottom flowed through the underbody structure, the air stream lost the attachment object and formed strong shear airflow. The peak value of negative vorticity occurred near the transmission, but the region size of negative vorticity decreased with yaw angle increased.

Figure 10.

The streamline and vorticity distribution w_y at the symmetric plane.

Figure 11 presents a comparison of streamline and vorticity distribution w_x of X plane. Due to which, the top vortex on the leeward side (defined as E) was more obvious, and the range of the vortex was expanded with the yaw angle increasing, the side force also increased. It could be found that the leeward side flow structures of model A and model B were similar when yaw angle was small. But the flow structure was significantly different at the bottom. A turbulence boundary layer would be produced at the bottom of model B and become more and more thick, which will lead to a rising pressure and a more and more slow velocity at the bottom. An anti-clockwise vortex (defined as F) was generated with the yaw angle increasing, which was induced by up-wash shear flowing from the bottom, and the diffusion range expanded with rising yaw angle. These changes in flow structure resulted in increased aerodynamic forces. Furthermore, a component force along Y axis was generated because of the underbody structure, which was impacted by oncoming flow.

Figure 11.

The streamline and vorticity distribution w_x of X plane at x = 2.2 m.

The vorticity at the bottom of the car with underbody structure changed obviously. There were two positive vorticity regions at the bottom of model B. The one was near the left tie, and another was near the rear drive axle. The vortex region was getting bigger and bigger as the yaw angle increased. There was also a negative vorticity region between the two positive vorticity regions. The positive and negative vortices overlapped frequently and the turbulent flow was complex.

Flow velocity under the vehicle on XY-section above the ground when the vehicle under different yaw angle is visualized in Figure 12. The results revealed that there was a vacuum pocket in its wake. The flow attached to the bottom surface of underbody and up-wash shear flowed along it. There was a significant contrast in the velocity on the bottom of model A and model B. The main vortices at the bottom of the two models were all obvious, and the size of the vacuum pocket of model B was quite larger. In addition, there were many small vortexes at the bottom of model B, and the flow field was complex. These flow effects occurred over the bottom of the body and then affected the mechanisms responsible for the formation of the underbody vortex, which is very important to the aerodynamic force coefficient, and the wake flow field of model A and model B were quite different. Hence, the aerodynamic force of model B went bigger.

Figure 12.

The velocity distribution of Z plane at Z = –0.16 m.

The velocity on the bottom of both model A and model B decreased gradually along the flow direction, as showed in Figure 13. The velocity at the front of the bottom was consistent. But the airflow of model B at the bottom lost attachment near the transmission, and step reduction of velocity occurred. The velocity increased gradually in the rear area of the transmission, and the varying tendency was consistent with the model A, while the velocity of model B on the bottom was much smaller. It would lead to larger pressure, and the lift force of model B was much larger, as described in Figure 9.

Figure 13.

The velocity at the downstream of the bottom along the flow direction (yaw angle = 15°).

Figure 14 presented the steady-state flow structure of underbody in the upward view. The vortex structure past the car model visualized by using the Q-criterion in this work and represented basically the second invariant of the velocity gradient tensor. The important observation from Figure 14 was that the vortex structure at the bottom and the tail of the car was quite different. The vortex direction kept the trend of side wind direction when the yaw angle varied. There were much more complex vortices at the bottom of model B. Traditionally, the trail vortices region of model B was much larger. The yaw angle was smaller, and the difference was more obvious. These phenomena lead to differences in aerodynamic forces, as described in Figure 6.

Figure 14.

Normalized Q-criterion isosurfaces at different yaw angles.

The computational results are evaluated according to the vortices generated by the flow around the bottom of the car, as it is shown in Figure 15. As yaw angle increased, the vortices generated by the flow around the bottom of the car enhanced, especially the vortex of leeward side. The underbody structure had great influence on the wake vortex when yaw angle was small. The tail vortex of the model B was obviously enhanced, and the energy dissipation was slow. When the yaw angle became larger, the vortices of model A and model B generated by the flow around the bottom of the car showed a high similarity from location, scale, and morphology, but the vortex of leeward side was obviously enhanced. It was well known that the wake vortex of model B diminished much more slowly and the wake vortex core was closer to the left side of the body. Therefore, the model B with detailed underbody structure has greater aerodynamic drag and aerodynamic side force.

Figure 15.

Streamlines around the bottom of car at back view.

Figure 16 shows that the TKE on the cross-stream when yaw angle was 24°, and the results revealed that the turbulent strength diminished gradually with distance. It is well known that the maximum TKE is located at the vortex core. It could be found that there were two trailing vortex at the back of model A. One was smaller and the other was larger, but both diminished quickly. There was a trailing vortex at the back of model B and the turbulent strength was larger. The wake vortex of model B diminished much slowly.

Figure 16.

Turbulent kinetic energy (TKE) on the cross-stream (yaw angle = 24°).

Conclusion

The effect of underbody structure on aerodynamic force and flow structure was examined on the basis of the simulation and experiment. The wind tunnel data proved the accuracy of the numerical model. The study shows the aerodynamic characteristics is closely related with the underbody structure.

The aerodynamic behavior of underbody structure in crosswind conditions was rather obvious. The aerodynamic force increased, especially the lift force acted on the rear axle, which would reduce the traction of real wheels and affect the running stability. It should be taken into consideration when studying the stability of high-speed vehicle in crosswind.

Both the wake vortex and flow structure of underbody would be changed. The wake vortex was closer to the body because of the uneven underbody structure, which was induced to increase force on the surface of body, and the wake vortex diminished much more slowly. The underbody structure, such as frame, suspension, transmission system, exhaust system, fuel tank, and spare tires and so on, was impacted directly by high-speed airflow, and there existed a zone of positive pressure. It is observed that the velocity of underbody decreased gradually along the flow direction, and the pressure of underbody will be increased.

In this study, only one vehicle with underbody structure was analyzed, but aerodynamic behavior of underbody structure was similar. The results can be applied to other cars and the increment may be different depending on the shape of the vehicle.

Footnotes

Handling Editor: Ruey-Jen Yang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was supported by the National Natural Science Foundation of China (grant nos.: 61805207, 51875186 and 51775395), and the education and scientific research projects of young and middle-aged teachers in Fujian Education Department (grant no.: JAT170414), and the Natural Science Foundation of Fujian Province (grant nos.: 2017J01723).

ORCID iD

Zhiqun Yuan

References

Batista

Perkovič

A simple static analysis of moving road vehicle under crosswind. J Wind Eng Ind Aerod 2014; 128: 105–113.

Suh

Yoon

Numerical investigations of the crosswind stability of the Korean light tactical vehicle during airlift. J Mech Sci Technol 2017; 31: 1067–1072.

Zhu

Zhigang

Fluid–structure interaction study of three-dimensional vehicle model under crosswind. Adv Mech Eng 2015; 7: 1–10.

Huang

Feng

et al . Transient aerodynamics simulations of a road vehicle in the crosswind condition coupled with the vehicle’s motion. Proc IMechE, Part D: J Automobile Engineering 2017; 232: 583–598.

Dongchen

Yicun

Qiang

et al . Modeling and simulation study on crosswind stability of the high-speed bus. Noise Vib Worldw 2011; 42: 44–50.

Kee

Rho

Kim

et al . High speed driving stability of passenger car under crosswind effects. Int J Automot Techn 2014; 15: 741–747.

Gohlke

Beaudoin

Amielh

et al . Shape influence on mean forces applied on a ground vehicle under steady cross-wind. J Wind Eng Ind Aerod 2010; 98: 386–391.

Tsubokura

Nakashima

Kitayama

et al . Large eddy simulation on the unsteady aerodynamic response of a road vehicle in transient crosswinds. Int J Heat Fluid Fl 2010; 31: 1075–1086.

Volpe

Ferrand

Da Silva

et al . Forces and flow structures evolution on a car body in a sudden crosswind. J Wind Eng Ind Aerod 2014; 128: 114–125.

10.

Ferrand

Forces and flow structures on a simplified car model exposed to an unsteady harmonic crosswind. J Fluid Eng 2014; 136: 249–256.

11.

Huang

Feng

CJ.

Coupled analysis of unsteady aerodynamics and vehicle motion of a passenger car in crosswind condition. J Appl Fluid Mech 2017; 10: 625–637.

12.

Gorre

Prasad

Kumbhar

et al . Underbody drag reduction study for electric car using CFD simulations. SAE technical paper 2015-26-0211, 2015.

13.

Huminic

. CFD study concerning the influence of the underbody components on total drag for a SUV. SAE technical paper 2009-01-1157, 2009.

14.

Yan

Qiao

Zhao

et al . Comparative study on external flow simulation of passenger car with complex underbody and flat underbody. Appl Mech Mater 2013; 380–384: 69–72.

15.

Yang

Shen

et al . Improved research of automobile underbody flow field based on aerodynamic lift. J Cent South Univ (Sci Technol) 2013; 44: 4063–4068.

16.

Zhu

ZW.

Flow between the train underbody and trackbed around the bogie area and its impact on ballast flight. J Wind Eng Ind Aerod 2017; 166: 20–28.

17.

Paz

Suárez

Gil

Numerical methodology for evaluating the effect of sleepers in the underbody flow of a high-speed train. J Wind Eng Ind Aerod 2017; 167: 140–147.

18.

Mulligan

Barimani

Green

SI.

Full-scale freight train underbody aerodynamics with application to track spraying. Int J Rail Transport 2016; 4: 151–166.

19.

Z-Q

G-M.

Analysis on the driving stability of high-speed container truck in crosswinds. Trans Beijing Inst Technol 2014; 34: 250–254.

20.

Tian

Zhang

Zhu

et al . Aerodynamic drag reduction and flow control of Ahmed body with flaps. Adv Mech Eng 2017; 9: 1–17.

21.

Shih

T-H

Liou

Shabbir

et al . A new k-ε eddy-viscosity model for high Reynolds number turbulent flows: model development and validation. Comput Fluids 1995; 24: 227–238.

22.

TienPhuc

ZhengQi

Zhen

Numerical simulation of the flow field around generic formula one. J Appl Fluid Mech 2016; 9: 443–450.

23.

Wang

Aerodynamic drag reduction of vehicle based on free form deformation. J Mech Eng 2017; 53: 135–143.

24.

Yang

Zheng

Huang

et al . Aerodynamic drag reduction method of vehicle body based on non-smooth surface and vortex interference. Chin Mech Eng 2016; 27: 982–988.

25.

Chen

Huang

et al . Research on aerodynamic characteristics of vehicle with an improved LRN k-e turbulence model. Chin Mech Eng 2015; 26: 2550–2555.

26.

Wang

Xin

et al . Numerical and experimental investigations on the aerodynamic characteristic of three typical passenger vehicles. J Appl Fluid Mech 2014; 7: 659–671.

27.

Gohlke

Beaudoin

Amielh

et al . Experimental analysis of flow structures and forces on a 3D-bluff-body in constant cross-wind. Exp Fluids 2007; 43: 579–594.