Study on aerodynamic characteristics and running safety of two high-speed trains passing each other under crosswinds based on computer simulation technologies

2.1. Computational model of aerodynamics

The DES (detached eddy simulation) method based on the Menter $k$-$\omega$ two-equation turbulent flow model was used to solve the Navier-Stokes equation [20-22]. Basic ideas of detached eddy simulation are as follows: a Reynolds time-averaged method is adopted on boundary layers near the wall faces, and small-scale pulsations are simulated by a turbulent flow model; in regions far from the wall face, large eddy simulation is used to simulate detached eddy. The DES equation based on Menter $k$-$\omega$ is as follows:

In the equation, $t$ is time, $\rho$ is density, $k$ is turbulent kinetic energy, $x_j$ is a directional coordinate, and $j=$ 1, 2, 3, which are directions in length, width and height in succession; $u_j$ is an airflow velocity component, $P_k$is a turbulent flow item, ${\sigma }_{\omega }$, ${\sigma }_{\omega 2}$, ${\sigma }_k$, $\beta$, $\gamma$ are empirical constants, $\omega$ is a specific dissipation rate of turbulent flow, $l_{k-\omega }$ is a turbulent flow scale parameter; $F_1$ is a switching function which represents the minimum distance between an eddy and the wall face, where in the near wall regions, $F_1$ approaches 1, and the model is approximate to a $k$-$\omega$ model; near the boundary layer edges, $F_1$ approaches 0, and the model is transformed into a $k$-$\epsilon$ model; ${\mu }_l$is a laminar flow viscosity coefficient, ${\mu }_t$is an eddy viscosity coefficient, and ${\mu }_t=\mathrm{m}\mathrm{i}\mathrm{n}\left[\rho k/\omega ,a_1\rho k/\mathrm{\Omega }{F}_2\right]$, where$\mathrm{\Omega }$is the absolute value of vorticity, $a_1=$ 0.31, the mixed function is as follows:

where: $y$ is the minimum distance between the first layer of meshes and the wall face. In the DES method, $l_{k-\omega }$ is substituted by $\mathrm{m}\mathrm{i}\mathrm{n}(l_{k-\omega },C_{DES}\mathrm{\Delta })$, where $\mathrm{\Delta }=\mathrm{m}\mathrm{a}\mathrm{x}(\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z)$ is the maximum edge length of elements. The constant is $C_{DES}=(1-F_1)C_{DES}^{k-\epsilon }+F_1{C}_{DES}^{k-\omega }$, where the constant item $C_{DES}^{k-\epsilon }=$ 0.61and $C_{DES}^{k-\omega }=$ 0.78. In this way, on the boundary layer near the wall face, the $\omega$ value is very large, the turbulence kinetic energy $k$ value is limited, $l_{k-\omega }$ is far smaller than the mesh scale, the SST turbulent flow model takes effect, and the Reynolds time-averaged method is adopted; at the position far from the wall face, the $\omega$ value is very small, and the changed model acts as a sub-grid Reynolds stress model of large eddy simulation when $l_{k-\omega }$ increases to $C_{DES}\mathrm{\Delta }$.

The high-speed train is a thin and long structure. The computational amount will be very large if the numerical simulation is conducted on flow fields of a complete train. With a certain distance away from the head train, flow fields on the train middle part basically have a stable structure. In this paper, a formation model including head train-mid train-tail train was adopted, where the head train and the tail train had the same shape. Meanwhile, in order to avoid excessive meshes, the train was simplified into a geometric body composed of smooth curved faces. Detailed features such as pantographs and bogies were not taken into account. The simplified model of the high-speed train is shown in Fig. 1.

The computational domain of two high-speed trains passing each other under crosswinds is shown in Fig. 2. The train length $H=$ 78 m was taken as the basis. Therefore, the computational domain was 8.0 H long, 2.5 H wide and 0.5 H high, the distance between two trains was 100 m, the linear distance among two tracks was 5 m, and the distance between the train and the ground was 0.2 m. The left cross section cdhg of the high-speed train was the crosswind inlet boundary and set as the velocity inlet condition, the crosswind velocity was 15 m/s, and the high-speed trains slipped and moved at the constant speed of 350 km/h (slippage mesh method was used to handle the train motion problem, and “interface pair” was used to handle motion parts and stationary parts at the mutual interfaces); other cross sections efgh, aehd and abcd of the high-speed train were the outlet boundaries and set as pressure outlet, where the size was 1 standard atmospheric pressure; the cross section bfgc right above the train was set as a symmetric boundary condition; the train surface was set as a fixed boundary, namely a non-slippage wall face boundary condition; the ground aehd was set as a fixed ground.

Fig. 1Geometric model of the high-speed train

Fig. 2Computational domain of two high-speed trains passing each other

Two opposite running trains were included in the flow field region. Two trains had the same running speeds. In order to solve problems of multi-body region and asymmetry, this paper used a partitioned computation method to divide the computational flow field of two high-speed trains passing each other into 3 parts, as shown in Fig. 3. The observation train, passing train and their nearby flow fields were region 1 and region 2, respectively. These two regions were movable according to expected directions and speeds, respectively. The ground and the flow field far from trains were region 3 which was a fixed part. All parts realized the relative motion by moving contact faces, and real-time data exchange was completed. Different parts had different mesh densities, so movable contact faces among these regions could not be overlaid completely. Therefore, information exchange should be conducted using moving grid technologies when the trains moved forward by one step. When the train moved forward, grids would move together with it, and region 1 and region 2 would be stretched on boundary faces of tail ends to form new grids. Grids were compressed and deleted on front ends. In this way, Shapes of moving contact faces among these regions would be kept unchanged. As a result, different regions of the flow fields could exchange information through moving contact faces. ICEM was used to divide meshes of the computational domain of high-speed trains. Tetrahedral meshes were used to divide the surface. Hexahedral meshes were used to divide fixed parts. Size of three-dimensional meshes was amplified according to a scaling factor. Tetrahedral meshes were adopted for parts far from the train. The maximum size of meshes on the train surface was 80 mm. The maximum size of spatial solid meshes was 1500 mm. The total amount of meshes was about 42.35 million. Surface meshes of the head are shown in Fig. 4. Fig. 5 shows the computational domain meshes of two high-speed trains passing each other.

Fig. 3Requirements for the grid of two high-speed trains passing each other

Fig. 4Surface meshes of the head of high-speed trains

Fig. 5Computational domain meshes of two high-speed trains passing each other

a)

b)

In the model, resultant wind theories were used to simulate the inlet wind velocity, as shown in Fig. 6. The train was deemed to be stationary. Resultant vector was conducted to environmental wind velocity $W$ and the reverse velocity – $V$ of trains, so the resultant wind velocity $U$ was obtained; $\alpha$ was a wind direction angle; $\beta$ was a lateral deflection angle. Studied results show that the approximate handling method will not cause obvious errors in general [23-25].

Fig. 6Theories of resultant wind velocity

2.2. Multibody dynamic model of trains

In the model, LM tread was used to match the TB234460 steel track. Based on the equivalent Hertzian contact characteristics, the Fastaim algorithm based on rolling contact non-linear theories was used to compute creep force and creep force moment between wheels and tracks. It is assumed that the displacement vector of the train is as follows:

In the formula, $C=$ 1; $i=$ 1, 2; $j=$ 1, 2, 3, 4; $k=$ 1, 2, 3, 4, 5, 6, 7, 8. $X_C$, $X_{\alpha }$, $X_{\beta }$ and $X_{\gamma }$ are displacement vectors of train body, framework, wheel-set and axle box, respectively. Letters in the subscript brackets are number of corresponding components. Based on those mentioned hypotheses, the train-track coupling dynamic equation could be established, as follows:

In the formula, $M_v$ is a mass matrix; $C_v$ is a damping matrix; $K_v$ is a rigidity matrix; $F{e}_v$ and Fa_v are equivalent forces caused by track irregularity and aerodynamic loads under crosswinds. It is found that Eq. (5) has considered aerodynamic effects of the train, the track irregularity is an important factor which leads to train vibration and affects running safety performance. In this paper, the irregularity spectrum was taken as the track excitation. Aerodynamic loads borne by the train which ran under the crosswind environment were loaded into the dynamic model. The dynamic Eq. (5) was solved. In this way, dynamic characteristics of the high-speed train which ran under crosswinds could be obtained.

In this paper, SIMPACK software was used to establish a multi-body dynamic model of the high-speed train. The dynamic model was composed of 15 components including 1 train body, 4 wheel-sets, 2 frameworks and 8 rotational arms. The rigid train body, frameworks and wheel-sets had 6 degrees of freedom, namely vertical direction, longitudinal direction, horizontal direction, head shaking, nodding and lateral rolling; rotational arms had 1 degree of freedom, namely nodding; the dynamic model of each train had 50 degrees of freedom, and the dynamic model of the complete train had 150 degrees of freedom. Structures of each system were represented by equivalent spring, damping and mass blocks; the track system adopted a dual-mass (sleeper and track bed) three-layer (steel track, sleeper, track bed and road bed) track model with spring-damping vibration continuous distribution. The dynamic model of a train established by the multi-body dynamic software SIMPACK is shown in Fig. 7. Each component was deemed as a rigid body.

Fig. 7Multi-body dynamic model of the high-speed train

Modern steel tracks are not an ideal flat and straight state, where the real track spectrum should include vertical, horizontal and alignment irregularity spectrums. Due to these spectrums, interactions are caused between wheels and tracks, and geometric shapes of tracks will be changed, so that the tracks are randomly unbalanced. The stability of high-speed trains will be reduced. Therefore, this paper input vertical, horizontal and alignment irregularity spectrums to the multi-body dynamic model, as shown in Fig. 8. It is also shown in this figure that the amplitude difference of track spectrums in three directions was not large, so any track spectrum cannot be neglected. Therefore, the actual engineering should use the irregularity spectrum in three directions as the excitation.

Fig. 8Irregularity spectrums in three directions of high-speed trains

a) Vertical irregularity spectrum

b) Horizontal irregularity spectrum

c) Alignment irregularity spectrum

4.1. Pressure waves of two high-speed trains passing each other under crosswinds

Fig. 15 shows time-history curves of each measure point during two high-speed trains passing each other. Table 1 shows compared results of peak values of measure points. Analyzed results from Fig. 15 and Table 1 show that:

1) Before the head of the windward train reached measure points on the lateral window of the leeward train, pressures already started increasing. Then, pressures were increased quickly. When the nose tip of the windward train reached the measure point, a positive pulse and a negative pulse were caused, namely the head wave. After the maximum negative pulse was caused, fluctuations with constant amplitudes would be started till the nose tip of the tail train of the windward train passed the measure point when a negative pulse and a positive pulse were generated, namely a tail wave.

2) During two high-speed trains passing each other, the head wave of the pressure full wave was more than the tail wave; absolute values of the negative wave were more than those of the positive wave, which was more obvious in the tail train.

3) On the cross section, peak values of each measure point were characterized in that: peak value distribution of the head wave was 11 > 14 > 13 > 12; peak value distribution of the tail wave was 11 > 14 > 13 > 12 (other measure points on the cross section had the same distribution); obviously, pressure peak values changed more obviously on the train meeting side (namely windward side), the train bottom position ranked the second place, the non-train meeting side (namely leeward side) ranked the third place, and the train top position ranked the last.

4) Along the running direction, head wave and tail wave peak values of measure points in the same longitudinal direction were decreased, where the measure point ot-11 (near the lateral window) had the maximum peak value; the head wave peak value was 1486 Pa; the tail wave peak value was 1279 Pa, as shown in Table 1.

5) During two high-speed trains passing each other, the pressure region on the meeting side was more than that on another side; negative pressures with large-area appeared on the non-train meeting side, train top and bottom; fewer positive pressure regions appeared on the meeting side compared with another side. Obviously, breaking of the lateral window glass during two high-speed trains passing each other is caused by that the window glass would be sucked out by the meeting negative pressures, where the breaking is not caused by positive pressure impacts.

6) Positive wave amplitudes of head waves on the windward side were positive pressures, and negative wave amplitudes of head waves were negative pressures; during two high-speed trains passing each other under crosswinds, positive wave and negative wave amplitudes of head waves on measure points of the top, non-train meeting side and bottom were negative pressures. Therefore, the high-speed train had the large lateral force and head shaking force moment.

7) Positive wave amplitudes of tail waves on each measure point were negative pressures (except for measure points on the windward side), and negative wave amplitudes of tail waves were negative pressures; in tail waves, the measure point on the windward side had the largest pressure peak value 1279 Pa, the pressure peak value on the measure point of the bottom ranked the second place, the peak value of measure points on the non-train meeting side ranked the third place, and the measure point on the top had the minimum pressure peak value. It is thus clear that the head train and tail train were affected by crosswinds seriously, and they had the poorest running safety performance under crosswinds.

4.2. Aerodynamic forces (force moment) of two high-speed trains passing each other under crosswinds

After the flow field of high speed trains was computed, aerodynamic loads which acted on the high-speed train under crosswinds should be loaded into the multi-body dynamic model, and running safety of the high-speed train can be analyzed. According to force translation and equivalence principles, distributed pressures acting on the train surface were simplified to a point of the train. Therefore, concentrated force and force moment acting on this point were obtained. Through simplification, resistance, lateral force, lift force, overturning force moment, head shaking force moment and nodding force moment acting on each train under different wind direction angles, different wind velocities and different running speeds could be obtained. In those reported researches, aerodynamic forces were basically simplified into stable loads. For example, Liu [26, 27] has applied stable aerodynamic loads to the multi-body dynamic model.

Fig. 15Time-history curves of each measure point on cross sections

a) Measure points on cross section 1

b) Measure points on cross section 2

c) Measure points on cross section 3

d) Measure points on cross section 4

e) Measure points on cross section 5

f) Measure points on cross section 6

g) Measure points on cross section 7

However, using the stable load as the excitation and applying it to the multi-body dynamic model cannot satisfy actual situations. Therefore, in this paper, random aerodynamic loads were applied to the high-speed train, as shown in Fig. 16. During 0-5 s, the aerodynamic force was loaded linearly, so running of the high-speed train gradually tended to be stable. Aerodynamic forces acted within 5-18 s. Then, the force reached 0 through linear unloading.

Fig. 16Two different kinds of loading methods

Computational coordinates of aerodynamic forces and force moment during two high-speed trains passing each other are shown in Fig. 17, where the $X$ axis is the resistance direction; $Y$ axis is the lift force direction; $Z$ axis is the lateral force direction; lateral rolling force moment was rotated around the $X$ axis; head shaking force moment was rotated around the $Y$ axis; nodding force moment was rotated around $Z$ axis. Simplified centers of aerodynamic forces and aerodynamic force moments in this paper are shown in Fig. 17, where the center was 2.1 m high above the ground; the center of the head train was 12.5 m away from the windshield; the center of the mid-train was also 12.5 m away from the windshield; the center of the tail train was 12.5 m away from the windshield; the force moment center was arranged on the center face in the longitudinal direction.

Fig. 17Schematic diagram of the aerodynamic force and moment

Fig. 18 shows change curves of resistance, lift force and lateral force on the leeward train during two high-speed trains passing each other under crosswinds. It is shown in Fig. 18 that: (1) Resistance characteristics: During two high-speed trains passing each other, resistance changed violently on the head train due to crosswinds, opposite resistance was borne during tail train meeting, resistance of the mid-train did not change obviously throughout the train meeting, and the tail train bore the maximum resistance with the amplitude 7000 N. (2) Lift force characteristics: During two high-speed trains passing each other under crosswinds, lift forces of the head train changed violently, where it bore the downward lift force before two high-speed trains passing each other and bore upward lift force during tail train meeting; both the mid-train and the tail train bore the upward lift force. (3) Lateral force characteristics: Lateral force values caused by the head train and tail train had the largest effects, and the lateral force peak of head waves in the head train was more than that of tail waves; when the windward train head passed the leeward train, a positive lateral force peak was caused by the head train, mid-train and tail train of the leeward train firstly, and then the lateral force was sharply decreased to a negative peak value; when the windward train tail passed the head train of the leeward train, the leeward train generated a negative peak value at first, and then the lateral force was increased to a positive peak value sharply.

Fig. 18Time-history curves of aerodynamic forces of each train body

a) Comparison of resistance

b) Comparison of lift forces

c) Comparison of lateral forces

Fig. 19 shows time-history curves of lateral rolling force moments on the leeward train during two high-speed trains passing each other under crosswinds. It is shown in Fig. 19 that: When the head or tail parts of the windward train passed the leeward train, the lateral rolling force moment of the leeward train had 3 peaks. The head train of the leeward train generated a negative peak at first. Next, when the windward train head passed 1/4 train length, the lateral rolling force moment was increased to the peak sharply and then experienced a negative peak with the opposite direction. When the windward train tail passed the head train of the leeward train, the leeward train generated a peak value of upward lateral rolling force moment at first; then, the lateral rolling force moment was decreased to a negative peak sharply; next, the lateral rolling force moment was increased to a positive peak sharply. Lateral rolling force moments of the complete train experienced 6 direction changes.

Fig. 19Time-history curves of lateral rolling force moments of each train body

Obviously, during two high-speed trains passing each other, each train body experienced 6 impact force moments with outward, inward, outward, inward, outward and inward directions. During two high-speed trains passing each other, the head train had the largest lateral rolling force moment amplitude; the tail train ranked the second place; the mid-train ranked the last. Therefore, in the subsequent dynamic analysis, only the safety performance of the head train was analyzed.

4.3. Running safety of two high-speed trains passing each other under crosswinds

According to Reference [28, 29], dynamic evaluation indexes of the train include derailment coefficient, reduction rate of wheel load and overturning coefficient. Derailment coefficient refers to the ratio $Q/P$ of horizontal forces $Q$ to vertical forces $P$ acting on wheels at a moment, where the evaluation limit value of the derailment coefficient is $(Q/P{)}_{\mathrm{l}\mathrm{i}\mathrm{m}}=$ 2.0; the reduction rate of wheel loads is defined as $\mathrm{\Delta }P/\stackrel{-}{P}$, where $\mathrm{\Delta }P$ is the reduction number of wheels on one side, and $\stackrel{-}{P}$ is average wheel loads on left and right sides. The limit value $(\mathrm{\Delta }P/\stackrel{-}{P}{)}_{\mathrm{l}\mathrm{i}\mathrm{m}}=$ 1.2; while a train is running along a rail, the train would bear various types of horizontal forces including wind forces, centrifugal forces, gravity and horizontal inertia forces. Under impacts of these forces, the train may overturn. The limit value of overturning coefficient is $(D{)}_{\mathrm{l}\mathrm{i}\mathrm{m}}=$ 1.0.

Fig. 20 shows a comparison of derailment coefficient, reduction rate of wheel loads and overturning coefficient of each train body under the working conditions: crosswind velocity is 15 m/s and running speed is 350 km/h. It is shown in Fig. 20 that the head train was the most unsafe, where the derailment coefficient was 1.9, the reduction rate of wheel loads was 1.1 and the overturning coefficient was 0.98, all of which were smaller than the limit value. Therefore, the high-speed train still had a high safety performance under the working conditions.

Fig. 20Analysis on running safety of each train body