1 Introduction

Global warming has intensified the frequency and intensity of typhoons, posing significant challenges to the infrastructure development of coastal cities worldwide, including projects such as high-speed railways [1]. The continual expansion of coastal railway transportation networks has increased the vulnerability to damage from extreme weather events. The safe operation of high-speed railways is significantly impacted by extreme natural phenomena, including powerful winds and heavy rainfall caused by tropical storms, which have emerged as significant risk factors [2, 3]. As shown in Fig. 1, the rapid transitions between geographical units have rendered the tunnel-embankment transition section a vulnerable aspect of high-speed rail safety. Thus, it is the primary task to understand the mechanism of wind–rain and train and the key to ensure the safe operation of high-speed trains to analyze the nonlinear spatiotemporal variations of train flow around and wake, especially in the case of wind–rain coupling.

Fig. 1
figure 1

The high-speed train runs in the tunnel-embankment transition section under wind and rain

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Although many scholars have studied the flow around and wake of blunt bodies with regular shape in the early stage [4], the flow around and wake of high-speed trains have only been extensively studied by scholars through numerical simulation and wind tunnel tests in recent years. Bell et al. [5, 6] introduced a wind tunnel testing technique to assess the slipstream and wake of high-speed trains, investigating the transient characteristics of the wake through frequency distribution and probability distribution. Chen et al. [7] investigated the differences in flows on the windward and leeward sides of trains with different nose lengths under crosswind by detached-eddy simulations. Zhou et al. [8] utilized an improved delayed detached eddy simulation and overlapping grid technique to investigate the influence of crosswind velocity on the transient vortex structure, pressure distribution, aerodynamic loads on trains, and safety indicators. Based on Shear-Stress Transport (SST) k-w turbulence model and moving model tests, Wang [9] analyzed the flow field evolution of a high-speed train exiting a tunnel under crosswind conditions, and compared the aerodynamic performance of the head, middle and tail trains. The above scholars only studied the flow around and wake of high-speed train under pure wind field, but there are few reports under the wind–rain coupling.

In current research on two-phase flow involving wind and rain, the predominant approaches involve computational fluid dynamics (CFD) based on Eulerian multiphase (EM) models or conducting wind tunnel experiments with spray devices [10]. Wind-driven rain (WDR) refers to the phenomenon where raindrops, influenced by the force of the wind, exhibit horizontal movement or acquire horizontal velocity in the air. Rainwater intrusion onto building facades and the structural response to raindrop impact in typhoon conditions are frequently evaluated using the WDR simulation technique [11, 12]. Under the influence of WDR, attention often focuses on the flow fields around static structures such as floating wind turbines [13], bridges [14, 15], and residential buildings [16]. Some scholars have observed the aerodynamic effect of high-speed trains under the wind–rain coupling. By employing the Euler- Lagrange method with non-spherical raindrops, Yu et al. [17,18,19] extensively explored the aerodynamic impacts on high-speed trains under wind–rain coupling. The results indicate a strong correlation between aerodynamic loads and rainfall intensity as well as wind speed. Zeng and Shao [20,21,22] utilized the EM model and k-ε turbulence models to simulate trains on viaducts or flat grounds in the wind–rain environment. They conducted an analysis of the aerodynamic effects on trains with varying wind speed and rainfall intensity, evaluating the safety of train operations in the process. Gou et al. [23] expanded their research to explore the effects of wind and rain on the dynamic response system of the vehicle, going beyond just analyzing aerodynamic loads on trains. The results suggest a positive correlation between rainfall intensity and the dynamic response of trains, indicating that as rainfall intensifies, the dynamic response of trains also increases.

The scholars above have undertaken varying degrees of research on the aerodynamic load variations of high-speed trains and safety assessments under wind–rain coupled environments, achieving certain outcomes. However, some issues remain unresolved. Firstly, their research has primarily focused on the aerodynamic characteristics of trains, neglecting the nonlinear spatiotemporal characteristics of the wind–rain flow field when trains are in motion. In particular, the nonlinear characteristics of the flow around and behind trains under wind–rain coupling have yet to be unveiled [17, 20]. Secondly, in the transitional section between embankment and tunnel with multiple-unit conversion, the existence of a portal acceleration effect has been demonstrated [24, 25]. In comparison to a single train, the interaction of trains intersection at a tunnel portal exacerbates the evolution characteristics of the flow field [2, 26]. Therefore, investigating the spatiotemporal evolution characteristics of the flow field around trains in wind–rain coupled environments is deemed particularly crucial. With this objective in mind, the paper seeks to elucidate and explore the aforementioned domain. Specifically:

  • EM model and SST k-ω turbulence model are used to study the flow around and wake of high-speed trains under wind and rain flow field, highlighting the influence of rainfall intensity on flow field.

  • The wind and rain flow field characteristics of two scenarios, including slipstream velocity, slipstream pressure, turbulent flow characteristics, and water film distribution, are compared between a single train running on the embankment and trains intersection at the tunnel portal. The coupling relationship among wind, rain, and the train is analyzed from multiple perspectives.

  • The purpose of this study is to provide guidance for running safety in the transition section between embankment and tunnel under wind and rain, and lay a foundation for formulating effective weatherproof measures.

The remaining chapters of this paper are as follows: Sect. 2 presents the numerical methodology, Sects. 3 and 4 discuss the wind–rain flow field of a single train and trains intersection, respectively, and Sect. 5 presents conclusions.

2 Numerical model

2.1 Rain phase model

Currently, the simulation of WRD commonly employs the discrete particle model (DPM) and the Eulerian multiphase flow model. However, the large computational resource requirements of DPM hinder its wide application in large-scale engineering numerical models. To address this issue, this study opts for the Eulerian multiphase flow model, a choice previously utilized by researchers investigating the aerodynamic performance of trains in windy and rainy conditions [19, 21]. Given that raindrops larger than 6 mm often break into smaller droplets in the air, the rain phase's particle size is categorized into six groups to facilitate rain phase simulation. Each group's representative value is the intermediate particle size; for instance, the second group, with a particle size range of 1 to 2 mm, is represented by 1.5 mm. Each group is represented by its intermediate particle size. For instance, the second group spans particle sizes from 1 to 2 mm, with a representative size of 1.5 mm. The equations for the mass and momentum of the rain phase are denoted as Eqs. (1) and (2) [27], and the relationship with the volume fraction of the two-phase flow is denoted by Eq. (3).

$$ \frac{\partial \rho_w \alpha_k }{{\partial t}} + \frac{{\partial (\rho_w \alpha_k u_{k,j} )}}{\partial x_j } = 0, $$
(1)
$$ \frac{{\partial \rho_w \alpha_k u_{k,i} }}{\partial t} + \frac{{\partial (\rho_w \alpha_k u_{k,i} u_{k,j} )}}{\partial x_j } = \rho_w \alpha_k g + \rho_w \alpha_k \frac{3\mu }{{\rho_w D^2 }}\frac{{C_D {\text{Re}}_p }}{4}(u_i - u_{k,i} ), $$
(2)
$$ \sum_{k = 1}^n {\alpha_k } + \alpha_a = 1, $$
(3)

where \(\alpha_{\text{k}}\) and \(\alpha_a\) are the volume fraction of kth rain phase and air phase, respectively; \(\rho_w\) is the density of raindrop; \(u_{k,i}\), \(u_{k,j}\) is the velocity of the i and j directions of kth rain phase, respectively; g is the gravitational acceleration; \(\mu\) is the dynamic viscosity; \(D\) is the raindrop diameter; \(C_D\) is the drag coefficient; \({\text{Re}}_p\) is the relative Reynolds number.

Raindrop spectrum is related to rainfall intensity. The Marshall–Palme (M–P) distribution, initially proposed as a raindrop spectrum, serves as the basis for the modified Gamma distribution. To enhance the accuracy of rain phase simulation, this paper employs the modified Gamma distribution form, as illustrated in Eqs. (4) and (5) [28, 29].

$$ N{(}D{) = }N_{0} D^{2.93} \exp ( - 5.38I^{ - 0.186} D), $$
(4)
$$ N_{0} = 1.98 \times 10^{ - 5} I^{ - 0.384} , $$
(5)

where I is the rain intensity.

Key parameters for simulating WDR include rainfall intensity (I), volume fraction of rain phase (\(\alpha_k\)), and terminal velocity of the raindrop (\(V_t (D)\)). Given the susceptibility of coastal areas to typhoons, heavy rainfall can reach up to 400 mm/h in a brief timeframe [30]. Therefore, this paper takes no rain as a contrast, considering the two kinds of intensity, one is 200 mm/h, the other is 400 mm/h. Then, the volume fraction of rain phase with different diameters is derived by using Eqs. (6) and (7) through the determined rainfall intensity. Equation (8) [31] is then employed to ascertain the terminal velocity of the raindrop.

$$ \alpha_k = \frac{{If\left( {I,D} \right) \times 10^{ - 3} }}{V_t (D) \times 3600}, $$
(6)
$$ f(I,D) = \frac{{D^3 N(I,D)V_t (D{)}}}{{\int_0^\infty {D^3 N(I,D)V_t (D{)}dD} }}, $$
(7)
$$ V_t (D) = \left\{ \begin{gathered} 0,D \le {0}{\text{.03 mm}} \hfill \\ 4.323(D - 0.03);0.03{\text{ mm}} < D \le {0}{\text{.6 mm}} \hfill \\ 9.65 - 10.3e^{ - 0.6D} ;D > 0.6{\text{ mm}} \hfill \\ \end{gathered} \right.. $$
(8)

2.2 Turbulence model

Although improved delayed separation eddy simulation (IDDES) and large eddy simulation (LES) can better capture turbulence details, they are usually used in scale models because of strict requirements on wall mesh size and large computational resource consumption. Compared with SST k-ω turbulence model, Renormalization Group (RNG) k-ε turbulence model is not suitable for observing the development of train flow around and wake because of its weak ability in wall shunt simulation. The SST k-ω turbulence model is chosen for simulating wind and rain flow due to its effective performance and manageable computational resource requirements.

2.3 Geometry and boundary conditions

The three-dimensional CFD geometric model for wind–rain flow structures is presented in Fig. 2. The train model CRH380B is selected, with the bogies and pantographs disregarded during modeling for simpler grid generation. The train consists of head, middle, and tail sections. The dimensions of the train are as follows: length L = 25 m, width W = 3.3 m, height H = 3.9 m. Other dimensional features of the model are represented dimensionlessly by L. Two trains running in opposite directions are defined as an observation train (Obs. Train) and a reference train (Ref. Train) respectively, with a speed of 350 km/h. In order to ensure that WDR can hit the windward side of the train, the length, width and height of the calculation domain are 21.2 L, 7.2 L and 1.6 L, respectively. The length of the embankment section is 10.6 L, the slope rate is 1:1.5, and the road width is 0.55 L. The length of the tunnel section is 10.6 L, and it is a standard 100 m2 double-train single-hole tunnel with a hat oblique tunnel portal. The coordinate system originates from the nose of the Obs. train at its initial position, which is located 4.5 L away from the tunnel portal. The inflow and top surfaces of the calculation domain are designated as Velocity-inlet boundaries for wind–rain phase control; rail, embankment, train, and tunnel are set as No-slip wall boundaries; remaining boundaries of the calculation domain are specified as Pressure-outlet boundaries.

Fig. 2
figure 2

Numerical model: a geometric model and grid display; b monitoring scheme

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Figure 2b illustrates the distribution of slipstream velocity and pressure monitoring lines for trackside and platform heights as required by the Technical Specifications for Interoperability (TSI). The distances from the rail top to the trackside and platform are 0.2 m and 1.4 m, respectively. Three vertical sections (L1, L2, and L3) are positioned laterally, with distances of 2 m, 2.5 m, and 3 m from the Obs. Train in the lateral direction. Notably, L2 is aligned with the centerline of the track. Therefore, two heights and three planes define six monitoring lines, namely TH-L1 (L2, L3) and PH-L1 (L2, L3).

2.4 Grid strategy

The specific numerical model grid details are shown in Fig. 2a. The strategy of combining dynamic grid and static grid is used in the simulation of wind and rain flow train intersection. Interface is used to transfer information at the junction of two grid areas. The dynamic grid method is described in detail in Liu et al. [33]. To capture the fine flow around and wake of the train, the grid is locally encrypted in two zones (Zone I and Zone II) along the high-speed rail. Zone I has dimensions of 22 L × L × 0.5 L with a restricted grid size of 0.3 m, while Zone II has dimensions of 22 L × 0.7 L × 0.4 L with a restricted grid size of 0.1 m. The train surface grid size is 0.02 m, featuring 10 layers of boundary layer grids. The first layer, with a thickness of 8 × 10–4 L (y + ≈ 30), expands outward at a ratio of 1.2. The total number of grids in the model is 54 million. The second-order implicit transient is solved using the SST k-ω turbulence model. A time step of t* = 2 × 10–4 s (with a Courant number less than 1) is utilized, iterated 20 times per step. The calculations are performed on a supercomputer with 320 cores over a two-week period. The simulation parameters are summarized in Table 1.

Table 1 Simulation parameter introduction
Full size table

2.5 Research route and data processing

To reveal the evolutionary interaction between wind–rain flows and trains by analyzing the flow around and the wake of high-speed trains under different rainfall conditions, three rainfall intensity (I) classes (0, 200, and 400 mm/h) are chosen, with an inflow velocity (U∞) set at 20 m/s, named as Case 1, Case2 and Case3, respectively, as shown in Table 2. Two scenarios of train running are examined: one involving a train running on the embankment (Sect. 3), and the other depicting two trains intersection at the tunnel portal (Sect. 4). Turbulence intensity and Reynolds stress are important parameters to describe the instability and vortex effect in fluid motion. During the process of train running in wind and rain, the rain phase may affect the airflow state around the train, causing the train to be disturbed by external wind and rain, leading to increased train jitter and shaking, which affects the stability and safety of the train. Therefore, discussions are conducted on the distribution characteristics of the flow, turbulence indices (turbulence intensity, turbulence kinetic energy (TKE), and Reynolds stress), and water film distribution in the flow around and wake of train in this study.

Table 2 Scheme parameter
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In order to facilitate simplification, the aerodynamic pressure and velocity are treated non-dimension [34], which is defined as follows:

$$ C_p = \frac{P - P_0 }{{0.5\rho U_\infty^2 }} $$
(9)

where \(C_p\) is the pressure coefficient; \(P\) is the aerodynamic pressure; \(P_0\) is the atmospheric pressure; \(\rho\) is the air density.

2.6 Verification

To validate the accuracy of SST k-ω simulation in predicting the pressure distribution around the train, a moving model test conducted by Yang et al. [35] is modeled, as shown in Fig. 3a. The scale ratio of the moving model test is 1:16.8, featuring a semi-enclosed noise barrier model with dimensions of 24.4 m in length, 0.74 m in width, and 0.53 m in height. A simplified moving train model, resembling the CRH380B high-speed train prototype, consists of three carriages. The dimensions of the model are reduced to 4.7 m in length, 0.24 m in height, and 0.22 m in width. The train passes the noise barrier at a speed of 350 km/h. The pressure coefficient at verification point P1, located on the central wall surface of the noise barrier and 0.24 m above the track surface, is examined.

Fig. 3
figure 3

Slipstream pressure of verification: a modeling based on the test of [35]; b comparison of three grid resolutions and tests

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By adjusting the layers of the train's boundary layer mesh, three different grid resolutions—coarse (1100 million), medium (2400 million), and fine (3600 million)—are obtained. Figure 3b presents a comparison of the results between the three grid resolutions and the moving model test. It is observed from the figure that the trends in the SST k-ω simulation results for the three grid resolutions are generally consistent with the moving model test results, with slight variations in peak values. Specifically, compared to the moving model test results, the differences for coarse, medium, and fine grid resolutions are 5.8%, 7.5%, and 15.6% for \({\text{C}}_{{p - max}}\), and 5.5%, 8.1%, and 21.3% for \({\text{C}}_{{p - min}}\), respectively. In conclusion, relative to the coarse grid, the SST k-ω simulation with medium and fine grids more accurately reflects the pressure distribution around the train, with differences within 2%. Considering computational resources and efficiency, the medium grid resolution is employed in other simulations in this study.

3 Flow around and wake of the Obs. Train on the embankment

3.1 Characteristics of train flow around and wake

3.1.1 Slipstream velocity distribution

In the analysis moment of the train flow around and wake, the 1500t* is selected in the Sect. 3. At this point, with the distance between the tunnel entrance and the train's nose surpassing the train length, we can dismiss the impact of tunnel topography on the train's flow around and wake, and it is considered that only Obs. Train runs on the embankment at this time. Figure 4a and b illustrate the velocity distribution along the track height and platform height monitoring lines, respectively. Two zones are marked on the Figures: the flow development zone, encompassing from the train's head to the tail, and the wake propagation zone. Velocity is dimensionless, normalized to the incoming flow velocity (U∞).

Fig. 4
figure 4

Slipstream speed distribution of Obs. train in monitoring lines on the embankment: a velocity distribution of trackside height; b velocity distribution of platform height; c Maximum velocity of the flow development zone; d Maximum velocity of the wake propagation zone

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From Fig. 4a and b, it is evident that the velocity distribution on the same monitoring line is basically the same, decreasing as the lateral distance from the train increases. Figure 4c and d present the velocity peaks of the flow development zone and wake propagation zone under single train operating conditions. In the flow development zone, the velocity peak differences among the three cases are small, especially for Case2 and Case3. On vertical section L1, the velocity peak increases and then decreases with increasing rainfall intensity. For example, at monitoring line TH-L1, the differences between Case1 and Case2, Case3 are 3.6% and 3%, respectively. On vertical section L3, the velocity peak decreases with increasing rainfall intensity. For instance, at monitoring line PH-L3, the differences between Case1 and Case2, Case3 are -3.1% and -3.8%, respectively. In the Wake Propagation zone, the velocity peaks on all three vertical sections (L1, L2, and L3) increase and then decrease with increasing rainfall intensity.

Additionally, there are differences in the lateral decay rate of the slipstream velocity peak for the three conditions. In the flow development zone, compared to vertical section L1, the velocity peaks at trackside height for Case1, Case2, and Case3 on vertical section L3 decrease by 30.4%, 36%, and 36.5%, respectively, while the velocity peaks at platform height decrease by 34.2%, 40.1%, and 40.4%, respectively. In the wake propagation zone, compared to vertical section L1, the velocity peaks at trackside height for Case1, Case2, and Case3 on vertical section L3 decrease by 2.8%, 3.8%, and 5.1%, respectively, while the velocity peaks at platform height decrease by 3.5%, 4.5%, and 5.3%, respectively. Hence, it can be observed that rainfall accelerates the lateral decay of the slipstream velocity.

3.1.2 Slipstream pressure distribution

In Fig. 5a and b, the impact of three rainfall intensities on the pressure distribution of the Obs. Train's flow around and wake is compared using pressure at track and platform heights. The pressure distribution patterns along the same vertical section are generally consistent. Typically, as the train's nose passes, there is a rapid increase in pressure, followed by a quick decrease due to the streamlined shape of the front, inducing an acceleration effect on local airflow. Similarly, a second peak occurs at the rear of the train [36]. From vertical section L1 to L3, the pressure at both trackside and platform heights decreases with increasing distance from the train. Figure 5c and d provide the peak velocity profiles in the flow development zone and wake propagation zone for the Obs. Train. Pressure increases and then decreases with the intensity of rainfall. For instance, in the PH-L1 profile, the differences between Case1 and Case2, and Case3 are 15.1% and 10.9%, respectively, while the difference between Case2 and Case3 is only 3.7%.

Fig. 5
figure 5

Slipstream pressure distribution of Obs. train in monitoring lines on the embankment with different rainfall intensity: a–c trackside height; d–f platform height; c Maximum pressure of the flow development zone; d Maximum pressure of the wake propagation zone

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Furthermore, different rainfall intensity result in variations in the lateral decay rates of slipstream pressure peaks. In the Flow Development zone, compared to vertical section L1, the trackside pressure peaks for Case1 to Case3 decrease by 20.6%, 8.8%, and 8.8%, respectively, while platform height pressure peaks decrease by 24.1%, 6.3%, and 5.7%. In the Wake Propagation zone, relative to vertical section L1, the trackside height pressure peaks for Case1, Case2, and Case3 decrease by 33.3%, 16.2%, and 16.1%, respectively, while platform height pressure peaks decrease by 32.2%, 23.5%, and 23.7%. This indicates that rainfall can weaken the lateral decay of slipstream pressure.

3.2 Turbulence intensity and turbulent kinetic energy

In Fig. 6, the distribution of turbulence intensity and TKE along the monitoring lines at the trackside and platform heights during the passage of the Obs. Train on the embankment is presented under different rainfall intensity. The TKE is non-dimensionalized, and U∞ represents the incoming flow velocity. Figure 6a–c illustrate the turbulence intensity and TKE at the trackside height, while Fig. 6d–f depict the corresponding values at the platform height.

Fig. 6
figure 6

Turbulence intensity and turbulent kinetic energy distribution of Obs. Train in monitoring lines on the embankment with different rainfall intensity: a1–f1 turbulence intensity; a2–f2 turbulent kinetic energy

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From Fig. 6, it is evident that, under varying rainfall intensity along the same lateral line, the trends in turbulence intensity and TKE are generally consistent, particularly in the wake propagation zone where differences are negligible. With increasing rainfall intensity, differences emerge in the peak values of turbulence intensity and TKE in the flow development zone. For instance, at the position (x/H =  − 4.5) along the monitoring line TH-L1, compared to Case1, the turbulence intensity decreases by 1.7%, 6.7% for Case2 and Case3, respectively, and the TKE decreases by 8.7%, 13%. Similarly, at the position (x/H =  − 4.1) along the monitoring line PH-L1, compared to Case1, the turbulence intensity decreases by 4.4%, 4.7% for Case2 and Case3, respectively, and the TKE decreases by 8.6%, 9.7%. This also indicates that, relative to turbulence intensity, rainfall intensity has a greater impact on TKE.

3.3 Reynolds stress

Figure 7 shows the cross-section cloud maps of longitudinal, transverse, vertical Reynolds normal stress (\({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\), \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\) and \({{\overline{u_z^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\)) and Reynolds shear stress (\({{\overline{u_y^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\)) distribution on the leeward side of the middle train when Obs. Train runs on the embankment under different rainfall intensity. The Reynolds stresses have been non-dimensionalized, with U∞ representing the incoming flow velocity. As illustrated in Fig. 7, it is evident that \({{\overline{u_z^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\) significantly exceeds \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\) and \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\), indicating an anisotropic state on the leeward side of the high-speed train [37]. With increasing rainfall intensity, localized variations in the Reynolds stresses around the flow of the high-speed train occur. Regions undergoing changes are delineated with dashed lines in Case 1, forming contour lines marked as Zone I to Zone V. To facilitate the comparison of Reynolds stress distribution on the leeward side of the train under different rainfall intensity, the same dashed lines as in Case 1 are positioned at corresponding locations in Case 2 and Case 3.

Fig. 7
figure 7

Reynolds stress distribution of the Obs. Train on the embankment with different rainfall intensity: a \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\); b \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\); c \({{\overline{u_z^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\); d \({{\overline{u_y^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\)

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As shown in Fig. 7a, influenced by the surface attached boundary layer of the train, the airflow near it is faster, and \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\) is larger on the surface of the train. When crosswinds pass over the train, a recirculation zone forms on the leeward side, causing a change in the \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\). In Zone I and Zone II, rainfall weakens the velocity in the near-ground region but increases the velocity of the x-component on the leeward side of the train. However, the increase in rainfall intensity from 200 to 400 mm/h has almost no effect on \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\). As shown in Fig. 7b, rainfall increases the velocity of the y-component on the leeward side, causing an increase in \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\) in Zone III. As shown in Fig. 7c, under the action of rainfall, the recirculation zone \({{\overline{u_z^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\) on the leeward side of the train decreases. As shown in Fig. 7d, the impact of rainfall on the \({{\overline{u_y^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\) on the leeward side of the train is minimal, and the range of Zone V decreases. In conclusion, with the change from no rain to rain, the velocities of the x-component, y-component, and z-component on the leeward side of the train also increase. However, the effect of rainfall intensity increasing from 200 to 400 mm/h on the Reynolds stress is weak.

3.4 Distribution of water film

The distribution of rain phase is closely related to the nonlinear characteristics of high-speed train flow around and wake. In order to study the relationship between the train flow around and wake and the distribution of rain phase, Fig. 8 shows the distribution of water film under different rainfall intensity when Obs. Train runs on the embankment. The spatial distribution of the rain phase is represented by the isosurface of the volume fraction of the rain phase, and the dimensionless velocity U/U∞ is used for dyeing.

Fig. 8
figure 8

Distribution of water film of the Obs. Train on the embankment with different rainfall intensity: a1–c1 leeward side; a2–c2 windward side; a3–c3 Z + view of the wake area

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As shown in Fig. 8a1–c1, the distribution of rain phase on the surface of the head train is much larger than that of the middle train and the tail train. With the increase of rainfall intensity, the distribution range of rain phase on the surface of the embankment and the surface of the train becomes larger and larger. When the rainfall intensity increases to 400 mm/h, a water film extending from the roof to the bottom of the train is formed on the leeward side of the train. The width of the water film decreases from front to back, and the distribution range of rain phase in the wake area increases. When the rainfall intensity increases from 0 to 200 mm/h, the volume fraction of rain phase increases and the volume fraction of gas phase decreases correspondingly, resulting in the increase of slipstream velocity on the leeward side. When the rainfall intensity further increases to 400 mm/h, water film forms on the leeward side, weakening the velocity of the backflow area on the leeward side, resulting in a decrease in the slipstream velocity. As shown in Fig. 8a2–c2, there is little difference on the windward side under different rainfall intensity conditions. With the increase of rainfall intensity, the distribution range of rain phase on the roof of the train increases. As can be seen from Fig. 8a3–c3, with the increase of rainfall intensity, the thickness of water film on the embankment increases, and a water film perpendicular to the road surface is formed, which has an accelerating effect on the wake under the action of crosswind.

4 Flow around and wake of the train intersection at the tunnel portal

4.1 Characteristics of train flow around and wake

4.1.1 Slipstream velocity distribution

In Sect. 4, the moment when the middle trains of two trains intersection at the tunnel portal is selected as the analysis moment. Figure 9a and b present the velocity distribution along the monitoring line for both the track height and the platform height at the same location. Although the velocity distribution patterns along the same line are generally consistent, some complex fluctuations are observed compared to the single-train condition.

Fig. 9
figure 9

Slipstream velocity distribution of train intersection in monitoring line at the tunnel portal with different rainfall intensity: a velocity distribution of trackside height; b velocity distribution of platform height; c Maximum velocity of the flow development zone; d Maximum velocity of the wake propagation zone

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For the analysis conducted at the intersection position (x/H = 29.5), Fig. 9a and b highlight the velocity peaks at the intersection position with blue open circles. With increasing rainfall intensity, the velocities at the intersection positions on the trackside height monitoring lines TH-L1 and TH-L2 initially increase and then decrease, while the velocity at the crossing position on TH-L3 decreases. Conversely, the velocities at the intersection positions on the platform height measuring lines initially decrease and then increase. Moreover, compared to trackside height, the velocity variations at platform height are more significant.

Taking the vertical section L1 monitoring point for trackside height and platform height as an example, as rainfall intensity increases, the velocity changes at the intersection positions for TH-L1 in Case 2 and Case 3 are within 3% of those in Case 1. However, for PH-L1, the velocities at the intersection positions in Case 2 and Case 3 change by 23% and 10.7%, respectively, relative to Case 1. As the lateral distance between the monitoring line and the train increases, the velocity at the intersection position decreases. The peak velocity position in the flow development zone shifts from the middle train to the tail train for the trackside position, while for the platform position, it shifts from the head to the tail of the train.

Figure 9c and d respectively present the peak velocities at monitoring lines in the flow development zone and the wake propagation zone of the Obs. Train under the conditions of trains intersection at the tunnel portal. In the flow development zone, the peak velocities at the monitoring line L1 initially increase and then decrease with increasing rainfall intensity. For example, for the PH-L1 measuring line, the differences between Case 1 and Case 2, Case 3 are 13% and 12.5%, respectively. In the wake propagation zone, rainfall affects the lateral distribution of the peak velocities on the track height monitoring lines. From TH-L1 to TH-L3, the peak velocity in Case 1 gradually increases, while in Case 2 and Case 3, it gradually decreases.

Similarly, there are differences in the lateral variation rate of the slipstream velocity under the three conditions at the intersection position. In the flow development zone, compared to vertical section L1, the peak velocities for trackside height in vertical section L3 decrease by 36.1%, 28.6%, and 26.1% for Case 1, Case 2, and Case 3, respectively, while platform height peak velocities increase by 14.7%, 2.6%, and 2.3%. In the wake propagation zone, compared to vertical section L1, the peak velocities for trackside height in vertical section L3 increase by 9.4%, − 9.2%, and − 8.3% for Case 1, Case 2, and Case 3, respectively, while platform height peak velocities decrease by 40%, 23.8%, and 28.9%. This indicates that under rainy conditions, the lateral variation of slipstream velocity during train crossing is more pronounced, suggesting that rainfall can weaken the lateral attenuation of slipstream velocity during train crossing in a wind–rain environment.

4.1.2 Slipstream pressure distribution

Figure 10a and b compare the influence of three kinds of rainfall intensity on the pressure distribution of the flow around and wake when the train intersects at the tunnel portal by using the pressure measured by the trackside height and platform height respectively. Compared with a single train, the slipstream pressure of a train intersects increases significantly. The pressure distribution patterns are generally consistent along the same vertical section, but in comparison to Case1 and Case3, the positive peaks around the train's head in Case2 decrease, and the negative peaks increase. Figure 9c and d present the peak velocities along the monitoring lines in the flow development zone and wake propagation zone, respectively, during the train intersection at the tunnel portal. In the flow development zone, the pressure peak in Case2 is smaller than in Case1 and Case3, and as the distance from the Obs. Train increases, the pressure peaks in Case2 and Case3 gradually decrease. However, Case1 shows a slight decrease followed by an increase in the pressure peak with increasing distance. In the wake propagation zone, the pressure peaks for all three cases gradually increase with distance from the Obs. Train.

Fig. 10
figure 10

Slipstream pressure distribution of train intersection in monitoring line at the tunnel portal with different rainfall intensity: a trackside height; b platform height; c Maximum pressure of the flow development zone; d Maximum pressure of the wake propagation zone

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Similarly, there are differences in the lateral variation of slipstream pressure under different rainfall intensity for the intersection conditions. In the flow development zone, compared to vertical section L1, the trackside height pressure peaks for Case1, Case2, and Case3 at vertical section L3 decrease by 0.3%, 5.7%, and 3.5%, respectively, while the platform height pressure peaks decrease by 0.6%, 6.5%, and 4.1%, respectively. In the wake propagation zone, compared to vertical section L1, the trackside height pressure peaks for Case1, Case2, and Case3 at vertical section L3 increase by 13.4%, 21.5%, and 20.6%, respectively, while the platform height pressure peaks increase by 8.5%, 13.3%, and 13.4%, respectively. This indicates that rainfall accelerates the lateral variation of slipstream pressure during the train intersect.

4.2 Turbulence intensity and turbulent kinetic energy

Figure 11 illustrates the distribution of turbulence intensity and TKE along monitoring lines at the trackside and platform heights when trains intersect at the tunnel portal under different rainfall intensity. Figure 11a–c represent turbulence intensity and TKE at the trackside, while Fig. 11d–f show corresponding values at the platform height. Compared to the single-train scenario, rainfall intensity has a more pronounced impact on turbulence intensity and TKE when trains intersect at the tunnel portal.

Fig. 11
figure 11

Turbulence intensity and TKE distribution of train intersection in monitoring line at the tunnel portal with different rainfall intensity: a1–f1 turbulence intensity; a2–f2 TKE

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From Fig. 11, it is apparent that the basic trends along the same monitoring line are similar, but there is a significant difference in the amplitude of fluctuations. In the wake propagation zone, the turbulence intensity and TKE of Case1 are noticeably higher than those of Case2 and Case3. For instance, on monitoring line TH-L1, relative to Case1, the peak turbulence intensity decreases by 20.5% and 18.5% for Case2 and Case3, respectively, and the TKE peak decreases by 36.7% and 33.8%. On monitoring line PH-L1, the turbulence intensity decreases by 21.4% and 20.4% for Case2 and Case3, respectively, and the TKE peak decreases by 38.2% and 37.1%.

Within the flow development zone, the differences in turbulence intensity and TKE between trackside and platform heights are more pronounced under the influence of rainfall intensity. For example, as shown in Fig. 11b1 and b2, the differences in peak turbulence intensity and TKE among the three Cases in TH-L2 are within 10%, while in Fig. 11e1 and e2, the differences in turbulence intensity and TKE peaks among the three Cases in PH-L2 reach around 40%.

4.3 Reynolds stress

Figure 12 presents cloud maps of longitudinal, lateral, vertical Reynolds normal stress, and Reynolds shear stress on the leeward side cross-section of the middle train when trains intersection at the tunnel portal under different rainfall intensity. Compared to a single train, the influence of rainfall intensity on the Reynolds stress distribution during train intersection is more pronounced.

Fig. 12
figure 12

Reynolds stress distribution of the train intersection at the tunnel portal with different rainfall intensity: a \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\); b \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\); c \({{\overline{u_z^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\); d \({{\overline{u_y^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\)

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In Fig. 12a, as the two trains intersection at the tunnel portal, \({{\overline{u_x^{\prime} u_x^{\prime} }} / {U_{_\infty }^2 }}\) in the region between the two trains is larger due to the influence of the boundary layer of both trains. The x-component velocity at the top of both trains and the leeward side of the Ref. train initially increases and then decreases, indicating that the velocity of this x-component at that position increases and then decreases with increasing rainfall intensity. In Fig. 12b, with increasing rainfall intensity, the region between the two trains gradually decreases in \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\), and \({{\overline{u_y^{\prime} u_y^{\prime} }} / {U_\infty^2 }}\) of the leeward side of the Ref. train first increases and then decreases. In Fig. 12c, compared to a single train, the trains intersection at the tunnel portal more effectively block the crosswind. The crosswind flows around the trains, forming a recirculation zone on the leeward side. This region expands compared to the conditions with a single train. With the increase of rainfall intensity, the \({{\overline{u_z^{\prime} u_z^{\prime} }} / {U_\infty^2 }}\) of backflow area on the leeward side decreases first and then increases. In Fig. 12d, the z-component on the leeward side of Ref. train in Case 1 and Case 2 is significantly larger than in Case 3.

4.4 Distribution of water film

Figure 13 shows the distribution of water film under different rainfall intensity when trains meet at the tunnel entrance. As can be seen from Fig. 13, with the increase of rainfall intensity, water film is attached to the surface of embankment, tunnel and train, especially the distribution of rain phase on the roof of the train increases significantly. Compared with single-trian condition, the crosswind is accelerated at the tunnel portal, and the rain phase distribution of train flow around and wake increases significantly when the train intersects. The distribution of water film on the lee side of Obs. Train is small, which forms a relatively closed area with the water film on the windward side and roof of Ref. train. With the increase of rainfall intensity, the sealing degree of this region is greater, which weakens the influence of slipstream action of Ref. train on the flow around Obs. Train, thus slowing down the lateral attenuation of the slipstream speed of Obs. As can be seen from Fig. 13a3–c3, the distribution of rain phase in the train wake area increases with the increase of rainfall intensity, the air phase distribution decreases, and the range of rain film formed increases.

Fig. 13
figure 13

Distribution of water film of the train intersection at the tunnel portal with different rainfall intensity: a1–c1 leeward side; a2–c2 windward side; a3–c3 Z + view of the wake area

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5 Conclusions

Based on the EM model, SST k-ω turbulence model, and slip grid technique, the study compares and analyzes the nonlinear spatiotemporal characteristics of the flow around the train and its wake when running on an embankment and passing through a tunnel portal under different rainfall intensity. The main conclusions are as follows:

  1. 1.

    Different rainfall intensity has a weak influence on the velocity distribution of train flow around and wake, but has an impact on the peak of velocity. Under single-train condition, with the increase of rainfall intensity, the transverse velocity attenuation on the leeward side increases. Under the condition of intersection of two trains at the tunnel portal, there are differences in the peak of slipstream velocity under different rainfall intensity, and the transverse velocity changes on the leeward side are more drastic under no rain.

  2. 2.

    The influence of different rainfall intensity on the pressure distribution of train flow around and wake is weak, but it has influence on the peak of slipstream pressure. Under single train condition, the peak value of the slipstream pressure is obviously smaller than that under rainy condition, and the transverse pressure decays faster on the leeward side. Under the condition of intersection of two trains at the tunnel portal, the increase of wake pressure peak is accelerated by rainfall.

  3. 3.

    The influence of rainfall intensity on turbulence intensity and turbulence kinetic energy is basically the same. Under single-train condition, different rainfall intensity has a weak influence on the turbulence intensity and turbulent kinetic energy of the train flow around and wake under single-train condition. When the train meets at the tunnel portal, different rainfall intensity has a significant influence on the turbulence intensity and turbulent kinetic energy of the train flow around and wake. Rainfall significantly reduces the turbulence intensity and turbulence kinetic energy of the train's wake. Compared with turbulence intensity, turbulent kinetic energy is more sensitive to rainfall intensity.

  4. 4.

    Under single-train conditions, the transition from no rain to rain increases the Reynolds stress on the leeward side of the train. However, the impact of rainfall intensity on the Reynolds stress is marginal when the rainfall intensity increases from 200 to 400 mm/h. The influence of rainfall intensity on the Reynolds stress on the leeward side becomes more pronounced when the train intersects the tunnel portal.

  5. 5.

    With the increase of rainfall intensity, the thickness and distribution range of water film in the space increase. In addition, compared with the scene where single train is running on the embankment, the thickness and distribution of water film in the space increase significantly when trains intersection at the tunnel portal. Under single train condition, as the rainfall intensity increases from 0 to 200 m/h, the rain phase on the leeward side and wake of the train increases and the gas phase decreases, resulting in the increase of the leeward side and wake velocity. When the rainfall intensity further increases to 400 mm/h, the water film formed on the leeward side of the train will weaken the velocity. When the trains intersect at the tunnel portal, a relatively closed area is formed between the trains on the windward side and the water film, which will weaken the influence of the slipstream of the trains on the leeward side.

In future research, it is necessary to conduct field tests at the entrance of high-speed railway tunnel in wind–rain prone areas to conduct numerical simulation of more realistic boundary conditions. It is worth noting that a series of problems brought by rainfall, such as uneven settlement of the base, dry/wet state of wheel-rail interface, etc., are also a promising and important topic on the influence of high-speed train wheel-rail dynamic response.