Effect of RANS Turbulence Model on Aerodynamic Behavior of Trains in Crosswind

The motion of the airflow around trains can be described by a continuity equation, a momentum equation, and an energy equation. The general governing equations are as follows:where ρ is the density, t is the time variable, \(\varvec{\varPhi}\) is a general variable, which can represent u, v, w, T, and the others, V is the velocity vector, Γ is the general diffusion coefficient, S is the generalized source term. The left side of the equation is the transient term and the convection term, and the right side of the equation is the diffusion term and the source term.

When a train runs at a high speed, the flow around the train can be treated as a three-dimensional incompressible flow process. A reasonable turbulent transport equation must be selected for the numerical simulation. The RANS simulation is currently the most widely used turbulent numerical method. The Reynolds stress term in the RANS equation is unknown, resulting in the unclosed equations. The eddy viscosity model introduces the turbulent viscosity μ_t based on the Boussinesq hypothesis and establishes the relationship of Reynolds stress with respect to the average velocity gradient. According to the number of differential equations that determine the turbulent viscosity μ_t, the eddy viscosity model is divided into zero-equation model, one-equation model, and two-equation model [27]. The one-equation model is mainly Spalart–Allmaras. Typical two-equation models include the standard k-ε, RNG k-ε, Realizable k-ε, SST k-ω, and standard k-ω. In this paper, the one-equation and 5 two-equation models commonly used in engineering are selected for comparison. The standard k-ε, RNG k-ε, Realizable k-ε, SST k-ω, standard k-ω, and Spalart–Allmaras turbulence models are respectively represented by SKE, RNG, RKE, SST, SKW, and SPA for simplicity.

Whether it is the standard k-ε model, RNG k-ε model or Realizable k-ε model, it is effective for fully developed turbulent flows, that is, these models are high Reynolds turbulence models, and they can only be used to solve the flow fully developed. In the near-wall region, the turbulent flow is not fully developed. In the region closer to the wall, the flow may be in a laminar flow state. Therefore, it is difficult to use the previous high Reynolds model for the calculation in the near-wall region, and special processing should be used. A wall function method or a k-ε model with a low Re number can be used. The wall function method is actually a set of semi-empirical formulas used to directly relate the physical quantities on the wall to the unknowns, in conjunction with the high Re number k-ε model. A standard wall function is used when the dimensionless wall distance y+ is in the range of 30 and 200, and a reinforced wall function is used when y + is close to 1. For the k-ω turbulence model, y+ is guaranteed to be close to 1.

The model is an ICE2 train, as shown in Figure 1(a), which consists of a head car and a part of the tail car. The tail car is mainly used to ensure the flow field of the head car and between the head and tail cars. There is a 5 mm gap to ensure the separate measurement of the head and tail forces. The numerical simulation calculation model is consistent with the wind tunnel test model [28]. The scale size of the train model is 1:10, the total length of the scaled train L = 3.552 m, the width B = 0.302 m, the height H = 0.358 m, and the streamlined length L_head = 2.656 m, the incoming flow is shown in Figure 1(b). Taking the center of the section as the origin, it is divided into four parts according to the angle, as shown in Figure 1(b). 4 parts are the upper surface (0°‒45°, 315°‒360°), leeward surface (45°‒135°), bottom surface (135°‒225°) and windward surface (225°‒315°). Figure 2 illustrates the speed vector diagram, where U_t is the train’s running speed; U_w is the crosswind velocity; U_R is the air speed relative to the train; β is the yaw angle. The experimental data were obtained from the Bombardier Transportation Company of the Zhukovsky Central Aerodynamics Institute in Russia in 2003 in the T-103 open-air jet tunnel [28].

The size of the computational domain should satisfy the requirement that it will not affect the flow field around the train. The computational domain shown in Figure 3 is established. The gap between the train and ground is 0.0503 m, and the length, width, and height of the computational domain are 16 m, 12 m and 4 m, respectively. The entrance plane in front of the train is 4 m from the nose of the head car, and the distance between the center line of the train and the left and right sides is 4 m and 8 m, respectively.

The boundary conditions are set as follows: the two entrances of the computational domain are specified as the inlet boundary conditions with a speed of 60.62 m/s and the crosswind velocity of 35 m/s. The two exits are set as the pressure outlets, and the ground and train surfaces are fixed wall boundaries. The top surface of the computational domain is symmetry, as shown in Figure 3. The air density is 1.205 kg/m³ and the kinematic viscosity is 1.81 × 10⁻⁵ Pa∙s. The SIMPLE algorithm of pressure-velocity coupling is used, and the second-order scheme is chosen for the discretization for all variables.

The pressure coefficient c_p, aerodynamic side force coefficient c_s and lift force coefficient c_l are defined as:where \(p\) is the static pressure, \(p_{\infty }\) is the reference pressure, \(\rho\) is the air density; \(u_{\infty }\) is the incoming flow velocity, F_s and F_l are the aerodynamic side and lift forces, respectively; A is the characterized area of the train.

In order to simulate the flow around the train better, the flow field is multi-layered and refined. The grid size is gradually increased to ensure a fine grid. The refined region is shown in Figure 4. Since the fluid variable in the near wall region has a large gradient, a fine mesh near wall region can basically determine the true physical phenomenon of the rear wall. A boundary layer is created on the surface of the train to simulate the flow near the wall. The first layer of the boundary layer has a height of 0.01 mm, a growth rate of 1.2. There are 12 layers in total, ensuring that y+ is close to 1. After selecting the reasonable mesh size for meshing, the mesh of the flow field around the train model is shown in Figure 4. The mesh in the boundary layer of the train surface is shown in Figure 5.

The accuracy of the numerical results is related to the number and quality of the mesh, therefore, the mesh independence is studied firstly. Three sets of grids were divided by changing the size of the train components. The SST k-ω turbulence model is used for the numerical calculation. The side force coefficients c_s obtained using different grids are shown in Table 1. The relative error of the side force coefficients is within 3%. When the number of meshes increases from 23.81 million to 29.85 million, the relative error in the side force coefficient is less than 2%. It can be seen that the further increase of the mesh has little effect on the side force coefficient, therefore, the grid with a number of 23.81 million is chosen for the remaining numerical calculations.

When the train is running in a crosswind environment, its surrounding flow field is shown in Figure 6. When the airflow is blocked by the train, it is divided into two parts sourced from the windward side of the train, a part of the airflow bypasses the roof and the other flows from the bottom of the train (see Figure 6(a)). A large number of eddies are formed on the leeward side of the train, and the flow field is extremely complicated (see Figure 6(b)).

Figure 7 shows the streamlines at different cross-sections of the train using SST k-ω turbulence model. The corresponding cross-sectional location is shown in Figure 1(a). When the steady airflow under certain conditions bypasses the train, a vortex is formed from the surface, that is, the boundary layer is detached, and the airflow over the roof and bottom is collected on the leeward side of the train, generating a lot of vortices. At different cross-sections, the number and scale size of vortices are different. When the air flows over the roof and the bottom of the vehicle, the shedding eddy is likely to occur on the leeward side of the train. It can be seen from different sections that as the airflow develops downstream, the scale of the vortex shedding becomes larger and larger, and the vortices are gradually far from the vehicle body. There are many large-scale vortices on the leeward side of the train. These vortices consume energy and generate a large suction pressure on the leeward side of the train.

The surrounding flow field obtained using different turbulence models are shown in Figure 8. Due to the similarity, only the flow field distribution at the cross-section x₅ = − 2.06 m is given. Different turbulence models have different simulation capabilities for the flow field. As shown in Figure 8(e), the SST k-ω turbulence model captures four typical vortices, defined as V_c1, V_c2, V_c3, and V_c4, respectively. Three turbulence models including RNG, SKW and SST have the ability to capture the above vortices. The other three turbulence models do not capture the V_c1 vortex, and the SKE turbulence model also has poor recognition of several other vortices. The V_c3 vortex is not obtained using RNG turbulence model, and the V_c3 and V_c4 vortices identified by SKW turbulence model are interconnected.

In order to further analyze the flow field, Figure 9 shows the comparison of the velocity distributions at different cross-sections. When the airflow is blocked by the train, the flow velocity is significantly reduced, and the airflow speed increases significantly when the airflow bypasses the roof and the vehicle bottom. As there are many vortex structures on the leeward side of the train, the flow velocity in the above regions is relatively low. Obviously, the calculation results show that the velocity distribution obtained using different turbulence models are different. The SKE turbulence model performs poorly compared with the other turbulence models. The other turbulence models show higher gradients in velocity. 4 turbulence models including RNG, SKW, SST and SPA show a lower velocity near the leeward side of the train, and RNG and SST turbulence models present a higher resolution of the velocity gradient.

Figures 10 and 11 show the flow pattern on the train surface including the leeward side and the bottom. The airflow branches off from the windward surface of the train, some flows over the roof and the others over the bottom of the train, as shown in Figure 6(a). There are many separation lines (red) and reattachment lines (black) on the surface of the train, mainly including separation lines S1, S2, S3 and reattachment lines A1, A2, A3, as shown in Figure 10(e). For the upper surface, airflow is separated at the separation line S1, and reattached at the reattachment line A2, and then separated again from the separation line S3. The airflow adheres to the surface of the train body at the reattachment line A2, and then separates at the separation line S3, and reattaches again at the reattachment line A1, and the airflow flows downstream to the separation line S1. The underneath airflow is separated at the separation line S2 and reattached at the reattachment line A3. The flow pattern around the bogie is more complicated, and the airflow has multiple separation and reattachment lines. The separation and reattachment of the airflow at the front end of the train will form bubbles. The air bubbles will fall off and develop to the downstream along the train. Once the air bubbles fall off, new air bubbles will be generated. When these air bubbles flow backward, the airflow on the leeward side of the train will be intensified, forming a large vortex.

The simulation results of the separation lines (S1 and S3) and the reattachment line (A1) for SKE and RKE are poor. RNG and SPA turbulence models predict a higher position of the separation line S3, and RNG turbulence model also has a higher predicted position for the reattachment line A2. However, the SPA turbulence model has a lower predicted position for the reattachment line A2. The three turbulence models including RKE, SKW and SST have similar predictions for the reattachment line A2. The length of the separation line S3 predicted using SKW turbulence model is significantly larger than the other models. The position and length of these surface flow lines represent the position and the size of the region for the airflow varied from the two-dimensional surface to the three-dimensional flow field. Obviously, different turbulence models have different predictions in surface flow.

Figure 12 shows the pressure distribution around the train at the cross-section x₄ = − 1.57 m. From the previous flow field analysis, it is only obvious that the results obtained using different turbulence models are different. 3 turbulence models including RNG k-ε, standard k-ω and SST k-ω show lower negative pressure on the leeward surface of the train, and the pressure gradient on the leeward surface is larger, while the other three turbulence models show a higher negative pressure, and a smaller pressure gradient.

Figure 13 shows the comparison of the surface pressure coefficient c_p and the experimental ones on different cross-sections of the train. The pressure coefficient on the windward surface and the top surface of the train at all cross-sections for different turbulence models are in a good agreement with the experimental results. The leeward surface also presents a reasonable match in pressure with the experimental data, indicating the rationality of the numerical method. As there are some differences in geometry between the numerical model and the experimental model, such as the bogie, the cylindrical support, and so on, a slight difference is observed on the bottom surface of train for the numerical results and experimental data.

At the cross-section x₁ = − 0.107 m, the different turbulence models show a large difference at the bottom of the train due to the exist of the spoiler. It can be seen from Figure 13(b) that the bogie has a greater influence on the airflow underneath the train, and it shows a higher suction pressure than the experimental value. Due to the presence of the bogie, there is no experimental data for comparison at the cross-section x₃ = − 0.50 m. Compared with the three cross-sections near the streamlined nose, the pressure on the latter three cross-sections are more intuitive than the experimental data due to the influence derived from the spoiler, the bogie and the cylindrical support. It can be seen from Figure 13(a) and (b) that the maximum negative pressure appears on the upper part of the leeward surface, and the maximum suction pressure appears on the windward surface of the streamlined part, indicating that the aerodynamic forces are concentrated in the streamlined part of the train. It can be seen that the pressure on the three cross-sections x₂ = − 0.25 m, x₃ = − 0.50 m and x₄ = − 1.57 m are different in a certain for different turbulence models. These areas are relatively less affected by the spoiler, bogies and cylindrical supports. It can be seen from Figure 13(b) that SST turbulence model exhibits the largest suction pressure, which can be confirmed from the previous analysis of flow field. Therefore, SST turbulence model shows a more accurate prediction in the flow separation in the boundary layer. Meanwhile, the pressure obtained using SST turbulence model agrees well with the experimental one at the other cross-sections. At the non-streamlined cross-section (x₃ = − 0.50 m), the suction pressure on the upper part of the leeward surface for SKE turbulence model is much lower than that of the other turbulence models, and SST and RNG turbulence models exhibit a higher suction pressure. On the bottom surface at the cross-section x₄ = − 1.57 m, SST and SKW turbulence models agree well with the experimental values. Comprehensive comparison of the pressure on all cross-sections, SST turbulence model shows a better prediction on the surface pressure and also the flow field.

The difference in the side force coefficient c_s and lift force coefficient c_l between the numerical results and the experimental value [28] is studied. The results are given in Table 2. For the side force coefficient, the error is within 10% compared with the experimental value. Compared with the experiment results, the error for SST turbulence model is 0.5%, and for SKE turbulence model is 8.06%, followed by RNG with an error of 4.53%.

For the lift force coefficient, SST shows a more accurate result with an error of 21.8%, followed by RNG model, the error is 26.3%, and SKE predicts the worst. The deviation could be induced by a considerable geometric difference between the numerical and the experimental models. Some researchers have studied the aerodynamic performance of ICE2 trains [13]. The relative error in the lift force coefficient between the numerical result and the experimental ones is from − 31.1% to − 47.6%. In this study, it provides a more accurate result with an error of − 21.8%.

In the streamlined part of the head car, when the airflow transitions from the top of the train to the leeward surface, the airflow separates and a negative high-pressure zone is formed at the separation point. On the leeward surface at the cross-section x₁ = − 0.107 m, the different negative pressure is generated using different turbulence models. The pressure predicted using SST is the largest, which results in the closest forces to the experimental ones. At the cross-section x₂ = − 0.25 m, the pressure for SST turbulence model is also the largest, and that for SKE model is also the smallest, which matches the deviation. For the SKE turbulence model, the flow assumed is completely turbulent regardless of the role of molecular viscous, the viscous effect dominates the near wall of the train, and the turbulent development is not sufficient, and the calculation produces distortion. In the transition zone of the curved surface and the near-wall area of the surface, SKE is distorted by the low Reynolds number, and the pressure on the surface of the fitted body is small. It is shown in Table 2 that the SKE model shows the largest error, and SST model gives a minimum error. At the cross-section x₄ = − 1.57 m, SST turbulence model gives the closest prediction in pressure on the bottom surface of the train, followed by SKW. Meanwhile, at the cross-section x₆ = − 2.49 m, the pressure obtained using RNG and SST is close to the experimental one. It can be concluded from the above analysis that the SST and RNG models have better predictions in the lift force coefficient.