# Numerical Study of the Unsteady Aerodynamic Performance of Two Maglev Trains Passing Each Other in Open Air Using Different Turbulence Models

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Geometry Model

^{+}value demand of IDDES, two 1/16th scale MT models were used in this study. The marshalling length of MTs in operation is currently 2–3 cars to achieve rapid transit of the MT and reduce energy dissipation [22]. Therefore, consistent with the MT model by Luo et al. [12], a two-car grouped model per train, namely the head and tail cars, was used in this study, as presented in Figure 2. The train height H, defined as the characteristic length, was 0.24 m (3.84 m at full scale). The total length and width of the MT model were 8.36 H and 0.75 H, respectively. The line spacing is 1.04 H. Moreover, the clearance between the train and track was extremely small (0.005 H), which may have had a greater impact on the aerodynamic performance of the train, specifically in the region near the bottom of the train, as illustrated in Figure 2. Hence, the track in the numerical simulation could not be neglected.

#### 2.2. Monitoring Points

#### 2.3. Computational Domain and Boundary Conditions

_{∞}was set to zero. The ground and two rails were set as the stationary walls. The dimensions of the overset domain were 19.53 H (length) × 1.28 H (width) × 1.43 H (height), and all faces were overset boundary conditions. To ensure full development of the wake flow fields, the computational domain was extended by 30.7 H backward from the tail nose of the train. Zone 2 and Zone 3 were moved in opposite directions at 200 km/h. In Zone 2, Train A travelled in the positive x-direction, while in Zone 3, Train B travelled in the negative x-direction.

^{5}, which is larger than the minimum Re (0.25 × 10

^{5}) recommended by CEN (2013) [24] for the scale-model test.

#### 2.4. Numerical Method

_{RANS}, l

_{LES}, and l

_{hyb}are the lengths of the hybrid, RANS, and LES models, respectively; ${\tilde{f}}_{d}=\mathrm{max}\left\{\left(1-{f}_{dt}\right),{f}_{\mathrm{B}}\right\}$, where f

_{dt}and f

_{B}are the delay and empirical blending functions, respectively, and f

_{e}is an elevating function. We selected the IDDES method based on the SA and SST k–ω models to compare the difference in capturing flow fields based on the one-equation model (SA) and the two-equation model (SST k–ω). In the SA model, ${l}_{\mathrm{RANS}}={d}_{\mathrm{w}}$, where d

_{w}is the distance to the wall, and ${l}_{\mathrm{LES}}={C}_{\mathrm{DES}}\psi \mathsf{\Delta}$, where C

_{DES}is the empirical constant and ψ is a low-Reynolds number correction. In the SST k–ω model, ${l}_{\mathrm{RANS}}=\sqrt{k}/\left({C}_{\mu}\mathsf{\omega}\right)$, where k is the turbulent kinetic energy and ω is the turbulent dissipation rate. The sub-grid length scale Δ is defined as follows:

_{w}is equal to 0.15. h

_{max}is the maximum size of the cell in three directions, or wall distance, and h

_{wn}is the mesh step in the direction normal to the wall.

_{t}calculated as follows:

^{−5}s, which was smaller than that reported previously [1,31]. After the initial development of the flow, the two MTs were operated with a total time step of 2070. For each time step, the maximum number of iterations was set as 30, and the residuals of all physical equations were below 1 × 10

^{−4}. Less than 1% of cells in the computational domain had a Courant–Friedrichs–Lewy (CFL) number greater than one. According to Krajnovic [32], a minor violation of CFL conditions has a minimal effect on the flow field. In this study, five cases (including three turbulence models, as well as the coarse and fine mesh of the SST-IDDES model) were performed at the National Supercomputing Centre in Wuxi, China.

#### 2.5. Mesh Generation

#### 2.6. Mesh Sensitivity

_{P}is defined by Equation (4), where ρ is the air density (1.225 kg/m

^{3}), P is the static pressure, P

_{∞}is the reference pressure (equal to zero), and u

_{tr}is the train speed (55.56 m/s). The results of grids with different densities were computed using the IDDES method based on the SST k–ω model to validate the grid sensitivity. The time-history pressure coefficients of monitoring points No. 3 and No. 8 on the train surface for different density grids are presented in Figure 7. The peak-to-peak pressure coefficient (ΔC

_{P}) is shown in Figure 7a. For monitoring point No. 3, the difference in ΔC

_{P}between the coarse grid and fine grid was found to be 4.1%, and that between the medium grid and fine grid was 0.5%. For monitoring point No. 8, the difference in ΔC

_{P}between the coarse grid and fine grid was 3.1%, and that between the medium grid and fine grid was 0.6%. The differences in ΔC

_{P}between the medium and fine grids were found to be rather small.

_{tr}is the train velocity. The resultant velocity predicted by three meshes is consistent each other before the tail nose passing, while it shows a large deviation between the coarse and medium as well as fine meshes when wake flow passing.

**Ω**is the rate-of-rotation tensor, and

**S**is the rate-strain-tensor. Figure 9 shows the instantaneous iso-surfaces of Q = 20,000 s

^{−2}of three sets of meshes at the t = 0.08 s position, coloured according to the resultant velocity U. The V

_{2}− V

_{2}′ and V

_{1}− V

_{1}′ regions in the figure indicate that the fine grid can capture more small-scale vortices than the medium and coarse grids with the same Q value. In the near-wake region, the number of small-scale vortices captured by the coarse grid is significantly smaller than that captured by the fine grid, whereas the difference in the flow structures between the medium and fine grid was found to be rather small. Combing with the mesh independence study on the pressure, the resultant velocity and the Q criterion, the medium grid was found to have sufficient resolution for the purpose of the present study.

## 3. Results and Discussion

#### 3.1. Experimental Validation and Transient Pressure

^{4}Hz.

_{P}

_{min}, the positive peak pressure coefficient ΔC

_{P}

_{max}, and peak-to-peak pressure coefficient ΔC

_{P}simulated by the three turbulence models are slightly different from the experimental results. Among them, the differences in ΔC

_{P}at monitoring point No. 5 between SST, SA−IDDES, and SST−IDDES and the experimental data were 1.2%, 1.7%, and 0.7%, respectively, and the differences in ΔC

_{P}at monitoring point No. 17 between SST, SA−IDDES, and SST−IDDES and the experimental data were 2.9%, 3.2%, and 3.6%, respectively. Thus, the simulated results are in good agreement with the experimental data, and the numerical simulation can be considered sufficiently reliable.

_{P}

_{min}, ΔC

_{P}

_{max}, and ΔC

_{P}were higher in the streamlined head and tail nose regions than in other regions of the train surface, and the difference in the pressure variation amplitude in the middle region of the train surface was found to be small. The maximum value of ΔC

_{P}for these monitoring points was measured at point No. 1. For monitoring points along the vertical distribution in the middle of the train surface, the absolute values of ΔC

_{P}

_{min}and ΔC

_{P}first increased and then decreased, and ΔC

_{P}

_{max}increased with increasing height of the monitoring points. At these four measurement points, the maximum value of ΔC

_{P}was measured at point No. 10.

#### 3.2. Velocity Profiles

_{1}, t

_{2}, and t

_{3}, as shown in Figure 13. Figure 14 presents the flow fields of resultant velocity U at positions t

_{2}and t

_{3}. As shown, the vortex structures of the intersection side interact, leading to a decrease in their intensity. Among them, the SA−IDDES and SST−IDDES methods can capture extra small-scale vortices, whereas the SST k–ω cannot capture these small-scale vortices. For quantitative analysis, two horizontal sampling lines were extracted in the train wake at positions t

_{2}and t

_{3}, as shown in Figure 14. Figure 15 shows the U values at the four lines (Figure 14). In contrast to the case of a single train, the U value on the intersection side is significantly smaller than that on the non-intersection side when the two MTs passing each other. Furthermore, the U value calculated with the SST k–ω method is significantly lower than that calculated with the SA−IDDES and SST−IDDES models.

#### 3.3. Wake Flows

^{−2}simulated with different turbulence models at positions t

_{1}and t

_{3}coloured using the resultant velocity U. Figure 16a presents the iso-surface of Q at position t

_{1}. A pair of twin counter-rotating longitudinal vortices (V

_{1}and V

_{2}or V

_{1}′ and V

_{2}′) were observed for all three cases. However, the iso-surface of Q calculated by different turbulence models was found to be significantly different. Among them, the SST k–ω model could not capture small-scale vortices. However, a large number of small-scale vortices were observed in the SA−IDDES and SST−IDDES models, and more small-scale vortices were captured with the SST−IDDES than with the SA−IDDES, particularly in the near-wake region. Figure 16b shows the iso-surface of Q at position t

_{3}. The V

_{1}−V

_{1}′ region was induced by the interaction of two wake vortices on the intersection side. With the SST k–ω model, the interaction between the two wake vortices in this region was strong and resulted in an ‘entanglement’ phenomenon. A similar phenomenon was also observed in the simulations using the SA−IDDES and SST−IDDES models. Furthermore, the flows in this region exhibited strong turbulence. For vortices V

_{2}and V

_{2}′, the SA−IDDES and SST−IDDES models resulted in the formation and development of the vortex structures, and their intensity gradually decreased until they dissipated with the increasing distance from the tail car. However, because the RANS model dissipated the turbulent eddy viscosity, it could not resolve the small vortices Li, et al. [43]. Therefore, we did not observe small-scale vortices at distances from the tail car larger than 7.58 H in the SST k–ω model.

_{2}and t

_{3}. The vertical planes were coloured by the dimensionless longitudinal velocity (U

_{x}= u/u

_{tr}). The wake vortex of a single train travelling without crosswind is symmetrically distributed [17,18,22]. However, Figure 17 shows that the interaction between vortices V

_{c}

_{1}and V

_{c}

_{1}′ on the intersection side causes vortices V

_{c}

_{1}and V

_{c}

_{2}(or V

_{c}

_{1}′ and V

_{c}

_{2}′) to be asymmetric. This phenomenon is observed with three turbulence models. The two vortices V

_{c}

_{1}and V

_{c}

_{1}′ on the intersection side interact, causing their intensity to decrease and their width to be much smaller than that of the vortices on the non-intersection side. To provide a more quantitative comparison of the size of longitudinal vortices, the height of the vortex on the intersection side was denoted as h

_{0}, and the height and distance of the vortex core on the intersection side were denoted as h

_{1}and w, respectively, which is similar to the definition of Wang et al. [29], as shown in Figure 17b.

_{2}. It is evident that the turbulence model has minimal effect on wake vortices V

_{c}

_{2}and V

_{c}

_{2}′ on the non-intersection side. However, vortices V

_{c}

_{1}and V

_{c}

_{1}′ on the intersection side obtained by different turbulence models are significantly different. Figure 18 compares the h

_{0}, h, and w values for the different models at different moments. In Figure 18a, it can be observed that the differences in vortex core height h computed by different models are small, while height h

_{0}of vortex V

_{c}

_{1}′ and distance w between the two vortex cores are significantly different. The height h obtained with the SST−IDDES model is 10% greater than that obtained with the SA−IDDES model, and the shortest and longest distances w of the two vortex cores are obtained with the SST−IDDES and SA−IDDES, respectively, with a difference of 10.3%.

_{3}. As the distance from the tail car increases, the vortex size increases and the interaction between vortices V

_{c}

_{1}and V

_{c}

_{1}′ becomes stronger, leading to a greater change in their shape. The height and width of the vortices are found to be significantly larger at position t

_{3}compared to position t

_{2}. Vortices V

_{c}

_{2}and V

_{c}

_{2}′ on the non-intersection side with the different turbulence models are found to be rather similar. Nevertheless, the h

_{0}, h, and w values are significantly different, as illustrated in Figure 18b. The highest and lowest values of h

_{0}were obtained with SST k–ω and SA-IDDES, respectively, with a difference of 11.8%, and the highest and lowest values of h were obtained with SST−IDDES and SA−IDDES, respectively, with a difference of 17%. Furthermore, the highest and lowest values of w were obtained with SST−IDDES and SST k–ω, respectively, with a difference of 30.9%.

#### 3.4. Aerodynamic Forces

_{d}, C

_{s}, and C

_{l}are the drag, side, and lift force coefficients, respectively; ρ is air density; and u

_{tr}is train speed. F

_{d}, F

_{s}, and F

_{l}are the drag, side, and lift forces, respectively. A MT running in the positive direction of x coordinates was used to analyse the aerodynamic force.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Transient pressure of monitoring points for different density mesh: (

**a**) No. 3 and (

**b**) No. 8.

**Figure 8.**Time-history curves of resultant velocity with different mesh at y = 0.25 m and z = 0.125 m (namely point g in Figure 3).

**Figure 9.**Instantaneous iso-surface of the second invariant of the velocity gradient Q = 20,000 s

^{−2}, coloured according to the resultant velocity, under different mesh density conditions at t = 0.08 s: (

**a**) side view and (

**b**) top view.

**Figure 10.**Results of numerical simulation and moving model test: (

**a**) photograph of physical test model; (

**b**) schematic of numerical model. Time-history curve of pressure coefficient for monitoring points (

**c**) No. 5 and (

**d**) No. 17.

**Figure 11.**Pressure variation amplitude of measuring points (MPs) on the train surface along (

**a**) the longitudinal and (

**b**) vertical direction of the train.

**Figure 12.**Time-history curves of resultant velocity with different turbulence models at y = 0.375 m and z = 0.188 m (namely point d in Figure 3).

**Figure 14.**Resultant velocity U distribution in the wake region simulated by different turbulence models at different positions: (

**a**) t

_{2}and (

**b**) t

_{3}. (The distance from the section plane to the track is 0.125 H.).

**Figure 15.**Resultant velocity U in the wake region obtained with different turbulence models at different positions: (

**a**) line ①, (

**b**) line ②, (

**c**) line ③, and (

**d**) line ④. (The black dashed line represents the symmetry plane of the train).

**Figure 16.**Iso-surface of Q (Q = 20,000 s

^{−2}), coloured according to the resultant velocity U, under different turbulence model methods at (

**a**) t

_{1}and (

**b**) t

_{3}.

**Figure 18.**Parameters h

_{0}, h

_{1}, and w obtained with different turbulence models at (

**a**) t

_{2}and (

**b**) t

_{3}.

**Figure 19.**Aerodynamic force coefficient: (

**a**) resistance of the head car; (

**b**) resistance of the tail car; (

**c**) side force of the head car; (

**d**) side force of the tail car; (

**e**) lift force of the head car; (

**f**) lift force of the tail car. (NN, NT, and TT denote the intersection of the train noses (t = 3.37 × 10

^{−2}s), intersection of the train nose and tail of other train (t = 5.18 × 10

^{−2}s), and intersection of the train tails (t = 6.99 × 10

^{−2}s), respectively.).

Grid | Coarse | Medium | Fine |
---|---|---|---|

Train surface | 0.016 H | 0.013 H | 0.01 H |

Wake refinement | 0.016 H | 0.013 H | 0.01 H |

Number of prism layers | 16 | 20 | 20 |

Number of cells (×10^{6}) | 27.17 | 45.75 | 68.39 |

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**MDPI and ACS Style**

Li, X.; Krajnovic, S.; Zhou, D. Numerical Study of the Unsteady Aerodynamic Performance of Two Maglev Trains Passing Each Other in Open Air Using Different Turbulence Models. *Appl. Sci.* **2021**, *11*, 11894.
https://doi.org/10.3390/app112411894

**AMA Style**

Li X, Krajnovic S, Zhou D. Numerical Study of the Unsteady Aerodynamic Performance of Two Maglev Trains Passing Each Other in Open Air Using Different Turbulence Models. *Applied Sciences*. 2021; 11(24):11894.
https://doi.org/10.3390/app112411894

**Chicago/Turabian Style**

Li, Xianli, Siniša Krajnovic, and Dan Zhou. 2021. "Numerical Study of the Unsteady Aerodynamic Performance of Two Maglev Trains Passing Each Other in Open Air Using Different Turbulence Models" *Applied Sciences* 11, no. 24: 11894.
https://doi.org/10.3390/app112411894