Mathematical modeling and sensitive analysis of the train-induced unsteady airflow in subway tunnel

Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

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Mathematical modeling and sensitive analysis of the train-induced unsteady airflow in subway tunnel Huan Zhang a, b, Chunguang Zhu a, b, Minzhang Liu a, b, Wandong Zheng a, *, Shijun You a, b, Bojia Li c, Peng Xue d a

School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, PR China National Engineering Laboratory for Digital Construction and Evaluation Technology of Urban Rail Transit, Tianjin 300072, PR China National Center for Quality Supervision and Testing of Solar Heating Systems (Beijing), China Academy of Building Research, Beijing 100013, PR China d Beijing Key Laboratory of Green Built Environment and Energy Efficient Technology, Beijing University of Technology, Beijing 100024, PR China b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Piston effect Dynamic mesh Bernoulli theoretical model Ventilation rate Multi-trains model

With the flourishing development of the subway construction, more and more attentions are paid to the subway tunnel ventilation. The piston wind caused by the running trains can not only improve the tunnel environment, but also strengthen the power of the tunnel ventilation. The majority of the current researches are based on similar experiments or CFD simulations, yet most of which lack universality. It takes long time to finish the research cycles and it is difficult to generalize the conclusions based on these researches. This paper is aimed to put forward a universal prediction formula to estimate the ventilation effect of piston wind. In this paper, a uniformly accelerated Bernoulli theoretical model is established firstly, which is validated by the experimental data and numerical simulations. Secondly, a passing through model is built with the verified theoretical method and the influence of the key parameters on the ventilation rate are analyzed. Thirdly, the grey correlation analysis is conducted to study each influencing factors, meanwhile, a general prediction formula on the ventilation rate is proposed, which is applicable to the most subway tunnels in China. Finally, the effects of the trailing distance between multi-trains on the tunnel ventilation is discussed.

1. Introduction With the flourishing development of the infrastructure constructions in China, more and more subway lines have been built in recent years. Subway has become one of the main means of transportation in China. At the same time, the air quality in the underground environment is becoming increasingly worse. Aarnio P et al. found that the air in a subway tunnel was approximately eight times more genotoxic and four times more likely to cause oxidative stress in human lung cells than that of the air on the urban street (Aarnio et al., 2005). As a result of the limitation of the tunnel wall, the expelled air cannot flow to the position area behind the train, which forms the piston wind. The piston wind plays an important role on the subway tunnel environment and it is a free power of the tunnel ventilation. In addition, it has a great influence on the tunnel fire situation. Therefore, more and more attentions are paid to the subway tunnel ventilation. For example, Gonz
alez and Vega researched the piston effect in longitudinal ventilation systems for subway tunnels through numerical modeling (Gonz
alez et al., 2014).

Juraeva researched the reduction of particle concentration with the piston wind in the subway tunnel (Juraeva et al., 2016). Wang et al. studied the characteristics of particulate matter (PM) concentrations influenced by the piston wind in Shanghai subway system (Wang et al., 2016). With the occurrence of several major fire accidents, fire safety and tunnel ventilation are increasingly concerned by scholars. Subway fires in tunnel could bring disastrous consequences to life and property (Li and Chow, 2003). The piston wind could control the smoke flow effectively. In the study of Xi et al., the relationship between the air velocity of piston wind and heat release rate were investigated, what's more, the temperature field around the train, and flame/smoke pervasion rule were studied as well (Xi et al., 2016). Theoretical calculations, experiments and computer simulations are three conventional ways to study the tunnel piston effect. The theoretical method is based on Bernoulli equation, which simplifies the piston effect into a one-dimensional model. The piston wind velocity could be solved by the simultaneous Bernoulli equation and the continuity equation. Theoretical modeling is complex, so quit few related studies are

* Corresponding author. E-mail address: [email protected] (W. Zheng). https://doi.org/10.1016/j.jweia.2017.09.005 Received 21 April 2017; Received in revised form 6 September 2017; Accepted 6 September 2017 0167-6105/© 2017 Elsevier Ltd. All rights reserved.

H. Zhang et al.

Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

Φ ε ρ SΦ r Ei Si E S MBE η η0 n

Nomenclature β A1 A v vs α a l L D P u k

the modified coefficient of uneven airflow the cross-sectional area of the train the cross-sectional area of the tunnel the air velocity of piston wind the air velocity of the return air in the annular space the blockage ratio of the train to the tunnel the acceleration of the train the length of the train the length of the tunnel The hydraulic diameter of the tunnel the wetted perimeter of the cross-section the velocity component turbulent kinetic energy

available. However, the theoretical model has the advantages of fast calculation and low cost. The one-dimensional calculation software SES (subway environment simulation program) is based on the theoretical method. Ke et al. optimized the design of subway environmental control system with SES and CFD method (Ke et al., 2002). Besides the theoretical studies, experimental researches are few because of the high cost and long research cycle. Most experimental studies are based on similar experiments (Zhou et al., 2014; Yang et al., 2016; Zhang et al., 2017). Consequently, quit few of the full scale experiments have ever been carried out (Liu et al., 2017). Chen et al. researched the effects of piston-effect and jet fan-effect on the tunnel ventilation in a 1/20th scale tunnels (Chen et al., 1998). Kim and Kim studied the velocity and pressure variations with time in a 1/20th scale of the subway tunnel (Kim and Kim, 2007). Gilbert et al. conducted an experimental investigation to establish how different configurations of confining infrastructure affect the transient slipstream velocities and maximum gust loads caused by a passing train in a 1/25th scale tunnel (Gilbert et al., 2013). Bai et al. conducted an experimental investigation on the flow field induced by a train urgently speeding to the rescue station in a 1/50th scale of a super-long railway tunnel (Bai et al., 2016). Lin et al. studied the piston effect influenced by draught relief shaft in a full scale underground subway system (Lin et al., 2008). Modic et al. experimentally and numerically studied the unsteady three-dimensional flow of the road tunnel and subway tunnel with a single track (Modic, 2003; Ogawa and Fujii, 1997). With the rapid growth of the advanced computer technology, the computational fluid dynamics (CFD) tools have been increasingly utilized to study the piston effect (Khayrullina et al., 2015). However, the CFD methods also have many disadvantages. Since the grid quantity of dynamic mesh model is huge and the research is usually directed at a particular tunnel. The three-dimensional dynamic mesh method could trace and record the distributions of the air velocity, the temperature and the pressure in the subway tunnels and ventilation ducts (Huang et al., 2013). Huang et al. conducted a series of investigation about the unsteady airflow induced by the subway train in a subway tunnel with natural ventilation ducts (Huang et al., 2010). A three-dimensional numerical model with the dynamic layering method for the moving boundary of a train was firstly developed. Camelli et al. researched the effects of multi-car trains with CFD tools as they arrived and departed in a network of interconnected stations at realistic speeds (Camelli et al., 2014). Yuan obtained the velocity and temperature field of the subway station and the optimized ventilation mode of the subway side-platform station through the CFD simulation (Yuan and You, 2007). Xue et al. measured and evaluated the effect of train-induced unsteady airflows in a subway tunnel using dynamic mesh method (Xue et al., 2014). Juraeva et al. conducted a numerical analysis of the ventilation system in a subway tunnel and confirmed the best position of an air-curtain to install

the time averaged velocity dissipation rate of turbulent kinetic energy molecular density the source term of Φ Pearson's product–moment correlation coefficient test value simulation value average value of test average value of simulation the Mean Bias Error the ventilation rate of piston wind the fitting result of the ventilation rate the position coefficient

(Juraeva et al., 2011). Both the experimental method and numerical simulation have the disadvantages of long research cycles and lacking universality. As to the studies on the train-induced piston wind, the piston wind is influenced by several factors, such as the speed of the train, the length of the train and the tunnel, the position of ventilation ducts, the blockage ratio and so on. Chu et al. focused on the influences of the tunnel length, the blockage ratio, the train speed and the intersecting location on the interactions of the aerodynamic waves generated by the trains with a series of numerical simulations (Chu et al., 2014). Choi and Kim evaluated the influence of the piston wind on the train nose length and the tunnel cross-sectional area. They found out that the aerodynamic drag decreased by approximately 50% when the cross-sectional area of the tunnel increased. It has drawn much more attentions than ever to find out the useful measures to improve the effect of the duct ventilation and to improve the ventilation quantity in the subway tunnels through the natural ventilation ducts (Choi and Kim, 2014). In addition, Cross et al. investigated the effect of altering the blockage ratio of an underground train upon the ventilating airflows driven by a train (Cross et al., 2015). However, few researches have ever focused on the coupling effects with the consideration of all the influencing factors, such as the train speed, the train length, the tunnel length, and the blockage ratio. Furthermore, there's a lack of one universal method to evaluate the ventilation effect of piston wind. This paper is aimed to propose a universal prediction formula to estimate the tunnel ventilation effect of piston wind for most of the subway tunnels in China. The universal evaluate method is based on the analysis of four main influencing factors. In order to achieve this goal, a mathematical model is established based on the Bernoulli theory and its results is compared with the experimental data and CFD simulation results. Based on the verified theoretical method, the four influencing factors of the tunnel ventilation rate are analyzed. Meanwhile, a grey correlation analysis is developed to evaluate the importance of the four influencing factors. In order to study the effects of trailing distance between multitrains on the piston wind, a theoretical model of two trains running in the same tunnel direction at the same time is built. The Bernoulli theoretical models and the prediction formula in this study are foundations for further analysis in the thermal field distribution of the subway tunnel. 2. Methodology 2.1. Description of the experimental study carried out by Kim et al (Kim and Kim, 2007) Since few full scale experiments can be cited. A highly cited similar experiment is adopted to validate the theoretical method. The experiment is conducted by Kim, and it is an experimental study on the train68

H. Zhang et al.

Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

Fig. 1. The dimensions of Kim's experimental layout.

induced unsteady flow of a subway tunnel model. Fig. 1 shows the dimensions of the experimental layout. The experimental model is 1/20th of a real subway tunnel and it is 39 m long, 250 mm high and 210 mm wide. The train model is 3 m long, 225 mm high and 156 mm wide. The blockage ratio of the train to the tunnel is approximately 0.67. The model train is towed by a drive motor equipped with an electrical inverter. The model tunnel is equipped with two velocity transducers and four pressure transducers. The velocity transducers are arranged at the inlet and the outlet of the tunnel. The separation distance of the pressure transducers is 7 m. The detailed layout of the measuring points is presented in Fig. 1. The test data collection interval of all the transducers is 0.1 s. The motion law of the train model is shown in Fig. 2. The maximum train speed is 3.0 m/s, the acceleration of the train is 1.0 m/s2 and the deceleration is 1.0 m/s2. The maximum Reynolds number is 4.9  104. 2.2. Theoretical model of the piston wind

Fig. 2. The motion law of the train model.

a

As the tunnel wall forms a space constrain, the forced air flow inside the tunnel generated by the moving train cannot completely flow to the rear of the train when the train runs in the tunnel. Meanwhile, a negative pressure vortex region is formed at the end of the train. As a result, some fresh air is sucked into the tunnel through the opening, forming the piston wind. And the test results have proved that the airflow in the tunnel changes with time as the train moving. With the analysis of the influencing factors on piston effect, the accuracy of the theoretical method should be validated firstly. A uniformly accelerated theoretical model is built base on the geometric parameters and the motion law of Kim's experiment in this section. All the theoretical analysis in this study are in terms of the theory of Bernoulli's equation for unsteady flow.

v21 P1 v2 P2 ∂v þ þ gz1 ¼ 2 þ þ gz2 þ hf þ β ∫ ds 2 ρ 2 ρ s ∂t

b

c

(1) d

where β ∫ s

∂v ∂t ds

refers to the inertia force, which represents the amount of

kinetic energy of fluid per unit mass changing over time. β is the modified coefficient of uneven airflow, and β  1 in this study. As shown in Fig. 3, the uniformly accelerated theoretical mode involves four processes: (1) The uniformly accelerated process; (2) The constant motion process; (3) The uniform deceleration process; (4) The decay process of airflow after the train stop. According to the continuity equation:

Fig. 3. (a) Schematic of the train accelerates. (b) Schematic of the train runs at a constant speed. (c) Schematic of the train slows down. (d) Schematic of the train stops.

①Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2:

A1 v1 dt ¼ Avdt þ ðA  A1 Þvs dt vs ¼

A1 v1  Av v1  v þ v1 ¼ A  A1 1α


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

(2)

(3)

➁Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3:

where v1 is the speed of the train. (1) The uniformly accelerated process:


l1 1 2 dvs ρv þ ρl1 P3 ¼ P2 þ ξ2 þ λ1 þ ξ3 2 s d dt 69

(4)

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

③Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4:

dvs a 1 dv where v1 ¼ at, vs ¼ atv 1α , dt ¼ 1α  1α dt . ➂Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4:


1 l34 1 2 dv ρv þ ρl34 P3 þ ρv2 ¼ Pa þ ξ4 þ λ 2 dt D 2


1 l34 1 2 dv ρv þ ρl34 P3 þ ρv2 ¼ Pa þ ξ4 þ λ 2 dt D 2

(5)

It can be obtained from the simultaneous equations:

It can be obtained from the simultaneous equations:

dv ¼ A þ Bv þ Cv2 dt

t 2 ½0; 3

dv ¼ A þ Bv þ Cv2 dt

(6)

k2 a t ð1αÞþ2l1 a k2 atð1αÞ 2 k1 Þð1αÞ where A ¼ 2ððl , B ¼ ðl12 þl , C ¼ 2ððlðk , 12 þl34 Þð1αÞþl1 Þ 34 Þð1αÞþl1 12 þl34 Þð1αÞþl1 Þ 2 2

34 þ ξ4 , k2 ¼ k1 ¼ ξ1 þ λ l12 þl D

where

l


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

v1 v dvs 1α , dt

(7)

1 2 dvs ρv þ ρl1 2 s dt


1 l34 1 2 dv ρv þ ρl34 P3 þ ρv2 ¼ Pa þ ξ4 þ λ 2 dt D 2

(9)

(10)

k v2 ð1αÞ

k1 ¼ ξ1 þ

þ ξ4 , k2 ¼

ð1αÞ

dv kv2 ¼ dt 2L

(16)

D¼

.

  ∂ðρΦÞ þ div ρ! u Φ  Γ Φ;eff gradðΦÞ ¼ SΦ ∂t

①Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2:

(17)

(18)

The standard k  ε equation can be described by:

ρ

(11)


 ∂ε ∂ ∂ε ε ε2 ¼ X þ C1ε ðC3ε þ Gk Gb Þ  C2ε ρ ∂t ∂xi ∂xi κ κ

(19)

2.3.2. Numerical simulation setup The PISO algorithm and the QUICK differencing scheme are employed for the numerical simulations. Since the grids are all built into structured grids, the dynamic layering method supplied by FLUENT code is adopted to the update of the volume mesh. The grid size is set to be 0.025 m in X direction (X direction is the moving direction of the train), while the size is set to be 0.0125 m in both Y and Z directions (Z direction is perpendicular to the ground, which is not included in the twodimensional model). The entire computational domain consist of 564,720 cells. The total time of unsteady simulation is 30 s and the time step of the calculations is 0.0025 s, which is adopted during the whole

➁Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3:


l1 1 2 dvs ρv þ ρl1 P3 ¼ P2 þ ξ2 þ λ1 þ ξ3 2 s d dt

4A P

The RANS equations can be comprehensive described by:

(3) The uniform deceleration process:


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

t 2 ½14; ∞Þ

2.3.1. Numerical model In order to validate the accuracy of the theoretical method in another way, two CFD simulations (a 2-D model and a 3-D model) are carried out. All the Geometrical models are made by Gambit 2.4.6 based on the size of Kim's experimental model. The characteristic length (D) of the twodimensional model is defined as Eq. (17). This study adopts FLUENT 6.3.26 to calculate the velocity field, the pressure field while the train is moving. The continuity equation, Reynolds-averaged Navier-Stokes (RANS) equations (Tominaga and Stathopoulos, 2017), and the standard k-ε turbulence model (Yang et al., 2015) are employed in this simulation. The hydraulic diameter:

k2 v1 ð1αÞ 2 k1 Þð1αÞ where A ¼ 2ððl12 þl2 341 Þð1αÞþl1 Þ, B ¼ ðl12 þl , C ¼ 2ððlðk , 34 Þð1αÞþl1 12 þl34 Þð1αÞþl1 Þ 34 λ l12 þl D

(15)

2.3. Numerical method

It can be obtained from the simultaneous equations:

t 2 ½3; 11


 l1 L  l1 1 2 dv ρv þ ρL þ ξ4 Pa ¼ Pa þ ξ1 þ λ1 þ λ 2 dt d D

(8)

1 dv 1α dt .

l ξ2 þλ1 d1 þξ3 2

l

ξ2 þλ1 d1 þξ3 . ð1αÞ2

l1 1 where k ¼ ξ1 þ λ Ll D þ λ1 d þ ξ4 .

where vs ¼ ¼ ➂Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4:

dv ¼ A þ Bv þ Cv2 dt

k2 ðv01 aðtt10 ÞÞð1αÞ ðl12 þl34 Þð1αÞþl1 ,

B¼

Bernoulli's equation for the relative motion between cross-section 1 and cross-section 4:

➁Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3:



k2 ðv01 aððtt10 ÞÞ2 ð1αÞ2al1 , 2ððl12 þl34 Þð1αÞþl1 Þ

(14)

(4) The decay process of airflow after the train stop:

①Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2:

l1 P3 ¼ P2 þ ξ2 þ λ1 þ ξ3 d

A¼

t 2 ½11; 14

2 k1 Þð1αÞ 34 , k1 ¼ ξ1 þ λ l12 þl þ ξ4 , k2 ¼ C ¼ 2ððlðk D 12 þl34 Þð1αÞþl1 Þ

ξ2 þλ1 d1 þξ3 . ð1αÞ2

(2) The constant motion process:




(13)

(12)

v0 aðtt Þv

a 1 dv 0 s where v1 ¼ v01  aðt  t1 Þ, vs ¼ 1 1α1 , dv dt ¼ 1α  1α dt , t1 is the beginning of the deceleration, v01 is the speed of the train through the uniform motion process.

70

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

simulation.

3.2. Modeling of the theoretical passing model

2.4. Data processing algorithms

As the above validation shows, the theoretical method is proved to be accurate enough. However, the above theoretical model is built based on the size of Kim's similar experiment. There are two deficiencies: (1) the size of the model is only 39 m, which is difficult to analyze the piston effect characteristic of full scale tunnel; (2) the motion lay of the model is based on Kim's experiment completely, which includes a uniformly accelerated process and a uniform deceleration process. As defined by the piston wind, air is confined by the tunnel walls to move along the tunnel, which is caused by a train passes through the tunnel. In reality, the movement of the train passes through the tunnel includes three processes as shown in Fig. 5: the process of the train entering the tunnel gradually, the process of the train moving on a constant speed in the tunnel and the process of the train leaving the tunnel gradually. The uniformly accelerated (decelerated) model cannot truly reflect the process of a train entering the tunnel. Therefore, it is necessary to build a theoretical passing though model on the basis of an actual full scale tunnel. To analyze the piston effect characteristic of full scale tunnel, the passing model is established based on the size of a typical tunnel. The length of the typical tunnel is 1500 m, the hydraulic diameter of the tunnel is 5.09 m, the length of the train is 120 m, and the blockage ratio is 0.41. And the passing speed of the train is 30 m/s.

As the verification, the Pearson's product–moment correlation coefficient (PPMCC) is used to test the correlation between experimental and simulated values (Puth et al., 2014). The PPMCC can be expressed as:

  Pn  i¼1 Ei  E Si  S r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn  2ffi Pn  i¼1 Ei  E i¼1 Si  S

(20)

The Mean Bias Error (MBE) is also used to test the accuracy of the simulation results (Zhang et al., 2016), defined as:

MBEð%Þ ¼ ð1=NÞ

n X jSi  Ei j i¼1

Ei

 100%

(21)

3. Results 3.1. Model validation Fig. 4 shows the comparison between the results of the theoretical calculation, numerical simulations and experimental measurement on non-dimensional u velocity (axial velocity), u/Ut_max of air flow with time at the inlet and outlet of the tunnel. As can be seen from the figure, the theoretical calculation results agree well with those of the experimental results, and followed by the three-dimensional numerical simulation results. The air velocity changes as the train accelerates or decelerates. The MBEs between the theoretical calculation results, numerical simulation results and the tested data is 21%, 35% (three-dimensional simulation) and 59% (two-dimensional simulation), respectively. Although the error between the calculation value and test value is above 20%, the correlation coefficient between the calculation results and tested data are 0.99 (theoretical calculation), 0.97 (three-dimensional simulation) and 0.93 (two-dimensional simulation), respectively. Because of the non-dimensional velocity (u/Ut_max) is a small numeric value, a little difference will lead to a great error rate. As the results of above analysis, the theoretical calculation model can accurately predict the overall trend of the piston effect. Considering the MBE and the correlation coefficient, the theoretical model is better than numerical simulation models. In addition, the computational speed of theoretical model is much faster than the numerical calculation. Therefore, all the following analyses are conducted based on the theoretical model.

(1) The process of the train entering the tunnel gradually: ①Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2:


l12 1 2 dvs ρv þ l12 ρ P2 ¼ Pa þ ξ1 þ λ1 þ ξ2 2 s d dt

(22)

1 dv 1 v dvs where vs ¼ v1α , dt ¼ 1α dt . ➁ Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3:


1 L  l12 1 2 dv ρv þ ρðL  l12 Þ P2 þ ρv2 ¼ Pa þ ξ3 þ λ 2 2 dt D dv ¼ A þ Bv þ Cv2 dt where k1 ¼

  l1 t 2 0; v1

k v2 ð1αÞ

1 1 A ¼ 2ðLð1αÞþl , 12 αÞ

l ξ1 þλ1 12 d þξ2 2

ð1αÞ

k1 v1 ð1αÞ B ¼ Lð1αÞþl , 12 α

(23)

(24) ðk1 k2 Þð1αÞ C ¼ 2ðLð1αÞþl , 12 αÞ

, k2 ¼ ξ3 þ λ LlD12  1, l1 is the train length and v1 is the

train speed.

a

b

c

Fig. 5. (a) Schematic of the train enters the tunnel. (b) Schematic of the train runs inside the tunnel. (c) Schematic of the train leaves the tunnel.

Fig. 4. Comparisons between the numerical and experimental results. 71

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

(2) The process of the train moving on a constant speed in the tunnel: ①Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2:


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

(25)

➁ Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3:


l1 1 2 dvs ρv þ ρl1 P3 ¼ P2 þ ξ2 þ λ1 þ ξ3 2 s d dt

(26)

Fig. 6. Comparison between the results of two models.

➂ Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4:


1 l34 1 2 dv ρv þ ρl34 P3 þ ρv2 ¼ Pa þ ξ4 þ λ 2 dt D 2 dv ¼ A þ Bv þ Cv2 dt

 t2

l1 L ; v1 v1

velocity of the two models is identical, but the characteristics of the growth and attenuation of the piston wind are significantly different. The MBE of the uniformly accelerated (decelerated) model is 22%. The error is unacceptable. Therefore, the uniformly accelerated (decelerated) model cannot accurately predict the actual changes of the piston wind when the train passes through a tunnel. The following analysis of the characteristics of the piston wind is all based on the full scale passing model. Different variables are changed to analyze the effects of the train speed, the train length, the tunnel blockage ratio and the tunnel length on the piston wind.

(27)

 (28)

k v2 ð1αÞ

k2 v1 ð1αÞ 2 k1 Þð1αÞ where A ¼ 2ððl12 þl2 341 Þð1αÞþl1 Þ, B ¼ ðl12 þl , C ¼ 2ððlðk , 34 Þð1αÞþl1 12 þl34 Þð1αÞþl1 Þ 34 þ ξ4 , k2 ¼ k1 ¼ ξ1 þ λ l12 þl D

l

ξ2 þλ1 d1 þξ3 . ð1αÞ2

3.3. Sensitive analysis on piston effect

(3) The process of the train leaving the tunnel gradually:

3.3.1. Effect of train speed on the piston effect Based on the parameters of the typical tunnel, 11 different train speed are adjusted to analyze the influence. The train speed is changed from 5 m/s (18 km/h) to 100 m/s (360 km/h), which represents the breakdown speed and the speed of high-speed rail. The results are shown in Fig. 7. With the increase of the train speed, the variation characteristics of the piston wind are significantly different. As shown in Fig. 7, the maximum piston wind and the duration time change immediately with the variation of the train speed. In order to compare the cumulative effects of different influencing factors on the piston wind, the ventilation rate is proposed. The results of the ventilation rate are presented in Fig. 8. The ventilation rate η, is defined as:

① Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2:


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

(29)

➁ Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3:


1 l23 1 2 dvs ρv þ ρl23 Pa ¼ P2 þ ρv2 þ ξ2 þ λ1 þ ξ3 2 2 s d dt dv ¼ A þ Bv þ Cv2 dt

 t2

L L þ l1 ; v1 v1

k2 v21 ð1αÞ

A ¼ 2ðl12 ð1αÞþl23 Þ,

where k1 ¼ ξ1 þ

λ lD12

þ 1, k2 ¼

B¼

(30)



v1 ð1αÞ l12k2ð1αÞþl , 23

n P ðvt ⋅Atunnel ⋅ΔtÞ

(31)

η ¼ t¼0 C¼

ðk2 k1 Þð1αÞ 2ðl12 ð1αÞþl23 Þ,

(34)

Vtunnel

l

ξ2 þλ1 23 d þξ3 . ð1αÞ2

(4) The decay process of airflow after the train left the tunnel: Bernoulli's equation for the relative motion between the inlet and outlet of the tunnel:


L 1 2 dv ρv þ ρL Pa ¼ Pa þ ξ1 þ λ1 þ ξ2 D 2 dt dv kv2 ¼ dt 2L

 t2

 L þ l1 ;∞ v1

(32)

(33)

where k ¼ ξ1 þ λ DL þ ξ2 . Fig. 6 shows the comparison between the uniformly accelerated (decelerated) model and the passing model. The maximum piston wind Fig. 7. Effect of train speed on the piston wind. 72

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

Fig. 8. Effect of train speed on the ventilation rate.

Fig. 10. Effect of train length on the ventilation rate.

With the train speed increasing from 5 m/s to 30 m/s, the ventilation rate is increased by 53%. The increase in ventilation rate is less than 21% while the train speed is greater than 30 m/s. Thus, 30 m/s is an economical train speed for tunnel ventilation. If the train speed is greater than 30 m/s, the increase of train speed has little effects on the ventilation rate. The logarithmic fitting formula is listed in Fig. 8. 3.3.2. Effect of train length on the piston effect Ten different train lengths are selected to analyze the influence of train length on the piston wind. The train length is changed from 20 m (1 carriage) to 200 m (10 carriages), which represents the cases of a locomotive passing through the tunnel and a super long subway train passing through the tunnel. The results are shown in Fig. 9. From Fig. 9, the duration time of the piston wind is almost unchanged with the increase of train length. But the maximum piston wind varies significantly with the variation of train length. With the train length increasing from 20 to 200 m, the ventilation rate of each train passing through the tunnel increases from 0.45 to 0.80, which is increased by 78%. Thus the train length has a great influence on the amount of piston wind. The effect of the train length on the ventilation rate is shown in Fig. 10. The exponential fitting formula is also listed in the figure.

Fig. 11. Effect of blockage ratio on the ventilation rate.

3.3.3. Effect of blockage ratio on the piston effect In order to study the influence of blockage ratio on the piston wind, the blockage ratio is changed from 0.1 to 0.9. Since the windward area of the train cannot be changed, the hydraulic diameter of the tunnel is changed from 2.38 m to 20.87 m. Fig. 11 presents the effect of the blockage ratio on the ventilation rate. The ventilation rate of each train passing through the tunnel decreases firstly and then increases with the growth of the blockage ratio. A quartic polynomial fitting is performed to

Fig. 12. Effect of tunnel length on the piston wind.

quantify the effect of the blockage ratio on the ventilation rate. When the blockage ratio is 0.59, the ventilation rate is 0.68, which is the minimum. Since, the blockage ratio of a typical tunnel is 0.41. Therefore, decreasing the blockage ratio, which is to increase the cross-sectional area of the tunnel, will improve the ventilation rate. The polynomial fitting formula is also listed in Fig. 11. 3.3.4. Effect of tunnel length on the piston effect The tunnel length is another important factor affecting the unsteady piston wind. Fig. 12 shows the variation characteristics of the piston wind. In the case of the constant train speed 30 m/s, the longer the tunnel

Fig. 9. Effect of train length on the piston wind. 73

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78 Table 2 The comparison of the calculated results with the fitting results.

Fig. 13. Effect of tunnel length on the ventilation rate.

Table 1 The fitting coefficients. A

B

C

D

E

0.48

0.07

0.23

157.54

9.24

length is, the longer it takes for the train to pass through the tunnel. Thereafter, the duration of the piston wind will change as the tunnel length increases. However, when the tunnel length is relatively short, the maximum size of the piston wind is greatly influenced by the tunnel length. Fig. 13 shows the simulation results of the tunnel ventilation rate when the tunnel length increases from 500 m to 6000 m. Its effect on the ventilation rate is significant. A decrease of 84% of the ventilation rate generates while the tunnel length increases from 500 m to 6000 m. Furthermore, the exponential fitting formula is listed in the figure. 3.3.5. The linear fitting of the prediction formula on the ventilation rate and the grey correlation analysis In the previous study, the influence of the four factors on the ventilation rate is studied, and the characteristic curves are fitted. In this section, we will combine the four influencing factors to fit the universal fitting formula. The form of the fitting formula is given in Eq. (35), and the fitting coefficients are listed in Table 1. As the fitting formula is based on the results of full scale theoretical passing model. The formula can predict the train speed range of 5 m/s to 100 m/s, the train length range of 20 m–200 m, the blockage ratio range of 0.1–0.9 and the tunnel length range of 500 m to 6000 m. The applicability of the fitting formula is listed below the formula, which is applicable to most subway tunnels in China. Table 2 shows the comparison of the theoretical calculated results with the fitting results. The MBE between the theoretical calculated results and the fitting results is 5%, and the correlation coefficient is 0.97. Thus, Eq. (35) is a good predictor of the ventilation rate of one subway train passing through the subway tunnels in China. The results of grey correlation analysis are shown in Table 3.

η0 ¼ A þB⋅lnðvÞ þ C⋅l0:2483 þ D⋅L0:733  þE⋅ α4  2:0255α3 þ 1:6073α2  0:6038α v 2 ½5m=s; 100m=s l 2 ½20m; 200m α 2 ½0:1; 0:9 L 2 ½500m; 6000m

case

v (m/s)

l (m)

α

L (m)

η

η0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

5 10 20 30 40 50 60 70 80 90 100 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

120 120 120 120 120 120 120 120 120 120 120 20 40 60 80 100 120 140 160 180 200 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0.47 0.57 0.66 0.71 0.75 0.78 0.80 0.82 0.84 0.85 0.87 0.45 0.55 0.60 0.65 0.68 0.71 0.74 0.76 0.79 0.80 1.24 0.93 0.79 0.72 0.69 0.68 0.71 0.79 1.05 1.64 0.97 0.71 0.58 0.49 0.43 0.38 0.35 0.32 0.30 0.28 0.26

0.31 0.66 0.71 0.74 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.46 0.56 0.62 0.66 0.70 0.74 0.77 0.79 0.82 0.84 0.95 0.81 0.75 0.74 0.73 0.73 0.76 0.86 1.08 1.65 0.99 0.74 0.60 0.50 0.44 0.39 0.36 0.33 0.30 0.28 0.26

Table 3 The grey correlation analysis. v

l

α

L

0.65

0.71

0.74

0.47

train has been researched. However during the peak hours of the subway, the running interval is about 2 min, and the design speed of subway train is 100 km/h. The trailing distance between two adjacent subway trains is 3.3 km. With the length of newly built subway tunnels increasing and the time interval between the two adjacent subway trains shortening, the probability of two trains running in the same tunnel direction at the same time increases. Such as the Zhongliangshan tunnel in the Chongqing (in southwest of China) Metro Line 1 is 4.33 km long, and the Laoshan subway tunnel in Qingdao (in southeast of China) is 4.58 km long. Thus, it is necessary to study the piston effect of two adjacent subway trains running in the same tunnel direction at the same time. This section will discuss the piston effect caused by multi-trains. The theoretical mode involves seven processes as shown in Fig. 14: ➀ the first train enters the tunnel; ➁ the first train moves on a constant speed in the tunnel; ➂ the second train enters the tunnel; ➃ the two trains move at a constant speed in the same tunnel direction; ➄ the first train leaving the tunnel; ➅ the second train moves at a constant speed in the tunnel; ➆ the second train leaves the tunnel, and the decay process of airflow after the two trains all leave the tunnel. The theoretical model for processes ➀, ➁, ➅, ➆ are the same with the model of single train, and the

(35)

4. Discussion 4.1. The theoretical model of the piston wind caused by multi-trains In the previous sections, the characteristics of the piston effect for one

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78




l2 P3 ¼ P2 þ ξ2 þ λ1 þ ξ3 d

a



1 2 dvs2 ρv þ l2 ρ 2 s2 dt

(42)

Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4 is:

b

P3 ¼ P4 þ λ

l34 1 2 dv ρv þ l34 ρ dt D 2

(43)

Bernoulli's equation for the relative motion between cross-section 4 and cross-section 5 is:


 l1 1 2 dvs1 ρv þ l1 ρ P5 ¼ P4 þ ξ4 þ λ1 þ ξ5 2 s1 d dt

c

(44)

Bernoulli's equation for the relative motion between cross-section 5 and cross-section 6 is: Fig. 14. (a) Schematic of the second train enters the tunnel. (b) Schematic of two trains running inside the tunnel. (c) Schematic of the first train leaves the tunnel.


1 l56 1 2 dv ρv þ ρl56 P5 þ ρv2 ¼ Pa þ ξ6 þ λ 2 dt D 2

Bernoulli's equations for the other processes are defined as follows: For Process 3, the second train enters the tunnel: When the first train moves to nL in the tunnel (n < 1 ), the second train is assumed to enter the tunnel. Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2 is:


l12 1 2 dvs2 ρv þ l12 ρ P2 ¼ Pa þ ξ1 þ λ1 þ ξ2 2 s2 d dt

It can be obtained from the simultaneous equations:

dv ¼ A þ Bv þ Cv2 dt

l23 1 2 dv ρv þ ρl23 dt D 2

l1 P4 ¼ P3 þ ξ3 þ λ1 þ ξ4 d



1 2 dvs1 ρv þ l1 ρ 2 s1 dt

k3 ¼

(37)

t2

  nL nL l2 ; þ v1 v1 v2

k4 ¼ ξ6 þ λ lD56  1.

(47)

Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3 is:

(38)


l2 1 2 dvs2 ρv þ l2 ρ P3 ¼ P2 þ ξ2 þ λ1 þ ξ3 2 s2 d dt

(48)

Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4 is:

(39) P3 ¼ P4 þ λ

l34 1 2 dv ρv þ l34 ρ dt D 2

(49)

Bernoulli's equation for the relative motion between cross-section 4 and cross-section 5 is:

(40)


l45 1 2 dvs1 ρv þ l45 ρ Pa ¼ P4 þ ξ4 þ λ1 þ ξ5 2 s1 d dt

ðk1 v22 þk2 v21 Þð1αÞ

2 þk2 v1 Þð1αÞ where A ¼ 2ðl12 þl1 þð1αÞðl23 þl45 ÞÞ, B ¼ l12ðkþl11vþð1αÞðl , 23 þl45 Þ   l23 l12 l1 k1 þk2 k3 λ D ð1αÞ ξ1 þλ1 d þξ2 ξ3 þλ1 d þξ4 C ¼ 2ðl12 þl1 þð1αÞðl23 þl45 ÞÞ, k1 ¼ ð1αÞ2 , k2 ¼ ð1αÞ2 , k3 ¼ ξ5 þ λ lD45  1.

(50)

It can be obtained from the simultaneous equations:

For Process 4, the two trains move at a constant speed in the same tunnel direction: Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2 is:


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

l

ξ4 þλ1 d1 þξ5 , ð1αÞ2


1 l12 1 2 dv ρv þ l12 ρ Pa ¼ P2 þ ρv2 þ ξ1 þ λ 2 dt D 2

It can be obtained from the simultaneous equations:

dv ¼ A þ Bv þ Cv2 dt

(46)

For Process 5, the first train leaves the tunnel: Bernoulli's equation for the relative motion between cross-section 1 and cross-section 2 is:

1 dv 1 v dvs1 where vs1 ¼ v1α , dt ¼ 1α dt . Bernoulli's equation for the relative motion between cross-section 4 and cross-section 5 is:


1 l45 1 2 dv ρv þ ρl45 P4 þ ρv2 ¼ Pa þ ξ5 þ λ 2 dt D 2

  nL l2 L þ ; v1 v2 v1

ðk v2 þk v2 Þð1αÞ

Bernoulli's equation for the relative motion between cross-section 3 and cross-section 4 is:




t2

2 2 3 1 2 v2 þk3 v1 Þð1αÞ where A ¼ 2ðl2 þl1 þð1αÞðl , B ¼ l2 þlðk , 12 þl34 þl56 ÞÞ 1 þð1αÞðl12 þl34 þl56 Þ   l34 l2 k2 þk3 k1 k4 λ ð1αÞ ξ þλ1 d þξ3 k1 ¼ ξ1 þ λ lD12 þ 1, k2 ¼ 2 ð1αÞ C ¼ 2ðl2 þl1 þð1αÞðl12 þlD 34 þl56 ÞÞ, 2 ,

(36)

1 dv 2 v dvs2 where vs2 ¼ v1α , dt ¼ 1α dt . Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3 is:

P2 ¼ P3 þ λ

(45)

dv ¼ A þ Bv þ Cv2 dt

 t2

L L þ l1 ; v1 v1

 (51)

ðk v2 þk v2 Þð1αÞ

ðk2 v2 þk3 v1 Þð1αÞ 3 1 where A ¼ 2ðl2 þl2452þð1αÞðl , B ¼ l2 þl , 12 þl34 ÞÞ 45 þð1αÞðl12 þl34 Þ   l34 l l2 k2 þk3 k1 λ D ð1αÞ ξ2 þλ1 d þξ3 ξ4 þλ1 45 þξ l12 5 d . C ¼ 2ðl2 þl45 þð1αÞðl12 þl34 ÞÞ, k1 ¼ ξ1 þ λ D þ 1, k2 ¼ ð1αÞ2 , k3 ¼ ð1αÞ 2

(41)

Bernoulli's equation for the relative motion between cross-section 2 and cross-section 3 is:

4.2. Effect of the trailing distance The position coefficient n represents the position of the first train arrives when the second train enters the tunnel. Eight different position 75

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

Fig. 15. Results of the piston wind caused by multi-trains.

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Journal of Wind Engineering & Industrial Aerodynamics 171 (2017) 67–78

3. The effects of the trailing distance between multi-trains on the piston wind is discussed according to the multi-trains model. With the position coefficient n increasing from 1/15 to 16/15, the ventilation rate is increased by 23%. A cubic polynomial fitting is performed to quantify the effects of position coefficient at the ventilation rate, which is applicative when two trains running in the same tunnel direction at the same time. The results show that a longer tracing distance is beneficial to improve the ventilation rate. The calculation method mentioned in this paper is also applicable to ordinary railway tunnels and high-speed railway tunnels. Acknowledgment All authors are grateful for the financial support provided by the Science and Technology Project of Urban-Rural Construction Commission of Tianjin Municipality (No. 2016–18) and the China Postdoctoral Science Foundation (No. 2016M600189).

Fig. 16. Effect of multi-trains on the ventilation rate.

coefficients are selected to analyze the influence of the piston wind. The position coefficient is changed from 1/15 (the second train enters the tunnel on the first train's heels) to 16/15 (the first train just left the tunnel, the second train enters the tunnel), which represents different trailing distances. The results are shown in Fig. 15. As summarized in Fig. 15, the maximum piston wind is basically unchanged with the increase of position coefficient n. But, the duration time of the maximum piston wind decreases as the position coefficient n increases. However, the integral duration time of the piston wind increases as the trailing distance increases. The ventilation rate of the multi-trains passing through the typical tunnel is given in Fig. 16. The polynomial fitting formula is also listed in the figure. With the position coefficient n increasing from 1/15 to 16/15, the ventilation rate of multi-trains passing through the tunnel increases from 0.84 to 1.03, which is increased by 23%. A cubic polynomial fitting is performed to quantify the effects of position coefficient n on the ventilation rate. The fitting formula is suitable to predict the ventilation rate of two trains running in the same direction in the long subway tunnel. What's more, a longer tracing distance is beneficial to improve the ventilation rate.

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5. Conclusion In order to validate the accuracy of the theoretical method, a uniformly accelerated (decelerated) model is established based on a highly cited experiment firstly and the results of the theoretical model are compared against the experimental data and CFD simulation results. Then, a full scale passing model is built to assess the influencing factors of the ventilation rate of one train passing through the typical subway tunnel based on the verified theoretical method. A universal prediction formula is proposed to estimate the tunnel ventilation effect of the piston wind for the most subway tunnels in China. Finally, the theoretical model of two trains running in the same tunnel direction at the same time is built to analyze the effects of the trailing distance between multi-trains on the piston effect. The following conclusions of this study are drawn: 1. The theoretical model is accurate enough to predict the overall trend of the piston effect. The theoretical calculation results are consistent with those of the experimental results, and followed by the threedimensional numerical simulation results. The correlation coefficient between the theoretical calculation results and tested data is 0.99. 2. The full scale passing model is built to analyze the influencing factors of the piston effect based on the verified theoretical method. The grey relational analysis results show that the blockage ratio and the length of the train have the greatest impact on the ventilation rate. What's more, a universal prediction formula on the ventilation rate is proposed, which can be applied on most of the subway tunnels in China. 77

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