Large Eddy Simulations of a typical European high-speed train inside tunnels

Ben Diedrichs and Mats Berg Royal Institute of Technology (KTH), Dept. of Aeronautical and Vehicle Engineering, Div. Railway Technology, Stockholm,

Sweden

Siniša Krajnovi Chalmers, Dept of Thermo and Fluid Dynamics, Gothenburg, Sweden

Copyright © 2004 Society of Automotive Engineers, Inc.

ABSTRACT

This article presents our results of external aerodynamics, obtained with Large Eddy Simulations (LES), about a typical European passenger-stock inside tunnels. The focal points are the aerodynamic forces and their typical frequencies applied to the tail. Two train lengths and three tunnels are employed in the study to model the conditions of double and single-track bores. Owing to the relatively high numerical cost associated with LES for external train aerodynamics we could only afford sufficient spatial grid resolution on our shortest train. The flow simulations confirm the existence of coherent structures alongside the body that give rise to continuously propagating pressure disturbances. These disturbances with a relatively small amplitude and high spatial frequency cannot affect the ride comfort. Still, they are found to influence the flow separation about the tail, which is regarded as one of the candidate mechanisms to impair the ride comfort and running stability.

INTRODUCTION

The importance of external aerodynamics for high-speed rolling stock is augmented as a consequence of increased speed, distributed propulsion along with lighter and longer lead and tail vehicles. As far as the aerodynamics is concerned the applied forces and their typical frequencies factor with the flow speed squared and linearly, respectively. These basic facts in conjunction with the discrete natural frequencies of the vehicles, including the kinematic frequency due to wheel conicity, indicate that critical forcing can arise at some distinct train speeds. Consequently, aerodynamic phenomena that are undiscovered or under control at lower speeds can quickly manifest themselves under these critical conditions. One such criticality is forced oscillations of the tail vehicle in a train set, which is the scope of the current work. These oscillations can be strong enough to disrupt the passenger comfort. They may also have implications on

the stability, safety and mechanical wear. We emphasize that the current problem is much dependent on the factor of aerodynamic forces over train weight, which has increased for modern passenger stocks, especially for non-motorized trailing units and Double Decker. As the traditional suspension systems are designed to reduce the influence of track irregularities by using enough soft springs and small dampers this conflicts with situations where forcing is applied directly to the body shell. In Japan, tail oscillation is in fact a real problem for the Shinkansen trains, c.f. [1] - [7] in particular when they travel inside double-track tunnels that impose non-symmetric flow conditions about the train. The origin of the periodic motion is yet claimed to be an open issue and is thus not fully understood. [1] investigated the problem for the experimental train STAR 21 under real conditions, mainly in open air. It was found by studying the flow with smoke from an airplane, that vortices are shed with a frequency of 3-5 Hz at the speed of 315 km/h. This corresponds to a non-dimensional frequency of St = 0.12 to 0.2 (St = Strouhal number is based on the train height and cruising speed). Spectra of pressure transducers about the tail indicated peak fluctuations at St numbers of 0.16 and 0.184, which thus are in the range of the smoke visualizations. Hot wire probes picked up a peak frequency at St = 0.148. An accelerometer that was fixed to the floor of the tail coach registered a peak signal at about 1.5 to 2.0 Hz. When moving the train through a tunnel the response, correlating the pressure differences of two sensors on opposite sides of the tail with that of the accelerometer signal, was amplified. Note: It is more relevant to use non-dimensional frequencies for the aerodynamics but, we will revert to frequencies in Hertz as far as the vehicle dynamics is concerned. [2] presents results from numeric flow simulations for a train that is 2½ vehicles long. Results obtained in open air and inside a double-track tunnel are compared, indicating substantially larger disturbances inside the tunnel. The larger fluctuations are explained with earlier separation on the train side closest to the tunnel wall. A

spectral analysis of the yawing moment discerned a small augmentation towards St = 0.1. Although, this is lower compared to what was measured for the STAR 21 train, it is in the range of the natural frequencies for comparable vehicles, see Table 1. The study was supplemented by simulating the flow about a train tail with flat sides inside a similar tunnel. Results showed that the flat side on the tunnel wall side (that vehicle side closest to the tunnel) successfully suppressed the unsteadiness of the separation lines, therewith mitigating the aerodynamic oscillations. However, the rather coarse computational grid should encourage further studies to investigate the issue of grid independence. Table 1. Natural frequencies and their damping ratios for the ICE 2 end coach. Results are obtained with the Multi Body Simulation tool described in [9].

Excitation mode Frequency [Hz] Damping [%]

Lower Sway 0.50 21.2 Body Yaw 0.75 62.2 Body Bounce 0.75 4.7 Body Pitch 0.90 5.4 Upper Sway 1.30 22.7

The motivation to understand and reduce the origin of pressure disturbances is explained by the fact that some of the Shinkansen trains run up to about 50% inside tunnels, c.f. the Sanyo line [3]. In the latter work, real measurements of lateral vibrations on the vehicle ahead of the tail coach for the WIN 350 train are reported. That vehicle holds the current collector on the roof that is covered, for aeroacoustic purposes, with a somewhat large casing mounted towards the rear. The measurements show that the aerodynamic forcing is substantially higher inside tunnels and is associated with a peak frequency of about 3 Hz, that corresponds to St = 0.14 at 270 km/h. Train meetings inside tunnels caused higher preferred frequencies and aerodynamic forces. This can readily be explained with dimensional analysis, owing to the increased flow velocity past the train. Vehicle oscillations in [3] are recognized to be caused by separation phenomena about the tail, but also by continuous pressure fluctuations along the vehicle side closest to the tunnel wall. Measurements showed that the RMS values of pressure fluctuations amplify downstream geometrical discontinuities such as inter-car gaps. Measurements also revealed that peak values of the periodic angular acceleration were as high as 0.1 rad/sec2 caused by a yawing moment of some 25 kNm. At a position of, say, 10 m away from the vehicle’s midpoint this corresponds to a maximum lateral acceleration of 1 m/sec2 and a typical side force of 2.5 kN. At the frequency of 3 Hz this acceleration amplitude would give a ride comfort index of Wz = 3.6 (Wertungzahl). Wz describes the human perception to acceleration motions. For a discussion of various ride comfort evaluation methods see [8]. In this reference the ride comfort associated with Wz = 3.6 is described as unpleasant to passengers for which prolonged exposure is intolerable.

In addition, [7] presents results from real pressure measurements conducted on WIN 350 in a double-track tunnel, supplemented by a numerical study. The experimentally obtained RMS values of the pressure disturbances showed a continued amplification alongside the train to 150 m from the nose, where the fluctuations stabilized. Both measurements and calculations showed that pressure fluctuations are more pronounced near the tunnel wall side. However, the somewhat coarse computational grid utilized in this study and the absence of presenting a grid independence test should again encourage continuing studies. This is emphasized since the spatial frequency of the calculated pressure disturbances were substantially higher than what was measured. Mitigation effects of the yawing motion using dampers between the Shinkansen vehicles are described in [4]. [5] and [6] explain the improvements that follow using a semi-active suspension technique on trains with vehicle tilt. In the latter it is explained that the tilt systems can make the vehicles more susceptible to vibrations applied to the body, especially as low frequency motions are damped less effectively. The European joint research project RAPIDE [10] carried out flow simulations with an unsteady RANS (Reynolds Average Navier Stokes) solver about Deutche Bahn AG’s high-speed train ICE 2. They compared the cases of running inside a tunnel and in open air. A body length of 40 m in full scale was used. Flow properties at the inlet cross section 40 m upstream the very tail, to give a total train length of 210 m, were extracted from a prior steady state calculation. In contrast to the Japanese findings, that project concluded that the aerodynamic forces about the tail vehicle of ICE 2 is nearly five times smaller than what is prescribed by [11] and [12], which relate to passenger comfort measures of continuous disturbances. The project also reported small differences of the aerodynamic forces in open air and inside a double-track tunnel. These results therefore give a diverse picture compared to the observations made for the Shinkansen trains. The relatively high Reynolds number (Re = flow inertia forces over viscous forces) used in the simulations with steady inlet boundary condition together with the shortcomings of the RANS methodology, likely explain why the pressure disturbances found about the Shinkansen trains could not be reproduced. However, the force frequency was established to 1.36 Hz at 35 m/s to give St ~ 0.15, irrespective if running inside the tunnel or in open air. This matches the findings for the STAR 21 train running in open air. [13] continued with the vehicle dynamics study employing the aerodynamic data from [10]. Wz for the lateral acceleration above the trailing bogie was calculated to merely 2.2. High-fidelity aerodynamic simulations of the flow about trains are indeed and will be in the future a grand challenge as far as computing resources are concerned. For example, [14] points out that the number of grid points required to successfully resolve the near wall behavior of the flow past a simple flat plate scales with Re1.8. Increasing the Reynolds number an order of magnitude therewith requires nearly 65 times the number

of near wall cells. Further, about sharp trailing edges the flow changes within the distance equal to the kinematic viscosity over wall velocity [15], wherefore the grid length needs to be shorter than that distance adjacent to separation lines. An overview of transient flow simulation techniques for vehicle aerodynamics is found in, for example, [16]. Amongst these methods flow simulations employing LES are carried out by [17] about a bus-shaped body (Ahmed body with a zero slant angle) at Re = 2.1⋅105 based on the height and incoming flow velocity. Computing times of the order of several weeks on a modern main frame with a few dozens of central processing units are reported. In that study results from three grids with 1.8, 2.1 and 4.5 million fluid cells are compared. Small coherent structures along the vehicle body that have not been revealed before are studied. A comparison is made of the time averaged aerodynamic coefficients of drag and lift for the various grids. Between the two finest grids the lift and drag force differed by 7% and 4%, respectively. An oscillating side force applied to the body with a dominating frequency of St = 0.22 was recorded. That frequency matched the shedding of wake vortices in the experimental study [18] that used a Re = 1.6⋅106. Encouraged by the study [17], we undertake a series of LES calculations for simplified train bodies based on the German ICE 2. Our most accurate calculation with the shortest train is thought to be a good candidate for a reference to pursue other less expensive simulation techniques.

Governing equations for LES LESs are here carried out with the commercially available flow simulation software for parallel computing STAR-CD/HPC from CD-Adapco Group Inc. That code is based on finite volume method for solving the incompressible filtered Navier-Stokes equations in conjunction with the sub grid scale (SGS) model of Smagorinsky [19]. The latter model is appreciated for its simplicity and relatively low computational cost. The governing equations for the resolved scales of the flow are calculated with the filtered incompressible Navier-Stokes equations and continuity,

i

i

jijiji

j

i uxp

uuuuuux

ut

ij

21 ∇+∂∂−=

−+∂∂+

∂∂ ν

ρτ

, (1)

0=∂∂

i

i

xu . (2)

Time integration is carried out with the 2nd order accurate Crank-Nicholson scheme. Convective fluxes are discretized with MARS (Monotonic Advecting Reconstruction Scheme) developed by CD-Adapco Group Inc, using the highest blending factor of unity. The latter introduces a minimum of upwinding and maximizes the resolution sharpness. MARS is a 2nd order accurate bounded scheme, which is multidimensional gradient limited. In comparison with the Central Difference

Scheme, it is more expensive to use. In addition, it was found to give somewhat closer agreement with experimental results, predicting the flow about a surface mounted cube [16]. The SGS stress tensor denoted in Eq. (1) is,

jijiij uuuu −=τ . (3)

Overbar denotes low pass filtering on grid level. Hence,

iu is the resolved velocity field in three dimensional space and time. The Smagorinsky model that treats the SGS stresses is based on a traditional eddy viscosity approach,

.32

32

2

ijSGSi

j

j

iSGS

ijSGSijSGSij

kxu

xu

kS

δν

δντ

+

∂∂+

∂∂⋅−

=+⋅−=

(4)

The SGS properties of kinetic energy and turbulent viscosity are modeled as follows,

.

,2

2

ijijsSGS

ijijkSGS

SSc

SSck

∆=

∆=

ν (5a-b)

Dissipation of turbulent energy on sub-grid scale is also defined as,

( ) ,/ 2/322/3ijijkSGS SScCkc ∆=∆= εεε (6a-b)

where the constants used in the simulations are given the following standard values,

.3/2

2/32

2

2

,5.1

,0425.0)2/3(2

,203.02

sk

k

ks

k

ccc

C

Cc

c

−

−

=

=

==

==

ε

π

π

(7a-d)

Hence, cs, which is really a function of space, time and the flow properties, is here treated as a constant.

3 V=∆ where V is the cell volume is the characteristic filter width of the implicit space filtering associated with the so called top hat filter,

∆≤−∆

= otherwise ,0

2/' if ,/1)',( ii

iixx

xxG (8)

The filtered velocity field, in explicit notation, is obtained with the following integral operation,

')',()'( iiiii dxxxGxuu Ω= , (9)

where Ω is the entire flow domain. Eqs. (5) and (6) for the SGS turbulent properties indicate that halving the grid size in all directions reduces the SGS turbulent kinetic energy and dissipation four times. To achieve this

the computational effort for flow simulations about vehicles increases roughly speaking 16 times (8×2 in space and time)!

Geometry, numerical characteristics and boundary conditions The geometry of the computational domain is depicted in Figure 1. The train shapes are based on Deutsche Bahn AG’s high-speed train ICE 2. As already mentioned that train was used in [10] and in e.g. [20] to study effects of crosswind with CFD. In this work the running gear, front plough, inter-car gaps and protruding equipment are omitted for simplicity. In the simulations, the train height measured from the ground plane, body width and the ground clearance are h = 3.817 m, 0.7912 h and 0.0608 h, respectively. Two train lengths are used, viz. 6.55 h (25 m, typical one vehicle length) and 26.20 h (100 m, typical four vehicle lengths). The intention with the longer train is to compare results with the study of [7]. Since we could not afford sufficient grid resolution for any of the longer train geometries these results should be regarded as essentially qualitative. In front and behind of the trains the domain extends 8 h and 16 h, respectively. Re number based on h and the incoming flow speed U is 1.01⋅105. Important to the accuracy of the flow predictions are the non-dimensional wall distances, which are calculated from,

.3,2,1

,/*

===∈

∆=∆=+

zyxi

xu

xx walliii ν

ρτν (10)

The surface shear stress wallτ is obtained from the simulation. u* and ν are the friction velocity and molecular kinematic viscosity, respectively. ρ is the density of the air. On a perfectly flat parallel plate

zyxixi ,, ; ∈+ typifies the wall distances in the streamwise direction, normal to the surface and transverse to the flow, respectively. ix∆ denotes the corresponding surface grid spacing. We have attempted for the shorter train, i.e. grid C in Table 2, to achieve time-averaged values with analogous definition confined to,

+1x < 150, +

2x < 1 and +3x < 40, (11)

on most of the body, c.f. the requirements given in [21]. Figure 2 shows the surface grid about the front-end. To satisfy the CFL condition, that is CFL < 1, for our finest grid we needed a time step of ∆t = 0.0042 units (time is made non-dimensional with U and h throughout this work). This gave maximum and average CFL numbers of about 2.82 and 0.217. Less than some 0.03% of the cells had a CFL number exceeding unity. Larger time steps caused the calculation to diverge. The calculations with that fine grid alone, took more than two months to complete on a 20 CPU main frame with the capacity of some 20 GFLOPS.

A uniform and steady inlet flow speed is used. Turbulence kinetic energy at the inlet is set to 1% of the train speed. The train is still whilst the tunnel walls are moving with the velocity of the fluid at the inlet. Neumann conditions are applied to the flow variables at the outlet surface of the tunnel domain. We do not attempt to resolve the near wall behavior of the tunnel walls due to the fact that they are rough surfaces anyway. A total number of five simulations are carried out with grids tagged and listed in Table 2. For the grids A, B and C, the train’s mid point is displaced 0.5895 h laterally, so the distance between the tracks is 4.5 m in full scale. In view of the limitations mentioned above it is only the results derived with grid C that we consider accurate enough for quantitative assessments. Results of grids A and B are included to show numerical disparities for this type of calculation and for qualitative assessments for longer train bodies. Despite the deficient number of grid points, our coarsest grid is comparable with the grids used in the train simulations described in the introduction. Grids D and E provide too a qualitative picture of the flow inside single-track bores with varying R, the ratio of cross-sectional areas of the train and tunnel.

Figure 1. Tunnel geometry definitions. Origin of the co-ordinate system with axes x, y and z is aligned with the train’s nose, mid tunnel and on the ground, respectively.

Figure 2. Grid C: Surface grid of front-end and tunnel.

ht L

rt

wt

Table 2. Listing of computational grids and pertinent geometry factors scaled with the body height h. Grid L wt ht rt Train

length Number of cells (body

surface cells) ×106 A 50.30 2.23 0.85 1.11 26.20 2.44 (0.053) B 50.30 2.23 0.85 1.11 26.20 4.13 (0.085) C 30.91 2.23 0.85 1.11 6.55 6.37 (0.091) D 50.30 1.31 0.85 0.65 26.20 3.38 (0.070) E 50.30 1.83 1.14 0.92 26.20 4.34 (0.072)

Coherent flow structures Inside tunnels the mean relative pressure along the train is linearly decaying from the nose towards the tail, as shown in Figure 6. The greater the ratio of R the steeper the pressure changes. Visualizing small pressure fluctuations for long trains for situations where R is large is therefore difficult. It is easier to visualize the flow features in terms of vorticity, which is obtained by taking the curl of the velocity field. Figure 3 plots the instantaneous longitudinal vorticity fields of ωx in three cross sections. For the long train these cross sections are chosen at [–3.27, –13.10, –22.27] h from the nose, whilst for the shorter train they are chosen at [–1.57, –3.27, –4.98] h. The results confirm the findings in [3] and [7], that coherent

disturbances spontaneously arise about the body shell. Our calculations though, with a different geometry, indicate that they originate at the upper and lower corners on the tunnel center side. In addition, the vorticity plots really convince us that grid independence for at least grid A has not been achieved. For the latter grid this is most obvious comparing results at the cross section −22.27 h, see Figure 3. At that location, grid B predicts flow disturbances all along the tunnel center side of the body in contrast to that of grid A. Figure 4 plots the vorticity distribution about the trains inside the single-track tunnels. In these situations vortices are clearly more absent but, for the smallest tunnel they are found above the roof towards the tail.

x = -3.27 h x = -13.10 h x = -22.27 h

x = -1.57 h x = -3.27 h x = -4.98 h

Figure 3. Longitudinal vorticity ωx for grids A (top three figs), B and C (bottom three figs).

x = -3.27 h x = -13.10 h x = -22.27 h

Figure 4. Longitudinal vorticity ωx for grids D and E.

Figure 5 plots snapshots of the pressure coefficient Cp on the body surfaces obtained with the grids A, B and C. This again shows the existence of pressure fluctuations that originate at the upper and lower right corners. Closest to the front these disturbances are stronger about the roof in contrast to those near the ground. On the tunnel wall side of the train, the calculations do not predict vortices. For the cases where the train is placed non-symmetrically in the tunnel, the vortices that originate at the upper right corner gently spread to the tunnel wall side along the roof and along the tunnel center side of the vehicle. This can also be seen in Figure 3. Figure 6 plots instantaneous surface pressure distribution, Cp, alongside and downstream the train near the upper and lower corners obtained with the grids A, B and C. Apart from the qualitative similarities, that vortices are predicted about these corners, quantitatively the results differ as far as the amplitude and spatial frequency of these flow features are concerned. The most conspicuous finding is that the more we refine the grid the shorter in length become these pressure disturbances. Therewith the importance to fulfill the

requirements pointed out in Eq. (11) is stressed. It is also observed that grid A predicts the flow features about the lower corner to amplify first after some 9.2 h from the nose. In grids B and C they have larger amplitude already near the nose.

a)

b)

c)

Figure 5. Instantaneous surface pressure distributions, Cp, for a) grid A (nose and tail), b) grid B (nose and tail) and c) grid C.

The maximum strength, peak to peak, of these pressure fluctuations is ∆Cp ~ 0.05. This corresponds to the real relative pressure of merely 180 Pa at 280 km/h. In the near wake the spatial fluctuations at that instant in time are nearly four times greater and the spatial frequency is significantly lower too. Hence we conclude, at least for

the current geometries, that the fluctuations due to the vortices have insignificant effect on the vehicle’s motions. Rather, it is more likely that the flow separation about the tail with its larger amplitude and lower frequency is responsible for the crucial forcing.

−32.74 −26.2 −22.27 −13.1 −3.27 0

−1

−0.8

−0.6

−0.4

−0.2

0C

p

Distance from front

Mesh A

−13.1 −3.27

−0.6

−0.59

−0.58

−0.57

−0.56

−0.55

−0.54

−0.53

−0.52

−0.51

−0.5

Cp

Distance from front

Mesh A

−32.74 −26.2 −22.27 −13.1 −3.27 0

−1

−0.8

−0.6

−0.4

−0.2

0

Cp

Distance from front

Mesh B

−13.1 −3.27

−0.6

−0.59

−0.58

−0.57

−0.56

−0.55

−0.54

−0.53

−0.52

−0.51

−0.5

Cp

Distance from front

Mesh B

−13.1 −6.55 −3.27 0

−1

−0.8

−0.6

−0.4

−0.2

0

Cp

Distance from front

Mesh C

−4.98 −3.27 −1.5719

−0.6

−0.58

−0.56

−0.54

−0.52

−0.5

−0.48

Cp

Distance from front

Mesh C

Figure 6. Cp. Instantaneous pressure coefficient. Solid line: Roof corner. Dotted line: Lower corner. The right column of figures is close ups.

Figure 7. Propagation of pressure disturbances alongside the body. An alternating color table is used to emphasize the pressure variations. First and last images are at times t = 219.05 and t = 225.34. Images are incremented ∆t = 1.05. z = 0.262 h. Figure 7 depicts pressure disturbances in the range Cp = [–0.6, –0.5] at z = 0.262 h above the ground. The images are confined to the last 13.1 h of the body for grid B. The

seven consecutive times are spaced ∆t = 1.05 units. The dotted trace-lines indicate motion with time. Studying the propagating pressure disturbances, it is found that they affect the flow separation and consequently the force spectrum of the tail. Figure 8 shows a snapshot, from grid C, of the near wake pressure field at z = 0.262 h above the ground. The figure illustrates that the shedding of vortices from the center side along the dotted line resembles the regularity of the coherent structures. On the tunnel wall side the regularity is weaker. This indicates that flow separation, at least those attributed to the higher frequencies, is affected by the coherent structures that develop further upstream the body. Our prefatory attempts with Detached Eddy Simulations, not shown here, that were unable to resolve the coherent structures, did not resolve eddies in the near wake with such regularity. This implies that numerical methods that fail to resolve these flow features would also fail to predict the proper force spectra. It is also observed that occasionally some structures coalesce to cause longer structures with more power to influence the characteristics of the separation, c.f. the development along the right most trace line in Figure 7. Coalescence of structures for the long train occurred towards the tail. This raises the question if and to what extent, their propagation speed is dependent on the pressure amplitude, their spatial extension and proximity to adjacent flow features. In addition, we raise the hypothesis that coalescing vortices could cause lower aerodynamic frequencies for long enough bodies, as the probability of structures to merge would be higher for longer bodies. In that sense the problem of tail vehicle oscillations is train length dependent. These questions are left unanswered for now.

Figure 8. Instantaneous pressure distribution, Cp, in the near wake obtained with grid C viewed from above, z = 0.262 h above the ground. On the tunnel wall side of the tail, vortices are shed with different frequencies, which we saw already in Figure 7. This is obvious viewing the whole animation sequence as a film strip. In Figure 9 we have extracted Cp as a function of time for 15 time units to show just this. The probe locations are listed in Table 3. Figure 10 displays PSD spectra of the pressure data for grid C that are partly plotted in Figure 9. The spectra are based on 204 time units. Figure 9 also demonstrates that the amplitude of the pressure fluctuations caused by the shedding and growth of vortices are some factors larger about the tunnel wall side of the vehicle. Further upstream the very tip, viz.

0.786 h, the pressure amplitude and preferred frequencies change significantly. The smallest frequency at that point is considerably lower in comparison with what is recorded in the wake. The ripples overlaid on the pressure data at the tunnel center side of the vehicle, most obvious for grid B, are caused by the passing coherent structures. As no vortices are present on the tunnel wall side of the vehicle the corresponding data is much smoother.

The spectra prove that the probes upstream the tail have a lower frequency content. This great variance of the pressure characteristics partly explains why pressure measurements to reveal exact frequencies of tail oscillations are difficult. It can also be observed that the frequency content on the tunnel side is somewhat richer.

a) 388 390 392 394 396 398 400 402 404 406

−0.1

−0.05

0

0.05

0.1

388 390 392 394 396 398 400 402 404 406−0.7

−0.675

−0.65

−0.625

−0.6Grid A

Cp le

ft, Cp rig

ht

388 390 392 394 396 398 400 402 404 406−0.1

−0.05

0

0.05

0.1

Cp le

ft − C

p right

Time b)

388 390 392 394 396 398 400 402 404 406−0.4

−0.2

0

0.2

0.4

0.5

388 390 392 394 396 398 400 402 404 406−0.9

−0.8

−0.6

−0.4

−0.2

0

0.1

Grid A

Cp le

ft, Cp rig

ht

388 390 392 394 396 398 400 402 404 406−0.4

−0.2

0

0.2

0.4

0.5

Cp le

ft − C

p right

Time

c) 226 228 230 232 234 236 238 240 242

−0.1

−0.05

0

0.05

0.1

226 228 230 232 234 236 238 240 242−0.7

−0.675

−0.65

−0.625

−0.6Grid B

Cp le

ft, Cp rig

ht

226 228 230 232 234 236 238 240 242−0.1

−0.05

0

0.05

0.1

Cp le

ft − C

p right

Time d)

226 228 230 232 234 236 238 240 242−0.4

−0.2

0

0.2

0.4

0.5

226 228 230 232 234 236 238 240 242−0.9

−0.8

−0.6

−0.4

−0.2

0

0.1

Grid BC

p left, C

p right

226 228 230 232 234 236 238 240 242−0.4

−0.2

0

0.2

0.4

0.5

Cp le

ft − C

p right

Time

e)106 108 110 112 114 116 118

−0.06

−0.04

−0.02

0

106 108 110 112 114 116 118

−0.64

−0.62

−0.6

−0.58

−0.56Grid C

Cp le

ft, Cp rig

ht

106 108 110 112 114 116 118

−0.06

−0.04

−0.02

0

Cp le

ft − C

p right

Time f)

106 108 110 112 114 116 118

−0.6

−0.4

−0.2

0

0.2

106 108 110 112 114 116 118

−0.8

−0.6

−0.4

−0.2

0Grid C

Cp le

ft, Cp rig

ht

106 108 110 112 114 116 118

−0.6

−0.4

−0.2

0

0.2C

p left −

Cp rig

ht

Time

Figure 9. Time histories of Cp in the near wake at z = 0.262 h. Dash dotted line: y = 0.985 h; Dotted line: y = 0.194 h; Solid line: Pressure difference of the two above, according to the scales on the right axes. a) and c) x = -25.41 h. b) and

d) x = -26.20 h. e) x = -5.76 h f) x = -6.55 h.

0 0.1 0.2 0.3 0.40

0.5

1

Probe 1: Tunnel wall side upstream the tail

PS

D: L

inea

r

St

1.)1.) = 0.0510

2.)

2.) = 0.0655

3.)

3.) = 0.0946

4.)

4.) = 0.13835.)

5.) = 0.2111

0 0.1 0.2 0.3 0.4

0

0.5

1

Probe 3: Tunnel center side upstream the tail

PS

D: L

inea

r

St

1.)

1.) = 0.03642.) 2.) = 0.0874

3.)

3.) = 0.2403

0 0.1 0.2 0.3 0.40

0.5

1

Probe 2: Tunnel wall side downstream the tail

PS

D: L

inea

r

St

1.)1.) = 0.6771

0 0.1 0.2 0.3 0.4

0

0.5

1

Probe 4: Tunnel center side downstream the tail

PS

D: L

inea

r

St

1.)

1.) = 0.0073

2.)

2.) = 0.03643.)

3.) = 0.0655

4.) 4.) = 0.0801

5.)

5.) = 0.1602

6.)

6.) = 0.1966

7.)

7.) = 0.22578.)

8.) = 0.32039.)

9.) = 0.4951

Figure 10. Spectra of pressure recorded with the probes of Table 3 for grid C.

Table 3. Probe locations. Probe Vehicle side x/h y/h z/h

1 Wall -5.76 0.985 0.262 2 Wall -6.55 0.985 0.262 3 Center -5.76 0.194 0.262 4 Center -6.55 0.194 0.262

Table 4. Time averaged values of CSide and CLift.

Grid <CSide> <CLift> Time steps and time units for

averaging

Total number of time steps and

time units A 0.0503 0.0787 28821 302 38821 406 B 0.0409 0.1419 25510 266 29058 304 C 0.1400 0.1264 68111 286 74111 310 D -0.0080 0.1099 10957 57 14357 76 E -0.0018 0.0593 8319 35 12819 53

Table 5. Min, max and standard deviation of the force coefficients regarding the last 2.62 h of the tail.

Grid CSide: min, max, σσσσ CLift: min, max, σσσσ A -0.0628 0.2101 0.0385 -0.0464 0.1689 0.0331 B -0.1049 0.1657 0.0427 0.0292 0.2619 0.0389 C 0.0552 0.2395 0.0316 0.0106 0.2169 0.0303 D -0.3216 0.1868 0.0951 -0.1915 0.2714 0.0855 E -0.089 0.0962 0.0306 2) 2) 2) 1) -0.06 0.06 0.0424 0.165 0.185 0.0071

1) Estimated data from figure in [10]. 2) Insufficient amount of data.

Aerodynamic tail forces Table 4 provides average values of the side and lift coefficients for the last 2.62 h about the tail. The non-dimensional coefficients are obtained by scaling the forces with the dynamic head pressure, ρU2/2, and with the cross sectional area of 0.686 h2 (10 m2) according to standard practice [22]. Ideally <CSide> (time average) obtained with the grids D and E should be zero, but the averaging suffers from too few time steps. We find the lift coefficient to be most susceptible to grid refinements, probably because the precise roaming of separation lines over tail surfaces is difficult to establish numerically, along with the fact that essentially all lift is generated in the vicinity of the tail. The mean value of the side force changes too but to a lesser extent, viz. 19%, in regards to the results obtained with grids A and B. This again confirms that grid independence has not been achieved. Figure 11 and Figure 12 show the time histories of the force coefficients for the last 2.62 h of the tail. The complete histories, for the grids C, D and E seem to indicate that the smaller the ratio of R the earlier the forces stabilize about their mean values. For grid A at least 40 time units were needed, which is equivalent to nearly 4000 time steps. This means that the flow has passed 1.6 train lengths. Grid C needed more than 20 time units and hence 6000 time steps, which for the shorter train equals some 3 train lengths. These figures are comparable to what was found in [23] studying the unsteady flow over a half cylinder in the proximity of a ground plane. Figure 11e and Figure 11f plot the relative side force coefficient about the mean values for the wall and center sides of the 2.62 h tail for the grids B and C. They prove that the largest aerodynamic excitation of the tail arise on the tunnel wall side. Table 5 lists the minimum, maximum and standard deviation of the force coefficients for the respective simulations. In terms of real forces at typical cruising conditions it is found that substantial aerodynamic excitation takes place. A change of the side coefficient of 0.1 at the maximum speed of the current train, which is 280 km/h, corresponds to 3.6 kN. This is comparable to the total drag force of the lead unit in open air conditions. For the grids A, B and C we register intermittently peak to peak values of nearly double and for grid D nearly four times that value, see Figure 11 and Figure 12. From a dimensional view and the use of continuity equation, the aerodynamic forces are expected to scale with,

2

1

0

0

1

1/11/1

~

−−

RR

FF

. (12)

F1 > F0 are the forces corresponding to the blockage ratios R1 < R0. Eq. (12) predicts the forces to be nearly 6 times greater in tunnel D compared to that of tunnel E. If we revert to the ratio of standard deviations in Table 5, the calculations rather indicate a factor closer to three. The amplitude of the side force inside our largest tunnel agrees fairly well with what was found in

[10], despite that they concluded small differences for the situations running inside the tunnel and in open air. The results in terms of the roll moment amplitude are quite comparable too. We registered typical peak-to-peak values of the moment coefficient to ~|0.06| for grid C, made non-dimensional with the leverage of h.

a)40 60 80 100 120 140

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Grid A

Time

Forc

e co

effic

ient

s

b)40 60 80 100 120 140

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Grid B

Time

Forc

e co

effic

ient

s

c)40 60 80 100 120 140

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Grid C

Time

Forc

e co

effic

ient

s

d)20 50 100 150 200 250 300

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Grid C

Time

Forc

e co

effic

ient

s

CSide

CLift

e)228 230 232 234 236 238 240

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08Grid B

Time

Forc

e co

effic

ient

abo

ut m

ean

f)106 108 110 112 114 116 118

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1Grid C

Time

Forc

e co

effic

ient

abo

ut m

ean

Wall sideCenter sideTotal

Figure 11. Aerodynamic coefficients of the last 2.62 h of the tail as a function of time for the grids A, B and C. a), b) and c) data confined to 100 time units, d) complete history for grid C. e) and f) side forces on the wall and center sides of the tail obtained with grids B and C.

0 10 20 30 40 50 60 70 80

−0.2

0

0.2

0.4

0.6

0.8

Grid D

Time

Forc

e co

effic

ient

s

0 10 20 30 40 50 60

−0.2

0

0.2

0.4

0.6

0.8

Grid E

Time

Forc

e co

effic

ient

sC

SideC

Lift

Figure 12. Aerodynamic coefficients of the last 2.62 h of the tail as a function of time for the grids D and E.

Frequency analysis of the aerodynamic tail forces For aerodynamic forces to have implications on the vehicle’s motions, its frequency content should peak near the natural frequencies of ditto when traveling close to the top speed. Hence, the ranges of interest based on the frequencies in Table 1 that we regard critical for passenger stocks segmented into high-speed 200-250 km/h and very high-speed 250 – 330 km/h, are St ~ [0.03, 0.09] and St ~ [0.02, 0.07], correspondingly. Intermediate frequencies of say, 0.05, which has a period of 20 time units, would therefore probably require longer time histories than what is obtained herein for a

statistically justified resolution of the low frequency content. Albeit, Figure 13 shows the linear spectra of the time histories from Figure 11, normalized with the greatest peaks. The spectra of the grids A, B and C are based on 301, 266 and 284 time units, respectively. In general the forcing elucidates a rich content below St ~ 0.5 with crests in the critical ranges. The spectra of the grids A and B do not give a congruent picture in terms of quantitative agreement. Despite, qualitatively both these spectra are indicating a few low frequencies that are fairly close. One peak though matches exactly in regards to the side force, viz. at St ~ 0.06, and lies within the critical ranges defined above. This is in agreement with the findings for the shorter train, where a prominent peak near St ~ 0.06 is found. Figure 14 follows-up on our earlier discussion regarding the different forcing amplitudes of the tunnel wall and center sides of the tail. It does so by plotting the respective frequency spectra of the side force that are based on 204 time units. Again it is shown that the tunnel side is subjected to forces that have richer frequency content. Figure 13 is however dominated by frequencies related to the tunnel wall side of the vehicle tail. For the grids D and E far too few time steps are utilized to assess the frequency content. However, estimates with the data at hand for the side forces indicate that grids D and E have salient peaks, at least, about St ~ 0.5 and 0.15, respectively. Hence, for the grids C and E we notice the concordant finding of the peak value at St ~ 0.15 with that of [1] and [10], explained in the introduction. The spectra for grid D indicate a peak frequency substantially higher in comparison with all the other cases. This is readily explained with the effectively higher flow speed about the train in concurrence with the linear scaling of frequencies with the flow speed. From a dimensional view, we expect the real frequencies to scale with,

−−

1/11/1

~1

0

0

1

RR

ff

. (13)

01 ff > if 01 RR < . This indicates that the tail frequencies for grid D should be some 2.5 times greater compared to that of grid E. From the calculations the scaling is predicted to ~ 0.5/0.15 ~ 3.3. For the lift force obtained with grids A, B and C the peaks with lowest frequencies are about 0.012, 0.03 and 0.06, and may therewith potentially affect the body bounce mode. Earlier we saw that the non-symmetric flow on the two sides of the body gives rise to different forcing frequencies. This leads us to another hypothesis, that beating phenomena can cause even lower frequencies owing to the aerodynamic forcing. For clarity consider two periodic forcing frequencies, 21 and ΩΩ , of equal amplitude that are added. Let,

221 Ω+Ω=Ω and Ω+Ω−=Ω−Ω=Ω−Ω=∆Ω 21

21

2.

(14)

By adding these frequencies together the beating or interference amplitude can be expressed as,

)cos()sin(2 ttA ∆Ω⋅Ω . (15)

The closer the two frequencies are the smaller the enveloping frequency ∆Ω . This means that if two forces of equal amplitude are applied to the left and right side of the tail, one with half the frequency of the other, then the composite force is made up of the frequencies ¼ and ¾ of that of the highest. Under real conditions the forcing is of course a compound of a number of sources all with different amplitudes and frequencies. When combined a richer spectrum will arise.

0 0.1 0.2 0.3 0.40

0.4

1

PS

D: L

inea

r: C

Sid

e

St

1.)

1.) = 0.02912.) 2.) = 0.0524

3.)

3.) = 0.0641

4.)

4.) = 0.0874

5.)

5.) = 0.0990

6.)

6.) = 0.1165

7.)

7.) = 0.1398

8.)

8.) = 0.15739.)

9.) = 0.1864

10.)

10.) = 0.203811.)

11.) = 0.2446

12.)

12.) = 0.3029

13.)

13.) = 0.3320

14.)

14.) = 0.3961

15.)

15.) = 0.4193

Grid A

0 0.1 0.2 0.3 0.4

0

0.4

1

PS

D: L

inea

r: C

Lift

St

1.)

1.) = 0.0116

2.)

2.) = 0.02913.)

3.) = 0.0641

4.)

4.) = 0.0990

5.)

5.) = 0.1165

6.)

6.) = 0.1340

7.)

7.) = 0.16898.)

8.) = 0.2097

Grid A

0 0.1 0.2 0.3 0.40

0.4

1

PS

D: L

inea

r: C

Sid

e

St

1.)

1.) = 0.0116

2.)

2.) = 0.0641

3.)

3.) = 0.1747

4.)

4.) = 0.1922

5.)

5.) = 0.2271

6.)

6.) = 0.25637.)

7.) = 0.2796

8.)

8.) = 0.3436

Grid B

0 0.1 0.2 0.3 0.4

0

0.4

1

PS

D: L

inea

r: C

Lift

St

1.)

1.) = 0.0116

2.)

2.) = 0.03493.)

3.) = 0.0641

4.)

4.) = 0.0815

Grid B

0 0.1 0.2 0.3 0.40

0.4

1

PS

D: L

inea

r: C

Sid

e

St

1.)

1.) = 0.04732.) 2.) = 0.0619

3.)

3.) = 0.09104.)

4.) = 0.1383

5.)

5.) = 0.2075

6.)

6.) = 0.2803

7.)

7.) = 0.3422

8.)

8.) = 0.4659

Grid C

0 0.1 0.2 0.3 0.4

0

0.4

1

PS

D: L

inea

r: C

Lift

St

1.)

1.) = 0.0073

2.)

2.) = 0.02913.)

3.) = 0.0364

4.)

4.) = 0.0473

5.)

5.) = 0.0546

6.) 6.) = 0.0837

7.)

7.) = 0.1019

8.)

8.) = 0.1383

9.)

9.) = 0.1638

10.)

10.) = 0.1784

11.)

11.) = 0.2111

Grid C

Figure 13. Frequency spectra of the force coefficients of the last 2.62 h of the tail for the grids A, B and C

regarding the time histories of Figure 11.

0 0.1 0.2 0.3 0.40

0.4

1

PS

D: L

inea

r: C

Sid

e

St

1.)

1.) = 0.0146

2.)

2.) = 0.0510

3.)

3.) = 0.0655

4.)

4.) = 0.0946

5.)

5.) = 0.1383

6.)

6.) = 0.2111

7.)

7.) = 0.2330

8.)

8.) = 0.2767

9.)

9.) = 0.3422

10.)

10.) = 0.371311.)

11.) = 0.4805

Grid C: Tunnel wall side

0 0.1 0.2 0.3 0.4

0

0.4

1

PS

D: L

inea

r: C

Sid

e

St

1.)

1.) = 0.00732.) 2.) = 0.0874

3.)

3.) = 0.1747

4.)

4.) = 0.2403

Grid C: Tunnel center side

Figure 14. Frequency spectra of the side force coefficients of Figure 11f.

Conclusions and summary LES are employed to predict the flow about a simplified ICE 2 train of various lengths inside double and single-track tunnels. From the results presented in the preceding sections the following conclusions are drawn: LES are today feasible for studying the flow about

simplified rolling stocks, particularly inside tunnels where the walls pose a natural confinement to the domain size. If one extrapolates the astonishing velocity with which the ratio of computer performance vs. cost has developed, LES and similarly demanding methodologies will in the near future be more competitive rather than full scale tests for a variety of train external aerodynamic problems.

Our calculations confirm the existence of coherent flow structures, for our simplified models of ICE 2, adjacent to the train’s surfaces. Non-symmetric flow conditions prevailing inside double track tunnels promote these flow structures. Their relative small amplitude and high spatial frequency prove that they have insignificant effect on the vehicle’s motion. Still, they influence the flow separation and thus the force frequencies applied to the tail. Numerical modeling that fails to regenerate these disturbances would also fail to predict proper spectral characteristics. In a broader sense, accurate enough results for other applications concerning vehicle aerodynamics and aeroacoustics could hinge on the success to deal with these pressure fluctuations.

Coalescing of coherent structures is observed towards the end of our longest body. In conjunction with the fact that these disturbances have great enough amplitude to affect the separation about the tail, it is plausible to expect train length dependence.

The computational grids must meet, at least, the high demands listed in Eq. (11) to accurately resolve the flow features. Coarser grid spacing might yet be able to resolve some qualitative aspects.

Frequency spectra of the aerodynamic forces applied to the tail display a number of peak frequencies. The peaks with lowest frequencies are in the vicinity of e.g. the critical body yaw mode at relatively high operational speeds.

It is shown that the vehicle side of the tail facing the nearest tunnel wall, inside double track tunnels, is subjected to relatively larger aerodynamic side forces.

In view of the qualitative results obtained inside the single-track bores, the blockage ratio R has a great impact on both the forcing amplitude and tail frequency. From dimensional analysis it can be predicted for conventional sized trains and tunnels, that halving R more than doubles the preferred frequencies and factors the forces with about six times. However, our calculations predict both the forces and frequencies to amplify about three times.

It is pointed out that interference of various forcing frequencies, under e.g. non-symmetric flow conditions posed by double track tunnels, can contribute to substantially lower frequencies.

A natural continuation of this work is to investigate the tail vehicle’s response to the force histories calculated herein in the context of vehicle dynamics. However, we have already started on our next project task, to calculate the flow about a Shinkansen train model with pertinent tunnel geometry to compare tail forces and generation of coherent structures with that of the ICE 2 train. Also in future studies the effects on the frequencies regarding the sloping of the tail, c.f. [24], would be an interesting topic to pursue. Of further interest is to establish possible influence on the aerodynamics caused by the motion of the vehicle body. Vortex shedding frequency is affected by the body’s motion and can be related to a phenomena referred to as lock-in described in [25], which effectively reduces the force frequencies. Eventually the full aeroelastic problem, c.f. [26], that combines aerodynamics with vehicle dynamics for trains would pose a great and interesting challenge. ACKNOWLEDGMENTS

This research work is financed by Banverket (Swedish National Railway Administration) under contract no. S02/850/AL50 and by Bombardier Transportation Sweden AB.

REFERENCES

[1] Kohama. Y., Yoshikawa. T. and Okude. 1994. M. Wake characteristics of a High Speed Train in relation to tail coach oscillations. Vehicle Aerodynamics Conference, Louvhborough Univ., UK. [2] Suzuki, M., Maeda, T. and Aria, N. 1995. Numerical simulation of flow around a train. Proc. of the IMAC-COST Conference on Computational Dynamics, pp. 311-317. [3] Ishihara. T., Utsunomiya. M., Sakuma. Y. and Shimomura. T. 1997. An investigation of lateral vibration caused by aerodynamic continuous force on high-speed train running within tunnels. Proceedings of World Congress on Railway Research, pp. 531-538. [4] Fujimoto. H. 1995. Lateral Vibration and decreasing measure of it on Shinkansen train. QR. Vol. 36. No. 3. Sep. 95. [5] Sasaki. K. 2000. A lateral semi-active suspension of tilting train. QR of RTRI, Vol. 41, No. 1. [6] Sasaki. K., Kamoshita. S. and Simomura. T. 1996. Development of field results of semi-active suspension for high-speed train. RTRI. Report, Vol. 10, No. 5, pp. 25-30. [7] Suzuki, M. Unsteady Aerodynamic Force Acting on High Speed Trains in Tunnel. QR of RTRI, Vol. 42, No. 2, May. 2001. [8] Andersson. E. Berg. M., and Stichel. S. 2000. Spårfordons dynamik. Royal Institute of Technology in Stockholm, Sweden. Div. Railway Technology. [9] Persson, I.: GENSYS – User Manual. DEsolver AB, Östersund 2003, www.gensys.info

[10] RAPIDE Workshop held in Cologne, Germany, 29th November 2001. Funded by the European Commission under the Industrial and Materials Technologies programme (Brite-EuRam III). [11] ENV 12299: Railway Application – Ride comfort for passengers – Measurements and Evaluations. European Standard. [12] prEN 14067-1, 2002. Railway Application –Aerodynamics – Part 1: Symbols and units. European Standard. [13] Stichel. S. 2001. Parametric study: Dynamics Wake tail excitations from DB.Train passing excitation from SNCF Influence of Vehicle Parameters. Reference: 3A2/011030-1-TR. Project funded by the European Commission under the industrial and Material Technology Program (BriteEuRamIII). Contract number: BRPR - CT97 – 0603. Project number: BE97 – 4089. [14] Krajnovi. S. 2002. Large-eddy simulation for computing the flow around vehicles. Dept. Thermo and Fluid Dynamics, Ph.D. Thesis. ISSN 0346-718X. [15] Bradshaw. P. 1994. Turbulence: The chief outstanding difficulty of our subject, Experiments in Fluids, Vol. 16, pp. 203-216. [16] Perzon. A. and Davidson. L. 2000. On CFD and transient flow in vehicle aerodynamics. SAE 2000-01-0873. [17] Krajnovi. S and Davidson. L. 2003. Numerical study of the flow around a bus-shaped body. ASME: Journal of Fluids Engineering, Vol. 125, pp. 500-509. [18] Bearman, P. W. 1997. Near wake flows behind two- and three-dimensional bluff bodies. Journal of Wind Engineering and Industrial Aerodynamics, 69-71:33-54. [19] Smagorinsky. J. 1963. General circulation experiments with the primitive equations. Monthly Weather Review, 91(3): 99-165. [20] Diedrichs. B. 2003. On CFD modeling of crosswind effects for high-speed rolling stock. Proc. Instn Mech. Engrs Vol. 217 Part F: J. Rail and Rapid Transit. [21] Piomelli. U. and Chasnov. J.R. 1995. Turbulence transition modelling. Lecture notes from ERCOFTAC/IUTAM Summer school held in Stockholm, 12-20 June, 1995. ISBN 0-7923-4060-4. [22] EN 14067-1. 2003. Railway Applications – Aerodynamics – Part 1: Symbols and units. European standard. [23] Kumarasamy. S. and Barlow. J. B. 1996. Unsteady flow over a half cylinder in proximity to a stationary moving wall. SAE 960682. [24] Morel. T. 1980. The effect of the base slant on flow in the near wake of an axisymmetric cylinder. The Aeronautical Quarterly, Vol. 31. [25] Murakami. S. and Mochida. A. 1995. On turbulent vortex shedding flow past a 2D square cylinder predicted by CFD. Journal of Wind Engineering and Industrial Aerodynamics. 54/55, pp. 191-211. [26] Tamura. T. 1999. Reliability on CFD estimation for wind structure interaction problems. Journal of Wind Engineering and Industrial Aerodynamics 81, pp. 117-143.

CONTACT

Ben Diedrichs KTH, Div. of Railway Technology Teknikringen 8 SE-100 44 Stockholm ben.diedrichs@se.transport.bombardier.com