Ten of the best – a personal choice of Train Aerodynamics papers from 2021


In January 2020 and January 2021, I posted quite lengthy blog posts that attempted to collate all the published papers in train aerodynamics over the previous year  – see here for the 2020 post and here and here for the two part 2021 post. . These were intended as supplements to the book “Train Aerodynamics – Fundamentals and Principles” published in 2019. These blog posts have been quite widely read. At the time of writing (mid-January 2022) the 2020 post has had 190 views and the two parts of the 2021 post 129 and 70 views. It had been my intention to do something similar for the papers published in 2021. However, I have changed my mind on this, and instead will take a different approach in this post. My reasons for this are twofold.

  • The number of papers in the field continues to proliferate and, quite frankly, many of them are of poor quality. This seems to be driven by the need, in some jurisdictions, for research students to publish papers in order to be awarded a PhD. This inevitably encourages a low standard of output. Also, I have noticed an increasingly disturbing trend, whereby when a paper is rejected by one of the higher quality journals, it is submitted in much the same form to other journals with less impact.  I have seen a number of such papers sent to me to review by different journals – and on two occasions in 2021 I have been sent the same paper by three journals. Obviously I have little influence on how researchers submit papers, other than through the normal reviewing process, but there seems to me no reason to give such papers the benefit of a mention in any comprehensive annual compilation.
  • The use of CFD techniques in train aerodynamics, which is proliferating at the same rate as the number of papers, is giving me increasing concern. CFD techniques ranging from RANS to LES are exceptionally useful tools in all fluid dynamics research and the same applies in the train aerodynamics field. But they are as much tools as any physical model tests and need to be used and interpreted very carefully. There are many investigators who do just that, including colleagues in my own institution. However, I fear that that is not always the case Specifically, the use of such techniques is in many circumstances becoming divorced from practical reality. There is a tendency to apply quite high level, but inflexible, CFD methods (such as IDDES) to look at quite trivial problems where much simpler methods could have given equivalent answers over a wider parameter range. And in the consideration of the results from these calculations, there is often little appreciation of the uncertainty that is attached both to the CFD results themselves (for example I have seen the percentage changes in predicted drag given to two decimal places) or in relation to full scale reality, where the uncertainties are multiplied by an order of magnitude or more.  Further the results of the CFD calculations are often massively over-analysed. For example, in studies of cross wind effects on trains, I have come across papers where the predicted wake systems are analysed in very great detail, with little realisation that any such systems cannot exist in reality due to the (unsimulated) large scale turbulence in the approach flow field – as of course is the case with many wind tunnel tests. The same can be said of the analysis of many other applications. Again, there is little I can do to influence these trends, but I see no reason to publicise such work any further in blog posts.

In the light of such developments, in this post I will not present a comprehensive compilation of all the train aerodynamics papers from 2021 but will rather choose a much smaller number (ten in total) which I believe are of particular significance and likely to influence the field in the future. These are spread across the range of train aerodynamic applications including train drag studies, trains in tunnels, crosswind effects and emerging issues. The choice of what to include is of course to some degree subjective and mirrors my own interests, but I hope that readers find it of interest.

Train drag studies

On the influence of Reynolds number and ground conditions on the scaling of the aerodynamic drag of trains. Tschepe et al (2021)

Very often the effect of Reynolds number on train drag measurements or calculations is broadly ignored provided that the Reynolds number is “high enough”. This is of course not adequate, as the skin friction component of drag must vary with Reynolds number throughout the parameter range – see for example my historically rather crude analysis of the problem from 1991. This paper, drawn from the doctoral work of Tschepe) presents the results of a thorough experimental and analytical investigation into this effect, using the results from water towing tank experiments. These experiments are quite novel and deserving of attention in their own right. The three-dimensional nature of the train boundary layer is clear, and the effect of ground roughness (ie sleepers and ballast) is shown to be of some importance (see also my blog post here). A simple analytical approach, based on flat plate theory, allows a correction method to be developed for extrapolating low Reynolds number results to full scale conditions.

A field study on the aerodynamics of freight trains Quazi ei al (2021)

This paper presents the results of full-scale measurements of the pressure drag of a freight container during a typical journey. As such it provides a basic benchmark for further studies. The technique is of interest in its own right, but the basic result, that, despite the container not having other containers immediately in front and behind it, the drag coefficient is much lower than that found in other full-scale, physical model and numerical calculations is of considerable interest. The authors suggest that this is because of the container position much further down the train than in other measurements, as well as other modelling issues. The results perhaps give pause for thought about the measurement of train drag from wind tunnel tests or CFD calculations.

Tunnel aerodynamics

Influence of air chambers on wavefront steepening in railway tunnels. Liu et al (2021)

The phenomenon of micro-pressure waves (sonic booms) emitted from tunnel portals has been much studied in recent years. These are caused by the steepening of the train nose pressure wave as it passes along the tunnel, resulting in a steep wave at the tunnel exit that is not wholly reflected with some energy being transmitted out of the tunnel in the audible frequency range. The standard method for the amelioration of such effects is through the use of tapering tunnel entry portals, that reduce the initial (and thus the final) steepness of the waves. Such portals can be quite long and extend some way out of the tunnel, and indeed can be quite expensive. This paper investigates an alternative to such portals – the distribution of air chambers along the length of the tunnel that in principle reduces the steepening of the pressure wave. Using a relatively straight forward gas dynamics analytical model, the authors show that suitably designed chambers can remove the dependence of the exit wave on the steepness of the inlet wave. Guidance is given for appropriate chamber volumes and the resistance of the connectors between the chambers and the tunnel. Overall, the method has much potential for future tunnel design.

Virtual homologation of high-speed trains in railway tunnels: A new iterative numerical approach for train-tunnel pressure signature. Brambilla et al (2021)

The standard methodology to investigate the passage of pressure waves along tunnels is to use full-scale measurements to measure the pressure wave system on train entry, and then to use data from those measurements to predict the pressure wave along the length of the tunnel using one dimensional gas dynamics methods. The latter can be run many thousands of times to investigate a range of operational conditions. Clearly the required full-scale tests are expensive and complex. Recently some full CFD calculations of the flow along tunnels have been published using sliding grids, which are again highly complex and computer resource requirements limits their use to just one or two conditions. This paper presents a combined methodology where CFD calculations using a standard fixed grid are carried out to measure the pressure characteristics at train inlet to the tunnel, and these are then used in one dimensional methods. The methodology has been validated against an extensive full scale data set. Its relative cheapness and flexibility means that it has the potential to become widely used within the industry.

Semi-empirical model of internal pressure for a high-speed train under the excitation of tunnel pressure waves. Chen et al (2021)

This paper looks in detail at the development of internal pressures within train cabins in tunnels. Using a combination of commercial CFD and finite element analysis, together with simple models of internal ventilation flow, the authors looked at pressure changes due to body deformation, pressure transmission through gaps in the train envelope and transmission through the air ducts of HVAC systems. Body deformation has little effect (unsurprisingly in my view) with the balance between gap and duct transmission varying depending on the degree of opening of the latter. Whilst the analysis is complex, the results should be of interest in describing a methodology that could ultimately be applied quite straightforwardly in design.

Pressure fluctuation and a micro-pressure wave in a high-speed railway tunnel with large branch shaft. Okubo et al (2021)

This paper describes an extensive experimental programme using a moving model facility that looked at the micro-pressure waves that occur as a result of the junction between the main tunnel and large branch tunnels with similar diameter (which would be used for passenger evacuation). The results are skillfully interpreted through the use of analytical models and show that in some instances the micro pressure wave emitted from the branch tunnel can be of greater magnitude than that omitted from the main tunnel. Both the physical and analytical modelling methodology have potential use for the design of complex branching tunnel systems.

Trains in crosswinds

Influence of the railway vehicle properties in the running safety against crosswinds. Heleno et al (2021)

I include this paper with some temerity, as I am named as an author – albeit the last one. However, my role was very minor, and mainly involved discussions on some technical details and proof reading the final draft (although they all contribute to my long term aim of getting to 200 journal publications before my demise!). This paper considers the effect of various railway vehicle properties on the overturning risk of a rail vehicle. It uses realistic vehicle dynamic and track roughness models and generates realistic time series of wind speed from wind statistical parameters. It is more rigorous in its modelling than the current method used in the CEN code, which uses a very simplified wind gust model. A thorough parametric analysis of the various vehicle parameters is carried out. In my view the major point to emerge is the lack of sensitivity of the calculated overturning wind speeds and safety risk to variations in the train suspension parameters. In principle this could lead to much simpler models for the CEN safety assessment than are used at present, where full dynamic modelling is required. This is personally satisfying as I have been arguing this very point for the last 10 to 15 years – see the discussion in this post from 2020.

Emerging issues

CWE study of wind flow around railways: Effects of embankment and track system on sand sedimentation. Horvat et al (2021)

I include this paper because it contributes to what I believe to be an important emerging issue as railways are developed in arid conditions – sand sedimentation over railway tracks. It is a straightforward CFD study of flow patterns over different railway track geometries that calculates wall shear stresses and used these to define potential regions of erosion and sedimentation. It lays the foundation for future work – possibly to integrate sediment modelling into the CFD calculations.

Investigation on flow field structure and aerodynamic load in vacuum tube transportation system. Zhong et al (2021)

This paper is a detailed CFD analysis of the flow around vacuum tube vehicles using IDDES techniques. Because of the enclosed nature of the vehicles and the well-defined geometry, this is a case where one would expect good accuracy from such calculations. Also of course the issues cannot be easily addressed by physical modelling techniques. Both subsonic and supersonic flows are considered, the nature of the flow field elucidated, and vehicle drag calculated. The results form a useful addition to the publicly available body of knowledge about the flows around such vehicles that can be used in further development of the concept. That being said, it is my firm view that, fascinating as the aerodynamics of the system might be, vacuum tube systems will not meet with wide adoption due to simple operational constraints – primarily the low capacity in comparison to conventional high speed rail systems.


Railway applications – Aerodynamics – Part 7: Fundamentals for test procedures for train-induced ballast projection. CEN (2021)

This is not a paper, but rather the latest offering from the CEN working group on Aerodynamics that looks at the issue of ballast flight beneath high-speed trains. It contains a wealth of information of the issues involved, economic aspects of the damage caused by ballast flight, current national practices and possible ways forward in terms of homologation. It is well put together and forms a very useful basis for further work in the field.

Pollutants, pathogens and public transport – ventilation, dispersion and dose


The ventilation of buses and trains has come to be of some significance to the travelling public in recent years for a number of reasons. On the one hand, such vehicles can travel through highly polluted environments, such as urban highways or railway tunnels, with high levels of the oxides of nitrogen, carbon monoxide, hydrocarbons and particulate matter that can be drawn into the passenger compartments with potentially both short- and long-term health effects on passengers. On the other, the covid-19 pandemic has raised very significant concerns about the aerosol spread of pathogens within the enclosed spaces of trains and buses. There is a basic dichotomy here – to minimise the intake of external pollutants into vehicles, the intake of external air needs to be kept low, whilst to keep pathogen risk low, then high levels of air exchange between the outside environment and the internal space are desirable. This post addresses this issue by developing a common analytical framework for pollutant and pathogen dispersion in public transport vehicles, and then utilises this framework to investigate specific scenarios, with a range of different ventilation strategies.

The full methodology is given in the pdf that can be accessed via the button opposite. This contains all the technical details and a full bibliography. Here we give an outline of the methodology and the results that have been obtained.


The basic method of analysis is to use the principle conservation of mass of pollutant or pathogen into and out of the cabin space. In words this can be written as follows.

Rate of change of mass of species inside the vehicle = inlet mass flow rate of species + mass generation rate of species within the vehicle – outlet mass flow rate of species– mass flow rate of species removed through cleaning, deposition on surfaces or decay.

This results in the equation shown in Box 1 below, which relates the concentration in the cabin to the external concentrations, the characteristics of the ventilation system and the characteristics of the pollutant or pathogen. The basic assumption that is made is of full mixing of the pollutant or pathogen in the cabin. The pdf gives full details of the derivation of this equation, and of analytical solutions for certain simple cases. It is sufficient to note here however that this is a very simple first order differential equation that can be easily solved for any time variation of external concentrations of pollutant generation by simple time stepping methods. For gaseous pollutants, the rate of deposition and the decay rate are both zero which leads to a degree of simplification.

Box 1. The concentration equation

The pdf also goes on to consider the pollutant or pathogen dose that passengers would be subjected to – essentially the integration of concentration of time history – and then uses this in a simple model of pathogen infection. This results in the infection equation shown in Box 2. Essentially it can be seen that the infection risk is proportional to the average concentration in the cabin and to journey length.

Box 2. Infection equation

The main issue with this infection model is that it assumes complete mixing of the pathogen throughout the cabin space and does not take account of the elevated concentrations around an infected individual. A possible way to deal with this is set out in the pdf. Further work is required in this area.

Ventilation types

The concentration and infection equations in Boxes 1 and 2 do not differentiate between the nature of the ventilation system on public transport vehicles. Essentially there are five types of ventilation.

  • Mechanical ventilation by HVAC systems
  • Ventilation through open windows
  • Ventilation through open doors
  • Ventilation by a through flow from leakage at the front and back of the vehicle (for buses only)
  • Ventilation due to internal and external pressure difference across the envelope.

Simple formulae for the air exchange rates per hour have been derived and are shown in Box 3 below. By substituting typical parameter values the air exchange rates are of the order of 5 to 10 air changes per hour for the first four ventilation types, but only 0.1 for the last. Thus ventilation due to envelope leakage will not be considered further here, although it is of importance when considering pressure transients experienced by passengers in trains.

Box 3. Ventilation types

Scenario modelling

In what follows, we present the results of a simple scenario analysis that investigates the application of the above analysis for different types of vehicle with a range different ventilation systems, running through different transport environments. We consider the following vehicle and ventilation types.

  • An air-conditioned diesel train, with controllable HVAC systems.
  • A window and door ventilated diesel train.
  • A bus ventilated by windows, doors, and externally pressure generated leakage.

Two journey environments are considered.

  • For the trains, a one-hour commuter journey as shown in figure 1, beginning in an inner-city enclosed station, running through an urban area with two stations and two tunnels, and then through a rural area with three stations (figure 1).
  • For buses, a one-hour commuter journey, with regular stops, through city centre, suburban and rural environments (figure 2).

Results are presented for the following scenarios.

  • Scenario 1. Air-conditioned train on the rail route, with HVACs operating at full capacity throughout.
  • Scenario 2. As scenario 1, but with the HVACs turned to low flow rates in tunnels and enclosed stations, where there are high levels of pollutants.
  • Scenario 3. Window ventilated train on rail route with windows open throughout and doors opened at stations.
  • Scenario 4. As scenario 3, but with windows closed.
  • Scenario 5. Window, door and leakage ventilated bus on bus route with windows open throughout and doors opened at bus stops.
  • Scenario 6. As scenario 5, but with windows closed.

Details of the different environments and scenarios are given in tables 1 and 2.  Realistic, if somewhat arbitrary levels of environmental and exhaust pollutants are specified for the different environments – high concentrations in cities and enclosed railway and bus stations and lower concentrations in rural areas. The air exchange rates from different mechanisms are also specified, with the values calculated from the equations in Box 3. Note that, in any development of this methodology, more detailed models of the exhaust emissions could be used that relate concentrations at the HVAC systems and window openings to concentrations at the stack, which would allow more complex speed profiles to be investigated, with acceleration and deceleration phases.

Figure 1. The rail route

Figure 2. The bus route

Table 1. The rail scenarios

Table 2. The bus scenarios

The results of the analysis are shown in figures 3 and 4 below for the train and bus scenarios respectively. Both figures show time histories of concentrations for NO2, PM2.5, CO2 and Covid-19, together with the external concentrations of the pollutants.

For Scenario 1, with constant air conditioning, all species tend to an equilibrium value that is the external value in the case of NO2 and PM2.5, slightly higher than the external value for CO2 due to the internal generation and a value fixed by the emission rate for Covid 19.

For Scenario 2, with low levels of ventilation in the enclosed station and in the tunnels, NO2 and PM2.5 values are lower than scenario 1 at the start of the journey where the lower ventilation rates are used, but CO2 and Covd-19 concentrations are considerably elevated. When the ventilation rates are increased in the second half of the journey all concentrations approach those of Scenario 1.

The concentration values for scenario 3, with open windows, match those of Scenario 1 quite closely as the specified ventilation rates are similar. However, for Scenario 4, with windows shut and only door ventilation at stations, such as might be the case in inclement weather, the situation is very different, with steadily falling levels of NO2 and PM2.5, but significantly higher values of CO2 and Covid-19. The latter clearly show the effect of door openings at stations.

Figure 3. The train scenario results

Now consider the bus scenarios in figure 4. For both Scenario 5 with open windows and doors, and Scenario 6 with closed windows and open doors, the NO2 and PM2.5 values tend towards the ambient concentrations and thus fall throughout the journey as the air becomes cleaner in rural areas. The internally generated CO2 and Covid-19 concentrations for CO2 and Covid-19 are however very much higher for Scenario 6 than for Scenario 5.

Figure 5. The bus scenarios

The average values of concentration for all the scenarios is given in Table 3. The dose and, for Covid-19, the infection probability, are proportional to these concentrations. For NO2 and PM10 the average concentrations reflect the average external concentrations, and, with the exception of Scenario 4, where there is low air exchange with the external environment for part of the journey. The average concentrations for CO2 and Covid-19 for the less ventilated Scenarios 4 and 6 are significantly higher than the other. For Covid-19, the effect of closing windows on window ventilated trains and buses raises the concentrations, and thus the infection probabilities, by 60% and 76% respectively.

Table 3. Average concentrations

Closing comments

The major strength of the methodology described above is its ability, in a simple and straightforward way, to model pollutant and pathogen concentrations for complete journeys, and to investigate the efficacy of various operational and design changes on these concentrations. It could thus be used, for example, to develop HVAC operational strategies for a range of different journey types. That being said, there is much more that needs to be done – for example linking the methodology with calculations of exhaust dispersion around vehicles, with models of particulate resuspension or with models of wind speed and direction variability. It has also been pointed out above that the main limitation of the infection model is the assumption of complete mixing. The full paper sets out a possible way forward that might overcome this. Nonetheless the model has the potential to be of some utility to public transport operators in their consideration of pollutant and pathogen concentrations and dispersion within their vehicles.

The calculation of Covid-19 infection rates on GB trains


In a recent post I looked at the ventilation rate of trains without air conditioning and compared them with the ventilation rate of airconditioned trains. The context was the discussion of the safety of trains in terms of Covid-19 infection. For air conditioned trains, the industry accepted number of air changes per hour is around 8 to 10. For non-air conditioned trains with windows fully open and doors opening regularly at stations, I calculated very approximate values of air changes per hour of around twice this value, but for non-air conditioned trains with windows shut and thus only ventilated by door openings, I calculated approximate values of a of 2.0. On the basis of these calculations, I speculated that the non-air conditioned trains with windows shut probably represented the critical case for Covid-19 transmission. In that post however I was unable to be precise about the level of risk of actually becoming infected and how this related to ventilation rate.

The work of Jimenez

I have recently come across the spreadsheet tool produced by Prof. Jose Jimenez and his group at the University of Colorado-Boulder that attempts to model airborne infection rates of Covid-19 for a whole range of different physical geometries, using the best available information on pathogen transport modelling, virus production rates, critical doses etc. They base their  analysis on the assumption that aerosol dispersion is the major mode of virus transport, which now seems to be widely accepted (and as anyone who has been following my blogs and tweets will know that I have been going on about for many months). I have thus modified the downloadable spreadsheet to make it applicable to the case of a standard GB railway passenger car compartment.  A screen shot of the input / output to the spreadsheet is shown in figure 1 below.

Figure 1 Screen shot of spreadsheet input / output parameters

The inputs are the geometry of the passenger compartment; the duration and number of occurrences of the journey, the air conditioning ventilation rate; the number of passengers carried; the proportion of the population who may be considered to be immune; the fraction of passengers wearing masks; and the overall population probability of an individual being infected. In addition, there are a number of specified input parameters that describe the transmission of the virus, which the authors admit are best guess values based on the available evidence, but about which there is much uncertainty. The outputs are either the probabilities of infection, hospitalization and death for an individual on a specific journey or for multiple journeys; or the number of passengers who will be infected, hospitalized or die for a specific journey or for multiple journeys.

The spreadsheet is a potentially powerful tool in two ways – firstly to investigate the effect of different input parameters on Covid-19 infection risk, and secondly to develop a rational risk abatement process. We will consider these in turn below.

Parametric investigation

In this section we define a base case scenario for a set of input variables and then change the input variables one by one to investigate their significance. The base case is that shown in the screen shot of figure 1 – for a journey of 30 minutes repeated 10 times (i.e. commuting for a week);  80 unmasked passengers in the carriage; a ventilation rate of 8 air changes per hour; a population immunity of 50%; and a population infection rate of 0.2% (one in 500). The latter two figures broadly match the UK situation at the time of writing. For this case we have a probability of one passenger being infected on one journey of 0.096% or 1 in 1042. The arbitrariness of this figure should again be emphasized – it depends upon assumed values of a number of uncertain parameters. We base the following parametric investigation on this value. Nonetheless it seems a reasonable value in the light of current experience. The results of the investigation are given in Table 1 below.

Table 1 Parametric Investigation

The table shows the risk of infection for each parametric change around the base case and this risk relative to the base case. There is of course significant arbitrariness in the specification of parameter ranges.  Red shading indicates those changes for which the infection risk is more than twice the value for the base case and green shading for those changes for which the infection risk is less than half the value for the base case. The following points are apparent.

  • The risk of infection varies linearly with changes in journey time, population infection rate and population immunity. This seems quite sensible, but is effectively built into the algorithm that is used. 
  • Changes in ventilation rate cause significant changes in infection risk. In particular the low value of 2ach, which is typical on non-airconditioned vehicles with closed windows, increases the infection risk by a value of 3.5.
  • The effect of decreasing passenger number (and thus increasing social distancing) is very significant and seems to be the most effective way of reducing infection risk, with a 50% loading resulting in an infection risk of 28% of the base case, and a 20% loading a risk of 6% of the base case.
  • The effect of 100% mask wearing reduces the infection risk to 35% of the base case.
  • 100% mask wearing and a 50% loading (not shown in the table) results in a reduction of infection risk to 10% of the base case.

From the above, regardless of the absolute value of risk for the base case, the efficacy of reducing passenger numbers and mask wearing to reduce risk is very clear.

An operational strategy to reduce risk.

The modelling methodology can also be used to develop a risk mitigation strategy. Let us suppose, again arbitrarily, that the maximum allowable risk of being infected per passenger on the base case journey is 0.1% (i.e. 1 in a thousand). Figure 2 shows the calculated infection risk for a wide range of national infection rate of between 0.01% (1 in 10,000) to 2% (1 in 50). Values are shown for no mask and full capacity; 100% mask wearing and full capacity; and 100% mask wearing and 50 % capacity. It can be seen that the no mask / full capacity curve crosses the 0.1% line at a national infection rate of 0.2% and the 100% mask / full capacity line crosses this boundary at 0.6%.

Figure 2 Effect of national infection rate on infection risk, with and without mask wearing and reduction in loading

Consideration of the results of figure 2 suggest a possible operational strategy of taking no mitigation risks below an infection rate of 0.2%, imposing a mask mandate between 0.2% and 0.6% and adding a significant capacity reduction above that. This is illustrated in figure 3 below.

Figure 3. Mitigation of risk to acceptable level through mask wearing and reduced capacity.

As has been noted above the absolute risk values are uncertain, but such a methodology could be derived for a variety of journey and train types, based to some extent on what is perceived to be safe by the travelling public. Regional infection rates could be used for shorter journeys. Essentially it gives a reasonably easily applied set of restrictions that could be rationally imposed and eased as infection rate varies, maximizing passenger capacity as far as is possible. If explained properly to the public, it could go some way to improving passenger confidence in travel.

The calculation of train overturning risk – what type of wind tunnel tests should be used?

A Mark 3 coach – the GB benchmark vehicle

When considering the effect of crosswinds on a new train, an obvious first step is to obtain data on the aerodynamic force and moment coefficients, usually through the use of wind tunnel tests, with the forces and moment coefficients being measured for a range of yaw (wind) angles from 0 to 90 degrees. This process however is not quite as straightforward as it sounds. The conventional approach is to use static models in a low turbulence wind tunnel. This approach of course models neither the relative motion between the train and the ground, nor the effects of atmospheric turbulence. It does however have the merits of simplicity and convenience. The conventional argument often used to justify this approach is that for high-speed trains, the relative motion between the train and the wind leads to the train experiencing low levels of turbulence. Whilst this is the case to some extent, it is not a wholly adequate argument. Figure 1, from Train Aerodynamics – Fundamentals and Applications (TAFA), shows how the turbulence length scale, turbulence intensity and velocity shear relative to the train vary with train speed for a 90 degree cross wind. Values are given as ratios of the values when the train is stationary. It can be seen that even at 400 km/h, the train still experiences a turbulence intensity of around 30% of its stationary value, which one might expect to have a not insignificant effect on the flow around the train,.

Figure 1 Variation of relative values of turbulence intensity (black), turbulence length scale (red) and shear (green) with train speed for a 90 degree cross wind (from TAFA)

An alternative approach would be to use a wind tunnel simulation of the atmospheric boundary layer in which to measure the train forces and moments. This of course is only really applicable to stationary trains. On the basis of figure 1, I argued in TAFA that low turbulence wind tunnel tests would be best for train speeds greater than 200 km/h and atmospheric boundary layer tests would be best for train speeds below that value – but that of course represents rather a messy compromise. And both methods fail to address the issue of train / ground relative motion.

So what are the alternatives? The first might be thought to be the use of CFD to properly model both atmospheric effects and train / ground motion. However, the simulation of a realistic scenario requires complex CFD methodologies (usually DDES) with very complex domain boundaries that include the specification of atmospheric turbulence. The calculation of the flow field for just one yaw angle takes several weeks on supercomputer systems, and in reality CFD calculations of this type tend to mirror the low turbulence wind tunnel tests.

In physical model terms, two alternatives present themselves. The first is the measurement of cross wind forces and moments on a moving model rig such as the TRAIN Rig owned by the University of Birmingham.  Again, the experimental issues are formidable. The use of force balances within moving model rigs is not straightforward, and measurements of this type are usually made through the measurement of surface pressures with internal transducers, which because of transducer size and the need to carry out multiple runs to obtains stable average pressures requires multiple runs, with different pressure positions at any one yaw angle – a very tedious and complex process. An alternative would be to carry out conventional wind tunnel tests, but with a range of different turbulence simulations, each simulation being valid for one train speed only. The thought of such tests is enough to make wind tunnel operators consign it to the rubbish bin without much hesitation.

But the issue is important. Figure 2 shows three different sets of lee rail rolling moment coefficients for the Mark 3 coach, the GB benchmark vehicle that has run on exposed lines for many decades without incident. The three sets of coefficients are obtained from low turbulence wind tunnel tests; tests with an atmospheric boundary layer simulation with the coefficients formed from the mean values of measured forces and velocities; and those obtained from similar tests but with the coefficients formed from one-second peak values of forces and velocities (from Measurements of the cross wind forces on trains). The atmospheric boundary layer results are shown together with corresponding full scale results from field measurements on a real train. There can be seen to be significant differences between the three curves, particularly in the low yaw angle range which is important at high train speeds, with the low turbulence values being significantly above the atmospheric boundary layer values and the peak values being below the mean ones. If these coefficients are used to obtain cross wind characteristics (CWCs), which are plots of accident windspeed against vehicle speed, as outlined in another post and in TAFA chapter 11 and in a recent blog post, then the differences in acceptable windspeeds can be seen to be significant, particularly in the speed range around 200 km/h – see figure 3. Note that this plot shows train speeds of up to 400 km/h, which is wholly unrealistic for the Mark 3 coach – and certainly I wouldn’t care to be in one travelling at that speed! – but serves to illustrate the lack of agreement between the CWCs calculated using different moment coefficients. The difference in CWCs can be expected to make a significant difference to the calculation of accident risk, or to any operational restrictions that might be imposed, with the low turbulence results giving higher risk values and more severe restrictions.

Figure 2. Lee rail rolling moment coefficients for Mark 3 coach

Figure 3 Crosswind characteristics for Mark 3 coach

I have to admit this is a problem that I have been mulling over on and off for many years (which gives a rather sad picture of the life I lead I fear). My thoughts have been basically around the idea of how to obtain representative force coefficients to allow for the major effect of atmospheric turbulence at low train speeds and the much smaller effect at high speeds, perhaps by some interpolation of the low and high turbulence coefficients. This is not simple however, as there is no direct correspondence between variation of these coefficients with yaw angle and variation with train speed.

But there is perhaps another way – and that is to consider not the force and moment coefficients, but rather the CWCs shown in figure 3. It seems reasonable to me to assume that the most representative CWC would lie somewhere between the low and high turbulence characteristics, lying close to the ABL curve at low train speeds, and close to the low turbulence curves at high train speed. Thus figure 4 shows the CWCs formed from giving a variable weighting to the low and high turbulence curves at different train speeds, with a 100% weighting given to the low turbulence curves at a train speed of 0 km/h, and a 0% weighting at a train speed of 400 km/h, with a linear variation in between. More sophisticated weighting variations could be considered, but this approach is adequate for illustrative purposes. The two curves of figure 4 are for the interpolation of the CWCs calculated from the mean and peak coefficients with those obtained from the low turbulence coefficients. It can be seen that this approach significantly raises the CWCs in the mid speed range from the low speed values and will thus result in substantial risk reduction.

Figure 4. Interpolated cross wind characteristics

Up to now, I have referred to potential risk reductions in rather broad terms. It is however possible to put some numbers to these statements. Table 1 shows the accident wind speed at a train speed of 200 km/h for each of the above CWCs and the associated risk at the reference site as defined in my earlier post. For the original CWCs derived from the ABL coefficients, , the risks is of the order of 10-7 to 10-8, but for the CWC derived from the low turbulence conditions, the risk approaches 10-5 – almost two orders of magnitude greater. Whilst the absolute values of risk are quite arbitrary, it is clear that the use of the low turbulence characteristic would lead to a much more pessimistic (and perhaps unrealistic) risk assessment, and lower than necessary wind speed restrictions. The interpolated CWCs give values of risk of ca little less than 10-6, roughly midway between the atmospheric boundary layer and low turbulence values.

Table 1. Accident windspeeds and risk values for Mark 3 coach at a train speed of 200 km/h

So to conclude, the method outlined above gives a potentially realistic way of solving the problem of what type of wind tunnel test to use for train cross wind risk assessment. It requires two sets of wind tunnel experiments, one with low turbulence and one with an atmospheric boundary layer simulation, which is a more complex methodology than at present, but does not require extremely complex wind tunnel or CFD trials. The method results in lower values of calculated risk than would be the case using conventionally derived CWCs, and higher values of accident wind speeds.

Thoughts on the leakage characteristics of trains

HS2 train coming out of a tunnel


Most modern trains are “sealed” in that they are designed to minimise the leakage of air between the inside and the outside of the cabin. There are a number of reasons for this. Firstly HVAC systems require as little leakage as possible to be able to operate efficiently. Secondly, when a train passes through a tunnel at speed, it generates large pressure transients that can cause passenger aural discomfort and pain – sealing the train attenuates or even eliminates such transients in the train interior. However the sealing of trains is never perfect and some way of quantifying leakage is required, and then of calculating the internal pressure of trains for the types of external pressure field experienced as trains pass through tunnels, allowing for this leakage. This brief post looks at the standard methodology for doing this, which is essentially empirical, and compares it firstly with methods for the calculation of leakage in buildings used by ventilation engineers, which are based on the concept of an equivalent orifice, and secondly with a new method which models leakage paths as simple pipe flows. It is shown that the empirical model currently used is consistent with the new leakage pipe model, and the use of the latter enables some of the limitations of the current method to be more fully appreciated.

The railway methodology

The standard method for assessing how well a train is sealed is to pump air into the train to raise the internal pressure to a specified level and then simply to observe the decay of pressure when the pump is turned off. It is then assumed that this pressure decay follows the simple rate equation shown in equation (1), which can be solved to give the exponential decay expression of equation (2) The parameter T is a leakage time constant and can usefully be used to quantify the degree of sealing (Note that this is usually denoted by the Greek letter tau, but this website is unable to cope with Greek letters in the text). Tests are usually carried out for static trains but can in principle be carried out for moving trains, where one might expect the degree of sealing to be somewhat less than the static case due to the relative movement of different parts of the train envelope. Thus two values of the leakage time are often defined – Tstatic and Tdynamic.  The sealing criteria themselves vary somewhat around the world and are usually expressed in terms of a minimum time for the pressure to fall from one specified value to another. These criteria are usefully summarised in Niu et al (2020). The range of criteria effectively imply values of T of the order of 10 to 60 seconds. Once a value of T has been determined, equation (1) can be used in reverse to find how  the internal pressure varies for rapidly varying external pressures, such as when trains pass through tunnels. This usually requires a numerical solution to equation (1).

The calculation of leakage in buildings

Now the approach taken in the study of building ventilation is somewhat different. A number of authors, for example Harris (1990) developed an equation for the flow in and out of buildings with both major openings such as windows, and with a distributed minor leakage openings. For the case of leakage only, which is most analogous to the train case, the basic expression used in given in equation (3). This is effectively an equation for flow through an orifice, and the basic assumption is that the leakage area can be represented by an equivalent orifice. It is then assumed that the change in internal pressure is an adiabatic process, and thus equation (4) applies. I am not altogether sure why the process should be adiabatic rather than isothermal, but that is probably due to my lack of understanding of thermodynamics. Putting these equations together gives equation (5), which is equivalent to equation (1) except that the change in internal pressure is proportional to the root of the difference between the external and internal pressures, rather than being directly proportional.

Now the analysis that leads to equation (5) assumes that the orifice coefficient remains constant. However, at low orifice Reynolds numbers, the coefficient is known to fall significantly – see Johansen (1930) and the figure below. (I find I have rather a perverse pleasure at quoting a technical paper that is almost 100 years old!). This implies that equation (5) can only be valid when the pressure differences, and associated leakage flow rates, are quite high. To investigate this further, on the basis of the experimental results shown in the figure below, we take the discharge coefficient to vary with the square root of the Reynolds number as in equation (6). Here the Reynolds number is based on the leakage velocity v and the average diameter of a single leakage path d (which can be expected to have a very much smaller area than the overall leakage area A). After some manipulation this results in equation (7). This is of exactly the same form as equation (1), and the leakage time constant can be explicitly expressed as in equation (8).

Discharge coefficient results of Johansen (1930). The x axis is the square of the Reynolds number. Values are shown for the smallest values of the orifice diameter to pipe diameter used in the experiments

There is still however an issue in applying this equivalent orifice analysis to the case of train leakage. From equation (3) above, the velocity through the orifice is directly related to the pressure difference, with the energy loss being described by the discharge coefficient. For the values of pressure difference across building facades, which are of the order of tens of Pascals, typical values of the discharge coefficient of around 0.6, result in velocities through the orifice are of the order of 1m/s. The pressure differences between the inside and outside of trains in tunnels however can be up to 2 or 3Pa, and an orifice type analysis would give velocities of 40 or 50m/s, which is clearly unrealistic. The discharge coefficient method therefore does not give an adequate energy loss to the leakage flow for high pressure differentials. So this type of analysis may be applicable in building ventilation, but does not seem appropriate for the consideration of train leakage. Some other framework needs to be developed to give the railway methodology for calculating leakage something other than an empirical basis.

Leakage tube methodology

As an alternative, it is possible to conceive of the leakage paths on a train as a set of equivalent pipes. Equation (3) can then be written in the form of equation (9) – which is effectively Darcy’s law for flow through a pipe. The energy loss in the system is determined by the Darcy friction factor. After some manipulation one arrives at the expression of equation (10), which is equivalent to equation (5). Now for high Reynolds numbers (> around 2000) the Darcy friction factor will be constant, and the rate of change of pressure will be proportional to the root of the pressure difference between inside and outside the train – as in the orifice flow analysis. However at low Reynolds numbers the Darcy friction factor varies inversely with Reynolds number as shown. This results in equation (11), which gives the rate of change of internal pressure as being proportional to the difference between the external and internal pressures rather than the root of the difference i.e. a similar form to the empirical equation (1). An equivalent value of the leakage time constant can be derived – equation (12). The energy loss in the system is very much greater than for an orifice flow.


The analysis above suggests that the leakage pipe model might form a useful tool for the interpretation of the current empirical methodology. For a pipe flow, the boundary between the laminar flow range range (when the friction factor is a function of Reynolds number) and the turbulent flow range (when the friction factor is constant) is at a value of Reynolds number of around 2000. It is straightforward, using the above equations, to calculate the pressure difference that results in a Reynolds number of 2000 for different leakage geometries. Typical values are given in the table below. It can be seen that for leakage diameters between 0.75 and 1.5mm, the value of pressure difference for the transition from laminar to turbulent flow falls from 7.7 kPa to 1.2 kPa. Typical pressure transient in tunnels have maximum values of 2 to 3 kPa. There is thus a possibility that for larger leakage holes, the laminar flow pipe flow methodology (equation 12) might not be applicable and an equation of the form of equation 10 might need to be used. In this case, the concept of leakage time is similarly not valid. The table also shows leakage times for each of the cases considered, and these can be seen to fall between 15 and 500 seconds. These all fall within the range measured in experiments, and suggests equation (12) might be a useful method for relating geometric leakage characteristics to leakage time

Train Aerodynamics Research in 2020 Part 2

Part 1 of this review can be found here

Trains in tunnels

The most important flow parameter to be considered in a study of tunnel aerodynamics is of course the rapid change in pressure as trains pass through. A number of investigations in this area have reported in 2020. Perhaps the most significant is the full-scale investigation of Somaschini et al (2020). They measured both on track and on train pressure measurements on a high-speed Italian line. They showed the pressure transients caused by trains in tunnels were very sensitive to the initial flow conditions in the tunnel, and specifically the residual velocities caused by the passage of earlier trains. The on-train measurements consisted of the measurements of pressures around the train envelope, together with the internal pressures for both sealed and unsealed trains. The effects of train passings were also measured, and the effect of HVAC shuttering systems on internal pressures identified. This is a very substantial piece of work and provides much data that could be used for the verification of physical and numerical modelling methodologies in the future. Lu et al (2020) investigated pressure transients for trains crossing in a tunnel using RNG k-epsiilon CFD techniques and moving model experiments.  They used three of four coach trains in tunnels of varying length. The main thrust of the investigation was aimed at investigating the effect of changes in tunnel cross section. There was a respectable level of agreement between CFD and physical model tests, and the authors concluded that the optimal geometry for a reduction in tunnel section depends upon the point at which trains pass, which is of course very difficult to control in practice. Izadi et al (2020) used a simple moving model of circular train in tunnel and compared the results with standard RANS methods. Unsurprisingly there was good agreement. Although this work is in effect a repeat of work that was carried out in the 1970s and 1980s, it does have a novel aspect in that the effect of trains accelerating and decelerating was investigated.

The other major flow parameter of importance is of course the flow velocity, in the slipstream and wake of the trains. These have been investigated by two studies. Li et al (2020a) investigated the slipstreams caused by single and passing trains using URANS CFD calculations around eight coach trains passing through a tunnel roughly three times that length. Unsurprisingly they found that the slipstreams and wakes were highly complex varying both spatially and temporally. The highest velocities were in the train / wall gap or in the gap between passing trains as would be expected. Interestingly they found that the size of the longitudinal vortices in train wakes decreased as the train entered the tunnel and were constrained by the tunnel walls, although their vertical extent increased. Meng et al (2020) used IDDES CFD techniques to investigate the slipstreams and wakes in tunnel for trains with different nose shapes. A three-coach train geometry was used, with noses of variable length. It was found that the long nose shape reduced the slipstream velocities throughout the tunnel.

The reduction in strength of micro-pressure waves from tunnel outlets continues to be a topic of investigation. Luo et al (2020b) investigated this effect for mountainous terrain where there was no space for lengthy for entrance structures, looking instead at the use of cross passages near the tunnel inlet. Both moving model tests and CFD techniques were used, and good agreement was found. It was concluded, again perhaps unsurprisingly, that as many large cross passages near the tunnel entrance as possible had most effect on the strength of the MPW emitted from the tunnel. Saito and Fukuda (2020) investigated entrance stepped hoods of variable area with porous opening using acoustic theory and found that the optimal design could results in significantly shorter hoods than conventional designs.

The study of the aerodynamics of subway systems continues to develop with a number of investigations carried out. In particular there have been two full scale investigations reported. The first, by Hu et al (2020) measured airflow characteristics in the tunnels around a subway station and used the results to calibrate a network model. This model was then used to investigate the effect of different arrival and departure strategies on the air flow within stations.  The cooling load of train air flow was also investigated, in relationship to mechanical ventilation methodologies.   There were significant variations in ventilation characteristics as train operation varied, but the authors found it was possible to arrive at an optimized HVAC operation. Khaleghi and Talaee (2020) carried out full scale velocity measurements in a subway station with longitudinal ventilation of tunnels, with a novel air curtain system to control the ventilation flows within the station. The results were used to calibrate a CFD methodology, which was then used to investigate a range of ventilation and air curtain strategies studied.

Liu et al (2020b) used a standard k-omega SST CFD methodology to investigate a four-coach train accelerating to 120km/h as it left a station and entered a tunnel, and in particular made estimates of the time varying pressure and friction drag. As would be expected, the latter increased substantially on tunnel entry. Huang et al (2020) also used a standard RNG k-epsilon CFD methodology to investigate the loads on the surfaces of tunnels caused by the passage of a six-car subway train. The methodology was verified using equivalent moving model tests. The investigation showed that the loads were particularly sensitive to overall tunnel blockage and tunnel shape.

Finally, it is necessary to point out that the effect of air movement on the spread of fires in tunnels is not considered here. The interested reader is referred to Liu et al (2020c) and Peng et al (2020) for recent investigations.

Trains in crosswinds

Crosswind forces

One of the basic requirements for the study of trains in crosswinds is a knowledge of the crosswind induced forces. As pointed out in TAFA, the determination of these forces is not straightforward either experimentally or numerically. A number of authors have addressed some of these issues. Liu et al (2020d) investigated the optimum number of pressure taps on a train to obtain accurate forces and moments through pressure integration using DES methodology for a three car HST at yaw angles between six to thirty degrees, and compared their results with directly measured forces from wind tunnel tests. They found that an arrangement of 15 x 4 taps on each face of the train produced adequate results although the difference between the computed and measured force coefficient values was considerable (up to 10% for side force coefficient, and up to 20% for lift force coefficient. Interestingly they found that only between 2 and 4% of the forces were due to friction rather than pressure effects.  Huo et al (2020) investigated whether the trailing edge shape of dummy vehicle in crosswind tests (which is conventionally mounted behind the live vehicle) affected the measured forces and moments. A range of shapes were considered, from blunt ended to streamlined, using DDES-SST techniques. Little effect was found for yaw angles up to 45 degrees, but both side and lift force coefficients fell below the values for long trains at a yaw angle of 60 degrees, with the trailing edge shape making little difference. Li et al (2020b)  looked in detail at the choice of the RANS methodology embedded within the DES approach, an important issue that has not been much investigated in the past. In particular they investigated the adequacy of the one equation SA-DES approach and the two equation SST-DES approach as applied to a Class 390 train at a yaw angle of thirty degrees, for which wind tunnel data was available.  Both methods gave similar values and trends, of surface pressure but there were considerable differences in the predicted separation positions. Side force and rolling moment coefficients were similar, but lift force coefficient were very different. The authors concluded that SST-DES was the most appropriate to use.

CFD techniques were also used to investigate the effect of specific geometrical features on measured and calculated crosswind forces. Guo et al (2020b) used DDES to investigate the effect of bogie complexity on crosswind measurements and found that the rolling moment coefficients increased as bogie simulations became more complex, with a variation of around 20%.  Jiang et al (2020) carried out a DES investigation of the effect rail type in cross wind simulation. No rail, simple rail, complex rail simulations were  used. It was found that there was little effect on side force coefficients and rolling moment coefficients were only affected in the higher yaw angle range but lift force coefficients were significantly affected for all yaw angles. The results for the simplified and complex rail simulations were very similar. Maleki et al (2020) in their LES study of double stack freight in crosswind particularly investigated the effect of the gap between containers. They showed that variations in gap width had a significant effect on flow topology, which was highlighted through significant differences in mode shape appearing in a POD analysis. The flow structures that were observed included vortices from the leading windward corner of the container and longitudinal vortices from the top and bottom leeward corners. The authors were mainly concerned with the effect of crosswinds on drag, and their work illustrated the drag benefit of keeping the gaps between the containers small, which became more substantial as yaw angles increased.

Zhang et al (2020b) carried out a CFD analysis of the Chiu and Squire idealised train model at 90 degrees yaw and used various optimization schemes to optimize cross wind forces by geometric changes. They found that the changes had little effect on side forces, but that lift could be reduced by 20% by small sectional modifications. The work has little practical significance.

The investigations described above have, if only implicitly, been concerned with the crosswind forces on train due to normal, cyclonic winds. By contrast Xu et al (2020a), using DES simulations, investigated the forces on  a three-car train passing through a tornado simulation. The tornado was small in relation to the train, and there were significant scaling issues as in all such simulations. Forces were calculated for different vortex positions relative to the train, and whilst of high intensity were found to be transient and very localized. The overall representativeness of the simulated flow field in relation to real tornadoes must be questioned.

A number of investigations, usually CFD studies, have looked at crosswind forces on trains in the presence of different infrastructure geometries. Guo et al (2020c) used DDES techniques to study flow over embankments with and without trains. A three-car HST model was used, with embankments up to 7m high, with a simulation of an upstream power law profile. Both velocities and train forces and moments were measured for a range of different cases. The results are potentially very useful and need to be integrated with existing compilations of similar measurements. Wang et al (2020e) carried out a RANS study on a three-car HST to investigate the effects of ground clearance, typical embankments and viaducts and a truss bridge, at yaw angles of 30, 45 and 60 degrees. Results are presented for side and lift force coefficients for the different cases. Li and He (2020) carried out wind tunnel measurements of a train on a bridge with a ninety-degree wind and measured aerodynamic forces and moments for different angles of attack. As this angle varied across the range that might be expected in reality, significant variations in the forces and moments were observed. These results are valuable, although the authors recognize that strictly they are valid only for the bridge geometry that they studied. Zou et al (2020) used RANS SST to study the aerodynamic forces and moments on a three-car HST as it travelled into and out of an area on a bridge sheltered by a wind barrier. Very high unsteady forces were observed on both train and barrier at entry and exit. Yao et al (2020) carried out a similar RANS SST study of a train on a truss bridge and also found similar highly transient and unsteady forces. They also investigated the effect of angle of attack. Gu et al (2020) report a study of flow and forces behind corrugated wind barriers, with a wavy section of different types. Very large-scale high blockage wind tunnel tests were carried out on a train section at 90 degrees yaw, together with equivalent DES calculations. The forces on the train section varied significantly with barrier “bendiness”.

Two investigations have looked in detail at the crosswind forces on trains as they emerge from a tunnel onto a viaduct in complex terrain.  Deng et al (2020b) carried out a RANS study and found very rapid transients for all forces and moments with some significant overshoots of the equilibrium value. Wang et al (2020f) using SST k-epsilon methods looked at the effect of wind barriers at the tunnel bridge junction, comparing the transient forces with and without barriers.

Vehicle system modelling

Having determined the force and moment coefficients, the next step in addressing crosswind safety is an analysis of the vehicle / wind dynamic system. This requires some formulation to describe wind gusts. There are three basic approaches – the specification of gust magnitudes alone, the specification of a discrete gust shape, and the full stochastic representation of the wind. All three approaches were investigated by Yu et al (2019) whose used examples of all three methods within a generic MDF model for a high-speed train and derived cross wind characteristics for each. These results showed the relationship between the three methods and illustrated the arbitrariness in defining peak gust values.

Montenegro et al (2020a) investigated the effect of cross winds on a train bridge system subject to a stochastic wind field and calculated the forces on both train and bridge. Train bridge interactions were specifically allowed for and a stochastic track irregularity model was used. Three criteria were used to define crosswind characteristics – the rail lateral / vertical force ratios, wheel unloading, and the Prud’homme limit. The comparison with the CWC calculated from the TSI discrete gust methodology showed that the latter was conservative. The authors followed up this work in Montenegro et al (2020b) which investigated the adequacy of the TSI methodology for various bridge heights, and showed that it became progressively less accurate as the bridge height increased due to the fact that it involves a fixed, rather than variable, turbulence intensity. A revised TSI methodology with variable turbulence was proposed.  Montenegro et al (2020c) used this methodology to investigate different types of bridge construction. They showed that direct wind load on trains were more important than the loads transmitted from the bridge, and also looked at safe running speeds.

Yang et al (2020) investigated the train dynamic response on a tunnel / bridge system such as described above. A three-coach train model was used with a many degree of freedom mechanical model, together with RNG k-epsilon CFD calculations for the train forces.  CWCs were again derived, and the rail lateral / vertical force ratios and wheel unloading criteria were used to derive CWCs. Sun et al (2020) investigated an HST passing a wind break with a breach. URANS was used, together with a complex MDF model, and artificial wind gust shapes. It was found, unsurprisingly, that when the gust duration experienced by the train as it passed the breach was equivalent to the train suspension natural frequency, then large force and displacement transients were observed. Wu et al (2020) investigated the hunting stability and rail creep on curved track with a cross wind. As such they looked at the dynamic stability of the vehicle ride, rather than overall stability. They showed that hunting behaviour was changed significantly by cross winds

Two papers of particular significance are those by Wang et al (2020g) and   Liu et al (2020e) The former considered a stochastic simulation of wind as input to a closed form dynamic model that allowed only for major suspension effects.   A frequency response method was used that used wind spectra, mechanical transfer functions to obtain track contact force spectra. This enabled peak values to be calculated from a normal peak value analysis and CDFs of exceedances were derived. The second paper similarly adopted a simple model of the dynamic system, but used both spectral methods and discrete gust profiles, together with force and moment coefficients from CFD calculations and wind tunnel experiments to calculate train displacements and wheel forces. The method can also be used to study pantograph dynamics. Liu et al (2020f) followed on from this work to investigate overturning coefficients for different windspeed changes over different times and to look at a range of geometry changes.

Finally, the work of Xu et al (2020b) should be mentioned. This is a follow on from the work of Xu et al (2020a) for the effect of tornadoes on trains but extended to include a MDF dynamic model. It was shown that derailment was more likely than overturning for the cases considered, although it must be stressed that the realism of the tornado simulation is doubtful.

Miscellaneous wind effects

Follow on from earlier papers on braking plates discussed in the first post (Nui et al 2020a, b, c, d), Zhai et al (2020), using DES calculations over a simulated train roof looked at the effect of a cross wind with a yaw angle of ten degrees.  The unsteady forces on the raised plate were considered during the opening process. Unsurprisingly it was shown that crosswinds increased these forces significantly.

Takahashi et al (2020) investigated the unbalanced tension in the overhead in crosswinds of up to 30m/s. Measurements were made of wire movement in high winds and it was shown that flapping wires imposed significant loads on structures. Methods were derived to determine the frequency and amplitude of the wire movements for use in fatigue analysis.

Emerging issues

Work continues to some extent on evacuated tube transport. Niu et al (2020e) looked at the acceleration and deceleration of short tube vehicles through the sound barrier using IDDES-SST techniques, predicting values of drag, pressure and temperature. They validated their methodology against wind tunnel tests on wedge like shapes. Calculations were performed for s range of acceleration and deceleration profiles and the flow patterns for the two sets of profile were shown to be quite different. Zhou et al (2020b) looked at longer, more train-like vehicles using a 2D axisymmetric k-omega method and investigated the onset of the critical flow phase. Both investigations showed the overall complexity of the shock wave pattern around such vehicles at high speeds.

Although perhaps somewhat peripheral to train aerodynamics, interest continues in air quality in the railway environment. Islam et al (2019) report measurements of a short trial to measure gaseous and particulate pollutants around a railway station in India, that measured high values of PM2.5. Xu and Liu (2020) similarly measured high values of PM2.5 around a Beijing railway station. The former used a trajectory analysis that indicated the majority of the pollutants were from local sources, whilst the latter used the data to develop a spatial prediction model based on modal decomposition that allowed future particulate concentrations to be predicted.  Loxham and Nieuwenhuijsen (2019) present a review of particulate levels on underground railways from a variety of sources , and in particular look at the health effects of the measured pollutants. They concluded that the particulates produced by the operation of the railways themselves was more toxic than the ambient values, because of their metal content, although their health effects were unclear. Ren et al (2018) looked at the use of momentum sources in CFD calculations to represent vehicles, as a potentially more economic type of calculation than using a standard dynamic mesh around trains models. A simple slow speed moving model rig was used for validation purposes. A significant resource saving was indeed reported, and it was shown that the methodology could be used to predict particulate movement in tunnels, with moving trains causing more movement than stationary trains.

Finally, a number of papers present work that looks at train ventilation and air movement within train cabins. These were mainly concerned with the optimization of HVAC systems – Barone et al (2020) who developed a dynamic simulation methodology of HVAC for train trips that included weather effects; Li et al (2019) who conducted CFD calculations to model the flow over passengers in and HST cabin to determine thermal comfort and airflow velocities; Schmeling and Bosbach (2019) who carried out laboratory test on a mock-up of a train cabins with mannikins; and  Talee et al (2020) who measured airflow velocities in a long metro train with a through corridor while accelerating and decelerating. Batutay et al (2020) report measurements of PM2.5 and CO2 levels in train cabins on a subway line in the Philippines. High levels of PM were measured at times.

Final reflections

The two trends noticeable in the last review are again apparent – the large number of published investigations from a small number of Chinese groups, and the growing use of CFD techniques, and in particular the IDDES technique seems to be becoming the most favoured. Having read the papers collated in this year’s review I feel it worth quoting directly two of my concluding comments from last year as they still seem to me to be relevant. Firstly

…….. it seems to the author that there is a growing need for a small set of freely available well documented validation cases, ideally from full scale experiments for a range of train types, that investigators can use routinely to prove their (CFD) techniques. At the moment the validations used are somewhat ad hoc, and perhaps a more systematic approach would give greater confidence in the results, and also allow research papers to be reduced somewhat in length, as the details of the validation cases would not be required…..

And secondly.

….. it must be remembered that CFD simulations, in the same way as physical models, can only offer a simplified representation of the flow around full-scale trains, and need to be interpreted in this light. There is a tendency amongst some authors (and I name no names!) to quote numerical results to higher levels of accuracy than is either sensible or useful when the uncertainty of the full-scale situation is considered.  Just as with physical model tests, the role of the engineer in interpreting CFD results in terms of the reality of the operating railway is crucial……..

These comments still stand.

Train Aerodynamics Research in 2020 Part 1


The book “Train Aerodynamics – Fundamentals and Applications” (hereafter referred to as TAFA) was published in early 2019, but in reality took no account of any material published after June 2018. In January 2020 I posted a review of Train Aerodynamics research published in the latter part of 2018 and all of 2019 to update the material in TAFA. In this two-part post I do the same for material published in 2020.

It should be emphasized at the outset that, as in last year’s post, this collation cannot properly be described as a review, which requires some degree of synthesis of the various reports and papers discussed. This of course requires a number of papers addressing the same issue to be available to synthesise. Looking at papers from a short time period that cover a wide range of subject matter, this is not really possible, so what follows is essentially a brief description of the work that has been carried out in 2020, with a few interpretive comments.

In this post we consider the papers that address specific flow regions around the train as outlined in TAFA – the nose region, the boundary layer region, the underbody region and the wake region. In part 2 we consider specific issues – tunnel aerodynamics, trains in cross winds and a variety of other effects.  In the text, published references are linked directly to their DOI, rather than to a reference list.

The nose region

The major aerodynamic feature of the flow in the nose region is of course the large pressure fluctuation that occurs as the nose of trains pass an observer. The major practical issue arising from this is the loading on passing trains or structures next to the track.

A number of investigators have studied these loads, using full scale, physical model scale and CFD methods. The most common structures investigated were noise barriers of different types, and a range of data has been obtained that adds to the general database of train loading on such structures.   Xiong et al (2020)  report a series of full-scale measurements to investigate the loads on noise barriers on bridges caused by different types of high-speed train running between 390 and 420 km/h. The variation of pressure with position on the barrier was measured and the results compared with earlier data from other experiments and codes. Oddly, the variation of load with train speed was considered in a dimensional way and was shown to increase with the square of the train speed – unsurprisingly implying that the pressure coefficients were constant. Also, a Fourier analysis of the unsteady data was carried out, which was not appropriate as the loading was deterministic rather than stochastic. Zheng et al (2020) measured the vibrational characteristics of semi-enclosed sound barriers at full scale, consisting of a box over track with panels on one side and on half the roof and an open lattice structure on the other side. Measurements were made using accelerometers and pressure loads were not measured directly. They supplemented their data with RANS CFD and FE vibration analysis. The CFD was validated for a short train against earlier tests and used to predict loads for the full-scale case that were then used in the FE analysis. Good agreement was found with the full-scale measurements of accelerations. Interestingly the authors found that to predict the measured vibrations, it was not necessary to take into account the mechanical vibrations caused by the passing train – this is somewhat contrary to previous work and is probably a function of the rather rigid geometry of the semi-enclosed barriers. Du et al (2020) investigated the pressure loads on a range of geometries of low noise barriers caused by the passage of high-speed trains, using moving model tests. The loads due to both single and passing trains were measured. The effect of train speed was again investigated through looking at dimensional pressure values only – but in effect show a near constancy of pressure coefficient as would be expected.  Luo et (2020a) made similar moving model measurements on two coach Maglev trains passing noise barriers, and investigated various barrier geometries using IDDES simulations.  Unsurprisingly the authors found that the pressures on the barriers were well in excess of open air values. Slipstream values were also measured and calculated in the gap between the train and barrier.

Liang et al (2020a) and Liang et al (2020b) measured loads on a bridge over the track and on the platform screen doors in stations and the roofs of enclosed stations, using moving model tests and LES or IDDES CFD calculations.  Good agreement was found between the physical and numerical modelling in both cases. For the bridge case, loads were measured at different positions across the bridge, for different bridge heights. There were no observable Reynolds number effects on pressure coefficients and good agreement was found with the CEN data collation. For the station case, good agreement with the CEN correlations was found for the station roof measurements, but the data for the various platform screens was widely scattered about the CEN value.

Moving away from the consideration of pressure loads, Munoz-Paniagua and Garcia (2020) investigated the optimization of train nose shape, with drag coefficient as the target function, using a genetic algorithm and CFD calculations of the flow around a two coach ATM (Aerodynamic Train Model). The looked at a large number of geometric variables and concluded that the most important parameter to optimize for drag was the nose width in the cab window region. 30% decreases in drag coefficient were reported but it is not clear to me whether this relates to the nose drag or the drag of the whole train.

The boundary layer and roof regions

A number of studies published in 2020 investigated the boundary layer development along high-speed trains, often in association with wake flow investigations. Whilst most of these  used CFD techniques, one study, that of Zampieri et al (2020) describes a series of full scale velocity measurements around an 8 car, 202m long, ETR1000 travelling at 300km/h. Measurements were made at the TSI platform and trackside positions and profiles of longitudinal profiles of ensemble average velocities and standard deviations were obtained. The effects of cross winds were studied and found to be particularly significant toward the rear of the train, where a significant asymmetry in flow fields was observed. These tests were supplemented by a range of CFD calculations using various RANS turbulent models. Only moderate levels of agreement between the two techniques were found. In my view the most important aspect of this work is the establishment of a high-quality full-scale dataset for future use in CFD and physical model validation.

The CFD studies all used DES or IDDES techniques. That of Wang et al (2020a) looked at the effect of simulating rails in CFD simulations of a two-coach high speed train, and although mainly concerned with wake flows, does present some boundary layer measurements. Wang et al (2020b) investigated the difference between the use of conventional and Jacobs (articulated) bogies for a three coach high speed train. Again, it is mainly concerned with the effect on the wake, but it does show that the use of Jacobs bogies results in a thinner boundary layer on the side of the train and reduced aerodynamic drag. Guo et al (2020) describes an investigation into the effect of the inter-unit gap between two coupled three care high speed trains. As would be expected from recent full-scale tests, an increase in boundary layer velocities is observed in the vicinity of the gap, with a significant thickening of the boundary layer on the downstream unit. Liang et al (2020c) investigated the effect of ballast shoulder height on the boundary layer and wake development of a four-coach train. Little effect on boundary layer was observed either on the train walls or roof. Finally, Tan et al (2020) carried out IDDES calculations for 2, 4 and 8 car Maglev trains. Unsurprisingly the boundary layer grew to be thicker along the eight-coach train than along the others. The maximum slipstream velocity increased with train length at platform level but was greatest for the four-coach set lower down the train.

Work has also been carried out to investigate boundary layer development on freight trains. Bell et al (2020) describe a series of full-scale velocity measurements around six different loading configurations of multi-modal trains. Rakes of anemometers were set up at three locations along the track, that enabled boundary layer measurements to be made. As all the configurations were different the normal technique of ensemble averaging was not possible. Nonetheless much valuable information was obtained on mean velocities, turbulence intensities, length scale and velocity correlations along the track. It was found that the effect of cross winds was quite marked, with significant differences between the measurements on the two sides of the track. Also, it was found that in general the effect of gaps between containers was small, except for the larger gaps in the configurations. The paper of  Garcia et al (2020) looks in detail at various CFD techniques for predicting the flow around container trains. In particular it investigates the performance of the URANS STRUC-epsilon methodology and shows that it compares favourably with reference LES results, with a much lower resource use. As part of the analysis the paper presents calculations for boundary layers around single containers with gaps in front and behind.

In a series of five papers, a group from China has investigated “Braking Plates”, flat plates that are lifted into a vertical position on the train roof in order to increase aerodynamic drag and act as brakes. All the papers describe IDDES CFD investigation on various geometric configurations of high-speed trains.  Niu et al (2020a) calculated the forces on a two car train and showed that the increased drag was more significant when the plate was on the centre of the roof rather than in an intercar gap. Niu et al (2020b) contains very similar material but with more flow field detail around the plates and the vehicle more generally. Niu et al (2020c) looked at the interaction between rooftop equipment such as HVAC units and pantographs, and the highly unsteady turbulent wake behind the plates. These interactions were found to be small. Niu et al (2020d) investigated the behaviour of plates near the nose and tail of vehicles in a two coach train, and showed that those near the nose were more effective in increasing drag than those near the tail, with the latter significantly affecting the vortices in the train wake. Finally, Zhai et al (2020)  calculated the flow over the roof of the train only and studied in detail the highly unsteady flow field caused the raising and lowering of braking plates at zero and ten degrees yaw. Whilst this concept is interesting, more work is required to determine how multiple plates would work together on longer more realistic trains – are the drag benefits significant in terms of the drag of the whole train. Also, the question remains as to how effective they would be in an actual braking process. Work is required to model the slowing down on trains using both conventional and aerodynamic methods to find out the speed range over which the braking plates make a significant contribution to overall braking forces.

The underbody region

hhttps://doi.org/10.1177/0954409720960889(opens in a new tab)

Two studies have been reported that look at specific underbody flow effects, rather than the effect of underbody changes on the development of the wake which will be reported below. The first is by Jing et al (2020) who report an investigation using a wind tunnel model of the flow over a 1:1 section of ballasted track, together with k-epsilon calculations of the flow beneath a two-coach train. They specifically look at the pressure distributions on different types of ballast configuration and draw some conclusions about the “best” ballast configuration to reduce ballast flight. However, the unrepresentative nature of the wind tunnel tests, and the lack of any link between the observed pressure distributions and the mechanics of ballast movement does not enable one to have a great deal of confidence in these conclusions.

Liu et al (2020) describe some very innovative studies of water spray from train wheels, addressing the problem of ice accretion in cold climates. The IDDES technique was used to study the flow beneath a two coach HST with detailed bogie simulations and rotating wheels. Water droplet trajectories were modelled using Lagrangian particle tracking methods.  Regions where water spray impinged on the underbody and bogies, and were thus prone to ice accretion were identified. It was noted that spray impingement fell substantially as the train speed increased above 250km/h.

The wake region

A number of CFD studies of the wakes of high-speed trains have been published in 2020, mainly carried out with two or three coach high speed trains, using DES or IDDES techniques. All identified the major wake structure as a pair of counter-rotating longitudinal vortices. Most of the studies investigated the effect of different geometry changes on these structures. Zhou et al (2020a) investigated the difference between train simulations with and without bogies, and found the longitudinal vortices were wider when bogies were present. A tail loop vortex could also be seen that shed alternately from each side of the train with bogies present but shed symmetrically with no bogies. Wang et al (2020a) investigated the effect of rails in the simulation and showed that the effect of rails was to constrain the width of the vortices and to reduce the TSI gust values. Similarly Liang et al (2020c) investigated the effect of ballast shoulder height on the wake, and in general found that the higher the ballast shoulder the lower were the wake slipstream velocities, both in term of ensemble averages and TSI values. High ballast shoulders tend to lift the wake vortices upward and away from the TSI measurement positions.   Wang et al (2020b) describe an investigation of the difference in wake structures between Jacobs bogies and conventional bogies. The former results in a narrower wake and lower TSI slipstream velocities. Wang et al (2020c) examined the effect of different bogie configurations, including a wholly unrealistic no bogie case, but with bogie cavities. Unsurprisingly this case was shown to result in the largest slipstream velocities, but because of its unrealistic nature has no real meaning.  Guo et al (2020) looked at the effect of the gap between two three car units on the wake of the combination. They found that the wake was wider for a double unit than a single unit, presumably because of the increased thickness of the train boundary layer at the tail. Two further studies of the effect of underbody clearance are reported by Dong et al (2020a) and Dong et al (2020b). Both use the IDDES technique, the first on a four coach ICE3 model without bogie representation, and the second on a three coach ICE3 with realistic bogie simulation. In the first case the ground clearance is directly changed, whilst in the second case it is changed by adding panels of different thicknesses onto the track bed. Whilst there are some effects of ground clearance on drag and lift and on the nature of the boundary layer flow along the side of the train, the primary effect in both cases is seen in the wake, as the underbody flow and wake vortices interact in different ways. For the more realistic case of the second investigation, increased TSI slipstream velocities were observed as the gap width decreased.

Tan et al (2020) present the results of an investigation of the boundary layers and wakes on two, four and eight car Maglev vehicles, rather longer than the vehicles used in the above investigations. The wake structures were very different for different train lengths, with a significant decrease in the Strouhal number of the wake oscillation as the train became longer.

Finally, Wang et al (2020d) investigated the wake structure of a two-car high speed train as the Reynolds number increased from 5 x 105 to 2 x 107. They showed that the overall flow pattern, in terms of large-scale vortex structure, tail separation positions and wake Strouhal number, was little affected by Reynolds number, although as the Reynolds number increased, more and more smaller scale vortex structures could be seen.

Part 2 of this review can be found here.

Reducing train aerodynamic resistance through the use of slab track

Ballastless double track of the type "Rheda 2000" including concrecte slabs and ties/sleepers, rails, and drainage slits.

There are major efforts underway to “decarbonize” the GB rail network. One way of pursuing this goal is to reduce traction energy costs which would contribute to decarbonization either directly through the reduction in fossil fuel use, or indirectly through the reduction in the use of electricity produced from non-renewable sources. In this post,  I will attempt to show that the  train aerodynamic drag reduction due to the use of slab rather than ballasted track may result in significant fuel and energy savings for an entire train fleet that would contribute to the decarbonization agenda and that could radically change the overall business case for the installation of slab track, which is currently only used in specific circumstances. It will be seen that the argument is very speculative in places, but perhaps strong enough to warrant further investigation. We begin in the next section with an introduction to train resistance.

Train Resistance

The specification of train resistance is required for the assessment of energy consumption, train timing etc. Now train resistance is, very broadly, composed of mechanical (rolling) resistance and aerodynamic resistance, and is conventionally described by the Davis equation given in equation (1).

Equation 1

Here v is the train speed and a, b and c are constants. The first two terms are taken to be the mechanical resistance, and the last term is taken to be the aerodynamic resistance. The aerodynamic resistance is thus proportional to the square of train speed and becomes progressively more important as train speed increases.  The parameters a, b and c are usually obtained from coast down tests on (ideally) straight, level section of track, in which trains coast from top speed to zero and acceleration, speed and distance are measured. A quadratic curve is then  fitted to data. Typical examples of tests sites in the UK are given in figure 1 and a typical set of results in figure 2. Note that this figure and most of those that follow are taken from the recent book “Train Aerodynamics – Fundamentals and Applications” by myself and a number of colleagues. Note also that it is also possible to estimate the aerodynamic component of resistance from wind tunnel tests and CFD calculations, but there are significant technical issues (mainly due to the inability of both techniques to model full length trains) and thus in what follows we  consider only data from full scale measurements.

Figure 2 Typical results for Class 45 and 6 passenger coaches between Thirsk and Northallerton

Drag coefficient

The coefficient c is related to the aerodynamic  drag coefficient CD by the simple expression of equation (2).

Equation (2)

Here A is the frontal area of the train and r  is the density of air. The drag coefficient for a wide range of trains is shown in figure 3 (from ???).

Figure 3. Drag coefficient correlation

Very broadly, for any individual train class, the drag  coefficient in linearly proportional to train length, and can be represented by the simple form of equation (3).

Equation (3)

Here L’ is an effective train length (the length of the train minus the length of the nose and tail sections) and p is the wetted perimeter of the train envelope. The values of the parameters K1and K2 are given in table 1 for the train types shown in figure 3.

Table 1 Parameter values

Breakdown of Aerodynamic drag

Figure 4 shows how the components of aerodynamic drag for high speed trains from the work of two different authors. Whilst there is some variability between the results it can be seen that the drag of the underbelly and bogies contributes 20 to 50% of the overall drag and skin friction drag on the train side and roof contributes 30% to 40%. An important point to appreciate is that the underbody drag includes drag due to the track roughness – energy needs to be used to overcome the aerodynamic resistance of the track itself. This point does not seem to have been well appreciated in the past.

Referring back to equation (3), K2 is a friction coefficient for train, combining theeffect of skin friction on side and roof and bogie and underbody drag. As can be seem from figure 3, values of 0.004 are typical for high speed trains (but note the quality of fit is not terribly good).

Friction coefficients can be obtained directly from measurements of the velocity profile on the side and beneath the train and then fitting of logarithmic profile to the data. This process is somewhat difficult and subjective, but has nonetheless been attempted by a number of authors in the past. Table 2 shows the values for skin friction on the side of the train that have been obtained, and table 3 shows values for the underbody of trains.

Table 2 Skin friction coefficients
Table 3 Underbody friction coefficients

Typical values of the former are 0.0015 and  typical values for the latter for ballasted track are 0.03. The higher values for underbody coefficient are of course to be expected because of the roughness of the train underbelly. For slab track the one set of data available gives a significantly lower value of the underbody friction coefficient of 0.01.


If we assume that, for high speed trains, skin friction values of 0.0015 and underbody drag of 0.03 and assume that the former acts over 90% of the wetted perimeter and the latter over 10% these weights give a value of K2 of 0.00435 which is consistent with drag compilation value from table 1 of 0.004 and result in a drag coefficient of 1.4 for a 200m high speed train. If underbody drag reduced to 0.01 by use of slab track, the same calculation leads to drag coefficient of 0.81 – a staggering 40% decrease. A rule of thumb that is often applied is that a drag coefficient reduction of x% results in an energy saving of 0.4x% suggests 40 x 0.4 % which suggest a potential reduction in fuel use of 16%.

Now many assumptions have been made in the above analysis, perhaps the most significant being the value of friction coefficient for slab track, which is based on one set of experimental results only. Thus the argument that significant fuel cost reductions might be a possibility through the use of slab track more widely, is at best suggestive but I would suggest merits further investigation. The question arises as to whether such energy savings have the potential to change the business case for slab track, which is in general only currently used for very specific situations such as tunnels, poor ground conditions etc.  I would thus suggest a preliminary investigation that addresses the question of what reduction in drag coefficient would actually be required to change business case for slab track? As both infrastructure and trains would be involved, a system approach would be required here. If further investigation of the business case shows that it is worth pursuing these ideas, the next stage would be to conduct coastdown tests with the same train over ballasted and slab track. A long straight level section of slab track would thus be required. Does such a section of track exist in the UK?

Modelling of extreme wind gusts


This post addresses the issue of the use of what has become known as the “Chinese Hat” gust model. The use of this title has become increasingly problematic over recent years for obvious reasons, and I will no longer use it, but will instead refer to the “CEN extreme gust model” in what follows.

The CEN extreme gust model

In a number of situations in wind engineering, some sort of deterministic (as opposed to stochastic) gust model is required in order to determine structural response. One such case is in the determination of the risk of overturning of road or rail vehicles in high winds. A methodology of this type is set out in CEN (2018), where an extreme gust model is described.  This model was originally developed in wind loading studies for wind turbines as a time dependent gust to be applied to calculate wind turbine loading at one fixed location (Bierbooms and Cheng, 2002). As such, it is perfectly adequate and a good representation of an average extreme gust in high wind conditions.  In the methodology of CEN however, it is re-interpreted as a stationary spatially varying gust. This must be regarded as a very significant assumption for which, in my view, there is little justification. Nonetheless the formulation has proved useful practically and we begin by considering it in a little more detail.

For a wind normal to the track, the extreme gust formulation is given by equation (1) on Box 1. Note that the “characteristic frequency” of the gust is calculated from standard wind engineering methods for temporally, rather than spatially, varying gusts. Equation (1) is a generalised form of that given in CEN (2018) to remove some of the constants that tie the expression to a particular location and topography through specific values of peak factor and the turbulence intensity (the ratio of the standard deviation to the mean velocity). The time dependence is recovered through the passage of the train passing through this gust at a speed v = xt to give equation (2). It can be seen that the gust thus has a maximum value of (1+ peak factor x turbulence intensity) when t = 0 and decreases to unity for small and large times. It is symmetrical about t = 0. The velocity relative to the train is then found by the vector addition of this gust velocity with the vehicle velocity to give a time varying value.

To enable the gust profile to be specified, the characteristic frequency f is required. This is specified in equations (3) to (5). These equations are again in a more generalized form than given in CEN (2018), where a value of the upper limit of integration is fixed at 1 Hz, together with an implicit value of the turbulence length scale of around 75m. The genesis of the 4.18 factor is however not clear to me.  Equation (3) shows that the calculation of the characteristic frequency is thus based on the calculation of the zero-crossing rate of temporal fluctuations through the use of the velocity spectrum. Again, note that these parameters describe a time varying rather than a spatially varying velocity, and their use is not formally consistent with a spatially varying gust. From equations (3) to (5), it can be seen that the normalized characteristic frequency is a function of the normalized upper limit of integration. A numerical solution of these equations was carried out and the following empirical line fitted to the results for a value of the latter greater than 1.5 (which is the realistic range) – equation (6). From equations (2) and (6) we thus obtain equation (7). Although the overall methodology cannot be regarded as wholly sound, equation (7) does (in principal) significantly simplify its use and also allows the implicit wind parameters in the method to be explicitly defined.

Box 1 Equations 1 to 7

Is there a better methodology?

It can be seen from the above that the CEN  methodology thus does not fully describe a typical gust as seen by a moving train, which would vary both spatially and temporally, and can at best be regarded as an approximation, although its practical utility must be acknowledged. Ideally, if such an approach is to be used, a gust that varies both in space and time is really required.  Such a gust was used in the SNCF route assessment method of Cleon and Jourdain (2001), where the shape of the gust is appropriately described as a rugby ball. This method was however for very specific wind characteristics and does not seem to have found widespread use. Thus in this post, we investigate the possibility of developing a spatially and temporally varying gust, that can be expressed in a simple form (ideally similar to equation (2)) for practical use.

Towards a new model

In this section we will draw on experimental results for extreme gust characteristics in both temporal and spatial terms to construct a simple, if empirical model, that fulfills the function of the CEN (2018) model without the theoretical drawbacks.

We consider first the full-scale experimental data analysed by Sterling et al (2006) which used conditional sampling to determine the average 99.5th percentile gust profile for four anemometers on a vertical mast with heights between 1m and 10m. These results thus give the time variation in gust speed as the gust passes the anemometers. They showed that the gust profiles could be well approximated by the formula shown in equation (8) (Box 2). The parameter G in this equation is the equivalent of the peak factor multiplied by the turbulence intensity in equation (2) and for these measurements was 0.786.  n was -0.096, and the value of m depended upon whether t was greater or less than zero. For t < 0, i.e. on the rising limb, m was 0.1, whilst for t > 0, on the falling limb, m was 0.2. The gust shape was thus asymmetric with a maximum at t = 0.  This curve was a good fit to all the gust profiles throughout the height range. In what follows we will use a rather different curve fit expression to the same data, more consistent with that used in CEN (2018) – equation (9). It was found that the best fit value of b  was equal to 0.5 for all t, whilst the best fit values of a were 0.49 for the rising gust and 0.37 for the falling gust. This expression thus describes the temporal variation of wind speed as a gust passes through the measuring point

To describe the lateral spatial variation of the gust profile, we use the data of Baker (2001) who presents conditionally sampled peak events for pressure coefficients along a 2m high horizontal wall. This data allows the lateral extent of the gusts to be determined, from the variation of the time varying pressure coefficient divided by the mean value of the coefficient and then assuming that the gust velocity variation can be found from equation (10). The spatial variations of velocity were then fitted by a curve of the form of equation (11). g was found to be 6.16 and d was found to be 0.7.

On the basis of the above expressions one can thus write the expression of equation (12), which describes the variation of the gust velocity in both space and time. The movement of the train through the gust can again be allowed for by letting x = vt (equation (13)).

Box 2 Equations 8 to 13

Model comparison

Box 3 sets out the formulations of the CEN extreme gust model and the model derived here. In some ways they are similar in form, with an exponential formula that is primarily a function of normalized time. Whilst the CEN model is symmetric around t = 0, the new model has a degree of asymmetry because of the different values of the curve fit parameters for t < 0 and t > 0. However an examination of the new model suggest that the asymmetric term may be small, and thus Box 3 also shows an approximate version of the new model where this term is neglected.

Box 3 Model Summary

Figure 1 shows a comparison of these three models for the following parameter values – peak factor = 3.0; turbulence intensity = 0.25; train speed = 75m/s; mean wind speed = 25m/s; turbulence length scale = 75m, upper frequency of integration = 1.0Hz. It can be all three models are similar in form, showing a sharp peak at t = 0. The full and approximate forms of the new model are almost indistinguishable, showing that the approximation suggested above is valid. The main difference is that the CEN model has a much greater spread in time than the new model. This difference persists whatever input parameters are chosen.

Figure 1 Model Comparison

At this point it is necessary to consider again the genesis of the models – the CEN model resulted from an application of a time varying gust model as a spatially varying gust model, whilst the new model was developed based on measured temporal and spatial gust values. As such, I would expect the latter to be more accurate. The broad spread of the CEN gust may result from an application of the time varying along wind statistics to a cross wind spatial gust. Since it is known that that longitudinal integral scale is several times larger than the lateral integral scale, this would result in a wider spread of the gust than would be realistic. This is to some extent confirmed by the period of the two gusts – around 2s for the CEN gust and around 0.8s for the new model. For a train speed of 75m/s, this corresponds to gust widths of 150m and 60m – roughly approximating to the expected the longitudinal and lateral turbulence integral scales.

Concluding remarks

In this post I have looked again at the CEN extreme gust method and raised concerns about its fundamental assumptions. I have also developed an equivalent, but perhaps more rigorous, methodology based on experimental data for wind conditions at ground level. This strongly suggests that the CEN gusts are spatially larger than they should be, which suggests its long term use should be reviewed. However, when used to compare the crosswind behaviour of individual trains, rather than in an absolute sense, it is probably quite adequate.  


Baker C J, 2001, Unsteady wind loading on a wall, Wind and Structures 4, 5, 413-440. http://dx.doi.org/10.12989/was.2001.4.5.413

Bierbooms, W., Cheng, P.-W., 2002. Stochastic gust model for design calculations of wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 90 (11), 1237e1251. https://doi.org/10.1016/S0167-6105(02)00255-6.

CEN, 2018. Railway Applications d Aerodynamics d Part 6: Requirements and Test Procedures for Cross Wind Assessment. EN 14067-6:2018.

Cleon, L., Jourdain, A., 2001. Protection of line LN5 against cross winds. In: World Congress on Rail Research, Köln, Germany.

Sterling M, Baker C, Quinn A, Hoxey R, Richards P, 2006, An investigation of the wind statistics and extreme gust events at a rural site, Wind and Structures 9, 3, 193-216, http://dx.doi.org/10.12989/was.2006.9.3.193

Train crosswind performance – is there a “best” shape?

ICE 3 Velaro

This post arises out of a discussion with a number of colleagues on the issue of train overturning, in particular Mr Terry Johnson and Dr Dave Soper. Their (perhaps inadvertent) contribution to the development of the ideas set out below is gratefully acknowledged, although the responsibility for any inadequacies and errors must remain mine.

1. Introduction

In recent decades a great deal of research has been carried out to investigate the safety of trains in high cross winds, primarily to determine the wind speeds at which overturning will occur, and the risk of a wind induced accident (Baker et al, 2019). This usually takes the form of the determination of the aerodynamic forces and moment coefficient for a particular train, the use of these coefficients to determine the cross wind characteristic (CWC) – effectively a plot of accident wind speed against vehicle speed – and then some sort of risk analysis on the route over which the train will run. The first two steps are usually the concern of train manufacturers and are undertaken when the design of the train, at least in terms of overall shape and size, is fairly well advanced. The third step is usually the concern of the infrastructure operator. One question that is not often asked however is whether there is a “best” design for a train to minimise the risk of a wind induced accident, and thus to maximise safety. This has been addressed to some extent by a number of recent investigations that used a combination of CFD methods to calculate the forces and moments on a train, and optimisation methods to consider the effect of changes to train geometry. It is not however clear as to what should be the objective function of such optimisation – for example a number of different force or moment coefficients for a range of different yaw angles could be chosen. This post addresses this issue though an analysis of accident risk and investigates the aerodynamic parameters required to minimise this risk 

2. Aerodynamic force and moment coefficients

In a recent book (Baker et al, 2019) the author suggests a way of parameterising train aerodynamic force and moment data that seems to have a wide validity. This is set out in Box 1 below, in which the formulation for lee rail rolling moment coefficient is given, and is illustrated for a specific case. It applies equally well to side and lift force coefficient data. It can be seen that the form of the rolling moment / yaw angle curve is specified by four parameters – the coefficients at yaw angles of 30 and 90 degrees and exponent shape factors that describe the shape of curve, n1 and n2. Figure 1 shows a comparison of this methodology with side force coefficient data from the CEN codes (CEN, 2018) and the AeroTRAIN project (Paradot et al, 2015) as given in Baker et al (2019). All this data was obtained in a consistent way, with an STBR ground simulation in low turbulence wind tunnels. The agreement can be seen to be in general good and gives some confidence in the use of the parameterisation in what follows. The biggest discrepancy is in the transition region between the high and low yaw angle regimes, but it will be seen that this is not particularly critical to the argument that follows.

Box 1. Force and moment coefficient parameterisation
Figure 1 Parametrisation curve fit (from Baker et al, 2019)

3. Crosswind characteristics

The method used to specify the crosswind characteristic is also taken from Baker et al (2019) and is set out in Box 2. Using this methodology, the CWC are functions of n1 and n2, the ratio of the moment coefficients at yaw angles of 90 and 30 degrees, and what is defined as a characteristic wind speed, which is itself a function of train and track parameters. Box 2 gives the formulation for flat straight track, with a wind angle normal to the track – a fuller form can be found in Baker et al (2019). A comparison of this method with the results from CEN (2018) and Paradot et al (2015) is given in figure 2, again from Baker et al (2019). Agreement can be seen to be good, and this gives further confidence in the use of the methodology in what follows. 

Box 2 Calculation of cross wind characteristics
Figure 2 Cross wind characteristic curve fit (from Baker eta al, 2019)

Box 2 also indicates how the accident risk can be calculated for a specific reference site using the Weibull distribution to specify wind speed probabilities. We assume a section of railway of a specified length, with specified values of the Weibull parameters and a typical service pattern, and we then express the CWCs as a plot of train speed against the probability that a wind induced accident will occur in the section, rather than accident wind speed. This enables us to better address the question as to what is a “good” vehicle in cross wind terms, as it will highlight the relative importance in risk terms of different vehicle speed ranges. 

4. Analysis

Figure 3 shows the calculated CWC, expressed as both an accident wind speed plot and as a risk plot, for what we will take as our base case. The parameters for this case are shown in the figure. The plot of accident wind speed against vehicle speed shows a reduction in the former as the latter increases, as would be expected. There is a break in gradient, at the point of transition between the low yaw angle (at high speed) and high yaw angle (at low speed) formulations of Box 2. Figure 2 shows that this is quite typical of the calculated CWCs from Paradot et al (2015). The plot of site risk against vehicle speed shows an increase in risk with the vehicle speed. At the vehicle speed of 350 km/hr the logarithmic risk is around -8 (but remember that this absolute value is completely arbitrary). The risk falls by an order of magnitude as the speed decreases through the low yaw angle range to around 100 km/h, with an increased rate of fall for low speeds, where the high yaw angle formulation becomes relevant.

Figure 3 CWC for base case

Figures 4 to 7 show the effect on the CWCs of changing the parameters for the moment characteristics. As the low yaw angle exponent n1 is varied between 1.3 and 1.7, there are variations of about half a magnitude in risk for the higher train velocities, although this varies through the speed range. This parameter is typically around 0.9 to 1.1 for lorries, 1.2 to 1.4 for blunt nosed trains, 1.4 to 1.6 for streamlined trains, and 1.7 to 2.0 for trailing vehicles. As the high yaw angle exponent n2 is varied, the variations in accident velocity and risk are confined to the low speed range as would be expected, although here the variations in risk can be several orders of magnitude. As the lee rail rolling moment coefficient at 30 degrees is varied between 3 and 5, there can be seen to be very significant variations in both accident wind speeds and risk throughout the speed range. For variations in the lee rail rolling moment coefficient at 90 degrees only the low speed accident wind speeds and risk levels are affected as would be expected. From these graphs it can be concluded that the risk of an overturning accident will be reduced for high vehicle speeds if n1 increases and the lee rail rolling moment coefficient at 30 degrees decreases; and for low vehicle speeds if n2 increases (becomes less negative) and the lee rail rolling moment at 90 degrees decreases. Of the parameters the 30 degree coefficient produces most change in accident wind speeds and risk levels across the speed range, and is perhaps where most design effort should be concentrated.

Now let us consider specific trains. Table 1 shows, for the CEN (2018) and AeroTRAIN (Paradot et al, 2015) trains, the maximum train speed, the values of the four parameters that define the rolling moment characteristic, the characteristic velocity, and the risk at the maximum operating speed. Those shaded red indicate values that would increase risk significantly above the average, and those shown in green indicate values that would decrease risk significantly the average. It can be seen that of these trains the ICE3, IR and Silbering has the “best” values of rolling moment coefficient. For the ICE3 this is presumably due to the nose shape, resulting in low levels of lift and side force, and thus rolling moment. For the IR and Silberling however, these low values are due to the lack of underbody blockage, at least as modelled in the wind tunnel tests. The ICE3 values of n1 and n2 are around the average, whilst those for the IR and Silberling are low, and would thus tend to increase risk. The worst train in terms of rolling moment coefficients is the double deck M6BX. The IC4, RevCo and ZTER also have high values of the coefficient at 90 degrees.

Table 1 Performance of a range of trains

The risk at the maximum speed for the all trains, with one exception, is between -7.3 and -8.4 i.e. it varies by one order of magnitude. The ICE3, TGV, ZTER and IR have the lowest risk and the M6BX the highest for the standard site. This risk variation is perhaps less than would be expected, and is partly caused by the reduction in risk with the reduction in maximum operating speed. The outlier from the range of -7.3 to -8.4 is the ADR, which has a low value of -9.1, which is due to its high mass and high resulting characteristic velocity. 

Concluding remarks

From the above, it can be seen that for high speed trains, the aerodynamic parameter that most affects the overturning risk is the lee rail rolling moment coefficient in the low yaw angle range, characterised by the value at 30 degrees. In these terms the ICE3 shape is “best”. However this does not necessarily apply for lower speeds, when the higher yaw angle range becomes of importance. These points being made there are some important caveats.

  • The overturning wind speed and thus accident risk depends upon a range of parameters as well as the aerodynamic characteristics. Train mass is particularly important.
  • Similarly the infrastructure characteristics are important, and accident wind speed and risk will be affected by can’t and topography.
  • Perhaps most importantly, the level of risk is determined by the nature of the train operation itself – if speed limits are imposed in high winds, it is quite possible that the most important aerodynamic characteristics will move from those in the low yaw angle range to those in the high yaw angle range.

One further point is of interest. In Baker at al (2019) the head pressure pulse magnitudes and wake slipstream gust velocities are tabulated for orange of trains. Of those trains included, the Velaro (i.e. the ICE3) has both the lowest pressure pulse magnitude and the lowest slipstream gust velocities, suggesting that the nose / tail shape of this train has considerable aerodynamic advantages.


Baker, C., Johnson, T., Flynn, D., Hemida, H., Quinn, A., Soper, D., Sterling, M. (2019) Train Aerodynamics – Fundamental and Applications, Elsevier.

CEN, 2018. Railway applications — Aerodynamics — Part 6: Requirements and test procedures for cross wind assessment. EN 14067-6:2018. 

Paradot, N., Gregoire, R., Stiepel, M., Blanco, A., Sima, M. et al., 2015. Crosswind sensitivity assessment of a representative Europe-wide range of conventional vehicles. Proceedings of the Institution of Mechanical Engineers. Part F Journal of Rail and Rapid Transit 229 (6), 594-624.