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.

Some thoughts on ventilation and pathogen concentration build up

Modeling airflow scenarios in classrooms
Covid spread from CFD studies


Up till recently most attention had been focused on the spread of Covid-19 by near field transmission – being in close proximity to an infected person for a certain amount of time, and rather ad hoc social distancing rules have been imposed to attempt to reduce transmission. However, there is another aspect of transmission – the gradual build up of pathogen concentrations in the far field in enclosed spaces due to inadequate ventilation. The importance of this mode of transmission is beginning to be recognised – see for example a recent seminar hosted by the University of Birmingham. The main tool that seems to have been used for both near and far field dispersion is Computational Fluid Dynamics (CFD) – see the graphic above from the University of Minnesota for example. Now whilst such methods are powerful and can produce detailed information, they are very much situation specific and not always easy to generalise. This post therefore develops a simple (one could even say simplistic) method for looking at the far field build up of pathogens in an enclosed space, in a very general way, to try to obtain a basic understanding of the issues involved and arrive at very general conclusions.

The model

We begin with equation (1) below. This is a simple differential equation that relates the rate of change of concentration of pathogen in an enclosed volume to the pathogen emitted from one or more individuals via respiration and the pathogen removed by a ventilation system. This assumes that the pathogen is well mixed in the volume and is a simple statement of conservation of volume.

From the point of view of an individual, the important parameter is the pathogen dose. This is given by equation (2) and is the volume of pathogen ingested over time through respiration. The respiration rate here is assumed to be the same as that of the infected individual.

Equations (1) and (2) can be expressed in the normalised form of equations (3) and (4) and simply solved to give equations (5) and (6).

Equations (5) and (6) are plotted in figures 1 and 2. Note that an increment of 1.0 in the normalised time in this figure corresponds to one complete air change in the enclosed volume. It can be seen that after around three complete air changes the concentration of pathogen reaches an equilibrium value and the dose increases linearly, whatever the starting concentration. To the level of approximation that we are considering here we can write the relationship between normalised dose and time in the form of equation (7), which results in the non-normalised form of equation (8).

Assuming that there is a critical dose, the critical time after which this occurs is then given by equation (9).

Equation (9), although almost trivial, is of some interest. It indicates that the time required for an individual to receive acritical dose of pathogen is proportional to the volume of the enclosure and the ventilation rate. This is very reasonable – the bigger the enclosure and the higher the ventilation, the longer the time required. The critical time is inversely proportional to the concentration of the emission, which is again reasonable, but inversely proportional to the square of the respiration rate. This is quite significant and a twofold increase in respiration rate (say when taking exercise or dancing) results in the time for a critical dose being reduced by a factor of 4, or alternatively the need for ventilation rate to increase by a factor of 4 to keep the critical time constant. Similarly if there are two rather than one infected individuals in the space, then the respiration rate will double, with a reduction in the critical time by a factor of four.


Now consider the implications of this equation for two specific circumstances that are of concern to me – travelling on public transport (and particularly trains) and attending church services. With regard to the former, perhaps the first thing to observe is that there is little evidence of Covid-19 transmission on trains, and calculated risks are low. In terms of the far field exposure considered here, respiration rates are likely to be low as passengers will in general be relaxed and sitting. This will increase the time to for a critical dose. On modern trains there will be an adequate ventilation system, and the time to reach a critical dose will be proportional to its performance. Nonetheless the likelihood of reaching the critical level increases with journey time – thus there is a prima facie need for better ventilation systems on trains that undergo longer journeys than those that are used for short journeys only. For trains without ventilation systems (such as for example the elderly Class 323 stock I use regularly on the Cross City line) has window ventilation only, and in the winter these are often shut. Thus ventilation rates will be low and the time to achieve a critical dose will be small.

See the source image
Class 323 at Birmingham New Street

Now consider the case of churches. Many church buildings are large and thus from equation (9) the critical times will be high. However most church buildings do not possess a ventilation system of any kind, and ventilation is via general leakage. Whilst for many churches this leakage this can be considerable (….the church was draughty to day vicar….), some are reasonable well sealed – this will thus, from equation (9) tend to reduce the critical time. In this case too the respiration rate is important. As noted above the critical time is proportional to the respiration rate squared. As the rate increases significantly when singing, this gives a justification for the singing bans that have been imposed. - Wikimedia Commons
Church interior – Wikipedia Commons

The above analysis is a broad brush approach indeed, and in some ways merely states the obvious. However it does give something of a handle on how pathogen dose is dependent on a number of factors, that may help in the making of relevant decisions. To become really useful a critical dose and initial pathogen concentration need to be specified together with site specific values of enclosed volume, ventilation rate and expected respiration rates. This would give at least approximate values of the time taken to reach a critical dose in any specific circumstance.

Pollution, Covid and Trains

Voyager at Birmingham New Street

There has been a significant amount of research recently to investigate the air quality in railway stations. Perhaps the major study, with which I was very much involved, involved extensive measurements of the air quality at Birmingham New Street by colleagues at the University of Birmingham (Figure 1). Measurements were made of the oxides of nitrogen (NOX) and particulate matter (PM) and concentrations were measured that were considerably in excess of Environmental Health limits. Typical daily average results are shown in Figure 2. This work informed the efforts by Network Rail to improve the air quality at the station through an improved ventilation system. Further work was carried out by Kings College London and Edinburgh University, under an RSSB contract, to measure NOX and PM at Kings Cross in London and Edinburgh Waverley. Typical results are shown in figure 3 and although these results are not as extreme as the Birmingham measurements, do show some exceedances of environmental health limits. Between them, these three investigations have given a great deal of information on station air quality and informed methods for alleviating the worst of the effects.

Figure 1. Air quality measurements at Birmingham New Street
Figure 2. Daily pollutant levels at Birmingham New Street (red lines show EU limits)
Figure 3 Comparison of pollutant levels at New Street, Kings Cross and Edinburgh Waverley

However, that is not the whole story. There are growing indications that air quality ON trains is also very poor. A study on diesel commuter stock in Canada has shown high levels of ultrafine particles and black carbon within the passenger cabins (Figure 4). In 2016 the BBC reported the measurements made by their reporter Tim Johns  as he commuted into London, which again showed high particulate levels on diesel commuter trains, although not as high as in Black Cabs (Figure 5). Similarly, the BBC in 2019 reported a study by the Committee on the Medical Effects of Air Pollutants which showed very high levels of particulates on the London underground (Figure 6) which resulted in a strong response from the rail unions. These high levels are presumably due to two sources – diesel particulate emissions from trains being ingested into air conditioning systems, and also from ambient particulates in the dirty tunnels of the underground. The levels of particulates measured have significant implications for human health, particularly for those with respiratory conditions.  

Figure 4. Air Quality measurements on Canadian trains
Figure 5. Particulate measurements by BBC Reporter
Figure 6. BBC report on Underground particulate levels

Similarly, some work has been recently reported from Greece that shows elevated levels of both gaseous pollutants and particulate pollutants on diesel trains, both in excess of EU limits (Figure 7). Again this is presumably due to ingestion of diesel emissions by ventilation systems. Hopefully in the near future we will see the results of more quantitative investigations for the UK of on train NOX and particulate concentrations, and of work to investigate the ingestion of external pollutants, both from diesel emissions and dirty environments, by ventilation systems. However current indications are, that, care should be taken in using ventilations systems that draw external air into the train without the use of extensive filtering of the input.

Figure 7. NOX measurements on Greek train (red line is EU limit)

And then along comes Covid-19. The importance of high levels of ventilation on reducing pathogen concentrations and thus the risk of infection is becoming clear – se for example the recent seminar organized by the University of Birmingham. Ideally, very high (airline) levels of air exchange with the outside are required in internal environments, including trains and buses. An interesting illustration of this is provided by the publicity material in figure 8 produced by SNCF in France. I have seen nothing similar for the UK. There is an obvious dichotomy here between the need to reduce external air intake to minimize NOX and PPM ingestion and to keep internal levels of NOX and particulates at an acceptable level, and the need to increase ventilation rates to decrease pathogen levels. Both could be achieved by aggressive filtration of the air drawn through the train. However, this is likely to require major modification to existing trains in Britain, that won’t be cheap. I suspect train ventilation is going to become a major issue in the near future.

Figure 8. SNCF publicity material

Pedestrian, cyclist and road and rail vehicle safety in high winds

On March 23rd 2020 I was due to give a presentation with the above title to a Transportation Futures workshop at the University of Birmingham. Unfortunately the workshop has been cancelled because of the ongoing corona virus situation. Thus I am posting the slides I would have used here. In order that the file isn’t impossibly large for downloading, the slides are in handout form with the video clips removed.  A brief commentary follows.

  • Slide 1 – Introduction 
  • Slides 2 to 4 – these describe the Bridgewater Place incident in Leeds in 2013 in which a lorry blew over and killed a pedestrian that was the catalysts for much of the recent work that has been carried out. A report on the incident can be found here.
  • Slide 5 gives typical comfort and safety criteria – the red outline indicates the safety criterion of relevance here.
  • Slides 6 to 10 illustrate recent work on an EPSRC funded project entitled “The safety of pedestrians, cyclists and motor vehicles in highly turbulent urban wind flows” to investigate wind effects on people. This project involved wind tunnel testing, CFD analysis and the measurements on volunteers in windy conditions, which are reported here. Slide 7 shows a photo of Dr. Mike Jesson of the University of Birmingham who had responsibility for the work with volunteers. Measurements were made with shoe-mounted sensors to measure the volunteer’s walking pattern, and back-mounted sensors to measure acceleration. The results are shown in figures 8 and 9 and summarized in figure 10. The latter shows that at all gusts speeds above 6m/s stride “swing width” variation could be measured in some volunteers, where the volunteers subconsciously adjusted their stride to take account of crosswinds. The frequency of such events rose from around 40% at gust speeds of 6m/s to 100% at gust speeds of around 15m/s. Lateral accelerations of the torso first appeared at about 10m/s and reached a frequency of 100% at 17m/s. Actual instability of volunteers was only rarely recorded, but seemed to begin at gusts of around 15m/s. In general however, there was not enough data to draw firm conclusions. Perhaps typically for such measurements, the period of the project proved to be quite calm in wind terms overall. 
  • Slide 11 is a re-iteration of the safety criteria – all work of the type described above needs ultimately to be expressed in very, very simple terms to be useful.
  • Slides 12 to 14 show the limited work that has been carried out on the effect of cross winds on cyclist safety – wind tunnel and CFD work supervised by Prof Mark Sterling and Dr Hassan Hemida whose pictures are shown in figure 3, to measure the aerodynamic forces on cyclists in cross winds, and some full scale work carried out under the EPSRC project, together with associated calculations of cyclist behavior. This work suffered even more than the pedestrian measurements from lack of suitable wind conditions and the results must be regarded as inconclusive.
  • Slides 15 and 16 begins the discussion of road vehicles in cross winds, with the latter showing the wind speed restrictions on Skye Bridge.
  • Slides 17 to 19 illustrate the various methodologies for determining crosswind forces on road vehicles – full scale, wind tunnel and CFD. The former were carried out by Dr. Andrew Quinn, whose photograph is shown on Slide 17. These results lead to the curves of accident wind speed against wind angle shown on slide 19, which can be used to develop wind speed restrictions.
  • Slides 21 to 24 summarise the study of bridge wind speed restrictions described in another post here.  In finalizing restriction strategies operational conditions for specific bridges become very important, and in particular the ease or otherwise of restricting specific types of vehicle and not others.
  • Slides 25 to 29 briefly describe the wind effect on trains. Methods of determining the aerodynamic forces are illustrated in figure 27, where the University of Birmingham moving model TRAIN rigis shown. These results were obtained by Dr Dave Soper, whose photo is shown on the slide. These forces can be used to calculate the curve of accident wind speed against vehicle speed in slide 28. The practicalities of imposing speed restrictions are illustrated in slide 29.

The overall message of the presentation was that, although investigations to determine the underlying physical processes involved are very important, the translation of the results into practice needs to take account of the sometimes severe operation constraints. 

Vehicle restrictions during windy conditions on long span bridges

The Bridges

Around the UK, there area number of relatively long and high bridges across river estuaries, that all operate some sort of traffic restriction protocol in high wind conditions, to limit the risk of vehicle accidents. In this post, I will attempt to collate publically available information on these traffic restriction protocols to assess their similarities and differences.  It will be seen (surprisingly in my view) that this information is not at all easy to find and sometimes does not seem to be in the public domain. .

The bridges that will be considered are shown in Table 1, which gives name, location, construction type and length.  Pictures of them are given in figure 1. It can be seen that, with the exceptions of the Cleddau Bridge in South Wales and the Skye Bridge in Scotland, these are all over a kilometer long. The construction types vary, from concrete boxes on large numbers of concrete piers to long span suspension and cable stay structures. Only two bridges in the table have protection for vehicles against cross winds – the Prince of Wales (Second Severn) Bridge and the Queensferry Bridge in Edinburgh. All the bridges in the table have Wikipedia entries, which give further details of planning, construction and operation.

Table 1 The Bridges

Vehicle restrictions

The data for wind speed restrictions was found from a variety of sources – official documents, FOI releases, newspapers etc. The information that has been obtained is shown in Table 2. Most have a similar form, with different levels of restriction being used as the gust wind speed increases – vehicle speed limits, lane closures, restrictions to various classes of vehicles, and total closure. Most seem to base the wind speed values on local anemometers, although it is usually not clear where these are sited, and neither is the period of the gust given. Thus the values that are given are not strictly comparable with each other in absolute terms. 

Table 2 Wind speed restrictions 
(H- Headwind, C – Crosswind, * values are given in mph in table, but equivalent values in knots are used in practice)

From table 2 it can be seen that no data could be obtained for the Kessock Bridge, the Humber Bridge or the Prince of Wales (Second Severn) Bridge. With regard to the latter, vehicles crossing the bridge are shielded by wind fences and the bridge has not had to impose restrictions on traffic during its lifetime. Kessock probably has the same sort of traffic restriction strategy as the other Scottish bridges, as Transport Scotland operates a common approach. From press reports it seems that Humber has some sort of vehicle speed limit and high-sided vehicle restriction strategy, although it has not been possible to determine the wind speeds at which the different measures are put into place. . Also note that Queensferry has much higher values of wind speed for restrictions than the other bridges, again due to the fact that vehicles are protected by wind fences.

For the other bridges, there seems to be a general consistency in the information shown, with vehicle speed limits of either 30mph or 40mph imposed when the wind gusts over 35 to 50mph. Vehicle restrictions begin at gusts of around 45mph to 60mph, with double deck buses and high sided vehicles being restricted at the lower gust speeds. Further restrictions may be imposed on vehicles of different types, before overall bridge closure at wind speeds of 65 to 80mph. Some bridges use different gust speeds for cross winds and for headwinds. Orwell Bridge for example applies the crosswind criterion if the wind gust direction is from a sixty degree segment centred on the direction normal to the bridge. The Queen Elizabeth II Bridge at Dartford uses similar strategies to inform speed limits, lane closures, vehicle restrictions and bridge closure.

The restriction strategies depend very much on the nature of the traffic over the bridge and its location. For example, if only some vehicles are to be restricted, then some method of filtering them out and diverting them is required, which needs to take place at some distance from the bridge. Such procedures are in operation at Severn, Erskine, Humber and the Queen Elizabeth Bridges amongst others. Clearly ease of identification of vulnerable vehicles is required – see figure 2 for the Humber Bridge. Other bridges simply base their protocols on vehicle height eg 1.9m for Cleddau and 2.1m for Severn.

Description: Macintosh HD:Users:chrisbaker:Dropbox:Web site:Blog material:Crosswinds:Bridge closure:Humber.png
Figure 2 Humber Bridge High sided Definition

Orwell Bridge operates a very simple strategy, with different gust speed triggers for crosswinds and headwinds, leading to complete closure, without any restrictions for, say, high sided vehicles at lower wind speeds. This arises because of the urban nature of its surroundings, which makes vehicle filtering difficult. This has led to a considerable number of closures in recent years, and much public concern. Recently both numerical and wind tunnel studies have been carried out to investigate ways in which this strategy can be modified, perhaps through the use of speed limits, lane restrictions or barriers. The details of these studies have not been released to date but may prove of some interest. Studies to relax the restrictions on Skye Bridge have also been recently carried out following frequent closures and public complaints.

As can be seen, the various restriction strategies are in general quite simple and easy to operate. This inevitably means that they are conservative and largely based on the most vulnerable vehicle – usually unladen high sided vehicles. There are in fact methods available for discriminating between vehicle types and vehicle weights – see the recent paper by Baker and Soper (2019) for example. This gives a method for determining a curve of accident wind speed against vehicle speed for specific vehicle type and weight, based on which restrictions strategies for any particular vehicle can be determined. However operational constraints make the full utilisation of such methods difficult. Until such time as vehicle type and vehicle weight can be automatically determined by (say) remote visualisation techniques and dynamic weight determination, and vulnerable vehicles can be suitably diverted, then the use of simple methods such as those currently adopted will remain the best that can be achieved.

Specifying accident windspeed risk

This post is intended to start a discussion – and ideally identify what data might be available to address this problem further. The analysis presented is preliminary in nature, and could almost certainly be refined. I would really value a discussion of this with colleagues who read it.

Accident risk

In studies of road and rail vehicles in cross winds, some estimate of the risk of an accident is often required.  If the critical accident wind speed for a particular vehicle is known, then my approach in the past has been to use the probability distribution for the hourly mean wind speed (assumed to be a Weibull distribution) and the probability distribution for the turbulence fluctuations around this average (assumed to be a normal distribution) to calculate the percentage of time that this critical value is exceeded, through a convolution of the two distributions.   Additionally, when wind-warning systems are being developed, the question often arises as to what would be an appropriate mean wind speed at which to limit vehicle movements. This can be derived by calculating the percentage of time that the critical wind speed is exceeded from the probability distributions for turbulence fluctuations, for a range of mean wind speeds, and then choosing a value that has an acceptable level of risk.  

In some recent work that I have carried out for a particular client, it has become clear to me that this approach is not really adequate – an example of practical reality not always conforming with attractive theoretical approaches! Both road and rail vehicles require a gust to be above the critical value for a specific period of timebefore an accident occurs. This period of time is usually between 0.5s and 3s, the time it takes for a vehicle to actually blow over.  Thus in determining the risk of an accident what is really required is some idea of the number of times the critical wind speed is exceeded, N, for more than (say) T seconds for a particular mean wind speed U. This is not the same as the proportion of time for which the critical wind speed is exceeded, as some these exceedances will often last for less than seconds. If the probability of N for any particular is known, then this can be convoluted with the probability distribution for U to calculate the overall risk, or used to determine an appropriate value of U for wind warning systems.

To the best of my knowledge, the specification of the number of gusts N lasting greater than a specific time T for a particular mean wind speed has not been investigated in the past – but if any reader knows of such work, I would be glad to hear of it. In this post, I present the results of a preliminary investigation into this problem.

The data

In what follows, I will use two experimental wind datasets as follows. 

  • Data from that late 1990s obtained at the Wind Engineering field site at Silsoe Research Institute, and in particular two one-hour datasets (Silsoe 1 and Silsoe 2) with wind velocities measured at 10Hz at 3, 6 and 10m above the ground, for 10m wind speeds of 9.7 and 10.5m/s.
  • Data from Storm Ophelia in 2017, obtained from measurements at the top of the Muirhead Tower at the University of Birmingham, 72m above the ground, measured at 10Hz, for mean hourly wind speeds of 10.4, 12.5 and 13.8m/s (Birmingham 1, Birmingham 2 and Birmingham 3). With thanks to Dr Mike Jesson of the University of Birmingham for making this data available

The basic statistics for each hour of data is given in table 1.

Table 1. Wind characteristics

From this table it can be seen that the Silsoe site has a surface roughness length (determined from velocity profiles) typical of smooth rural environments (0.005m), with turbulence intensities (standard deviation / mean values) that are consistent with such an environment and which fall slightly with height. The Birmingham data was obtained at one point high above a suburban environments, and thus the surface roughness length cannot be determined from a velocity profile, but can be expected to be an order of magnitude or more higher than at the Silsoe site. The turbulence intensity is similar to that measured at Silsoe, although the measurements were made at a much greater height above the ground. For the Silsoe data the probability distributions of the data all show a positive skew, whilst the Birmingham data show both positive and negative skew values that are much closer to zero. Typical examples  of such distributions are shown in figure 1. The Silsoe near-ground distribution has a significantly longer upper tail, than the Birmingham values high above the ground, i.e. a significant skew towards the higher velocities. This may well be because of individual sweep events in the atmospheric boundary layer being more significant near to ground level. The normal distribution, which I have assumed in the past for my calculations, does not fit either dataset particularly well.

Figure 1 Wind Probability distribution

Analysis of exceedances

The approach to using this data has been to find, for each dataset, the number of exceedances for T= 0.5s, 1s and 3s gusts above a range of velocity levels above the mean. To enable comparison between the different datasets, these velocities are expressed in terms of standard deviations above the mean, denoted by X. The results are shown in figure 2 for the Silsoe data and figure 3 for the University of Birmingham data. The following comments can be made.

  • N falls as T increases, which is only to be expected. 
  • The value of X at which N falls to zero falls as T increases, as again is to be expected. This value is around 3 to 3.5 for the Silsoe data, and 2.5 to 3 for the Birmingham data, reflecting the form of the tail of the probability distributions discussed above.
  • For the Silsoe data, the results for the two datasets are very similar and there is an indication that N varies with  height above the ground. 
  • The Birmingham datasets also have similar results, and there is no discernable effect of wind speed in the data when plotted in this way.
Figure 2 Number of exceedances (Silsoe data)
Figure 3 Number of exceedances (Birmingham data)

Clearly the distributions of N have an upper limit. This can be characterized in two ways.

  • By the value of X for which the probability of the wind speed exceed  T/3600, X1
  • By the highest value of for which N>0, X2

Both these values of X are shown in table 2 for the various datasets.  It can be seen that there is some variability in the results, which is inevitable as we are dealing with the tails of the distribution where data becomes discontinuous. In general the values for Xare higher than those for X2, particularly for the near ground Silsoe data, suggesting that the use of simple probabilities rather than gust numbers may well significantly overestimate vehicle overturning risk. Both values fall as the time period increases as would be expected, and the values for the Silsoe data are significantly higher than for the Birmingham data, which again follows from the difference in probability distributions.  The equivalent values for X1 for a normal probability distribution are 3.64, 3.45 and 3.14, for T= 0.5, 1 and 3s respectively. It can thus be seen from Table 2 that the Silsoe values lie above the normal distribution values, and the Birmingham values lie significantly below them. 

Table 2. Upper limits of X

The data from figures 2 and 3 thus appears to be consistent and sensible, but the question then arises as to how this data can be parameterized to enable it to be used easily in calculations. After some trial and error analysis it was found that all the data for each site could be made to collapse around a single curve by plotting the combined variables NT and (X1-X)/X1 against each other. These variables seem sensible, as both are dimensionless, with the former giving a normalised value of number of exceedances, and the latter describing being the difference between specific gust velocities, and the value at which N must be zero. The results are shown in figures 4 and 5 for the Silsoe and Birmingham data respectively, using the measured values of X1 for each dataset. It can be seen there is much scatter, but the data collapse is reasonably good. The two sets of data do not however coincide, indicating the effects of the underlying shape of the probability distribution, and in particular the upper tails.

Figure 4. Analysis of Silsoe exceedance data
Figure 5. Analysis of Birmingham exceedance data

The region of most practical interest on these data collation is for a low number of events, since these represent conditions where the risk might be tolerable. Thus figures 6 and 7 thus show expanded versions of figures 4 and 5 for NT<50. It would quite possible to fit lines or curves to this data, although the best fit values would be different between the Birmingham and Silsoe datasets.

Figure 6. Expansion of figure 4 for low NT values
Figure 7. Expansion of figure 5 for low NT values

It would seem that if this method is to become useful in a predictive, rather more detailed information on near ground probability distributions is required for a variety of ground roughness conditions / heights above the ground etc., so that the variation in the exceedance curves of figures 4 to 7 can be more fully understood and an overall data collation be achieved. If any reader knows of systematic data for wind probability distributions, please let me know. 

Trains in crosswinds

A blog from a previous version of this website – written in 2017

At the time this blog was written, Storm Friederike has just passed over Western Europe and has resulted in a number of deaths and considerable traffic chaos. The BBC reported that

Deutsche Bahn had already suspended rail traffic in North Rhine-Westphalia (NRW), neighbouring Rhineland-Palatinate state and Lower Saxony, when it announced a Germany-wide suspension of long-distance trains. Any regional trains still running have cut their speed because of the strong winds. 

A spokesman said it was the “right decision” due to the risk of trees falling on overhead wires and on tracks.


The Dutch Railways (NS) and operator ProRail said overhead power lines had been damaged by the wind, as well as some railway tracks. An alert on the NS website said that “at most, only a few trains” would run throughout the evening.

Trains do occasionally blow over. The first recorded incident was on the Leven viaductin south Cumbria in 1904 when a wooden bodied train blew over on the embankment on the approach to a viaduct. A number of other incidents have occurred around the world in recent years, the latest being in Switzerland where a video has been posted online of a train in the process of being blown over – here and here.  Clearly as accidents of this type can have potentially very severe consequences they need to be in some was taken into account by train builders and railway operators in design and practice.  

The effect of cross winds on trains (and lorries to some extent) is a research topic that has stayed with me throughout my career.  My first involvement with the issue was when I worked for BR Research in the early 1980s in looking at the effects of high winds on the Advanced Passenger Train. The issue arose again when the Channel Tunnel was opened as the very light lorry carrying vehicles were found to be at risk of blowing over in ports. Then, with the advent of high speed trains in the 1990s, considerable effort has been devoted to developing a methodology to ensure that cross wind effects are taken into account in both design and operation  – in Europe, Japan, Korea, and most recently in China.  There is broad agreement on the methodology that is to be used. It consists of three parts.

  • An assessment of the aerodynamic loads on the train – usually in the form of graphs of aerodynamic forces and moments against wind angle.
  • The use of this aerodynamic data in some sort of mathematical model of the effects of wind on the vehicle under consideration to determine a graph of wind speed that will cause and accident against vehicle speed – usually referred to as a Cross Wind Characteristic or Characteristic Wind Curve (CWC).
  • The use of this CWC together with weather, route and operating data to determine the risk that the train will blow over on the route under consideration.

The design of trains usually considers only the first two steps and the CWC that is obtained is compared with reference CWCs in the train certification process. Train operators clearly need to know the output of the third stage, so they can design suitable risk alleviation systems – eg. slowing trains down, providing protection such as wind breaks etc.  

Each of the above steps can have varying levels of complexity. 

  • The assessment of aerodynamic loads can involve physical model tests of different types – using standard low turbulence wind tunnel tests, wind tunnel tests with a simulation of atmospheric turbulence or moving model tests. Embankments and bridges may or may not be modeled. Alternatively the loads can be determined by CFD calculations, again of varying levels of complexity, from simple RANS calculations, through to complex (and resource hungry) DES and LES calculations. 
  • The calculation CWCs needs a simulation of the wind – that can either be the specification of a simple gust velocity, the specification of a spatially and temporally varying ideal gust, or the full specification of a wind time history; and also a simulation of the vehicle system – either a simple one, two or three mass model or the specification of the suspension system with varying levels of complexity. More recently some authors have even used calculations that are coupled with the track dynamics and with the dynamics of a bridge that the train passes over.
  • Finally the determination of the risk requires detailed wind statistical information that is not always easily available, together with route topographical information – embankment heights, bridge geometries etc. 

To my mind one of the most important things about this three part process, and one that is not always appreciated, is that each component has a very different level of precision. The aerodynamic forces and moments can probably be specified to within 5%. The calculation methodology for CWCs, given specific values of the forces and moments, is highly accurate (say to within 1%), whilst the calculation of risk has massive inbuilt uncertainties because of the uncertainties in the meteorological information. Thus usually the risk of a wind induced accident can only be specified to within an order of magnitude i.e. 10-8or 10-9. Thus highly accurate determinations of CWCs are really pointless when the uncertainties in the risk calculations are considered. 

Having spent the last 40 years involved with this problem to some degree or other, I would thus like to make the following reflections.

  • The different aspects of the subject – fluid mechanics, vehicle dynamics, meteorology etc. – make for a fascinating intellectual mix, and have led to the development of a range of complex modeling and analytical techniques. For an academic these challenges are fascinating – but these intellectual challenges can sometimes result in the end points of the process (train certification and risk specification) to be forgotten. I am as guilty of this as anyone of course. As an academic I can argue that my work in this area has enabled progress in other research areas, as indeed it has, but the end goal shouldn’t be forgotten.
  • The current train certification methodology in the CEN code is essentially a comparative one with CWCs for particular trains being compared with CWCs for trains that are considered safe. As such, accurate values of accident wind velocities are not required, as long as they were derived in the same way as for the reference safe vehicle. The CEN code sensibly goes down this route, and specifies a simple type of wind tunnel test to obtain the force coefficients. However it requires a full multi-body dynamic simulation with an artificial gust simulation, with a complexity that seems inconsistent with the accuracy with which the aerodynamic forces and moments can be specified. 
  • The above multi-body simulation technique can, and has in the recent past, result in game playing that has no relevance to train operation or safety – by marginally changing the suspension parameters in an arbitrary way in the dynamic calculations so that the CWC is above the reference value and thus allowing the train to be certified. There must be doubts about any methodology that allows such things to happen.
  • Taking the above considerations a little further, there is an increasing tendency in published papers in this area to include as many complications as possible – multiple degrees of freedom of train, track and (if appropriate) bridge; coupling of train movement with the aerodynamic coefficients; very high resolution (and resource usage) CFD calculations. In my view the proper way to use such techniques is to carry out studies to determine the effects of such complex methodologies on overall aerodynamic forces and CWCs (almost always second order) and then to develop a much similar methodology that allows for them in an approximate way that is consistent with the accuracy of the overall process. Just because it is possible, using modern numerical techniques, to make complex calculations, it is not always sensible, or a proper use of resources, to do so. 
  • Finally there is the effect of operation that needs to be taken into account, which brings us back to where we began. In the recent storms, the German and Dutch railway authorities simply stopped train movement, because of worries about debris on the track or trees falling onto overhead wires – not because of worries about trains overturning, as the wind speeds were much too low for that. The same happens in the UK. When high winds are forecast Network Rail and the TOCs first impose a blanket regional 50mph speed limit, mainly so that trains have some hope of reducing speed when debris is blown onto the track. A major problem in this regard seems to be trampolines at the moment – see figure 2 below – and at higher predicted gust speeds of around 65mph, train operation is stopped completely. Also, very often, train movements are blocked by tree fall onto the overhead. Operational reality takes precedence over all the wind tunnel tests, CFD calculations and MDS modeling we can conceive of doing.

The study of the effects of high winds on trains is fascinating and alluring academically, and allows the use of a whole range of fun physical model experiments and challenging computational techniques. But a sense of perspective is required I think – to keep the various methodologies simple enough for reasonably routine use in train certification and route risk assessment; and not to forget the overriding importance of train operational considerations.

Air Quality at Birmingham New Street station

In 2019, Alice Hickman, whose PhD work I helped supervise, won the ICE Safety in Construction Medal for her work on evaluating air quality in railway stations. This work is described in two papers, Air quality in enclosed railway stations and Air Quality Evaluation at Birmingham New Street Railway Station. Alice, myself and another colleague made a presentation of this work at an ICE evening meeting. The slides of the meeting can be found here – but note that this is rather large file and may take some time to download.