A review of train aerodynamics Part 2 – Applications

Abstract This paper is the second part of a two-part paper that presents a wide-ranging review of train aerodynamics. Part 1 presented a detailed description of the flow field around the train and identified a number of flow regions. The effect of cross winds and flow confinement was also discussed. Based on this basic understanding, this paper then addresses a number of issues that are of concern in the design and operation of modern trains. These include aerodynamic resistance and energy consumption, aerodynamic loads on trackside structures, the safety of passengers and trackside workers in train slipstreams, the flight of ballast beneath trains, the overturning of trains in high winds and the issues associated with trains passing through tunnels. Brief conclusions are drawn regarding the need to establish a consistent risk based framework for aerodynamic effects.


Introduction
This paper is the second of a two part review of Train Aerodynamics. Part 1 began by giving a brief historical introduction to the subject and setting the bounds of the review -essentially constraining it to consider only wheel on rail vehicles, and excluding a small number of specialist areas such as train aero-acoustics. It then went on to consider the range of methodologies used in the study of the aerodynamics of trains -specifically full scale and model scale testing and CFD techniques -before considering in detail the flow field around trains. A number of different regions of the flow were identified, and the nature of the flow in each of these regions described. The effects of cross winds and constraints such as embankments and tunnels on these flow fields were also discussed at some length. This paper builds on this consideration of the flow fields around trains to consider a number of current applications of train aerodynamics. In sections 2 to 8 we consider a range of aerodynamic issues that are of current concern -aerodynamic drag and energy consumption (section 2); pressure loads on passing trains and trackside structure (section 3); the effects of slipstreams on waiting passengers and trackside workers (section 4); the flight of ballast and ice particles beneath trains (section 5); the effect of cross winds on trains (section 6); passenger comfort due to pressure transients in tunnels and the effect tunnel micro-pressure waves or sonic booms (section 7). A brief review is given of other actual and potential issues in section 8 and some concluding comments made in section 9. As in Part 1 of this paper what follows draws on the work of previous reviews (1) (2) (3), the CEN codes of practice (4) (5) (6) (7) (8) and the Infrastructure and Rolling Stock TSIs (9) (10).

Aerodynamic drag and energy consumption
Traditionally the overall resistance of a train to motion has been required by train designers in order to be able to specify the necessary power of the traction system, and, for electric trains, the power that is required from the electricity supply system. Whist such calculations are still required, in recent years the focus has changed somewhat, and there is an increasing use of train resistance equations in train simulators to attempt to minimise energy consumption through optimising speed profiles and similarly for timetable optimisation. Whatever the requirement, there is a need to be able to specify the overall train resistance. Conventionally this has been given by the Davis equation (11) which is given by (1) Here v is the train speed relative to the ground and V is the train speed relative to the air, and a, b 1 , b 2 and c are train specific constants. The first two terms are taken to be mechanical resistance terms, the third term is the air momentum drag due to the ingestion of air for cooling and air conditioning purposes, and the fourth term is taken to be the aerodynamic component -i.e. the aerodynamic drag increases with the square of wind speed. This implies that aerodynamic effects will become of increasing importance as vehicle speed increases, and indeed the aerodynamic resistance of trains dominates over mechanical resistance for conventional passenger trains at vehicle speeds greater than around 200 km/h, and at 250 km/h aerodynamic drag is around 75 to 80% of the total resistance (1). The constant c in the Davies equation can be related to the aerodynamic drag coefficient as follows. (2) where  is the density of air, A is a reference area (conventionally taken as 10m 2 , the approximate frontal area of the train) and C D is the drag coefficient -the non-dimensional drag in the direction opposite to the train direction of travel. We consider first the case of no cross wind i.e. V=v. The drag coefficient itself can, in principle, be divided into a number of components. The first is a pressure drag caused by the pressure differences on the nose and tail of the train and which can be expected to be a function of nose and tail shape -the length / height ratio l/h, degree of streamlining etc. The second component is caused by the pressure difference across the bogies and underfloor equipment and is, to a first approximation, a function of train length. The third component is a friction drag caused by the friction on the train side, roof and underbody. One might expect that this later component would be a function of the Reynolds number (Re) based on train length, and the overall "roughness" of the train -some sort of integral measure of the irregularities along the length of the train -say k/h. One might expect this last component of drag would be a function of Re -n , where n is of the order of 0.1 to 0.2 for a smooth train and close to zero for a rough train, based on conventional boundary layer theory. Such a formulation reveals one of the major difficulties in carrying out model scale tests to determine the aerodynamic drag coefficient -the effect of Reynolds number on the friction drag term, which for passenger trains can be expected to be the dominant effect (12). A further issue with conventional wind tunnel tests, made without a moving ground, is that the scale of the boundary layer along the ground plane becomes of the same order as train height towards the end of the train, resulting in an unrealistic flow field. A better simulation can be obtained using a moving ground plane, although this still suffers from the Reynolds number issue outlined above, and significant practical difficulties in supporting a long thin model at small heights above a moving ground (13). Recently work has begun on investigating the use of the moving model technique to obtain aerodynamic drag coefficient, by measuring the deceleration of models along the test track (14), and whilst these are showing some promise, issues remain concerning Reynolds numbers effects on the friction drag. These points being made however, standard wind tunnel tests can be useful in determining changes in drag due to variations in nose shape, addition of fairings etc and have been used in this way by a number of authors (15) (16) (17). Similarly there have been recent studies that have investigated the drag of components of trains using RANS CFD methodologies, including studies to opitmise vortex generator behaviour for reducing tail drag, and for opitmising container spacing (18) (19) (20).
It is because of the difficulties of carrying out physical model tests or CFD trials that the most reliable way of measuring aerodynamic drag remains the full scale coasting test (6), in which trains are allowed to coast to a rest without power from their top speed, and the velocity and distance travelled measured. In an ideal situation, these tests would be carried out on straight, level track, although usually there will be some slope on the track that has to be allowed for. A velocity versus train position function can then be derived, which, through the simple use of Newton's Law, can be converted into a velocity -acceleration / resistance function, which can make allowance for track gradients. A quadratic velocity curve is then fitted to this function and the various components of the resistance thus determined. An example of such a curve fit is shown in figure 1 (15). This procedure still depends upon the fundamental assumption that underlies the Davies equation, that the aerodynamic drag is wholly represented in the velocity squared term. The above discussion shows that this cannot be wholly true, as the expected Reynolds number dependence would reduce the exponent of the non-linear velocity term to somewhat below 2.0. Experience with such a methodology also suggests the derivation of the resistance versus velocity equation, which requires a differentiation of the velocity data to obtain the acceleration, is an inherently noisy process which can introduce significant errors, and the fitting of a quadratic velocity curve itself is far from straightforward, with the values of the constants being sensitive to the precise methodology that has been used.
An interesting variant to the above has been developed in Japan using a combination of wind tunnel tests, measurements of the pressure and velocity variations in running through tunnels and tunnel coasting tests to obtain an alternative estimate of overall resistance (see figure 2). This methodology does not however seem to have become widely used (21).
These points having been made, there have been a number of collations of passenger train aerodynamic drag over recent decades, and some of these are tabulated in table 1 below. It can be seen that in general the drag coefficient is most strongly influenced by train length as might be expected, although for a fixed train length, some trains show considerably lower drag coefficients than others. Freight trains, which are not considered in table 1, inevitably have much higher drag coefficients and are very configuration dependent (22).
In the design of trains it is naturally desirable to be able to predict train drag before the train is built and subjected to coast down testing. A number of nationally based methods exist for doing this, and these are well reviewed by (11) and compared to one another. The UK methodology for Electrical Multiple Units, the Armstrong and Swift method (23), gives the aerodynamic resistance component in the Davis equation as Here is the drag coefficient of the nose and tail, is the bogie drag coefficient, P is the train perimeter, L is the train length. l is the inter car gap length, is the number of trailer cars, is the number of power cars, is the number of bogies and is the number of pantographs. The first term represents the nose / tail drag, the second is the skin friction drag and the other terms are the repeating drag terms along the train. A comparison of the results of this method ith the results of coast down tests for the Class 373 Eurostar train are shown in figure 3 and it can be seen that the use of this methodology results in an over-prediction of the overall resistance (11).
The methods by which train drag can be reduced are in a sense obvious and have been known for many years. These include the following.
 The streamlining of the nose and tail. However note that for many trains this component of drag is relatively small and the law of diminishing returns applies as the degree of streamlining is increased. Data from Japanese investigators that shows the changes in drag coefficient for nose / tail length / height ratios greater than 2.0 is small (3).  figure 4 for a container train (20). To a first approximation, for small yaw angles, one may where is the yaw angle (in radians), and is the drag coefficient at zero yaw. Values of  are of the order of 0.5 to 1.0. In reference (1) it is estimated that for typical UK weather conditions, the effect of cross winds can add around 10% to the aerodynamic drag term. This effect will be considered further below.
As trains pass through tunnels, it is clear from Part 1 that the flow pattern around the train changes significantly. The energy losses in the flow also change significantly, with the flow between the train and the tunnel wall having to overcome friction on both, and the separation regions around the nose and tail of the train changing significantly. The pressure waves that are created in the tunnel, as they contain energy, also create an effective drag. Very broadly the longer the tunnel, or the greater the blockage ratio, the more the tunnel drag is dominated by friction drag. In two papers in the 1990s (25), Vardy argued that the aerodynamic drag within tunnels has to be considered as the sum of the pressure drag (which includes pressure wave drag) and skin friction drag, and that these two types of drag vary in different ways for different trains and different tunnels. He thus argues that, although an overall drag can be defined, it is not the fundamental parameter. He also makes the point that the area used to define the two components needs to be carefully defined. He defines the term "aerodynamic area" -the volume of space enclosed by the train divided by its length. In the CEN code (4) (5) the approach to allowing for these effects is much simpler and defines a tunnel friction  Determine the value of the train aerodynamic drag, and other resistances, and assign to each reasonable uncertainties based on the methods that have been used to derive them and thus specify probability distributions for these parameters.
 Specify a typical service pattern for the train, and identify operational uncertainties, such as speed restrictions, and their likely frequency of occurrence.
 Specify a mean wind speed and direction, and the variation of these parameters about the mean i.e. again specify a probability distribution.
 Carry out a large number of train simulator runs, with statistical realisations of the resistance terms, operating conditions and wind conditions, to find a probability distribution of the overall tractive effort and energy consumption.
 Take the design values of these parameters as, say, the mean plus two standard deviations of the probability distribution.
Such a process would give a context to any changes in drag coefficient that may result from design modifications, and would allow a proper cost-benefit analysis of such modifications to be carried out.

Loading requirement
The pressure fields around the trains ( In the author's view the setting of the limit values without reference to the effects that pressures cause is a misguided one, but perhaps made understandable by the split between the Rolling Stock and Infrastructure TSIs (9) (10), as some methodology is required that can be easily used in train authorisation processes. An earlier methodology in the UK, actually attempted to address this, through requiring measurements of train pressure pulses to be made on stationary trains on an adjacent track. A limit for the peak to peak pressure transient of 1.44kPa was used, based on tests carried out on the HST in the 1980s. The adoption of the TSI methodology represents, in the author's view, a significant retrograde move.

Design methodologies
The measurement of loads on structures is an essentially simple process. At full scale, surface pressure taps are installed on the surfaces to be tested, and measurements made with pressure transducers of the required range and frequency response to capture the peak to peak pressure loads. The CEN procedures (6) where x is the distance along the train and Y is the distance of the structure from the track centre With regard to loads on passing trains, moving model experiments that measured the load caused by an ETR500 train with different nose lengths passing a stationary ETR500 for different track spacings have been carried out (29). Typical results are shown in figure 8 for trains with different nose shapes.
It can be seen that, as would be expected, the blunter the nose shape the higher the loads, and that the loading falls off as the distance between trains increases. Note that the fall off has a power law exponent of between 1.0 and 1.6. This is contrary to the exponent of 2.0 that is specified in the description of the work. The difference is probably due to the fact that this exponent was assumed based on earlier work and the data then plotted in a form that fixed it, rather than allowing a free curve fit as above. The use of data such as this is that it can be used to determine acceptable track spacings for new lines, if an allowable peak to peak pressure transient level can be specified. We will turn to this aspect in the next section.

Loading limits
The loading data discussed in this section is required for two practical reasons. The first is to determine the regular repeated loads on trackside structures to enable ultimate and fatigue loading calculations to be carried out. The second is the loading on passing trains to ensure that the train displacements are acceptable to passengers and do not cause any risk to safety. For the former, the methodology would be to average the loading over a suitable loading length and to transform it into a suitable load effect (eg a bending moment at the base of a barrier). This is broadly the approach taken in CEN (6), where the load is specified as peak maximum and minimum values over a 5m length either side of the zero crossing point for a variety of structures. For vertical structures, it is given in the form of a moment weighted force that can be easily (if confusedly) converted into a base moment. This can be used to either give a maximum design loading for any structure, or through a consideration of train type, frequency and speed, converted into a fatigue load for a specific number of load cycles in a specific time period. It is not really possible to specify limit values in such a case however, as the design will take account of whatever loading is specified.
The second issue underlies the UK limit mentioned above. Anecdotal evidence suggest that this was adopted in a somewhat roundabout way as being small enough not to cause problems with coffee in cups on the tables in trains. However a rather more rigorous approach could be adopted along the following lines (suggested to the author by Richard Sturt of Arups (private communication).
 For a particular train, being passed by another train, using the experimentally determined pressure transients, calculate the 20m moving average force time history F(t) (assuming that one car of a train is 20m long, this represents the overall force on the car).
 Calculate the displacement of the train on its primary suspension, y(t), from the simple equation.
where M is the mass of the vehicle and s is the primary suspension stiffness. For an even greater level of simplicity the stiffness term can be omitted.
 In line with the comments made on other practical applications, such a calculation would best be carried out a large number of times, allowing for uncertainties in pressure time histories, train characteristics and operational characteristics, to calculate a probability distribution of displacement. A limiting value of (say) the mean plus two standard deviations of the displacement could then be compared with realistic values of what might be regarded as acceptable (perhaps one or two centimetres?).

The problem
It was shown in Part 1 that the air velocities in the boundary layers and wakes of trains can be significant, and there would seem to be every possibility that these could be dangerous for trackside workers and passengers waiting at platforms. A recent study of such accidents in the UK showed that In the 32 years since 1972, 16 incidents have been reported. Most have involved empty pushchairs although one incident contained a pushchair carrying a child. Minor injuries were sustained by members of the public in two incidents. Three people were almost swept of their feet in other incidents. However, no fatalities have occurred (30). Thus whilst the problem does not seem to be a major one, it is a factor that needs to be borne in mind in terms of train and infrastructure design, and will become of increasing concern as train speeds increase.

Design methodology
In Europe the current methodology for assessing the slipstream risk is outlined in the CEN code (5) and the Rolling Stock TSI (9). The basis of the method is the determination of a specific characteristic velocity. This has to be obtained from at least 20 full scale train passes with measurements being made at a specific trackside position and at a specified position on a platform. The velocity time histories from these train passes are then averaged with a one second moving average filter, and the maximum value of this averaged velocity obtained for each train pass. The characteristic velocity is then formed as the mean plus two standard deviations of these gust values. This characteristic velocities are then compared with limit velocities, specified as 22m/s for the trackside position (and thus of relevance to trackside workers) and 15.5m/s for the platform position (and thus for waiting passengers). Two points should be noted here however. Firstly within the TSI there are clauses to allow for measurements to be made at those positions historically used in the UK, with somewhat different limit velocities. Secondly a revision to both the CEN code and the TSI, based on the results of the recently completed AeroTRAIN project, is currently in preparation that eliminates the need for platform measurements, and bases both criteria on measurements made at different heights at the trackside position (31).

Gust measurements
The AeroTRAIN project mentioned above, measured slipstream velocities for a wide range of train types and formations, and was able to specify the characteristic velocity in a much more extensive

The effect of train slipstreams on people
The obvious comment to be made on the above methodology is that it is solely concerned with the slipstream velocities created by trains and does not make any allowance for individual human response. As such it is another example of the design methodology for trains being divorced from considerations of infrastructure and operation. In the RAPIDE project (33) However the results were somewhat inconclusive, with the cylinders seeming to respond to pressure fluctuations around the nose of the vehicle, and the dummies to velocity fluctuations in the wake, although direct correlation between forces and velocities was not found. On the basis of these results, little confidence could be placed in the use of such techniques, and these methods do not seem to have been pursued further. Now it is has been shown that the characteristic velocities themselves are subject to significant uncertainties (31), largely due to the underlying physics of the issue, that involves large scale turbulence in train boundary layers and wakes, but nonetheless these values are compared with deterministic limits. The argument can be made that the formulation of the characteristic velocity, as the mean plus two standard deviations of gust values, is a quasi-statistical description, and that the limits themselves are based on the statistical distribution of human reaction, but this is far from explicit and the derivation of these limits is not clear.
The response of a range of real individuals to different wind gusts has been studied in large scale wind tunnel tests, through a series of wind tunnel tests on a range of individuals, and this data used to calibrate models of human behaviour in typical gusts around trains (34). This experiment and analysis indicates that there is a wide range of human response, with females being more at risk than males to instability in slipstreams, and also suggest that the one second gust value adopted in the TSI is rather longer than it should be. Specifically however it allows a cumulative probability distribution of human stability in gusts of different types to be determined -see figure 11. In a recent paper (35) the author has proposed the following statistical, risk based methodology for addressing this issue a and b can be expected to be functions of the assumed gender breakdown of the exposed population.
 Through a convolution of train gust speed probability distributions and the cumulative probability distribution of human stability, calculate the risk of an accident (i.e. a person becoming unstable) for one train passing a particular location. After some manipulation this can be shown to be given by where √ √  Through operational considerations, determine the risk of an individual being present on a particular section of track when a train goes by.
 Thus obtain the overall accident risk for an accident to occur on a specific route.
It can be argued that it is more rationale to apply limits to the risk levels thus identified, rather than to the slipstream limit velocities themselves -in other words to include a proper consideration of risk within the design process. Clearly however further work is required to fully specify the cumulative distributions of human stability in gusts produced by a train, rather than the sharp edged gusts studied in (34).

The issues
The problem of ballast flight beneath trains is one that has made itself felt very forcibly over recent years, with a variety of (usually unpublicised and unpublished) events occurring on high speed lines, where ballast has been lifted from the track, seemingly by aerodynamic effects, and caused

Initiation of motion
The first question that needs to be addressed is what are the mechanisms that initiate ballast flight?
An obvious way to alleviate the problem would, of course, be to simply ensure that ballast The correlation of this parameter with the movement of grains is shown in figure 14.
In the author's view the multi-variate nature of the problem should not be neglected, and there is a need for further research on the effect on ballast flight initiation of the combined effects of track vibration, pressure transients and shear. It may well be that, for some trains and in some situations, one or other of these effects will dominate, such as in the various experiments described above where surface shear has been identified as being of importance, but a combined consideration is required to enable the phenomenon to be more fully understood, and for all the various effects described above to be put into a common framework. For example, the author and his co-workers recently carried out some exploratory experiments on a short train mounted upside down beneath a representation of the track on the moving model TRAIN Rig, in order to measure in detail the flow field "beneath" (in this case , above) the train, and in particular the correlations between pressures and velocities - figure 15. This enabled the instantaneous overturning moment on ballast particles at the bed to be calculated as the sum of shear (drag) and pressure (lift) forces (although assumptions need to be made concerning the relative contribution of these to the moments). Typical ensemble average results are shown in figure 15. The peaks at the front of the vehicle are dominated by pressure forces, whilst those in the wake of the vehicle by shear / drag forces. Whilst these results must only be regarded as preliminary, and the calculations of moment in some way as quite arbitrary, they do show that it is possible to have ballast moments that are influenced both by pressure and shear forces.

Flight of ballast
Once the ballast starts to move, it can either continue to "creep" Typical results are shown in figure 17. Numerical measurements were also carried out of ballast stone movement, and showed that the initial flight paths of ballast caused primarily by shear were low and flat, but when these impacted on other ballast, there was a certain possibility of much more extensive ballast movement, with much higher and more destructive flight paths.

Risk and mitigation
Based on the work of the AOA project, SNCF in France developed an outline risk assessment methodology that involves the determination of the "stress" on the bed caused by the passage of the train given by the mean and standard deviation of the parameter shown in figure 14 (essentially a surface shear stress), and the "strength" of the bed, given by the mean and standard deviation of the number of ballast particles moved at a particular "stress" (36) The probability of the movement of the ballast can then be calculated from a convolution of the two probability distributions. The former is of course a function of the train type and the latter a function of the track characteristics.
This joint consideration of the train and the infrastructure is perfectly consistent with the comments of earlier sections. In more recent work, the authors have taken this further to add into the calculation process the consideration of meteorological conditions and whether or not ice will form, and using this to determine early warning systems to reduce train speeds so that catastrophic ballast flight incidents do not occur (41).

TSI methodology
The where the movement of small pieces of ballast is an issue, as noted above, it would seem to the author that some restriction should be based on the magnitude of the suction peak beneath the front and rear of the train, although no consideration has been given to this at this stage. Secondly this approach is once again moving to a separation of the consideration of train and infrastructure effects. The reader will no longer be surprised to know that the author does not consider this desirable.

The problems
There are a variety of different issues relating to the effect of cross wind on trains. The first, which has received by far the most attention and will be the one discussed at length in this section, is the overturning of trains in cross winds. Such incidents are not as unusual as might first appear and, although the first incident can be traced back to the blowing over of a train on the approach to the Leven Viaduct in Cumbria in the UK in 1903 (42). More recent events have taken place in Japan (43), China (44) and Switzerland (45). This is an issue that needs to be taken seriously in both train design and route operation because of the major consequences of an accident. A detailed review of the issues involved up to 2009 was presented in (46). However there are other related problemsexcessive lateral force on tracks due to cross winds (47), gauge infringement (i.e. the vehicle being moved laterally so that it exceeds its maximum allowable displacement) (48) and displacement of the train pantograph with respect to the overhead wire, which can lead to dewirement and possible catenary damage (49). These issues will be discussed briefly in section 8.
The first serious study of the stability of trains in high winds seems to have been in connection with the development of the Advanced Passenger Train in the UK in the late 1970s. However the advent of high speed trains in many parts of the world means that this is an issue that has received attention across Europe and in the Far East. The current situation in Europe is that the Rolling stock TSI requires an assessment to be made of train stability in high winds for all new trains that will travel faster than 160 km/h (9). The methodology for this is given in the CEN code (8), although at the time of writing there is much debate about this methodology and some of the underlying assumptions it contains.

Outline of methodology
An outline of the methodology used in the train authorisation and route risk assessment processes is given in figure 18. addressing, which is the specification of the risk of an accident. In that paper the author proposes a simplified methodology which can be applied consistently in the train authorisation and route risk assessment procedure with a balance of uncertainties throughout the process. The arguments of that paper will not be repeated here, where we will rather dwell on more fundamental matters and issues arising from the current CEN methodology.

Aerodynamic characteristics
When trains are subject to a cross wind they experience three aerodynamic forces (drag, side and lift The standard simulation that has been adopted by CEN is the "single track ballasted rail" or STBR, which is shown in figure 19 (8). For low speed or stationary vehicles however it is necessary to model the atmospheric turbulence and shear using the standard methodology adopted in atmospheric boundary layer wind tunnels. Again this is an approximation that whilst absolutely valid for stationary vehicles, becomes less so as the vehicle speed increases. Finally experiments or simulations can be made with a vehicle moving across the ground. Experimentally this is very complex and has not often been attempted -see (55) for example. In CFD terms it is somewhat easier to achieve such a simulation by moving the floor of the computational domain. In the first instance we will consider the side and lift force characteristics only. Figure 20 shows the variation of these forces for the ETR500, TGV Duplex and the ICE3 train, from the data given in (8) vehicles (56). In the lower yaw angle range, these curves can be parameterised by curves of the form where n = 1.6 for streamlined passenger trains, 1.2 for non-streamlined leading vehicles and 1.7 for trailing vehicles. Table 4 shows a collation of the values of lee rail rolling moment coefficient at 40 degrees for the ETR500, TGV and ICE3 trains from the CEN code (8).  (58) (68) (69), (70). The latter are particularly important for route risk calculations where different types of infrastructure need to be considered.

Overturning calculations, gusts and CWC
The next stage in the process of calculating CWCs is to use a model of the vehicle wind system. As  (71). For such a situation this is an appropriate method. However in its use in the code, this has been transformed into a spatially varying gust through which the train passes, with the temporal characteristics being replaced by spatial characteristics through the spatial wind correlations, in a one to one fashion. This is not theoretically sound and the present author can see no justification for this approach. Ideally some sort of gust varying both spatially and temporally is required. Finally the most complex gust formulation is to generate the complete stochastic wind field as seen by the train as in (52). These investigators coupled this with the specification of an aerodynamic admittance that allows for the non-correlation of turbulence over the side of the train.
Whatever the approach use, CWCs of the form shown in figure 23 for the reference Class 1 vehicle in the Rolling Stock TSI can be obtained. This, as would be expected, shows a fall off in the wind speed required for an accident to occur as the vehicle speed increases. A similar curve is shown in figure 24 which shows a similar curve for the ETR 500, but calculated over a large number of realisations of a full stochastic approach, using simulated wind time series. It can be seen that there is considerable variation of the curve about the mean value (52).
As has been mentioned above, in a recent paper the author has proposed a greatly simplified methodology (54). In this methodology the simple rolling moment correlations outlined above are used, which allows the definition of parameter he describes as the characteristic wind speed, defined as where  is a parameter that allows for suspension effects and a range of other "real" effects and  is the track semi-width. This in turn is used, with a simple vehicle analysis, to generate generic and easily parameterised cross wind characteristics than can be generally applied. This generic cross wind characteristic is given by where u i is the wind speed for an incident to occur and , , , The simplified vehicle mode has been calibrated against a more complex approach and can include first order corrections for real effects, such as suspension effects, admittance effects and track irregularities.

Risk analysis
However they are derived the cross wind characteristics can then be used, together with meteorological data to find the probability that, at a particular section of track the wind speed will exceed the accident wind speed and an overturning incident will occur. A number of such methods (those used in Germany, France, Italy and the UK) were considered during the AOA project during which a comparative exercise was carried out (72). A detailed comparison is not given here, but as well as varying in the manner in which cross wind characteristics are derived, these methods also vary in the way in which meteorological conditions are derived and assessed, with some methods using existing wind data, modified for local effect by terrain factors such as those found in wind loading codes of practice, whist others use large scale CFD modelling of the orography around the route. The nature of the wind speed probability distributions, and the convolution of these distributions with the CWCs also takes on a number of forms.
The probability of the local wind speed exceeding the accident wind speed having been obtained, there are also considerable differences in how this information is used and interpreted. Some methods take into account train operations -how often and for how long trains will be in a particular section of route, the consequences of an accident, potential number of fatalities etc., whilst others simply focus on the train itself. The resulting risks are then considered either by comparing the risk with the risk of an accident on existing operations that are perceived to be safe, or by calculating an absolute risk level, and can be used to inform the construction of wind protection along the track or the development of traffic restriction strategies during strong winds.
Whatever the precise methodology that is used, it is found that the final risk value is firstly very sensitive to small changes in the input parameters, and can only really be specified to the nearest order of magnitude, and secondly, a very large proportion for the risk comes from just a few exposed sites on any particular route.
The simplified methodology of (54) describes above allows for such a risk analysis to be carried out.
For a particular section of track the overall risk of a fatality can be shown to be Hereand k are the parameters of the Weibull distribution for gust wind speeds, which are a function of the mean wind speed values and the turbulence intensity; u i is the wind speed at which an accident will take place (equation (12)  For the unsealed train, the differences between the criteria for single and double track tunnels are due to the fact that the double track criterion is for the worst case, whilst the single track criterion will be repeated for each train pass. For sealed trains, the longer term pressure differences become to be more of an issue.
It should be noted that these are essentially deterministic criteria, with the variability of human response allowed for by choosing upper limits of acceptable pressure transients. Another possible methodology would be to carry out an analysis similar to that adopted for the calculation of the risk of slipstream accidents, based on curves such as those shown in figure 25 (from (74), based on the results of (73)) which show the cumulative probability of human response for pressure transients of different levels, with different comfort criteria. For a double track tunnel the procedure would be as follows.
 Assume that the probability distribution of the maximum pressure transients, p, in a tunnel, for the complete range of train crossing points, is given by a normal distribution of mean m p and standard deviation s p . These values can be obtained using multiple runs of standard analysis software for predicting pressure transients in tunnels.
 Assume that the cumulative probability distributions for human response in figure 25 can be given by (15) From figure 25, a and b can be taken to be given by (16) where R is the pressure comfort rating.
 The probability that two trains will actually meet in the tunnel is given by where T is the length of the tunnel and v is the train speed,  is the timetabled time between the two trains entering the tunnel, and s D is the standard deviation of the delay of any one train from the timetabled arrival time.
 The risk that a specific comfort rating will be exceeded by any one passenger on any one train pass through the tunnel is then given by where √ √ This method can be directly used in the consideration of the design and operation of a specific tunnel, and overall route risk can be determined by summing the risk for individual tunnels along the route.

Micro pressure waves
The existence of micro-pressure waves, or sonic booms, at the exit to tunnels has already been mentioned in Part 1. Essentially these form because of the steepening of the initial pressure wave as it passes along the tunnel, and when it is reflected at the tunnel exit, some pressure fluctuations, of a considerably lower magnitude than those that pass up and down the tunnels, are radiated out from the tunnel outlet. This effect is most noticeable in long tunnels, where the pressure waves have a greater distance of travel in which to steepen, and for slab track (concrete) tunnels rather than ballasted tunnels, since the frictional damping of the pressure waves in the former is much lower than in the latter. The steepening is due to the faster speed at which the back of the wave front moves in comparison to the front of the wave front. This phenomenon is very well described in detail in (75). That paper describes methods for calculating whether or not such micro-pressure waves will occur, and if they do occur, methods for their alleviation. The former is straightforward at least in principle. Firstly the magnitude and gradient of the initial pressure rise can be calculated, either from standard formulae, from experiments, or from CFD calculations. These will be functions of tunnel blockage ratio and train speed. For short tunnels, these values of magnitude and gradient can be assumed to exist at the outlet of the tunnel as well as at the inlet. However for long tunnels the steepening of the wave along the length of the tunnel must be calculated -see figure 26. This is straightforward for slab track tunnels, but much less so for ballasted tunnels where reliable methods do not exist. However this is of little practical importance, as there are usually no problems with micro-pressure waves in ballasted tunnels. At the exit, acoustic theory can then be used to calculate the magnitude and frequency of the external micro-pressure waves.
In order to reduce these magnitudes the methods that are normally adopted are to modify the tunnel entrance, through the introduction of an area of decreasing cross section. This reduces the gradient of the initial pressure wave, and thus of the emitted micro-pressure waves. A typical construction -on the Finnetunnel tunnel in Germany -is shown in figure 27 (76). The pressure gradients can also be reduced by the lengthening of train noses as in recent versions of the Shinkansen train trains. The exit pressure gradients can also be modified by modifications along the tunnel -airshafts, refuges, and perhaps ballast modifications. Surprisingly modifications to the tunnel exit, such as ventilated exit regions have not been found to be effectives. Reference (75) also describes a number of attempts to use active devices -such as "anti-noise" generation, and the release of large quantities of air to disrupt the pressure rise in any passing wave. These have been shown to be effective, but suffer from the problem associated with all such active devices of lack of reliability during power outages etc.
This leaves the issue of what are the acceptable levels of radiated pressure at the tunnel exit. This issue has been considered in (77). The authors of that paper studied the entire process of entry wave steepening and radiation from the end of the tunnel, showing that the wave steepening calculations worked well, but the uncertainty in the solid angle over which the sound radiated outside the tunnel was large. Criteria for maximum sound levels were derived from an EU directive for C weighted sound levels. They suggest that within 25m of the exit from a tunnel tunnel, these levels should not exceed 115 dB(C) whilst the levels at the nearest properties should not exceed 75 dB(C).

Other issues
The preceding sections have outlined a range of railway aerodynamic issues that are of major current relevance. But they are by no means the only aerodynamic issues that exist that may be of importance in some circumstances. In Part 1 the scope of this paper was deliberately stated to exclude the subject of aero-acoustics. Acoustic issues, arising from both mechanical and aerodynamic effects, however can be of major concern in the development of new trains and infrastructure as it directly impinges upon those who live in the vicinity of railway lines. Further details of current work in this area can be found in (78). Similarly the scope of this paper was deliberately limited to exclude MagLev vehicles. Whilst there has been some recent MagLev developments, most notably in Germany (79) and China (80) this mode of transport has perhaps not lived up to its early promise. However the current plans for the ultimate replacement of the Japanese Shinkansen fleet by MagLev systems may result in the need for further work in this area.
More details of the aerodynamics of Maglev systems can be found in (2).
Some other issues that are of importance include.  Some work has been carried out to consider the loads on trains in very sudden gusts such as downburst and tornadoes, and in particular considered the overshoot of the cross wind forces above the mean values during the establishment of the flow (87).
 As noted in section 6 above there a number of other cross wind issues that are of concern, in particular wind effects on track lateral load and gauge infringement. The interested reader should refer to (46), (47) for further details.
 In recent projects, some concern has been expressed as to the effect of new trains on birds or small animals, either through direct collision with trains, or through major disturbance caused by the train slipstream. This problem can in principle be addressed using the methodology briefly described in section 5 for predicting the flight of ballast, although to predict the animal movement in train slipstreams, some realisation of unsteady wind fields is required, either through moving model experiments, or through unsteady CFD simulations.

Concluding Remarks
It can be seen from the material presented in earlier sections that the range of train aerodynamic problems is extremely diverse, and the range of methodologies required to address these problems is equally diverse. Perhaps the major point to emerge, and the one which the writer would wish to emphasise, is the essentially ad hoc nature in which may of the problems have been addressed, design limits specified etc. Often the nature of the flow field around the train that causes these problems is not fully considered and, more importantly, the limits are related to parameters that can easily be measured, rather than those that are actually of concern to the passenger or train operator. It would seem to the author that a much better approach would be to consider all these issues within a consistent risk analysis framework, that allocates risk levels to different issues that are consistent with the risks arising from other aspects of train operation, and to use these risks as design targets. This would allow a proper appreciation of aerodynamic risk in comparison to that from other sources. Whist such an approach has been adopted for certain aerodynamic effects in certain railway administrations, it is very far from universal. This issue has been addressed to some extent in this paper, but there is scope for much more work and development in this area.
It will be evident from the reference list that the author is indebted to a large number of industrial and academic colleagues and research students past and present for much of the information that is included in this paper, and it is simply not possible to name them all. Their contribution is gratefully acknowledged. However the author would specifically like to acknowledge the part played by Roger Gawthorpe, former Head of Aerodynamics Research at British Rail, and the one who first introduced him to the fascinating subject of train aerodynamics.

Figures from other sources
The

Notation
In a publication such as this, with much material taken from published sources, it is difficult to achieve complete consistency of nomenclature. Nonetheless the major notation used is shown below. On many of the figures, information is given in the caption on the parameters shown in the figures.
A Reference area a, b 1 Constants in equations (7) and (15) a 1 , a 2 , a 3 , a 4 Constants in equation (13) Drag coefficient Drag coefficient at zero yaw angle Bogie drag coefficient (equation (3)) Nose and tail drag coefficient (equation (3)) Force coefficient Lee rail rolling moment coefficient at yaw angle Ψ c Characteristic velocity (equation (14)) Generalised force f Constant in equation (14) (3)) Number of pantographs (equation (3)) Number of power cars (equation (3)) Number of trailer cars (equation (3)) n lee rail rolling moment exponent P Train perimeter (equation (3)) p Pressure transient R Train resistance (equation (1)) or comfort condition rank (equation (16)    Train end Figure 11 Cumulative distribution of the probability of human instability in sharp edged gusts (34) (The x axis shows the velocity of a wind tunnel produced gust, and the y axis shows the percentage of displaced subjects, for different gender and displacement categories; afacing the oncoming wind, b back to the oncoming wind, cside on to the oncoming wind, d-both genders)  (76)