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.
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.
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.
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.
The debris trajectory animations of Figures 6 to 11 were provided by Professor Mark Sterling, whose ability to use advanced EXCEL functions seems to be significantly greater than mine. His contribution is much appreciated.
In 2017 Mark Sterling and I published the paper “Modelling wind field and debris flight in tornadoes”, which described the integration of a tornado wind field model and the debris flight equations to look at the pattern of compact debris movement in tornadoes of different types. Typical results for falling and flying debris are shown in figure 1 below and give an indication of the complexity of the debris trajectories that were predicted.
Now whilst the tornado wind model that was used in the analysis was a considerable improvement over those that existed at the time, in that it gave a consistent three dimensional velocity formulation, it did however have one major drawback. This was the fact that the vertical velocity component was unbound and increased with height, albeit quite slowly. In a more recent paper in 2020 “The lodging of crops by tornadoes”, we developed an improved model, in which the vertical velocity peaked at a certain height and then decreased at greater heights. In this blog post I will briefly explore the use of this wind model to predict compact debris flight paths using the same methodology as in the first paper, and in doing so will illustrate the importance of the tornado model on debris trajectory prediction.
The tornado wind model
The expressions for the radial, circumferential and vertical velocities in the 2020 model are given in figure 2. Here the velocities are normalized by the maximum circumferential velocity and the radial and vertical distances by the radius at which the maximum velocity occurs. Note that this is different from the 2017 paper where the maximum radial velocity was used for normalization. The parameter K is related to what will be termed the swirl ratio S (the ratio of the maximum circumferential to maximum radial velocity) by a function of the parameter gamma, which is a shape parameter that affects the shape of the radial and vertical profiles. (Unfortunately this web template doesn’t support Greek letters, so I have to spell them out). Figure 3 shows typical velocity profiles for different values of this parameter. It can be seen that for gamma = 2, the peak of the vertical velocity is at the vortex centre, as in a typical single cell vortex, whilst for higher values it moves away from the centre becoming more like a two cell vortex (but note there is no downflow at the vortex centre in this case.
Debris flight equations
The equations for compact debris flight are given in figure 4. These are the same as in the 2017 paper, although expressed a little differently. The debris velocities (lowercase) in the three directions are again normalized by the maximum tangential tornado velocity. Two dimensionless parameter are identified – the Tachikawa number Ta that relates the flow force on the debris particle to its weight, and a tornado Froude number Fr. Different dimensionless parameters were used in the 2017 paper, because of the different reference velocity that was used
Putting together the velocity equations in figure 2 and the particle flight equations in figure 4, it can be seen that there are four parameter that define debris trajectories – the tornado parameters S, gamma and Fr, and the debris Tachikawa number Ta. In addition any one flight trajectory will be defined, at least in its early stages by the dimensionless values of the radius and height at its release point. If these six parameters are specified then the equations of debris flight can be solved in a straightforward manner. In what follows we define a base case situation as in figure 5, and then vary each of the parameters around this base case value. We present the results in the animations of figures 6 to 11.Each animation shows four plots – the trajectories projected onto a vertical plane through the tornado centre; the trajectories projected onto a horizontal plane; the trajectories in a rotating plane in the radial and vertical directions, and a plot of the variation of particle kinetic energy with time. The latter acts as a damage indicator of debris flight, but also clearly shows whether or not the solution converges or diverges with time. Note that the dimensionless time shown in the kinetic energy plots is proportional to the time of revolution of the vortex – a time of 2 pi corresponds to one vortex revolution.
Figure 6. Effect of variations in Tachikawa number
First consider the effect of changing Tachikawa number, Ta – see Figure 6. This represents changes in the nature of the debris. A low value of Ta represents heavy debris and vice versa. It can be seen that at low values of Ta, the debris tracks can reach significant heights and the debris undergoes a diverging motion when viewed in the radius / height plane, with a diverging kinetic energy oscillation. At some point in the trajectory the debris hits the ground and the energy falls to zero. The base case situation at Ta = 100 is still mildly diverging but the trajectory does not intersect the ground plane for the length of the calculation. As Ta increases further, the debris takes up a stable path in the radius / height plane travels around a small circular trajectory, with the kinetic energy converging to a stable value. This suggest that light debris can reach an equilibrium where it is held aloft by the tornado. The position around which the circular motion takes place is around a normalized radius of 1.3 and a normalized height of 0.9. The value of height is much less than calculated in the 2017 paper, reflecting the fact that the vertical velocity does not decrease indefinitely with height for the new model as it did in the old.
The effect of variations in Froude number is shown in Figure 7. The primary effect that increase in Fr has is to increase the centrifugal force on the debris. At low values, the trajectories are stable and similar to that of the base case. As the values increase above 1.0 the oscillations become larger due to the increased centrifugal forces and eventually become unstable, with the trajectories meeting the ground at high values.
The effects of variations in the Swirl ratio shown in Figure 8 are complex, with diverging trajectories (and ground impact) at both low and high values, and a region of stable trajectories between values of around 1.0 to 1.9. At low values the trajectories are destabilized by the high values of radial velocity, and at high values are destabilized by high values of the circumferential velocity.
The change in values of gamma from the one cell form of gamma = 2 to the quasi-two cell form of gamma = 4 shown in Figure 9 results in little change to the debris trajectories from the base case, although the oscillations in the kinetic energy fall as gamma increases.
Figure 11. Effect of variations in vertical starting position
The debris trajectories remain stable as the normalized radius varies between 1 and 1.9 but outside those limits the trajectories diverge and intersect with the ground (Figure 10). Similarly the trajectories are only stable for normalized values for height between 0.8 and 1.2 (Figure 11). Thus the starting point window for the trajectories to ultimately attain a stable form is quite small.
A number of points arise from the results presented above.
Even for the simple wind and debris flight formulation adopted, debris trajectories can be quite complex.
A comparison of the results obtained with the old and the new wind field model show very considerable differences, due to the different vertical velocity formulation. analysis reveals that the debris trajectories can be specified by a small number of debris and tornado parameters, with the Tachikawa number and the Swirl Ratio being the most significant.
There are regions within parameter space for which the debris trajectories become stable – i.e. the debris flies indefinitely.
This post addresses the issue of the use of what has become known as the “Chinese Hat” gust model. The use of this title has become increasingly problematic over recent years for obvious reasons, and I will no longer use it, but will instead refer to the “CEN extreme gust model” in what follows.
The CEN extreme gust model
In a number of situations in wind engineering, some sort of deterministic (as opposed to stochastic) gust model is required in order to determine structural response. One such case is in the determination of the risk of overturning of road or rail vehicles in high winds. A methodology of this type is set out in CEN (2018), where an extreme gust model is described. This model was originally developed in wind loading studies for wind turbines as a time dependent gust to be applied to calculate wind turbine loading at one fixed location (Bierbooms and Cheng, 2002). As such, it is perfectly adequate and a good representation of an average extreme gust in high wind conditions. In the methodology of CEN however, it is re-interpreted as a stationary spatially varying gust. This must be regarded as a very significant assumption for which, in my view, there is little justification. Nonetheless the formulation has proved useful practically and we begin by considering it in a little more detail.
For a wind normal to the track, the extreme gust formulation is given by equation (1) on Box 1. Note that the “characteristic frequency” of the gust is calculated from standard wind engineering methods for temporally, rather than spatially, varying gusts. Equation (1) is a generalised form of that given in CEN (2018) to remove some of the constants that tie the expression to a particular location and topography through specific values of peak factor and the turbulence intensity (the ratio of the standard deviation to the mean velocity). The time dependence is recovered through the passage of the train passing through this gust at a speed v = xt to give equation (2). It can be seen that the gust thus has a maximum value of (1+ peak factor x turbulence intensity) when t = 0 and decreases to unity for small and large times. It is symmetrical about t = 0. The velocity relative to the train is then found by the vector addition of this gust velocity with the vehicle velocity to give a time varying value.
To enable the gust profile to be specified, the characteristic frequency f is required. This is specified in equations (3) to (5). These equations are again in a more generalized form than given in CEN (2018), where a value of the upper limit of integration is fixed at 1 Hz, together with an implicit value of the turbulence length scale of around 75m. The genesis of the 4.18 factor is however not clear to me. Equation (3) shows that the calculation of the characteristic frequency is thus based on the calculation of the zero-crossing rate of temporal fluctuations through the use of the velocity spectrum. Again, note that these parameters describe a time varying rather than a spatially varying velocity, and their use is not formally consistent with a spatially varying gust. From equations (3) to (5), it can be seen that the normalized characteristic frequency is a function of the normalized upper limit of integration. A numerical solution of these equations was carried out and the following empirical line fitted to the results for a value of the latter greater than 1.5 (which is the realistic range) – equation (6). From equations (2) and (6) we thus obtain equation (7). Although the overall methodology cannot be regarded as wholly sound, equation (7) does (in principal) significantly simplify its use and also allows the implicit wind parameters in the method to be explicitly defined.
Is there a better methodology?
It can be seen from the above that the CEN methodology thus does not fully describe a typical gust as seen by a moving train, which would vary both spatially and temporally, and can at best be regarded as an approximation, although its practical utility must be acknowledged. Ideally, if such an approach is to be used, a gust that varies both in space and time is really required. Such a gust was used in the SNCF route assessment method of Cleon and Jourdain (2001), where the shape of the gust is appropriately described as a rugby ball. This method was however for very specific wind characteristics and does not seem to have found widespread use. Thus in this post, we investigate the possibility of developing a spatially and temporally varying gust, that can be expressed in a simple form (ideally similar to equation (2)) for practical use.
Towards a new model
In this section we will draw on experimental results for extreme gust characteristics in both temporal and spatial terms to construct a simple, if empirical model, that fulfills the function of the CEN (2018) model without the theoretical drawbacks.
We consider first the full-scale experimental data analysed by Sterling et al (2006) which used conditional sampling to determine the average 99.5th percentile gust profile for four anemometers on a vertical mast with heights between 1m and 10m. These results thus give the time variation in gust speed as the gust passes the anemometers. They showed that the gust profiles could be well approximated by the formula shown in equation (8) (Box 2). The parameter G in this equation is the equivalent of the peak factor multiplied by the turbulence intensity in equation (2) and for these measurements was 0.786. n was -0.096, and the value of m depended upon whether t was greater or less than zero. For t < 0, i.e. on the rising limb, m was 0.1, whilst for t> 0, on the falling limb, m was 0.2. The gust shape was thus asymmetric with a maximum at t = 0. This curve was a good fit to all the gust profiles throughout the height range. In what follows we will use a rather different curve fit expression to the same data, more consistent with that used in CEN (2018) – equation (9). It was found that the best fit value of b was equal to 0.5 for all t, whilst the best fit values of a were 0.49 for the rising gust and 0.37 for the falling gust. This expression thus describes the temporal variation of wind speed as a gust passes through the measuring point
To describe the lateral spatial variation of the gust profile, we use the data of Baker (2001) who presents conditionally sampled peak events for pressure coefficients along a 2m high horizontal wall. This data allows the lateral extent of the gusts to be determined, from the variation of the time varying pressure coefficient divided by the mean value of the coefficient and then assuming that the gust velocity variation can be found from equation (10). The spatial variations of velocity were then fitted by a curve of the form of equation (11). g was found to be 6.16 and d was found to be 0.7.
On the basis of the above expressions one can thus write the expression of equation (12), which describes the variation of the gust velocity in both space and time. The movement of the train through the gust can again be allowed for by letting x = vt (equation (13)).
Box 3 sets out the formulations of the CEN extreme gust model and the model derived here. In some ways they are similar in form, with an exponential formula that is primarily a function of normalized time. Whilst the CEN model is symmetric around t = 0, the new model has a degree of asymmetry because of the different values of the curve fit parameters for t < 0 and t > 0. However an examination of the new model suggest that the asymmetric term may be small, and thus Box 3 also shows an approximate version of the new model where this term is neglected.
Figure 1 shows a comparison of these three models for the following parameter values – peak factor = 3.0; turbulence intensity = 0.25; train speed = 75m/s; mean wind speed = 25m/s; turbulence length scale = 75m, upper frequency of integration = 1.0Hz. It can be all three models are similar in form, showing a sharp peak at t = 0. The full and approximate forms of the new model are almost indistinguishable, showing that the approximation suggested above is valid. The main difference is that the CEN model has a much greater spread in time than the new model. This difference persists whatever input parameters are chosen.
At this point it is necessary to consider again the genesis of the models – the CEN model resulted from an application of a time varying gust model as a spatially varying gust model, whilst the new model was developed based on measured temporal and spatial gust values. As such, I would expect the latter to be more accurate. The broad spread of the CEN gust may result from an application of the time varying along wind statistics to a cross wind spatial gust. Since it is known that that longitudinal integral scale is several times larger than the lateral integral scale, this would result in a wider spread of the gust than would be realistic. This is to some extent confirmed by the period of the two gusts – around 2s for the CEN gust and around 0.8s for the new model. For a train speed of 75m/s, this corresponds to gust widths of 150m and 60m – roughly approximating to the expected the longitudinal and lateral turbulence integral scales.
In this post I have looked again at the CEN extreme gust method and raised concerns about its fundamental assumptions. I have also developed an equivalent, but perhaps more rigorous, methodology based on experimental data for wind conditions at ground level. This strongly suggests that the CEN gusts are spatially larger than they should be, which suggests its long term use should be reviewed. However, when used to compare the crosswind behaviour of individual trains, rather than in an absolute sense, it is probably quite adequate.
CEN, 2018. Railway Applications d Aerodynamics d Part 6: Requirements and Test Procedures for Cross Wind Assessment. EN 14067-6:2018.
Cleon, L., Jourdain, A., 2001. Protection of line LN5 against cross winds. In: World Congress on Rail Research, Köln, Germany.
Sterling M, Baker C, Quinn A, Hoxey R, Richards P, 2006, An investigation of the wind statistics and extreme gust events at a rural site, Wind and Structures 9, 3, 193-216, http://dx.doi.org/10.12989/was.2006.9.3.193
Recently I have been considering the fundamental nature of a range of analytical models of tornado like vortices, and have written up my musings as an extended essay that can be downloaded at “Some musings on tornado vortex models”. In the essay I look at the class of tornado models that are solutions of the Navier-Stokes or Euler equations. It is clear that they all share a common analytical basis based on the assumption, either implicit or explicit, that the three velocity components (radial, vertical and circumferential) can each be specified by the multiple of two functions – one a function of radius only, and one a function of height only. Assumptions are made concerning the nature of one particular velocity component, and this assumption then allows the other components to be calculated from the continuity and momentum equations via the method of separating the variables. The recognition of this commonality allows a common analytical formulation to be developed that underlies all the models.
Those models that are solutions of the full Navier-Stokes equations (the Burgers-Rott, Sullivan and Vasistas et al models) derive velocity component formulae that are functions of Reynolds number. In the context of a full-scale tornado, this is a Reynolds number based on turbulence eddy viscosity rather than molecular viscosity. The assumptions required to obtain analytical solutions result in vertical velocities that are unbound with height and in some cases radial velocities that are unbound with distance from the vortex centre.
Those models that are solutions of the Euler equations (two by Baker and Sterling and two new models A and B) have, on the whole, rather more realistic formulations of the velocity components and, with one exception, all components for these models are bound in the vertical and radial directions. Instead of the Reynolds number, the velocity components are functions of constants of integration that relate to the Swirl ratio – the ratio of the maximum circumferential to radial velocities. As the circumferential velocity profiles in these models fall to zero at ground level in a reasonably realistic way, the boundary layer at the bottom of the tornado is modeled to some extent. The common analytical framework of these models allows, in principle, the derivation of a large number of different models, provided that they are of a form that allows the solutions to be obtained through simple integrations. However the drawback of such models is that the pressure is zero at the ground for all distances from the vortex core and thus the dip in pressure at the centre of tornadoes is not modeled. This is broadly a consequence of viscous effects not being properly modeled near the ground.
Whilst most of the models represent single cell tornado vortices, two of them – those of Sullivan and new model B – give solutions for two cell vortices. The essay shows that the Sullivan model, based on the Navier-Stokes equations, has a more general form than that given in the original paper and can model one-cell and two-cell vortices and the transition between them. New model B, based on the Euler equations is also able to model both sorts of vortex.
The essay concludes that further work is required in two areas. Firstly there is a need to develop methods that do not rely on the assumption that the velocity components are multiples of two functions – one of radius and one of height – as recent experimental data suggests that the vortex radius can vary significantly with height. Secondly, the tornado boundary layer needs to be modeled in a more satisfactory way than at present, and the essay suggest that this might be done through matching a viscous solution of the Navier-Stokes equation near the ground, with an inviscid solution from the Euler solution away from the ground. I may have more to say on this in the future.
This post outlines some of the results from the project “The safety of pedestrians, cyclists and motor vehicles in highly turbulent urban wind flows” funded by the UK Engineering and Physical Sciences Research Council. The work that is described below involved a number of colleagues, whose contribution to the project was significantly greater than mine, particularly Dr Zhenru Shu, Dr Mike Jesson, Dr Andrew Quinn, and Prof Mark Sterling. Their contribution is gratefully acknowledged.
The assessment of wind conditions around new buildings has become standard practice over recent years, either by wind tunnel testing or through the use of CFD calculations. The assessments usually concentrate on two aspects – the effect of wind conditions on human comfort and thus the usability of the area around the building; and the effect of high wind conditions on human safety and stability. It is with the latter that this paper is concerned. In general the criterion for assessing a site for pedestrian safety is based on a gust wind speed of a specified magnitude with a specified probability of occurring, that is deemed to be at the safety limit. Current UK practice is illustrated in Figure 1 below. There is a great deal of variability in the specification of this windspeed and the specification is usually based on largely subjective data from questionnaires etc. Following a fatality caused by high winds around a new building in the city of Leeds, a major research project was funded by the UK Engineering and Science Research Council to enable the University of Birmingham to investigate the safety of vehicles and pedestrians around high-rise buildings. This included full-scale wind measurements and the assessment of the ability of different wind tunnel and CFD techniques to replicate these measurements. In addition tests were carried out to make quantitative measurements of human response in gusty winds, using instrumentation mounted on volunteers. As will be appreciated by any reader who has tried to make full scale wind measurements of any type, the setting up of the experimental apparatus usually guarantees that strong winds will not occur, and the same phenomenon was observed for these tests. The two winter seasons that were available for these measurements had relatively few storms, and only two trials could be carried out. As a result, although some very interesting results were obtained and will be presented in what follows, they must be regarded as provisional and tentative. More work is required to obtain a fuller dataset of human response measurements of the type that are presented here.
2. The trials
The trials on the response of pedestrians to high winds were carried out on the campus of the University of Birmingham (figure 2). A walking route of length 63m was set up in the centre of the campus. Eight sonic anemometers were placed 2m above the ground at 9m intervals along the route. A reference anemometer was installed at the top of the nearby high rise Muirhead Tower. A reference anemometer was mounted at the top of the Moorhead Tower. All the anemometers sampled at 10 samples / sec, and data was recorded on an AntiLog data logger. Human response was measured using GaitUp Physilog (combined accelerometer and gyroscope) sensors. Sensors were attached to both feet of the subjects, and provided details of walking speed and stride parameters every second through GaitUp’s proprietary software. A third sensor was placed on the back of a safety jacket worn by the subjects and thus gave details of upper body acceleration.
Two trials were carried out – October 2017 during Storm Ophelia, and in February 2019 (figure 3. In total there were 15 subjects, with weights ranging from 54 to 110kg, and ages between 28 and 75. Each subject was asked to walk along the test route 10 times in each direction during which the gait and acceleration information was measured.
The overall wind conditions at the reference site on the Muirhead Tower are shown in figure 3 for the two test periods. It can be seen that in each case the wind is from the South-West (shown in longer term analysis to strongly be the prevailing wind direction), with gust speeds up to 18m/s
Before the data could be analysed, some data preparation was required. Firstly the gait data and accelerometer time series had to be synchronized with the anemometer time series of velocities and the raw accelerometer data was transformed into horizontal and vertical co-ordinates. The time series of velocity and direction relative to the subjects were then derived form the stationary anemometer data as the subject walked along the route. A histogram of gust speed distribution, as experienced by the volunteers, for the two trials is shown in figure 4.
Initial inspection of the data showed that there was very significant variability between each recorded walk along the track. This was in part due to the normal variation in wind conditions with higher gust speeds on some walks than on other, but it also seemed that the reaction of subjects varied both with time and between subjects. A typical set of results is shown in figure 5. The direction of travel of the subject is from 0 to 63m. The wind speed relative to the subject can be seen to have a maximum of around 12 m/s in this case (associated with the corner flow from an adjacent building). The horizontal and vertical accelerometer data show slight oscillations around the gust position gust with the former having an average value of zero, and the latter an average value of 1.0. Most of the gait measurements (cycle time, stride length, stride speed) revealed little change in behaviour as the subjects walked along the route, all remaining approximately constant along the walk in most conditions. The one parameter that did show variation was the swing width – the lateral variation of the foot during a stride cycle. In particular rapid changes in swing width were sometimes (but not always) observed as the subjects encountered gusts – see the graph for swing width gradient.
At the highest gust speeds that were recorded, there were three events where the subject became unstable to a variable extent. Figure 6 shows the experimental data for one such case. Here it can be seen that at the gust position there are significant vertical and horizontal acceleration responses, and all the gait parameters show a response at the event. The swing width response is again the most noticeable.
A somewhat more quantitative approach to the data is possible by looking at the various responses statistically. In what follows we consider the results from both trials, for all subjects, as one dataset. Figure 7 shows the percentage of such gusts in which the subjects showed a swing width response (with either the left or right swing width changing by more than 0.06m in one second) and acceleration response (where an acceleration response greater than 0.05g could be detected) or an instability response (with an acceleration response greater than 0.4g). In considering these results the low number of gust events in the upper velocity bands need to be considered, as does the subjectivity of the response limits used. These points being made, it can be seen that for even low speed gusts of magnitude less than 10m/s, around 50% of the gusts result in a swing width response (which are mostly unconscious responses not registered by the subject). The frequency of such responses rise rapidly for gust speeds above 10 m/s, and all gusts over 14 m /s show such a response. Acceleration responses become significant at gust speeds of about 10m/s, and are observed for all gusts above 16m/s. Instability responses begin to occur at gust speeds over 14m/s, although it should be noted here that only a very small number of such events (3) were observed.
4. Concluding remarks
The results for human response in gusts presented here suggest that three levels of response can be identified – swing width response , upper body acceleration response and instability response, with the frequency of each such response increasing with wind speed. However it must be emphasised once more that the number of bot high speed gust events and the number of subjects was too small for a valid statistical analysis to be carried out, and more data is required before firmer conclusions can be drawn.
This post arises out of a discussion with a number of colleagues on the issue of train overturning, in particular Mr Terry Johnson and Dr Dave Soper. Their (perhaps inadvertent) contribution to the development of the ideas set out below is gratefully acknowledged, although the responsibility for any inadequacies and errors must remain mine.
In recent decades a great deal of research has been carried out to investigate the safety of trains in high cross winds, primarily to determine the wind speeds at which overturning will occur, and the risk of a wind induced accident (Baker et al, 2019). This usually takes the form of the determination of the aerodynamic forces and moment coefficient for a particular train, the use of these coefficients to determine the cross wind characteristic (CWC) – effectively a plot of accident wind speed against vehicle speed – and then some sort of risk analysis on the route over which the train will run. The first two steps are usually the concern of train manufacturers and are undertaken when the design of the train, at least in terms of overall shape and size, is fairly well advanced. The third step is usually the concern of the infrastructure operator. One question that is not often asked however is whether there is a “best” design for a train to minimise the risk of a wind induced accident, and thus to maximise safety. This has been addressed to some extent by a number of recent investigations that used a combination of CFD methods to calculate the forces and moments on a train, and optimisation methods to consider the effect of changes to train geometry. It is not however clear as to what should be the objective function of such optimisation – for example a number of different force or moment coefficients for a range of different yaw angles could be chosen. This post addresses this issue though an analysis of accident risk and investigates the aerodynamic parameters required to minimise this risk
2. Aerodynamic force and moment coefficients
In a recent book (Baker et al, 2019) the author suggests a way of parameterising train aerodynamic force and moment data that seems to have a wide validity. This is set out in Box 1 below, in which the formulation for lee rail rolling moment coefficient is given, and is illustrated for a specific case. It applies equally well to side and lift force coefficient data. It can be seen that the form of the rolling moment / yaw angle curve is specified by four parameters – the coefficients at yaw angles of 30 and 90 degrees and exponent shape factors that describe the shape of curve, n1 and n2. Figure 1 shows a comparison of this methodology with side force coefficient data from the CEN codes (CEN, 2018) and the AeroTRAIN project (Paradot et al, 2015) as given in Baker et al (2019). All this data was obtained in a consistent way, with an STBR ground simulation in low turbulence wind tunnels. The agreement can be seen to be in general good and gives some confidence in the use of the parameterisation in what follows. The biggest discrepancy is in the transition region between the high and low yaw angle regimes, but it will be seen that this is not particularly critical to the argument that follows.
3. Crosswind characteristics
The method used to specify the crosswind characteristic is also taken from Baker et al (2019) and is set out in Box 2. Using this methodology, the CWC are functions of n1 and n2, the ratio of the moment coefficients at yaw angles of 90 and 30 degrees, and what is defined as a characteristic wind speed, which is itself a function of train and track parameters. Box 2 gives the formulation for flat straight track, with a wind angle normal to the track – a fuller form can be found in Baker et al (2019). A comparison of this method with the results from CEN (2018) and Paradot et al (2015) is given in figure 2, again from Baker et al (2019). Agreement can be seen to be good, and this gives further confidence in the use of the methodology in what follows.
Box 2 also indicates how the accident risk can be calculated for a specific reference site using the Weibull distribution to specify wind speed probabilities. We assume a section of railway of a specified length, with specified values of the Weibull parameters and a typical service pattern, and we then express the CWCs as a plot of train speed against the probability that a wind induced accident will occur in the section, rather than accident wind speed. This enables us to better address the question as to what is a “good” vehicle in cross wind terms, as it will highlight the relative importance in risk terms of different vehicle speed ranges.
Figure 3 shows the calculated CWC, expressed as both an accident wind speed plot and as a risk plot, for what we will take as our base case. The parameters for this case are shown in the figure. The plot of accident wind speed against vehicle speed shows a reduction in the former as the latter increases, as would be expected. There is a break in gradient, at the point of transition between the low yaw angle (at high speed) and high yaw angle (at low speed) formulations of Box 2. Figure 2 shows that this is quite typical of the calculated CWCs from Paradot et al (2015). The plot of site risk against vehicle speed shows an increase in risk with the vehicle speed. At the vehicle speed of 350 km/hr the logarithmic risk is around -8 (but remember that this absolute value is completely arbitrary). The risk falls by an order of magnitude as the speed decreases through the low yaw angle range to around 100 km/h, with an increased rate of fall for low speeds, where the high yaw angle formulation becomes relevant.
Figures 4 to 7 show the effect on the CWCs of changing the parameters for the moment characteristics. As the low yaw angle exponent n1 is varied between 1.3 and 1.7, there are variations of about half a magnitude in risk for the higher train velocities, although this varies through the speed range. This parameter is typically around 0.9 to 1.1 for lorries, 1.2 to 1.4 for blunt nosed trains, 1.4 to 1.6 for streamlined trains, and 1.7 to 2.0 for trailing vehicles. As the high yaw angle exponent n2 is varied, the variations in accident velocity and risk are confined to the low speed range as would be expected, although here the variations in risk can be several orders of magnitude. As the lee rail rolling moment coefficient at 30 degrees is varied between 3 and 5, there can be seen to be very significant variations in both accident wind speeds and risk throughout the speed range. For variations in the lee rail rolling moment coefficient at 90 degrees only the low speed accident wind speeds and risk levels are affected as would be expected. From these graphs it can be concluded that the risk of an overturning accident will be reduced for high vehicle speeds if n1 increases and the lee rail rolling moment coefficient at 30 degrees decreases; and for low vehicle speeds if n2 increases (becomes less negative) and the lee rail rolling moment at 90 degrees decreases. Of the parameters the 30 degree coefficient produces most change in accident wind speeds and risk levels across the speed range, and is perhaps where most design effort should be concentrated.
Now let us consider specific trains. Table 1 shows, for the CEN (2018) and AeroTRAIN (Paradot et al, 2015) trains, the maximum train speed, the values of the four parameters that define the rolling moment characteristic, the characteristic velocity, and the risk at the maximum operating speed. Those shaded red indicate values that would increase risk significantly above the average, and those shown in green indicate values that would decrease risk significantly the average. It can be seen that of these trains the ICE3, IR and Silbering has the “best” values of rolling moment coefficient. For the ICE3 this is presumably due to the nose shape, resulting in low levels of lift and side force, and thus rolling moment. For the IR and Silberling however, these low values are due to the lack of underbody blockage, at least as modelled in the wind tunnel tests. The ICE3 values of n1 and n2 are around the average, whilst those for the IR and Silberling are low, and would thus tend to increase risk. The worst train in terms of rolling moment coefficients is the double deck M6BX. The IC4, RevCo and ZTER also have high values of the coefficient at 90 degrees.
The risk at the maximum speed for the all trains, with one exception, is between -7.3 and -8.4 i.e. it varies by one order of magnitude. The ICE3, TGV, ZTER and IR have the lowest risk and the M6BX the highest for the standard site. This risk variation is perhaps less than would be expected, and is partly caused by the reduction in risk with the reduction in maximum operating speed. The outlier from the range of -7.3 to -8.4 is the ADR, which has a low value of -9.1, which is due to its high mass and high resulting characteristic velocity.
From the above, it can be seen that for high speed trains, the aerodynamic parameter that most affects the overturning risk is the lee rail rolling moment coefficient in the low yaw angle range, characterised by the value at 30 degrees. In these terms the ICE3 shape is “best”. However this does not necessarily apply for lower speeds, when the higher yaw angle range becomes of importance. These points being made there are some important caveats.
The overturning wind speed and thus accident risk depends upon a range of parameters as well as the aerodynamic characteristics. Train mass is particularly important.
Similarly the infrastructure characteristics are important, and accident wind speed and risk will be affected by can’t and topography.
Perhaps most importantly, the level of risk is determined by the nature of the train operation itself – if speed limits are imposed in high winds, it is quite possible that the most important aerodynamic characteristics will move from those in the low yaw angle range to those in the high yaw angle range.
One further point is of interest. In Baker at al (2019) the head pressure pulse magnitudes and wake slipstream gust velocities are tabulated for orange of trains. Of those trains included, the Velaro (i.e. the ICE3) has both the lowest pressure pulse magnitude and the lowest slipstream gust velocities, suggesting that the nose / tail shape of this train has considerable aerodynamic advantages.
Baker, C., Johnson, T., Flynn, D., Hemida, H., Quinn, A., Soper, D., Sterling, M. (2019) Train Aerodynamics – Fundamental and Applications, Elsevier.
CEN, 2018. Railway applications — Aerodynamics — Part 6: Requirements and test procedures for cross wind assessment. EN 14067-6:2018.
Paradot, N., Gregoire, R., Stiepel, M., Blanco, A., Sima, M. et al., 2015. Crosswind sensitivity assessment of a representative Europe-wide range of conventional vehicles. Proceedings of the Institution of Mechanical Engineers. Part F Journal of Rail and Rapid Transit 229 (6), 594-624.
Perhaps the oldest sport world record still current is that for “Throwing the Cricket Ball”, with the record being listed in Wisden’s Cricketers Almanack as 140 yards 2ft by Robert Percival on Durham Sands Racecourse around 1882. The length of the throw, and the inability of any others to throw that distance over the last 140 years, has resulted in considerable scepticism concerning its veracity and reliability. As a result of a recent newspaper article about Percival’s throw (Guardian 23/4/2019), the author began to consider whether it would be possible to actually calculate the flight of a cricket ball given certain assumptions about throwing speed and angle of throw and the like, and perhaps to come to some more quantitative conclusion about whether or not Percival’s throw was possible. This paper presents the results of these calculations, together with a historical survey of “Throwing the cricket ball” competitions, and an examination of the events (and in particular the weather) on the day the record was set.
We begin by setting out some of the background for the event at Durham Sands “around” 1882 (it will become apparent why quotation marks are used in what follows), give a brief discussion of the event itself, and then move on to discuss the results of flight trajectory calculations (in very broad terms) before coming to some sort of conclusion about whether Percival’s throw was possible.
Throwing the cricket ball as an activity has a long history. In 1792, Mark Richmond, gamekeeper to the Duke of Richmond, threw 119 yards at Goodwood Park to defeat the Earl of Winchelsea “who had never before been beaten” (Hampshire chronicle 3/6/1820). In the 1820s, contests were vehicles for wagers amongst gentlemen (Morning Chronicle 25/12/1822). As an athletics event it was popular at sports days in the mid- to late Victorian era, along with other events that sound strange to a modern ear, such a place kicking and drop kicking for distance and target throwing with a cricket ball at stumps between 20 and 50 yards away (for example, see the Luton Times and Advertiser 29/5/1855). However, throwing the cricket ball did not ultimately make it into the list of accepted sports for athletic events and its popularity waned. This is illustrated by the histogram of figure 1, which shows the number of mentions the phrase “Throwing the cricket ball” receives in a search of the British Newspaper Archive by decade from 1800 to 1950. This is hardly a valid statistical approach, since it depends upon the vagaries of press reporting, but is nonetheless illustrative. After around 1900, the event goes into sharp decline and by the middle of the century is confined to school sports days. It seems odd that such a simple throwing sport did not ultimately find favour at an international level, as it seems one of the most physically natural of all events and one can speculate on the reasons. Perhaps it was because throwing the ball is not really a stadium sport, as the throws are too long to conveniently fit within athletics tracks; or because it was not included as an Olympic sport.
Figure 1 Search results for “Throwing the cricket ball” in the British Newspaper Archive
The BNA is also useful in enabling us to get some idea of how competitions were conducted and how far a cricket ball could be thrown. The competitions usually involved between two and four throws per competitor, presumably from behind some sort of throwing line. Sometimes the throw was from the top of a barrel to ensure that there was no overrunning. On occasion, penalties in terms of a set number of yards were applied, presumably for overrunning the line, and some competitions were run as handicaps (Sporting Life 12/16/1878). There is even one record of a competition where the ball had to be thrown in the left hand, won with a throw of 38 yards, presumably with no natural left hand / arm users taking part (Sporting Life 25/10/1862). Figure 2 shows the winning lengths for throwing the cricket ball events between 1860 and 1900 from “The Sporting Life” published in London, but with a national reach, and for papers published in Edinburgh during the same period. This represents only a small proportion of all the newspaper reporting, but is at least geographically representative. In general, only those results from senior pupils in school sports; from University sports; from military competitions; and from Athletics Clubs have been used. Length of throw is given in yards, in deference to historical usage, although all other units in this article will be the S.I. units in which the author (an engineer) would normally work.
Figure 2 Length of throw from The Sporting Life and Edinburgh newspapers
The results, although again statistically rather suspect, are nonetheless illustrative. The London and the Edinburgh datasets are consistent with each other, with competition winning lengths through the period were around 80 to 110 yards. The school sports results tend to be at the bottom end of the range, and the student, military and athletic club results being at the higher end. There are a relatively few results above 110 yards, and the recorded limit seems to be around 120 yards. However there were a few reports of longer throws. A letter in the Dundee Evening Telegraph of 9/1/1889,reports that a Mr. Fawcett of Brighton College threw 126 yards 6 inches (or possibly 127 yards 4 inches – two figures are given). Much later, the Nottingham Journal of 18/3/1925 gave the information that, in 1873, W. H. Game of Oxford University threw 127 yards 1 foot 3 inches; in 1876. W. F Forbes threw 132 yards at the Eton College Sports; and in Dundee in 1882, A. McKellar threw 130 yards, 1 foot 6 inches. There is also the (almost inevitable) report of the omni-competent W G Grace’s prowess in this field, with a throw of 122 yards (Edinburgh Evening News 10/8/1895). Wisden itself lists two throws of similar distance to that of Percival – in 1872, Ross MacKenzie is said to have thrown 140 yards and 9 inches in Toronto, and on December 19 of that same year “King Billy the Aborigine” threw 140 yards at Clermont in Queensland.
Now let us consider the world record event itself. The Sportsman magazine in 1889, states that it took place in 1884 at Durham Sands Racecourse (Sportsman Magazine, 3/1889). However, Rayvern Allen as reported on Cricinfo, states that this is a mistake and that it took place on Easter Monday April 18thin 1882. Something has clearly gone wrong in the transmission of information however, as Easter Monday in 1882 was on 10thApril. It was however on the required date in 1881, and the event is duly mentioned in the report in the Durham County Advertiser of 22/4/1881. Durham Sands Racecourse, was, and is, a large stretch of level ground next to the River Wear in Durham. It is shown on a map from the 1860s in figure 3. It is basically oriented east to west along the river.
Figure 3 Durham Sands Racecourse in the 1860s (from Edina Digimap)
1881 was the first year of the Sands Sports and was bitterly cold (in the author’s experience, typical of an Easter Monday on whatever date it occurs in whichever century one might be in) with a moderate easterly wind. This will be seen to be of some significance in what follows. There was a significant crowd, but visibility of the events was poor, and there was only one small stand that was poorly occupied. In addition to the Sports “there were a good number of shows, roundabouts, shooting galleries etc, …while two quadrille bands provided unlimited pleasure to numbers of young people and dancing was freely indulged in”. There was a short and rather cramped 300 yard track that was used for a horse races – flat races for horses above 14 hands, for ponies below 14 hands, and a hurdle race for horses, all with an entrance fee and cash prizes for the winners and placed horses. For human competitors the events were a 220 yard flat race, quoits, high leap, 220 yard hurdle race, long leap, donkey race, pole leaping, put stone, one mile walking competition, 100 yards boys races, a mountebank race (!), an open flat race, and, of course, throwing the cricket ball. All had prize money for winners between 7s 6d and £1. The prize for throwing the cricket ball was the lower value. The results of the competition are simply stated as follows.
1stPercival, 2ndGnatt, 5 competitors
No throwing distances are given. It would seem that Percival was something of an expert in this event, and won many prizes, and thus supplemented his earnings as a miner quite well. At the time of the throw he was 25 years old. The census records give contradicting birth locations – Alston (1861/1871), West Auckland (1881/1891) or Northead (1901/1911). In 1881 he lived with his family in East Thickley in County Durham. In the years following he was often to be seen at open weight wrestling competitions and was thus clearly a strong and well-built individual. The Cricinfo report of Rayvern Allen’s work suggest that in October 1884 he won £10 in a wrestling competition at Durham Sands – hence the confusion about the date of the Throwing Event. The author has not been able to trace any reference to this, but Percival did win a best of seven falls wrestling match worth £10 against G Stockdale of Spennymoor, at Wood View Gardens, Tudhoe Grange in October 1884, so again there is possibly some confusion in the transmission of information (Durham County Advertiser 24/10/1884). He was married to Mary, and they had 6 children. In the mid 1880s and early 1990s he was firstly the professional at New Brighton CC and then groundsman to Liverpool Police Athletic Society. But by the early 1900s he was again a miner and died in South Shields in 1980 of broncho-pneumonia. There was no obituary.
In terms of the claim for a world record length, the Sportsman magazine in March 1889 stated that it took place on Easter Monday, 1884 (3 years too late) and “the throw was measured by the committee“. In 1897 Sporting Records was more skeptical writing “It has been claimed by R Percival that he threw 141 yards at Durham Racecourse in 1884, but this is regarded as so doubtful that few authorities even mention it.” Note that Percival himself seems to have been making the claim, and it was clearly contentious even at that stage. The record was not listed in Wisden until the 1908 edition. Also note there were other claims to the world record around at that time – on 8/11/1889 the Sporting Life reported that in Australia a certain “Crane” threw 128 yards 10½inches, beating the world record by 2 yards and 7 inches, in a competition with a touring American baseball team. Who set the “old” record, and who designated it as such, is not clear.
So it can be seen that while Percival’s claimed throw is very much above the run of the mill competition winning throws of around 80 to 110 yards of the period, there are a number of other recorded throws of rather greater distances, and Percival’s throw seems to be at the upper end of what was possible. However, its claim to be a world record has always been treated with skepticism. Can any more be said about the likelihood of him being able to make such a throw? We thus move on now to briefly consider the trajectory calculations. They are described in a little more detail in the Appendix for those who are interested. In simple terms the calculations use Newton’s laws to determine the trajectory of the cricket ball, allowing for air resistance and the somewhat peculiar aerodynamic properties of the cricket ball. The maximum distances are always achieved at an initial throwing angle of around 40 degrees (so the trajectory is rather like that of a javelin rather than the normal cricket throw). This results in trajectory heights of the order of 30 to 40 m. Figure 4 shows the maximum distances achieved against initial throwing speed for a new cricket ball and an old cricket ball, for no wind. Paradoxically the aerodynamic resistance of the latter is less than that of the former (just as dimpled golf balls have lower drag than smooth golf balls), and this is reflected in the distances travelled. To give some context to the throwing speeds, 40m/s (≈90mph) is the bowling speed of a current international fast bowler – but as the throwing angle and ball orientation needs to be precisely controlled, this is probably less than the maximum speed obtainable in a less controlled throw. Major League baseball players have been known to throw at up to 50m/s (≈110mph). From this figure one can conclude that, at least in still air conditions, an old ball and a high initial speed are necessary to approach the 140 yard mark. One might expect that it would be normal to use a used ball in such competitions rather than waste a new ball.
Figure 4 Calculated length of throw against throwing speed for old and new cricket balls
However, as noted above, on the day of Percival’s throw it was somewhat windy. The wind speed increases with height above the ground, and this effect has been modeled in the calculations using the same methods as would be uses in calculating the wind load on buildings in modern structural engineering codes of practice. Figure 5 shows the calculated contours of throwing distance for an old ball, plotted against initial throwing speed and wind speed at 10m above the ground, assuming a following wind. The annual average wind speed in England is of the order of 4 m/s at 10 m above the ground. It can be seen that wind speeds above the average can have a significant effect on the throwing distances at any one throwing speed. In particular a 6m/s following wind will allow a throw of 140 yards to be achieved with the same initial throwing speed as a 120 yard throw with no wind. These calculations show that the trajectory of cricket balls are much more sensitive to wind conditions than, say, javelin trajectories, largely because cricket balls are aerodynamically bluff rather than streamlined
Figure 5 Contours of length of throw plotted against initial throwing speed and wind speed at 10m height.
Now the wind conditions on the day of the throw can actually be quantified with some precision. The Durham University Observatory(figure 6), which has the second oldest sequence of continuous meteorological measurements in the world after the Radcliffe Observatory in Oxford, is just over 1 km away (see figure 3). Prof Tim Burt of the University Geography Department, who now has charge of the Observatory, has kindly provided the author with meteorological data for April 18th1881. Basically the wind speed that day, at the 10.00 observation, was 6 m/s from the north east. This is probably a mean value and one might expect gust wind speeds to exceed this. Presuming this was a direct following wind (and there is no indication of the throwing direction on the day, but somewhere in the east / north east quadrant is quite possible looking at the layout of the Sands) then this level of wind speed couldhave significantly assisted the throw, although, there can be no certainty on this.
So what then are we to conclude? Robert Percival was clearly one of the top throwers of the age judging by the number of competitions he won, and his wrestling activities suggest considerable innate strength. It seems to the author that, there is a prima facie case that he would have been capable of propelling the ball at the necessary speed for a 120m plus throw of an old cricket ball in still air. The conditions at the Sands on the day of the record were such that the winds may have given him considerable assistance. A throw of 140 yards seems a realistic possibility. That such a throw is possible is further confirmed in Wisden which lists a number unverified, throws of around the 140 yard mark in recent decades (in particular Ian Pont, in Cape Town in 1981 was said to have thrown 138 yards) and an unverified world record throw has appeared on Youtube.
Before we conclude, some other points come to mind.
Firstly the press reports tell us that on the day of the throw the conditions were quite chaotic at the Sands, with a considerable crush of people, and there was difficulty of finding space for the events themselves. This would hardly have made for accurate measurement of the throw and perhaps gives pause for thought as to the accuracy of the measurements.
Secondly, can a throw be described as a world record if it is heavily influenced by wind conditions, as the calculations suggest was the case on Easter Monday in 1881? This point of course is a general one that reflects on the actual integrity of all results for throwing the cricket ball as the required ball trajectories are quite high and can be expected to be affected by the wind even on relatively calm days.
Finally was there perhaps an anti-north, anti-working class bias in the reporting – most of the athletics reports between 1860 and 1900 concentrated on the activities of the public schools, the military and the southern clubs, and the exploits of a miner in an open Durham meeting would not be likely to gain great acceptance. Perhaps this is partly why there was a reluctance to accept the record?
Appendix. More details on the flight calculations
The author has reported calculations of cricket ball trajectory in normal play in “A unified framework for the prediction of cricket ball trajectories in spin and swing bowling”, and the method that was developed in that paper will be used here . The aerodynamics of cricket balls is quite complex and varies depending upon whether the ball is new or used. Basically there are three aerodynamic forces acting on the ball – the drag, the lift force due to the spin of the ball, and a side force due to differential separation on either side of the ball because of the presence of a seam. The force due to spin is only of relevance at low-ball speeds and will not be considered here. Similarly the side force relies on the seam of the ball having a fixed orientation to the flow, and a spin to stabilize it. Neither of these will be possible in a long throw, and thus this force will also not be considered. With regard to the drag force, the Reynolds’ number of the ball during flight means that the ball will pass through the “drag crisis” associated with the transition from laminar to turbulent separation. In the trajectory calculations it was assumed, based on earlier work, that for a new ball this occurs between Reynolds numbers of 1.8 x105and 2.2 x 105, whilst for old balls, it occurs between 105and 1.4 x 105. In both cases, the low Reynolds number drag coefficient was taken as 0.5 and the high Reynolds number coefficient as 0.3. For the trajectories calculated, the ball passed through the critical Reynolds number range both on the upward part of the trajectory, and on the downward leg.
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.
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.
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.
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.
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.
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.
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 T seconds. If the probability of N for any particular U 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.
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.
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.
Analysis of exceedances
The approach to using this data has been to find, for each dataset, the number of exceedances N 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.
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 X 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 X1 are 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 T 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.
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.
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.
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.