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On the impact of roof geometry on rain shelter in football stadiums

J. Persoon (a), T. van Hooff (a), B. Blocken∗ (a), J. Carmeliet (b,c), M.H. de Wit (a)

(a) Building Physics and Systems, Technische Universiteit Eindhoven, P.O. box 513, 5600 MB Eindhoven,

the Netherlands (b) Chair of Building Physics, Swiss Federal Institute of Technology ETHZ, ETH-Hönggerberg, CH-8093 Zürich,

Switzerland (c) Empa, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Building

Technologies, Überlandstrasse 129, CH-8600 Dübendorf, Switzerland

Abstract Apart from sports purposes, stadiums are also increasingly being used for concerts and other events with large spectator attendance. The main aspect for spectator comfort in open stadiums is protection from wind and rain. In the design of many existing stadiums however, rain shelter has insufficiently been taken into account because most roofs have been designed with vertical rainfall in mind, while no consideration has been given to the possibility of rain being blown onto the stands and spectators by wind (wind-driven rain). Adequate rain shelter should be addressed during the design stage of new sports and football stadiums. Currently, almost no information or guidelines on this matter are available. This paper presents an investigation of the impact of roof geometry on rain shelter for stadiums that consist of two separate roof-covered stands facing each other. 2D Computational Fluid Dynamics (CFD) simulations and Lagrangian particle tracking are performed to analyse the wind-flow pattern and rainfall distribution for seven generic stadium cross-section configurations and to assess the performance of each roof type. Although most existing stadium roofs are built with a light to medium upward slope towards the field, the analysis indicates that roofs with a downward slope of 13° provide significantly better rain shelter. The reason is not only the well-known trigonometric shielding effect. In addition, this roof type – as opposed to its counterparts – seems to restrict the extent of the primary vortex in the stadium and appears to generate a sufficiently strong counter-rotating secondary vortex below the roof that sweeps the rain away from the stands.

Keywords: Wind-driven rain; Driving rain; Wind flow; Rain trajectories; Rain shelter; Football stadium; Soccer stadium; Sports stadium; Computational Fluid Dynamics (CFD); Roof type; Guidelines 1. Introduction

Apart from sports purposes, stadiums are also increasingly being used for other activities such as concerts, outdoor movie festivals and other events with large spectator attendance. Spectator comfort is very important and it includes protection of the spectators at the stands from wind and rain. In the design of many existing stadiums however, rain shelter has insufficiently been taken into account because most roofs have been designed with vertical rainfall in mind, while often no consideration has been given to rain that is blown onto the stands and spectators by wind (wind-driven rain or WDR). This is reflected by the fact that the roofs often extend not much further than just above the separation between the stands and the field, which is the case in most stadiums worldwide. Some typical examples of stadiums with such roofs are illustrated in Fig. 1.

Insufficient shelter from rain is one of the main reasons why the bottom rows of many stadiums are unpopular and are often left untenanted. In some stadiums, this problem has been tackled by providing an excessively large roof overhang or by completely closing the stadium roof. However, several disadvantages are associated with these options, such as reduced lifetime of natural and semi-artificial grass covers due to insufficient daylight, insufficient CO2 supply and excessive dampness, insufficient smoke removal from the field and stands, excessive reverberation times, etc. For these reasons, open stadiums are preferred and a compromise

∗ Corresponding author: Bert Blocken, Building Physics and Systems, Technische Universiteit Eindhoven, P.O. box 513, 5600 MB Eindhoven, the Netherlands. Tel: +31 (0)40 247 2138, Fax: +31 (0)40 243 8595. E-mail address: b.j.e.blocken@tue.nl

PRE-PRINT of the article “Persoon J, van Hooff T, Blocken B, Carmeliet J, de Wit MH. 2008. On the impact of roof geometry on rain shelter in football stadia. Journal of Wind Engineering and Industrial

Aerodynamics 96(8-9): 1274-1293”

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has to be found between a roof that performs well in the above-mentioned issues but that also provides sufficient shelter from WDR.

WDR has received quite some attention in the past in other applications. Most recently, numerical simulation of WDR with Computational Fluid Dynamics (CFD) has been applied to investigate wetting patterns on building facades (Choi 1991, 1993, 1994a, 1994b, van Mook et al. 1997, van Mook 1999, 2002, Blocken and Carmeliet 2002, 2004, 2006, 2007, Tang and Davidson 2004) and to determine WDR distributions over small-scale topographic features such as hills and valleys (Arazi et al. 1997, Choi 2002, Blocken et al. 2005, 2006). Recent validation efforts of CFD for WDR studies for these applications (Blocken and Carmeliet 2002, 2004, 2006, Tang and Davidson 2004, Blocken et al. 2006) have provided confidence to extend CFD WDR simulation to other applications.

In this paper, CFD WDR simulations are performed to investigate the impact of roof geometry on rain shelter in football stadiums that consist of two separate roof-covered stands facing each other (Fig. 2). Examples of such stadiums are given in Fig. 1b and d. The geometry in Fig. 2 is also considered representative for stadiums with stands running along a larger part of their circumference but with two clearly separate roofs that are facing each other (Fig. 1a and c). Table 1 provides a non-complete list of football stadiums that fit one of these descriptions. In this study, simplified 2D calculations are made to approximately investigate the behaviour of WDR in cross-section αα’ (Fig. 2) with wind direction perpendicular to the stands.

In section 2, a classification of existing roof geometries is made from which seven generic stadium cross-sections are derived. Section 3 briefly describes the numerical models that include the Reynolds-averaged Navier-Stokes (RANS) equations and a turbulence model for the wind-flow pattern, and Lagrangian particle tracking for the raindrop trajectories. In section 4, model validation is briefly reported. The simulation results of wind flow and rain impact in the seven stadium configurations are presented and discussed in section 5. Finally, sections 6 (discussion) and 7 (conclusions) conclude the paper.

2. Generic stadium configurations

To reduce the extent of the study and to enhance its general character, a classification of existing stadiums based on roof geometry is made. From this classification, a set of seven generic stadium cross-sections is derived. The basic shape (without roof) of all cross-sections is taken from the DSB stadium of football club AZ in Alkmaar, the Netherlands. This stadium is characterised by a roof with a downward slope towards the field (Fig. 3a). The designers of the stadium chose this roof type because they expected that it would provide some more rain shelter than traditional roofs with a light, medium or large upward slope towards the field. The basic shape of the cross-section is given in Fig. 3b. The total height of the stand, without roof, is about 16 m.

The roof type classification with the seven roof types or categories is illustrated in Fig. 4, together with the corresponding generic model cross-section and a few examples of well-known stadiums with roofs belonging to each category. The seven roof types are: (1) a flat roof; (2) a flat roof with a signboard; (3) a flat roof with a backward extension; (4) a flat, elevated roof with a backward extension; (5) a curved roof with an upward slope; (6) a straight roof with an upward slope; and (7) a straight roof with a downward slope. Fig. 5 shows the dimensions of the different generic cross-sections. The 13° roof slope is based on the DSB stadium roof. Note that in all generic configurations, the roof overhang extends just beyond the first row of seats, such that vertical rainfall (rain without wind) does not wet the stands. 3. Numerical models

The steady-state wind-flow pattern over and in the stadiums is obtained by solving the 2D Reynolds-averaged Navier-Stokes (RANS) equations in combination with a turbulence model and an appropriate near-wall treatment, using the commercial CFD code Fluent 6.1.22 (Fluent Inc. 2003). Various turbulence models are available, ranging from the standard, RNG and realizable k-ε models to the more complex Reynolds Stress model. As turbulence modelling inherently imposes the need for model validation, the performance of several turbulence models for two geometries, that show some resemblance to the stadium geometry, is assessed in the next section. From this study the best turbulence model is selected and used for predicting 2D stadium flow. Near-wall modelling can be performed by low-Reynolds number modelling or by using wall functions. Low-Reynolds number modelling implies solving the near-wall flow in the entire boundary layer down to the wall, including the buffer layer and the viscous sublayer. This requires a rather high near-wall grid resolution. Wall functions on the other hand bridge the region between the wall and the logarithmic layer and allow the use of rather coarse grids near solid boundaries. Because of the high Reynolds numbers associated with wind flow around buildings, which imply a very thin viscous sublayer (down to a few mm or even a few microns in thickness), accurately resolving this layer would require very small cells near the wall which would yield an excessively large total amount of cells. Therefore, wall functions are used in this study. The standard wall

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functions by Launder and Spalding (1974) are employed with a sand-grain based roughness modification which requires additional care to limit horizontal inhomogeneity (Blocken et al. 2007).

Lagrangian particle tracking is performed which implies solving the equation of motion of a raindrop moving in a wind-flow field. In a flow field characterised by a mean velocity vector V

r, it is given by:

2

2Rd

2ww

w

dtrd

dtrd-V

4ReC

dρμ3g

ρρ-ρ rr

r=⎟

⎠⎞

⎜⎝⎛⋅⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛ (1)

where ReR is the relative Reynolds number (referring to the airflow around the raindrop):

dtrdV

μρdReR

r

−= (2)

and ρw is the density of the raindrop, ρ the density of the air, g the gravitational acceleration, μ the dynamic air viscosity, d the raindrop diameter, Cd the raindrop drag coefficient, rr the position vector of the raindrop in the xyz-space and t the time co-ordinate. More information on the implementation and parameters for Lagrangian particle tracking for raindrops can be found in (Blocken and Carmeliet 2004, 2006). 4. Model validation

Model validation refers to a systematic comparison between the numerical results obtained with the model and corresponding high-accuracy full-scale or reduced-scale measurements. Because such experimental data are often not available for complex geometries, one has to rely on data obtained for simpler configurations, the main flow features of which show some resemblance to the flow in the actual geometry. The model validation is then performed for the simple configuration, and the confidence that can be obtained from this validation is used to justify the application of the same model for more complex situations.

In this study, the experiments by Kovar-Panskus et al. (2002) in an idealised street canyon (cavity) are used (Fig. 6) because the flow in the cavity exhibits similar vortex structures as will be present inside the stadiums. Note however that there are also significant differences between the two configurations, as will be explained later in section 6. The experiments in the street canyon were conducted in a wind tunnel at a scale of 1:500 for nominally 2D cavities with a height (H) equal to 106 mm and a variable depth (W) in order to create cavities with several aspect ratios including W/H= 1 and 2 (Fig. 6). The neutrally-stratified approach conditions in the experiments were a turbulent boundary layer with height δ = 737 mm (δ/H = 6.95) characterised by a logarithmic vertical velocity profile with free-stream velocity Uref = 8 m/s, aerodynamic roughness length y0 = 0.3 mm, displacement height d = 1 mm and a friction velocity u*ABL = 0.4 m/s. In addition, the flow turbulence characteristics were provided. These profiles were measured just before the separation at the upstream cavity corner (Fig. 6), which is important for their proper use in CFD (Blocken et al. 2007). Because of this measurement position, these profiles are called “incident” flow profiles. Measurements of mean wind speed in the canyon were made along five vertical lines (Fig. 6).

The CFD simulations are performed at model scale, in a 2D computational domain with the same height as the wind tunnel (HWT = 964 mm). The domain is discretised with a structured grid consisting of about 25,000 control volumes. The grid resolution is based on grid-sensitivity analysis. The commercial CFD code Fluent 6.1.22 is used to solve the 2D RANS equations, the continuity equation and the turbulence model equation(s). A range of different turbulence models is tested: the Spalart-Allmaras model, the standard, RNG and realizable k-ε model, the standard and SST k-ω model and the Reynolds Stress model. In all cases, pressure-velocity coupling is taken care of by the SIMPLE algorithm, pressure interpolation is second order and second order discretisation schemes are used for both the convection terms and the viscous terms of the governing equations. The inlet mean wind speed profile is taken equal to the measured “incident” wind tunnel profile. The measured Reynolds stresses are converted to turbulent kinetic energy as input for the simulations. The turbulence dissipation rate profile is given by:

)y(yκ)(u(y)ε0

3ABL+

=∗

(3)

where y is the height co-ordinate, κ the von Karman constant (∼ 0.42) and u*ABL the friction velocity related to the logarithmic mean velocity profile. The top of the computational domain is modelled as a slip wall (zero normal velocity and zero normal gradients of all variables). At the outlet, zero static pressure is specified. Specific attention was paid to the problem of horizontal inhomogeneity (Blocken et al. 2007), which refers to the

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occurrence of unintended streamwise gradients in the vertical flow profiles of mean wind speed and turbulence quantities when they flow over a uniformly rough terrain without obstructions. This means that in the CFD simulation, there can be differences between the inlet profiles and the incident profiles. In the present case, the CFD incident profiles are those obtained at the location of the upstream cavity corner. For the present simulation, where we use the measured incident profiles as inlet profiles, the changes between these profiles have to be avoided because we want to reproduce the wind tunnel experiments as closely as possible. Therefore two measures are taken: (1) The upstream length of the domain was kept as short as possible, i.e. 106 mm, which is allowed because of there is almost no upstream disturbance caused by the cavity flow; (2) in the standard wall functions modified for roughness, the parameters kS (physical roughness height) and Cs (roughness constant) are taken to yield profiles that correspond as much as possible to the measured profiles. Therefore kS is determined from y0 and CS based on this equation (Blocken et al. 2007):

s

0ABLS, C

y9.793k = (4)

More details on this matter can be found in the above-mentioned reference.

From all turbulence models tested, the best results were obtained with the Reynolds Stress model (RSM). The results of this model and their comparison with the experiments are presented in two ways: as wind velocity vectors in the entire cavity (Fig. 7) and as profiles of the horizontal and vertical wind speed components along the five vertical lines along which the measurements were made (Fig. 8). Figs. 7c-d indicate that both the off-centre position of the primary vortex centre and the development of the small secondary vortex in the upstream corner are predicted with good accuracy. Although Fig. 8 reveals some discrepancies in the shear layer just above the cavity, the wind speed at the remaining positions is predicted quite well. Based on the quite good general outcome of the validation study with the RSM, this model is selected for the simulations of wind flow in the generic football stadiums in the next section.

Note that validation of the Lagrangian particle tracking model was not performed for two reasons: (1) Particle tracking involves numerically solving the raindrop’s equation of motion, which is more straightforward and to a lesser extent based on important assumptions, as is the case for turbulence modelling; (2) The results of particle tracking for raindrops have been validated on several occasions in the past (Blocken and Carmeliet 2002, 2004, 2006, Tang and Davidson 2004, Blocken et al. 2006). 5. Model application and results 5.1. Computational domain, grid, boundary conditions and solver settings

The dimensions of the 2D computational domain are L x H = 1100 x 500 m², yielding a blockage ratio of 4 %. For studies that focus on the wind-flow pattern only, the upstream length does generally not need to be larger than about 5H where H is the height of the building (Franke et al. 2004). For WDR studies however, a significantly larger upstream length is needed because the raindrops have to be injected outside the wind-flow pattern that is disturbed by the stadium, i.e. at a sufficient distance upstream and above the stadium. In this case, an upstream length of 400 m was needed, which is much more than 5H ≈ 125 m. Because of the irregular geometry of some of the stadium roofs, unstructured triangular grids are employed. The total number of cells for each stadium geometry is about 24,500.

Four sets of inlet boundary conditions are used. The first set corresponds to wind flow over a uniformly rough, grass-covered terrain (power-law exponent αP = 0.15) with a reference wind speed at 10 m height U10 = 5 m/s. The second has identical terrain features but U10 = 10 m/s. The third and fourth set represent wind flow over suburban terrain (αP = 0.20) with U10 = 5 m/s and 10 m/s, respectively. Turbulence intensity associated with αP = 0.15 ranges from 20% at 2 m height to 5% at gradient height. Turbulence intensity for αP = 0.20 ranges from 30% at 2 m height to 5% at gradient height. The inlet profiles of the mean wind speed ratio U/U10, turbulent kinetic energy k and turbulence dissipation rate ε for both terrains and for U10 = 10 m/s are given as solid lines in Figs. 9a-d. Because of the relatively large distance (400 m) between the domain inlet and the location of the stadiums, streamwise changes in the vertical wind speed profiles will occur upstream of the stadiums (horizontal inhomogeneity) which have to be limited and assessed. This is important because the simulation results that will be obtained will be representative of the incident flow profiles and not of the inlet flow profiles. To reduce the streamwise gradients, the roughness height kS was determined with Eq. (4) and the height of the wall-adjacent cells was taken larger than 2kS, as typically recommended by commercial CFD software employing sand-grain roughness wall function modifications (Blocken et al. 2007). Note that the value of kS was determined based on the assumption that a terrain with αp = 0.15 corresponds to approximately y0 = 0.03 m and that αp = 0.20 corresponds to about y0 = 0.1 m. It is important to report both the inlet profiles and the incident profiles, where the latter are obtained by conducting a CFD simulation in an empty 2D domain that has a similar grid

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distribution as the domains with the stadiums. Fig. 9 shows the degree of horizontal inhomogeneity for the different inlet profiles. The changes are most pronounced near ground-level and for the turbulent kinetic energy. Note that all simulations are performed with the same solver settings (SIMPLE, second-order schemes) as in section 4. 5.2. Wind-flow pattern

Fig. 10 displays the wind-velocity vectors for αP = 0.20 in all seven configurations. All configurations show a large primary vortex with a centre that is located near the downstream stand, similar to the vortex in the cavity with W/H = 2 that was used for model validation. This provides some confirmation of the suitability of the cavity configuration for model validation and turbulence model selection. For stadium configuration 4, the primary vortex is to some extent disturbed by the jet through the opening. The extent to which the primary vortex penetrates below the downstream roof is dependent on the roof geometry. This effect is significant for configurations 5 and 6, with the upward sloping roof. It is less pronounced for the configurations with the flat roofs and appears to be absent for configuration 7 with the downward-sloping roof. In most cases, a secondary (counterclock-wise rotating) vortex is present below both the upstream and downstream roof, which is driven by the primary vortex. Note that this is not the case for roof type 4 (Fig. 4), where the wind jet through the opening destroys the upstream vortex. No significant qualitative differences were found between the wind-flow patterns for U10 = 5 and U10 = 10 m/s and between the patterns for αp = 0.15 and αp = 0.20. The quantitative differences however will have a significant effect on the raindrop trajectories and rain shelter. 5.3. Raindrop trajectories and rain shelter

Fig. 11 displays trajectories for raindrops of 5 mm diameter, for the seven configurations and for the wind-flow field characterised by αP = 0.20 and U10 = 10 m/s. While the downstream stand is for a large part wetted in case of roof types 5 and 6, this is less pronounced for roof type 4, even less for roof types 1, 2 and 3, and almost no wetting occurs for roof type 7.

Fig. 12 shows the influence of raindrop size by illustrating trajectories of 0.5 mm, 2 mm and 5 mm raindrops for three selected configurations (Types 6, 1 and 7) and for αP = 0.20 and U10 = 10 m/s. Raindrops of 0.5 mm represent a median value for drizzle (rainfall intensity Rh < 0.5 mm/h), drops of 2 mm represent the median for a moderately heavy rain spell (Rh = 5 to 10 mm/h) and drops of 5 mm are contained in heavy rain spells (Rh > 20 mm/h). Fig. 12 shows that the larger the drops, and thus the higher their inertia, the less they are disturbed by the wind-flow pattern above and inside the stadium. Also, the larger the raindrops, the larger the part of the stand that is wetted. The behaviour of the raindrops near the downstream stand is significantly influenced by the roof type. Fig. 12a shows that 0.5 mm raindrops that impact on or near the stand have convex trajectories below the upward-sloped roof, almost rectilinear and vertical trajectories below the flat roof and concave trajectories below the downward-sloped roof. The last two roof types effectively shield the stand from small raindrops (drizzle). Fig. 12b shows that, even for larger raindrop sizes, the downward-sloping roof effectively shields the stands, while the other two roof types cannot prevent rain impact on the lower rows of the stands. Note that as the roof slope decreases, the convex shape of the raindrop trajectories also decreases. Finally, Fig. 12c confirms the trends observed in Figs. 12a and b for the largest raindrops.

To some extent, the superior performance of the downward-sloping roof and the poor performance of the upward-sloping roof could be expected. Indeed, if raindrop trajectories were straight lines, it is clear that a down-sloping roof would provide more shelter from rain than a flat roof, and even more than an upward-sloping roof. This is indicated in Fig. 13. Trigonometry yields the following expression for the percentage of the stand that is wetted:

tgγDH

HLL

SS

R

TOT

WET

+= (5)

where HR is the height of the roof edge, HS the height of the stand, DS the depth of the stand and γ the rain trajectory inclination angle, i.e. the angle between the raindrop trajectory and the vertical. With HR = 28.2 m, 20.4 m and 16.7 m for roof types 6, 1 and 7 respectively (see Fig. 5), HS = 16 m and DS = 26 m, LWET/LTOT is shown in Table 2 for γ = 10° and 20°. The angles 10° and 20° are chosen because they are to some extent representative for the inclination of the drop trajectories near the roof edge, in the U10 = 10 m/s flow pattern. It is clear that roof type 7 performs best according to Eq. (5). In this paper, the rain shelter effect that is predicted by this simple equation will be called the trigonometric shielding effect. This term is adopted from earth sciences in

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which a trigonometric model (assuming rectilinear raindrop trajectories) is often used to determine the WDR intensity on sloping soil surfaces (Fourcade 1942, Blocken et al. 2006).

Trigonometric reasoning is probably why the designers of the DSB stadium (Fig. 3a) chose a roof with a downward slope. However, the particular inclination of the roof in combination with the wind-flow pattern in the stadium appears to give rise to an important additional effect. Fig. 14 shows the details of the extension of the primary vortex into the region below the downstream roof and the interaction of this vortex with the secondary vortex below the roof. Fig. 14a illustrates that the primary vortex is not significantly restricted by the upward-sloping roof and extends a considerable distance below this roof. This is more clearly indicated in Fig. 14b that shows the horizontal velocity component along a vertical line below the edge of the roof. The extent of the primary vortex provides an extra driving force to the rain and sweeps it towards the stands (Fig. 14c). In Fig. 14d the primary vortex experiences a larger obstruction by the roof type and its strength below the roof is weakened, as indicated in Fig. 14e. Less rain is therefore swept towards the stands (Fig. 14f). Finally, the downward-sloping roof prevents the primary vortex from entering the region below the roof (Fig. 14g). As a result, the secondary vortex, which rotates in opposite direction, governs the flow below the roof edge. Figs. 14h and 14i show that this sweeps the raindrops that are falling passed the roof away from the stand. The behaviour of the primary and secondary vortices is also responsible for the observations for the other raindrop trajectories in Fig. 12, including the convex versus concave trajectories for d = 0.5 mm which clearly indicate the effect of the direction of the local wind vectors.

To indicate the importance of this effect compared to the trigonometric shielding effect, Table 2 compares the ratio LWET/LTOT obtained by the two models, the trigonometric model (with trajectory inclinations 10° and 20°) and the CFD model (for αp = 0.20 and U10 = 10 m/s). Comparing the values in Table 2 shows that the trigonometric model significantly underestimates the rain shelter for roof types 1 and 7.

Finally, Fig. 15 summarises the performance of all seven roof types in terms of rain shelter by plotting the percentage of the stand that is wetted for the four wind-flow patterns (U10 = 5 m/s, U10 = 10 m/s, αP = 0.15 and αP = 0.20). The order on the horizontal axis indicates decreasing performance. The roof with the downward slope is clearly superior in all circumstances investigated, followed by the different flat roof types. Roofs with upward slopes can experience considerable wetting, going up to even 50% for rain spells in moderately strong winds. 6. Limitations of the study

It is important to note the limitations of this study. The validation was conducted for idealized canyons (cavities), while the geometry of the 2D stadium model is quite different from a cavity because it is an semi-enclosed space that is separated from the exterior by walls. Furthermore, the cavity aspect ratios were limited to W/H = 1 and 2, while the stadium aspect ratio is about W/H = 4. In spite of these differences and although the aspect ratio of the structure influences the aspect ratio of the primary vortex, the essential features of the vortices in the cavities and the stadiums show some similarities. In both cases, the shear layer drives the primary vortex, which in turn generates secondary vortices, and the vortex centre is shifted downstream in both the cavity with W/H = 2 and the stadium configurations. The validation has been considered suitable to select the turbulence model in this study, but future research should focus on more detailed validation studies. These studies will include wind tunnel experiments with 3D generic stadium models.

Simplified 2D simulations were performed in the middle vertical cross-section of the stadiums. In reality, pronounced 3D behaviour will be present near the edges of the stands, and due to the limited length of the stands, the flow in the middle vertical cross-section might also show some deviations from 2D flow. Although the study in this paper provided an indication of the relative performance of the different roof types, further research is necessary and will include 3D simulations of wind-flow patterns and raindrop trajectories to study roof type performance. 3D simulations also allow investigating rain shelter for different wind directions and for different stadium geometries. Many stadiums consist of stands that run along the entire perimeter together with the roof structure. In such stadiums, the wind flow patterns will be quite different and more complex than those in the present paper. More research is therefore required before the validity of the present findings can be extended to other, more complex stadium configurations.

Finally, this study only included simulations for medium-rise stadiums, with a basic roof height of 20 m (Fig. 5). Although this height is representative for quite a large number of existing stadiums, many higher stadiums do exist. Further research should investigate the validity of the present conclusions for higher stadiums, because their different width-to-height ratio (aspect ratio) might lead to different vortex and raindrop behaviour. 7. Conclusions

The impact of roof geometry on rain shelter in football stadiums that consist of two separate roof-covered stands facing each other has been investigated with 2D Computational Fluid Dynamics simulations and Lagrangian particle tracking. The following conclusions are made:

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• The calculated wind-flow pattern in the stadium cross-section appears to consist of a large primary vortex with a centre shifted to the downstream stand, and a secondary vortex below the roof of the downstream stand.

• The horizontal extent of the primary vortex is limited by the roof construction. This effect is least pronounced for an upward-sloping roof, more pronounced for a flat roof and most pronounced for a downward-sloping roof, where the primary vortex does not enter the region below the roof.

• The downward-sloping roof prevents the primary vortex from entering the region below the roof. Instead, a counter-rotating secondary vortex occupies this space and sweeps the raindrops away from the stand. The upward-sloping roofs allow the primary vortex to significantly extend below the roof and to sweep the raindrops to the stands.

• Although the better performance of the downward-sloping roof compared to the flat roofs and upward-sloping roofs could be expected based on trigonometric reasoning, the CFD study has revealed an important additional effect, namely the action of the primary and secondary vortices on the raindrops entering the region near and below the roof.

• This study was performed with simplified 2D modelling and for stadiums of medium height. Further research is needed to investigate the validity of the present findings for different wind directions, for other stadium heights and certainly for other stadium geometries such as those with stands and roofs running along the entire perimeter.

Acknowledgements

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Tang, W., Davidson, C.I., 2004. Erosion of limestone building surfaces caused by wind-driven rain. 2. Numerical modelling. Atmospheric Environment 38(33), 5601-5609.

van Mook, F.J.R., de Wit, M.H., Wisse, J.A., 1997. Computer simulation of driving rain on building envelopes, in: Proceedings 2nd European and African Conference on Wind Engineering (2EACWE), Genova, Italy, 1059-1066

van Mook, F.J.R., 1999. Full-scale measurements and numeric simulations of driving rain on a building, in: Proceedings 10th International Conference on Wind Engineering (10ICWE), Copenhagen, Denmark, 21-24 June, 1145-1152.

van Mook, F.J.R., 2002. Driving rain on building envelopes, Ph.D. thesis, Building Physics and Systems, Technische Universiteit Eindhoven, Eindhoven University Press, Eindhoven, The Netherlands, 198 p.

9

a b

c d

Figure 1. (a) Grotenburg Stadium, Uerdingen, Germany; (b) Netanya Stadium, Netanya, Israel; (c) Gwangju World Cup Stadium, Gwangju, South-Korea; (d) Estadio Municipal de Braga, Braga, Portugal (© World Stadiums / www.worldstadiums.com)

Figure 2. Basic plan of football stadium (dimensions in meter) with indication of cross-section αα’.

10

a b

Figure 3. (a) Stand cross-section of the DSB stadium (AZ, Alkmaar, the Netherlands) with downward sloping roof (© Zwarts and Jansma Architects, reproduced with permission). (b) Basic stand configuration based on the DSB stadium.

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Stadium roof type 1: Flat roofAnfield Road, Liverpool Simplified model Examples

Celtic Park, Glasgow (UK)

De Euroborg, Groningen (NL)

Anfield Road, Liverpool (UK)

Stadium roof type 2: Flat roof with a signboardBayArena, Leverkusen Simplified model Examples

Stadium roof type 3: Roof with backward extensionVolkswagen Arena, Wolfsburg Simplified model Examples

Stadium roof type 4: Elevated roof with backward extensionEmirates Stadium, London Simplified model Examples

Stadium roof type 5: Curved roof

Stadio San Nicola, Bari Simplified model Examples

Stadium roof type 6: Ascending roof towards the fieldKoning Boudewijn Stadion, Brussels Simplified model Examples

Stadium roof type 7: Descending roof towards the fieldDSB Stadion, Alkmaar Simplified model Examples

De Vijverberg, Doetinchem (NL)

Fenixstadion, Genk (BE)

BayArena, Leverkusen (GER)

Ruhrstadion, Bochum (GER)

Veltins Arena, Gelsenkirchen (GER)

Volkswagen Arena, Wolfsburg (GER)

Estádio do Dragão, Porto (POR)

Stade de France, Paris (FR)

Emirates Stadium, London (UK)

Amsterdam Arena, Amsterdam (NL)

Stadio Delle Alpi, Turin (IT)

Stadio San Nicola, Bari (IT)

Nou Camp, Barcelona (ESP)

Herman Vanderpoorten Stadion, Lier (BE)

Koning Boudewijn Stadion, Brussels (BE)

De Kuip,Rotterdam (NL)

Old Trafford, Manchester (UK)

DSB Stadion, Alkmaar (NL)

Figure 4. Classification of stadium roof types into seven categories, the corresponding generic stadium cross-section models and examples of well-known stadiums belonging to each category.

12

Figure 5. Cross-sections of the generic stadium configurations with the different roof geometries (dimensions in meter).

Figure 6. Idealised urban street canyons tested in the wind tunnel experiments, with indication of the vertical lines along which the measurements were made.

13

Figure 7. Wind tunnel measurements and CFD simulation results for the two cavities: (a) Wind tunnel, W/H = 1; (b) CFD, W/H = 1; (c) Wind tunnel, W/H = 2; (d) CFD, W/H = 2.

14

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

X/W [-]

Y/H

[-].

SimulationMeasurement

0.09 0.3 0.5 0.7 0.9

U/Uref = 0.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

X/W [-]

Y/H

[-]…

.

SimulationMeasurement

0.16 0.3 0.5 0.7 0.84

V/Uref = 0.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

X/W [-]

Y/H

[-].

SimulationMeasurement

0.16 0.3 0.5 0.7 0.84

U/Uref = 0.5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

X/W [-]

Y/H

[-]…

.

SimulationMeasurement

0.09 0.3 0.5 0.7 0.9

V/Uref = 0.5

H

W

W/H = 2

H

W

W/H = 2H

W

W/H = 1

H

W

W/H = 1

a c

b d

Figure 8. Comparison between wind tunnel measurements and CFD simulation results for W/H = 1: (a) horizontal and (b) vertical velocity component; and for W/H = 2: (c) horizontal and (d) vertical velocity component.

15

0

100

200

300

400

500

0 0.5 1 1.5 2 2.5U/U10, k (m²/s²), (m²/s³)

y[m

]

0

100

200

300

400

500

0 0.5 1 1.5 2U/U10, k (m²/s²), (m²/s³)

y[m

]

0

5

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2U/U10, k (m²/s²), (m²/s³)

y[m

]

0

5

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2 2.5U/U10, k (m²/s²), (m²/s³)

y[m

]

a b

c d

ε

ε ε

ε

k

U/U10

ε

k U/U10

ε

k

U/U10ε

k

U/U10

ε

Inlet (x = 0 m)x = 400 m

Inlet (x = 0 m)x = 400 m

Figure 9. Comparison of inlet (x = 0 m) and incident (x = 400 m) vertical profiles of mean wind speed ratio U/U10, turbulent kinetic energy k and turbulence dissipation rate ε illustrating the occurrence of horizontal inhomogeneity in the empty computational domain for U10 = 10 m/s and (a-b) αP = 0.15; (c-d) αP = 0.20.

16

Type 1: Flat roof Type 2: Flat roof with a signboard

Type 3: Roof with backward extension Type 4: Elevated roof with backward extension

Type 5: Curved roof Type 6: Ascending roof towards the field

Type 7: Descending roof towards the field

a b

c d

e f

g

Figure 10. Wind-flow pattern for αP = 0.20 and for each stadium configuration.

17

Type 1: Flat roof Type 2: Flat roof with a signboard

Type 3: Roof with backward extension Type 4: Elevated roof with backward extension

Type 5: Curved roof Type 6: Ascending roof towards the field

Type 7: Descending roof towards the field

a

c

e

g

b

d

f

Figure 11. Trajectories of raindrops with diameter d = 5 mm in the U10 = 10 m/s and αP = 0.20 wind-flow pattern and for each stadium configuration.

18

Type 6 Type 1 Type 7p

p

p

Figure 12. Trajectories of raindrops with different diameters in the U10 = 10 m/s and αP = 0.20 wind-flow pattern and for three stadium configurations. (a) d = 0.5 mm; (b) d = 2 mm; (c) d = 5 mm.

Figure 13. Illustration of the trigonometric shielding effect for stadiums with (a) roof type 6, (b) roof type 1 and (c) roof type 7.

19

U/Uref = 0.2

0

5

10

15

20

25

30

4 8 12

Hei

ght[

m]

0

5

10

15

20

25

30

4 8 12

Hei

ght[

m]

U/Uref = 0.2

0

5

10

15

20

25

30

4 8 12

Hei

ght[

m]

U/Uref = 0.2a b c

d e f

g h i

Type

7

Typ

e 1

Typ

e 6

Figure 14. CFD simulation results for U10 = 10 m/s (= Uref) and αP = 0.15. (a, d, g) Detail of interaction between primary and secondary vortex near and below the stand; (b, e, h) Horizontal mean velocity component along a vertical line below the roof edge; (c, f, i) Trajectories of 5 mm raindrops as influenced by the local wind-flow pattern.

20

U10 = 5 m/sαP = 0.15

U10 = 10 m/sαP = 0.15

Figure 15 (a-b). Comparison of roof type performance in terms of the percentage of wetted stand length, for (a) αP = 0.15, U10 = 5 m/s; (b) αP = 0.15, U10 = 10 m/s; (c) αP = 0.20, U10 = 5 m/s; and (d) αP = 0.20, U10 = 10 m/s.

21

U10 = 5 m/sαP = 0.20

U10 = 10 m/sαP = 0.20

Figure 15. Continued. (c-d) Comparison of roof type performance in terms of the percentage of wetted stand length, for (a) αP = 0.15, U10 = 5 m/s; (b) αP = 0.15, U10 = 10 m/s; (c) αP = 0.20, U10 = 5 m/s; and (d) αP = 0.20, U10 = 10 m/s.