Modelling and 3D simulations of the dispersion of droplets and drops carrying the SARS-CoV-2 virus inside semi-conned ventilated spaces – Application to a public railway transport coach

7 Computational Fluid Dynamics (CFD) modelling and 3D simulations of the air flow and dispersion of droplets or 8 drops in semi-confined ventilated spaces have found topical applications with the unfortunate development of 9 the Covid-19 pandemic. As an illustration of this scenario, we have considered the specific situation of a 10 railroad coach containing a seated passenger infected with the SARS-CoV-2 virus (and not wearing a face mask) 11 who, by breathing and coughing, releases droplets and drops that contain the virus and that present 12 aerodynamic diameters between 1 and 1,000 µm. The air flow is generated by the ventilation in the rail coach. 13 While essentially 3D, the flow is directed from the bottom to the top of the carriage and comprises large to 14 small eddies visualised by means of streamlines. The space and time distribution of the droplets and drops is 15 computed using both an Eulerian model and a Lagrangian model. The results of the two modelling approaches 16 are fully consistent and clearly illustrate the different behaviours of the drops, which fall down close to the 17 infected passenger, and the droplets, which are carried along with the air flow and invade a large portion of the 18 rail coach. This outcome is physically sound and demonstrates the relevance of CFD for simulating the 19 transport and dispersion of droplets and drops with any diameter in enclosed ventilated spaces. As coughing 20 produces drops and breathing produces droplets, both modes of transmission of the SARS-CoV-2 virus in 21 human secretions have been accounted for in our 3D numerical study. Following these initial results, physical 22 and biological modelling will be extended to the mass transfer between the droplets and the ambient air and to 23 the fate of the virus throughout its transport and dispersion in droplets or drops. Furthermore, a model will be 24 developed to take into account the influence of a face mask on the production of

4 confined spaces. In the latter case, knowledge of the ventilation features is crucial for determining the space and In this perspective, the mode of transmission of the SARS-Cov-2 virus, which relies on droplets transported and 95 dispersed though the atmospheric environment and indoor spaces, at distances from less than one meter to a few 96 meters (and perhaps more), opens up possibilities for using Computational Fluid Dynamics (CFD). Generally 97 speaking, CFD enables choices to be made regarding efficient designs and developments in order to reach 98 desired results and to prevent or limit the adverse effects of critical situations. Therefore, CFD could find a 99 specific breakthrough application in response to the propagation of infectious agents in enclosed spaces. This is 100 precisely the objective of this paper, which aims to demonstrate the relevance of physical modelling and 101 numerical simulation using tried and tested CFD computer software to evaluate the transport and dispersion of 102 drops and droplets carrying the SARS-CoV-2 virus in human secretions and, at a later stage, to evaluate the 103 health consequences of the virus. In addition, the 3D numerical study intends to compare the Eulerian and 104 Lagrangian approaches for dispersion modelling. It is thus a matter of determining which methods are 105 appropriate for simulating the transport and dispersion of drops and droplets in an indoor environment. geometry and the ventilation system considered in this study seem to be very specific, and it is unclear if the 146 patients were actually modelled. Wang et al. 28

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In summary, while these studies represent valuable efforts in modelling and simulation, they are impaired by a 166 number of limitations in terms of the digital mock-up of the modelled confined space and, principally, of the 167 human beings populating this space, and in terms of the size range of the particles accounted for. In contrast, we 168 have opted to pay careful attention to the realism of the geometry, including the human beings, and to consider 169 virus-laden particles ranging in size over four orders of magnitude. The case study we have developed herein 170 corresponds to a public railway transport coach in which a passenger infected with the SARS-CoV-2 virus is With regard to particle size, it is important to recall the usual terminology. By definition, an "aerosol" designates solid or liquid particles suspended in the air. The droplets satisfying this criterion have an aerodynamic diameter in the range between some tenths of micrometers (µm) to some micrometers. The aerodynamic diameter gives Particles with aerodynamic diameters above some micrometers do not form an aerosol. By convention, we make 179 use of the word "droplets" for particles whose diameter is between 1 and 10 µm, whereas the word "drops" is 180 used for particles with a diameter between 100 and 1,000 µm.

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The following parts of the paper are dedicated to presenting the CFD study with the objective of proving the 183 capability of the numerical modelling to represent the air flow and the transport and dispersion of drops and 184 droplets in semi-enclosed ventilated spaces. A railway coach of the type used by millions worldwide for travel 185 on suburban transportation networks is taken as an example in our methodology. The results of this case study 186 are useful not only for similar situations, but can also be directly transposed to any other semi-confined 187 ventilated place, as discussed later on. A summary of the results regarding the airflow and the dispersion of 188 drops and droplets is proposed hereafter. These results lay the foundation for a discussion about the findings of 8 205 Figure 1 shows the single-level rail coach taken as an example throughout the numerical study. The dimensions 206 of the carriage are 15.5 m in length, 2.5 m in width and 2.6 m in height. The original high-precision 3D data of 207 the rail coach geometry and internal layout have been processed from the www.turbosquid.com site and 208 transformed in order to generate a 3D mesh for CFD computations. The coach is occupied by passengers 209 represented by humanoid manikins selected from the www.traceparts.com site and shown in Figure 2.    Figure 3 illustrates the geometry of the rail coach occupied by the passengers. The carriage is subdivided into 226 compartments 1 and 3, which are occupied by the passengers, and compartment 2, from which people get into 227 and out of the train. In the study, the occupation rates of compartments 1 and 3 by seated passengers are 228 respectively 92% and 100%. These figures chosen by the modellers could be varied to examine the influence of 229 the rail coach occupation rate on the distribution of the drops and droplets secreted by the passengers.

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An unstructured CFD mesh was generated from the geometry with tetrahedral cells in order to fit the complex 232 internal geometry of the train and of the passengers, as can be observed on Figure 4. The minimum cell size is 1 233 centimetre near the mouth of the manikins, and 3 centimetres on the body of the manikins and on the seats. The 234 maximum size of the cells is between 5 and 10 centimetres on the internal walls of the rail coach. The mesh 235 consists of 4 million cells in total, a number that was proved to satisfy the convergence of the flow field.    In order to model the air flow in the carriage, one has to implement a ventilation system as with any other semi-247 confined space. The ventilation in the mock-up of the carriage is operated in the same way as for an actual rail 248 coach. It is described briefly hereafter and in more detail in the "Methods" section. While the ventilation system 249 considered in the mock-up of the carriage is quite common, it may be different in other rail coaches. Still, it 250 would not be a major issue to take account of alternative blowing and extracting air vents corresponding to 251 different models of carriages.

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The ventilation system is organised by zones corresponding to the volumes defined in Figure 3. We assume that

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Simulation of dissemination events in the rail coach 3D numerical simulations give unlimited opportunities for studying scenarios of the dissemination of pathogenic 319 biological agents such as the SARS-CoV-2 virus inside a rail coach. In our case, we decided to consider a brief 320 cough and the normal respiration of a passenger assumed to be infected with the virus. In our exploratory 321 computations, the passengers did not wear individual protective masks. The contaminated individual who is 322 coughing was assumed to be seated in compartment 1 of the carriage in one of the two seats shown in Figure 10.

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We also took into account a contaminated individual who was breathing and occupying the position of the green 324 manikin in Figure     summary, the release of droplets and drops due to a cough is a single event of short duration, and the speed of 337 the air carrying the droplets or drops leaving the mouth is high in comparison with the air velocity around the 338 manikin. On the contrary, the release of droplets due to exhalation is repeated at each cycle of respiration, and 339 the speed of the air carrying the droplets is just slightly higher than the air velocity around the manikin. In the 340 test cases involving a cough, droplets and drops of four different aerodynamic diameters (1, 10, 100 and 1,000 Furthermore, a realistic number of droplets or drops (in order of magnitude) was released from the mouth of the 343 infected passengers, either 10,000 particles of each size during the cough or 1,000 particles for each exhalation.

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The three dissemination events reported here (two coughs and one cyclic exhalation) were considered 346 independently. As for the air flow, the transport and dispersion simulations were performed with the CFD model 347 referred to as Code_SATURNE. As far as the dispersion modelling is concerned, two approaches may be 348 employed: Eulerian or Lagrangian. In our study, both approaches were used with two simultaneous aims: first, to 349 compare the results and verify their similarity, and second, to contribute to the development of appropriate 350 methods that are generally applicable to the dissemination of infectious agents in confined, ventilated spaces. In

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This sub-section describes the results obtained for the brief cough of passenger 1, who is assumed to emit 10,000 365 droplets of 1 µm in diameter in 0.5 second. Figure 11 shows the 3D distribution of the micrometric droplets in the inner space of the rail coach. Videos were 368 produced in the framework of this study in order to effectively illustrate the dynamic nature of these results. In

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In the same time interval, the droplets tend to diffuse and dilute in the space between the passengers.

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It should be noted that the Eulerian and Lagrangian dispersion results illustrated at six different instants in Figure   399 11 are remarkably similar, not only for this test case, but for all situations studied. This is a reassuring output that 400 reinforces the potential conclusions drawn from this research based on numerical simulation.

Dynamic behaviour of the droplets and drops generated by a cough
An outstanding feature of the CFD modelling developed in the context of this research is its ability to capture the 404 inherent differences in the aerodynamic behaviour of particles depending on their diameters. Once more, it was 405 verified that the Eulerian and Lagrangian approaches to dispersion led to analogous results and related conclusions. For the sake of concision, only Lagrangian simulations are reported hereafter. This sub-section 407 describes the results obtained for the brief cough of passenger 2, who is assumed to emit 10,000 droplets of 408 either 1, 10, 100 or 1,000 µm in diameter, in 0.5 second.

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After t = 10 seconds, the micrometric droplets spread on both sides and above the passenger seated opposite to the spreader. They become diluted in the space around this passenger, while also moving along the flow imposed 420 by the ventilation system, and gradually reach the seat row next to the group of four seats including the spreader.

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With variations compared to coughing passenger 1, the droplets emitted by passenger 2 have more trajectories    3.5 s after the beginning of the cough. Figure 15 shows the 3D distribution of the drops of 1,000 µm in diameter in the inner space of the rail coach.
these instants are considerably shorter than those considered for the 1 µm and 10 µm droplets, and even for the 467 100 µm droplets, revealing extremely different aerodynamic characteristic times. In Figure 15,

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This numerical research has been carried out with the first principal goal of demonstrating the feasibility of 559 properly accounting for the dynamic processes. This method has required many assumptions to be made. As 560 numerous the hypotheses may be, however, there exist opportunities to remove them, offering scientific 561 perspectives for this work that are enumerated hereafter:

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 In this study, the air flow turbulence is modelled using a Reynolds-averaged Navier-Stokes k-epsilon 563 model. At the cost of increased computational resources, turbulence could be accounted for using large-

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 It would also be a valuable option to model some biological aspects related to the SARS-CoV-2 virus 576 by benefiting from information provided by specialists focusing on this particular virus. For instance, depending on his or her stage of the disease would be crucial quantitative information for estimating the 579 likelihood of healthy passengers becoming infected. One could also model the depletion of infectivity 580 where appropriate, and more generally the fate of the virus in droplets and drops as they dry up, 581 whether they are suspended in the air or deposited onto accessible surfaces.

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 The manikins integrated in the numerical 3D mock-up could be rendered more humanlike and 583 animated. For example, we could alternate manikins of diverse sizes representing male and female 584 adults or children. In this study, we have modelled the mouth of the passenger assumed to be infected 585 with the SARS-CoV-2 virus in order to make him or her breathe out and cough. In the next step, we 586 could easily model the nose of all manikins to make them inhale air and droplets carrying the virus.

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Furthermore, real spectra of droplets and drops should be used as source terms for the breathing out and

Methods 635
The central tenet of this numerical research is to exploit a proven, reliable CFD tool to replace experiments in the 636 real world. This strategy is appropriate insofar as the computational tool operated in the study has been thoroughly validated for simulations of the dispersion of aerosols in laminar or turbulent flows. This study is

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The dissemination events considered in the numerical study originate either from a cough or from exhalation.

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While in both cases droplets or drops are expectorated by the passenger, these events are associated with distinct 723 source terms. The cough leads to a single brief release. Of course, more than one cough, as occurs with a 724 coughing attack, could be considered, with several coughs simulated one after the other. In contrast, exhalation 725 leads to a periodic release related to the respiration cycle. The initial impulse of the expectoration is much higher 726 for the cough than for the exhalation. Yet, in either case, the impulse is directed orthogonally to the mouth of the 727 passenger, with an angle of 15° beneath the horizontal direction. Another difference between coughing and 728 exhalation is the size of the particles produced. While coughing may lead to a full spectrum of droplets and 729 drops, breathing out produces micrometric droplets. Regarding the cough, we decided to consider particles 730 separately over a wide range of sizes, from 1 µm to 1,000 µm in aerodynamic diameter. In a further stage of this 731 research, it would be interesting to adopt a more realistic spectrum produced in the event of a cough. It is also 732 worth noting that other dissemination events such as sneezing or speaking could be envisaged. Indeed, these 733 events are quite close to coughing and breathing out, respectively. Table 2 and Table 3 Table 2. Characteristics of a cough disseminating droplets and drops of different sizes.

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Type of dissemination event A cough is a single expectoration from the mouth.
Duration of coughing 0.5 s Table 3. Characteristics of the exhalation disseminating micrometric droplets.

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Type of dissemination event Breathing out is an intermittent expectoration from the mouth.  Table 4 with the duration of the simulations as the 745 prominent information.
746 747 Table 4. Main features of the aeraulics and dispersion of the droplets and drops considered in the simulations.