Numerical model for cough‐generated droplet dispersion on moving escalator with multiple passengers

Abstract To investigate the motion of virus‐laden droplets between moving passengers in line, we performed numerical simulations of the distribution of airborne droplets within a geometrically detailed model similar to an actual escalator. The left and right sides and the ceiling of the escalator model were surrounded by walls, assuming a subway used by many people every day with concern to virus‐laden droplets. Steps and handrails were incorporated in the model to faithfully compute the escalator‐specific flow field. The ascending and descending movements of the escalator were performed with 10 or 5 passengers standing at different boarding intervals. To resolve the unsteady airflow that is excited by a moving boundary consisting of passengers, steps, and handrails, the moving computational domain method based on the moving‐grid finite‐volume method was applied. On the basis of the consideration that the droplets were small enough, droplet dispersion was computed by solving the equation of virus‐laden droplet motion using a pre‐computed velocity field, in which the flow rate of a cough, diameter distribution, and evaporation of droplets are incorporated. The simulation resolved the detailed motion of droplets in flow, and therefore, we were able to evaluate the risk of viral adhesion to following passengers. As a result, we found that the ascending escalator had a higher risk of being exposed to virus‐laden droplets than the descending escalator. We also reported that the chance of viral droplet adhesion decreases as the distance from the infected person increases, emphasizing the importance of social distancing.


| INTRODUC TI ON
The pandemic caused by the novel coronavirus COVID-19 is currently ongoing throughout the world, and careful measures are still required to curb significant loss of life and unprecedented economic loss. Fluid droplets expelled from infected individuals by respiratory events have attracted much attention in research because droplet transmission is considered to be one of the most critical mechanisms responsible for the rapid spread and continued circulation of infection among humans. 1 According to a recent study, airborne transmission is the main route for the spread of COVID-19. 2 Multiple studies on saliva droplets have demonstrated that thousands of virus-laden droplets were expelled from the mouth of an infected person through breathing, coughing, and sneezing. [3][4][5] Bahl et al. 6 used a high-speed camera to monitor virus-laden droplets expelled during coughing and sneezing. Asadi et al. 7,8 measured the rate of droplet release during human conversation and discussed the risk of spread.
They demonstrated that viral droplets emitted during human speech could be significantly dispersed, and they discussed the potential spread of infection by asymptomatic patients. The basic mechanism of inhalation of particles in the air and adhesion of particles in airways was summarized by Inthavong. 9 Focusing on the airborne infectious route, virus-laden droplets expelled from the mouth or nose during coughing, sneezing and conversation have the potential to remain suspended in the air for a long time and travel a long distance. Khosronejad et al. 10 16 investigated the effect of cough jets on flow fields and droplet transport characteristics in an airliner cabin section. Talaat et al. 17 studied aerosol transmission and intervention measures on a Boeing 737 cabin zone at different passenger capacities with and without sneeze guards. The effect of cough jet and ventilation was investigated in an elevator, which is considered to have a very high risk of infection due to it being a narrowly confined space. 18,19 During the pandemic, people tended to refrain from using elevators within a confined space to reduce infectious risk, preferring to use escalators within an open space. In particular, at stations and subways, the use of escalators is essential for traveling long height differences and many people use them with low-risk awareness.
However, the airborne transmission risk of escalators has not been fully understood; passengers can be very close together during busy hours such as commuting, long escalators installed at major terminal stations are semi-opened with confined spaces on the left, right, and top. Moreover, considering dispersion characteristics of droplets expelled from an infected person that remain suspended for a long time, passengers may come into contact with virus-laden droplets floating in a distant place as they move on the escalator. Therefore, it is important to predicate the movement of infectious droplets following passengers for a long time and understand the impact on distant passengers and appropriate boarding position. Li et al. 20 investigated the effects of escalator slope and speed on the dispersion of cough-generated droplets from a passenger in regard to steady flow with a Reynolds-averaged Navier-Stokes solver. They observed the difference in wake structures behind the person to significantly impact droplet dispersion: these were defined as "downwash" for the ascending escalator, "upwash" for the descending one.
The result showed that droplets expelled from an infected person on a descending escalator could travel long distances at the height of the person's head, and a person following behind was at risk. Wang et al. 21 studied the dispersion of cough-generated droplets from a person going upwards or downwards with a laboratory experiment in a water tunnel, using mannequin models similar to the one used by Li et al. 20 They used a virtual particle simulation based on the velocity data from the experiments performed to study the effect of the initial position on particle concentration. The result showed that unsteady fluctuations of the airflow could result in the clustering of droplets, in contrast to droplet dispersion in a steady flow field.
They suggested greater social distancing for people riding descending escalators and state that it is critical for a person on an escalator to cough or sneeze with their head down so that the dispersed droplets are entrained into the wake region toward the ground. In both studies, however, the effect of escalator geometry on airflow and droplet dispersion was not considered with a simple escalator model. Instead, the escalator was modeled as a moving floor and confined spaces around the escalator were not considered. This is very different from real escalator environments, where a complex

Practical implication
• Computational investigation of air flow provides unsteady behavior of the dispersion of virus-laden coughgenerated droplets on an escalator, which consists of complex moving boundary conditions.
• Effects of traveling direction of an escalator and boarding position of passengers on the exposure risk are quantitatively examined and shows the exposure risk is higher for the ascending cases than the descending ones.
• Taking a farther distance from the infected person (social distancing) is important to reduce the risk of exposure. flow field is generated by a dynamic combination of moving steps, moving handrails, and static side panels. Also, only a single passenger was included for risk evaluation of infectious droplets and the interaction with other passengers was ignored in the previous studies. 20,21 In this paper, we investigate the effect of cough-generated droplets expelled from an infected person on other passengers riding the same escalator. A geometrically detailed escalator model is used with steps and handrails moving in ascending and descending directions.

| NUMERIC AL APPROACH
To solve this dynamical simulation, the flow fields are computed by solving the Navier-Stokes equations with the moving-grid finite-

| Governing equations for fluid flow
In this study, the airflow in the escalator is computed first. The computed flow field is then used to track the movement of virus-laden droplets. The continuity equation and the three-dimensional uncompressed Navier-Stokes equations used to compute the airflow are as follows: where E a , F a , G a are advection vectors, E v , F v , G v are the corresponding viscous stress vectors, u, v, w are the corresponding velocity components in the x, y, z directions, respectively, p is pressure, and Re is the Reynolds number. Each variable is nondimensionalized as follows: ( % ) represents a dimensional quantity. L 0 is the characteristic length, Ũ 0 is the characteristic velocity, ̃ a is the density of the air (1.2 kg/ m 3 ), and ̃ a is the viscosity of the air (1.8 × 10 −5 Pa • s).

| Discretization method for moving objects
The moving-grid finite-volume method, 22 which is applied to unstructured grids, is used to solve the governing equations for computations with moving boundary conditions. Therefore, the entire computational domain moves along the escalator. In this method, the control volume is considered in four-dimensional space (x, y, z, t) to satisfy the geometric conservation law for space and time. By applying the divergence theorem on the space-time unified control volume Ω, Equation (2) can be written as follows: where ñ = %ñ x , %ñ y , %ñ z , %ñ t is the outward unit normal vector in the The fractional step method 24 is used to solve these equations. The Lower-Upper Symmetric-Gauss-Seidel (LUSGS) method 25 is used to determine the pseudo-time stepping of the equations for velocity.

The Poisson equation for pressure is solved using the Bi-Conjugate
Gradient Stabilized (Bi-CGSTAB) method. 26 The in-house code was used to simulate the various moving objects. [27][28][29] The quantitative validation is also presented in Appendix A in Appendix S1.

| Motion equations for airborne virusladen droplet
In this paper, the motion of airborne droplets was simulated using one-way coupling, and the effect of droplet movement on the flow field was ignored due to very small virus-laden droplets. The droplets expelled from the mouth of a passenger consist of saliva. In addition, it is assumed that all droplets are spherical and their interactions with each other are negligible. The equation of droplet motion is defined as follows: where, d is the density (998 kg/m 3 ), V is volume, r is radius, q d is velocity vector of the droplet, and g is the gravitational acceleration vector.
q a is the airflow velocity at the center position of the droplet, which has second-order accuracy based on gradient reconstruction in a computational cell. Here, considering the effect of contaminants in saliva such as proteins, the boundary between air and water is treated as a no-slip condition. 30 As a rigid sphere, the following equation is used for the drag coefficient 31 : where t is time, D is diffusion coefficient of water vapor, RH is relative humidity (60%), R v is gas constant of water vapor, and T is temperature (27°C). The saturated vapor pressure, e s , can be calculated as where T 0 is base temperature (0 • C) and L is the latent heat of vaporization of water. The saturated vapor pressure at the temperature of T 0 = 0 is e s (0) = 6.11 × 10 2 Pa. In this paper, the minimum radius is set as the equilibrium state in which the droplets do not evaporate any more. The ratio of the equilibrium radius to the initial radius of the droplets is reported by Marr et al. 33 as a function of the humidity.

| Model geometry and simulation conditions
In this paper, a detailed escalator model is used for numerical simulations. The angle of inclination of the escalator is 30 degrees. On the basis of general specifications, 34 the escalator moves at constant speed 0.5 m/s. Figure 1 shows the escalator model geometry:  Table 1. Figure 2 shows the computational grids for (a) ascending with 10 passengers (CASE I), (b) ascending with 5 passengers (CASE  Figure 2E shows a detailed view of grids around the passenger head. The generated grids have a minimum grid size of ~2 mm on the mouth surfaces where a cough jet is exhaled. To ensure that the obtained results are independent of the grid resolution, the effect of the grid size is preliminarily tested with a mesh spacing of ~6 mm (coarse), ~2 mm (medium), and ~ 1 mm (fine) on the mouth surfaces. As described in Figure 3, we confirmed that the spatially averaged kinetic-energy does not vary significantly between the medium and fine meshes. The difference in the welldeveloped state was smaller than 2%, even though the resolution differs by a factor of 2. Therefore, we use the medium mesh in this study. We have also conducted the quantitative validation by considering the flow around a sphere at Re = 100, which is a wellknown configuration, and confirmed that the present method can achieve to simulate the flow at moderate Reynolds number. Details are provided in Appendix A and Figure A1 in Appendix S1. Notably, owing to the presence of boarding passengers, the shape and number of grids are slightly changed. The total number of grid cells used The leading person coughs once after 20 s. To simulate the cough jet exhaled from the mouth, the flow rate measured from a spirometer test is used as a boundary condition. Figure 4 shows the flow rate of the cough ejected from the passenger's mouth. 35 Human respiratory events produce droplets with typical sizes in the range of 10 −1 μm-10 3 μm. 4,7,36,37 This wide range of droplet sizes results in complex particle patterns in the cough flow. To investigate realistic droplet motions, we use the frequency distribution of the diameter of the droplets expelled from the mouth by coughing shown in Figure 5A. This particle distribution was used in the authors' previous study. 12 Figure 5B shows the time variation of cough-generated droplet radius r based on Equation (10) with a minimum radius. The total number of droplet particles expelled from the mouth in a cough is 10 900. 12 The number of droplets generated per unit time is proportional to the flow rate. These droplets are uniformly distributed on the mouth surface of the infected person as the initial location.
The initial velocity is set to be the same as the moving velocity of the escalator.
Next, to quantitatively assess the risk of exposure to infectious droplets, we measure the number of droplets adhered to the head and body of each passenger, where the droplets that penetrate surfaces are considered to be adhered. In this risk evaluation, apart from the droplet dispersion simulation, the number of droplets released by coughing is set to be 100 times larger (i.e., 1 090 000 droplets) to achieve better accuracy. The effect of the riding position (i.e., distance from the infected passenger Δ) on the risk is investigated.

| Estimation of infection probability
To estimate the risk of infection quantitatively, we adopt the doseresponse model used by Bale et al. 38 In the model, the probability of infection is written as follows: where is a factor that accounts for the effect of the variant strain or vaccination, N is the total number of virions inhaled, and N 0 is the number of viral particles leading to infection. In this study, where v is the viral load or viral density (copies/m 3 ), and v 0 d is the total volume of droplets attached to the passenger's body. We set v = 7 × 10 6 copies/ml, as used in the literature. v 0 d is computed by using the droplet size at the time of ejection from the mouth of infected person because the viral load contained in the droplet does not vary during evaporation.   20,21 The flows separate at the tip of the steps, and the magnitude of recirculation (vorticity) decreases as the distance from the first passenger increases due to an augmented flow entrainment. For 5 passengers with a wider riding distance (see Figures 6B,F), in addition to the flow separation at the head, the flow separates at the hip and reattaches at the back; however, the magnitude of vorticity decreases toward the passenger at the back of the queue. A large recirculation is formed in the wider distance between passengers due to a reduced downwash effect. For the descending case with 10 passengers (see Figures 6C,G), flow separates at the hip and reattaches at the head while the separated flow from the head flows backward. This is similar to the experimental result in Wang et al. 21 For the descending escalator (CASE III and IV), in the spaces between passengers, upwash blowing is observed. As a result, the flow separates from the head and advects over the heads at the further back position. For 5 passengers (see Figures 6D,H), a large recirculation region is observed behind the first passenger.

| Dispersion of virus-laden droplets
After the escalator moved for 20 s to ensure the fully developed flow around the passengers, the leading passenger ejected droplets from their mouth by coughing in the horizontal direction. Figure 7 shows the temporal distribution of airborne droplets at t = 0.2, 1.0, 2.0, 4.0 s after coughing for all four cases. Droplets are colored by their radii: red indicates large droplets, while blue represents small droplets. A strong jet is formed due to the coughing, and droplets emanate from the mouth. The jet strength (at t = 0.2 s) does not appear to be affected by the direction of movement. Once the coughing is completed, the jet starts to form a vortex ring, 39 and this is more prominent in the downwards movement ( Figure 7C,D). As expected, large droplets fall faster than small ones due to the gravity and inertia force from flow fields in accordance with Equation (7). Note that the discontinuous dispersion of falling droplets is due to the discrete distribution of initial droplet size used in this simulation (see and entrained into a vortex ring traveling over the heads of passengers. Figure 8 shows the droplet dispersion 30 s after the coughing. In the ascending cases, small (r < 10 −6 ) and medium (10 −6 < r < 10 −4 ) size droplets remain suspended around the passengers. Note that in the ascending case with 5 passengers, a large number of droplets are still moving with passengers. This is because just after coughing, droplets move downward into steps where the air flows in the same direction as the passengers' traveling due to stirring by the convex shape of the step. On the contrary, only medium-sized droplets float in the descending case because small droplets are entrained into a vortex ring and passed over the passengers' heads. Therefore, the suspended height of cough-generated droplets has an impact on long-term exposure time of airborne droplets to passengers. Figure 9A shows a typical snapshot of droplets attached to passengers (CASE III). From the figure, the number of attached droplets decreases as the distance from the infected person (passenger ID 1) increases. Figure 9B shows the temporal evolution of the number of attached droplets for CASE III; the number significantly depends on the distance from the infected person and varies in orders of magnitude. The number of attached droplets increases with time but reached a plateau when t > 40 s. Figure 9C-E display the temporal evolution of the number of attached droplets in CASE I, III, and IV, respectively. In the ascending cases, CASE I (10 passengers) has a higher risk of contact with droplets than CASE II (5 passengers), whereas people in the latter case are exposed to droplets for a long time due to the large recirculation area and stagnated flow near steps. The possible difference between CASE I and CASE II is the height of the high concentration of droplets; the droplet cloud remains suspended around the area above the waist of the coughing person with 10 passengers but falls into the inseam of the coughing person with 5 passengers. In other words, the risk for passengers increases mainly due to the interaction with a high-concentrated droplet cloud rather than prolonged exposure to droplets entrained by recirculation and stagnation near steps. For the descending cases, no significant effect of riding distance is found although CASE IV (5 passengers) has a lower exposure risk than CASE III (10 passengers).

| Risk evaluation by droplets adhesion to passengers
Next, we compare the risk of contact with virus-laden droplets between the ascending and descending cases: the descending cases have a lower risk than the ascending ones because most droplets expelled from the mouth entrained into a vortex ring and passed over the heads of passengers due to the upwash, as shown in Figure 7.
The growth pattern of exposure risk over time is different between the ascending and descending cases; the number of attached droplets rapidly increases for the ascending cases, whereas it gradually increases for the descending ones. This results from an interaction between passengers and the high-concentrated droplet cloud. The present CFD study shows that the risk of exposure to virus-laden droplets is higher in the ascending cases than descending ones, However, the major factor in the increase in droplet exposure is contact with high-concentration droplet clouds. Early droplet clouds are greatly affected by the intensity and expelled angle of a cough. In this study, a direct factor in the low risk for "descending escalators" is that the droplets expelled farther forward (upward) by a cough and did not descend. The cough condition was a steady flow of 6.5 m/s In other words, wake interference was not computed. In addition, Second 74% 24% 13% 6.5% somewhat due to the unsteady fluid motion, a similar trend is observed from all simulations with different cough timings. Figure 9F shows the relationship between the horizontal distance from the in- increases. This trend shows the importance of physical distancing as the World Health Organization recommends. 40 Here, the slope of the graph is almost −1, which means doubling the distance from the infected person reduces the amount of droplet adhesion by half.
This relationship is analogous to the stationary solution of the diffusion equation in spherical coordinates, even though a droplet is not a continuum and its motion is discretely computed herein. This scaling law will be a key consideration to assess the risk of viral droplet contact. However, as shown in Figure 9F, under a number of cases, in last passenger by a factor of 25; A passenger near the infected person faces a higher risk of infection. For this reason, the importance of social distancing must be emphasized. In this paper, the isothermal condition was adopted, we regard that this study estimates the risk of infection on the safe side. Incorporating natural convection into the simulation would predict a lower exposure risk because the updraft around a passenger's face keeps away infectious droplets.
Finally, the effect of coughing direction on the rate of droplet adhesion was investigated. We demonstrated that coughing direction has a critical impact on viral droplet adhesion and, as a result, the infected passenger on the escalator is recommended to cough toward the left or right so that following passengers can avoid contact with concentrated droplets expelled. Although the present simulations require large computational resources, numerical simulation is a powerful tool to resolve the strong unsteady nonlinear fluid dynamics. We hope that our paper will motivate the need to assess the risk of viral droplet adhesion via airborne droplets.

AUTH O R CO NTR I B UTI O N S
Masashi YAMAKAWA conceived the project. Ayato TAKII performed the numerical simulations and analyzed the data with assistance from Masashi YAMAKAWA, Atsuhide KITAGAWA and Tomoaki WATAMURA. Ayato TAKII wrote the paper. All authors contributed discussing the results and providing feedbacks.

ACK N OWLED G M ENTS
This work was partially supported by JST CREST Grant Number JPMJCR20H7, by JKA through its promotion funds from KEIRIN RACE, and by JSPS KAKENHI Grant Number 21K03856.

CO N FLI C T O F I NTE R E S T
The authors declare that they have no known competing financial interests or personal relationships that could influence the work reported in this paper.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of study are available from the corresponding author upon reasonable request.