The Coronavirus Disease 2019 pandemic has become an unprecedented global challenge for public health and the economy. As with other respiratory viruses, coronavirus is easily spread through breathing droplets, particularly in poorly ventilated or crowded indoor environments. Therefore, understanding how indoor environmental conditions affect virus transmission is crucial for taking appropriate precautions. In this study, the effects of different natural wind-driven ventilation conditions and ambient relative humidities (RHs) on the cough droplet transmission in an indoor environment are investigated using the large eddy simulation approach with Lagrangian droplet tracking. The simulations show that the velocity and temperature of droplets significantly decrease in a short time after ejection. This feature for droplet velocity and temperature is more pronounced at smaller inlet wind speed (Vin) and larger Vin or lower RH, respectively. Wind-driven ventilation plays a crucial role in affecting the horizontal transmission distance of cough droplets. Under strong natural ventilation conditions (Vin = 4.17 m/s), cough droplets can spread more than 4 m within 1 s, whereas they can only travel within 2 m under weak ventilation with Vin = 0.05 m/s. The results confirm that the social distancing of 2 m is insufficient, while revealing that proper ventilation control can significantly remove virus-laden droplets from indoor air. We believe that there is no absolute safe social distancing because the droplet transmission and dispersion are mainly controlled by the local environmental conditions, and for safety, we recommend wearing a face mask and maintaining good indoor ventilation to reduce the release of potentially virus-laden droplets into the air.

The Coronavirus Disease 2019 (COVID-19) pandemic has presented an unprecedented global challenge to both public health and the economy. As reported by the Centers for Disease Control and Prevention (CDC), the COVID-19 could be spread in three major routes: close contact, droplet transmission, and airborne or aerosol transmission.1 The contact transmission is primarily through direct contact with an infectious person, such as touching during a handshake, or with a contaminated surface.1 The droplet transmission is mainly via virus-laden droplets (with a diameter larger than 100 μm) that form in the respiratory organs of an infected person and are exhaled from the nose and mouth during sneezing, breathing, coughing, and talking.1,2 The aerosol transmission is associated with small virus-laden respiratory droplets (with a diameter smaller than 100 μm) that can remain suspended and transported in the air over a long time and distance.1,2 Among these three transmission routes, the latter two routes are regarded as the major routes of virus propagation.3 Therefore, a comprehensive understanding of the airborne behaviors of respiratory droplets is critical to mitigating the risk of infection and breaking the transmission chain of viruses.

Experimental measurements and numerical simulations are two commonly used methods to investigate the transmission behaviors of respiratory droplets.4 Experimental investigations often involve the use of particle image velocimetry (PIV) system and high-speed cameras to capture the size and distribution of respiratory droplets, as well as their motion trajectories, to gain insight into the hydrodynamic process of respiratory events. For instance, Zhu et al.5 utilized a PIV system to study the transport characteristics of saliva droplets produced by coughing in a calm indoor environment. They found that during each individual cough, more than 6.7 mg of saliva droplets were expelled at speed of up to 22 m/s and that these droplets could travel distances exceeding 2 m. Xie et al.6 conducted a series of experiments to measure the number and size of respiratory droplets produced from the mouth of healthy individuals during talking and coughing with and without a food dye. These measurements showed that the average size of droplets was about 50 100μm and more droplets seemed to be generated when a food dye was used. Busco et al.7 captured the flow characteristics of human sneezing using high-speed cameras and pointed that the sneezing ejection lasted about 0.5 s and mainly contained a heterogeneous mixture of moist air and saliva droplets. By studying the hydrodynamics of violent respiratory events such as coughing and sneezing, Bourouiba et al.8 found that small droplets (less than 50 μm in diameter) could remain suspended in the air for a time long enough that could reach a height of 4 6 m to contaminate ventilation systems. Verma et al.9 used qualitative visualizations to examine the performance of masks equipped with exhalation valves. Visualizations revealed that masks leaked a large number of droplets through the exhalation valves, which significantly reduced their effectiveness as a means of source control. Overall, experimental measurements can provide reliable results and improve our understanding of virus transmission. However, when compared to numerical simulations, experimental measurements are less flexible in investigating the effects of different environmental conditions and boundary conditions on the physical transport mechanisms and their couplings in the complex virus aerosol transmission.4 

Concerning the numerical simulations, Reynolds-averaged Navier–Stokes (RANS) equations are commonly applied to study the airborne transmission of respiratory droplets. For example, Blocken et al.10 used the RANS to investigate whether a first person moving nearby a second person at a distance of 1.5 m or beyond could cause droplets to transfer to the second person. To reduce the risk of infection, they suggested that the second person avoid waking or running in the slipstream of the leading person and maintained a distance of 1.5 m in staggered or side-by-side arrangement, or kept a larger social distancing. Li et al.11 and Feng et al.12 used RANS approach to investigate the effects of different wind speeds and relative humidities (RHs) on social distancing during human coughing in an outdoor and enclosed environments, respectively. Li et al.11 reported that the travel distance for a 100 μm droplet could be up to 6.6 m under a wind speed of 2 m/s. Feng et al.12 emphasized the necessity of a social distancing longer than 1.83 m due to the complex real-world environment ventilation conditions. Using the similar numerical approach, Wu et al.13 studied the effects of seating arrangements and air conditioning on the spread of sneezing droplets in cafeteria environments. Pal et al.14 evaluated the infection risks associated with two distinct scenarios (i.e., when students were infected and when the teacher was infected) in a classroom. Li et al.15 examined the impacts of environmental conditions (e.g., changes in ventilation location, ventilation rate, and relative humidity) and variations in dental patient's breathing rate on droplet transmission during dental service. More recently, the large eddy simulation (LES) has been employed to study respiratory droplet propagation since it predicts the flow features more accurate than RANS.16 For instance, Khosronejad et al.17,18 used LES to study the transmission of saliva droplets during human coughing with and without a face mask. They found that a non-medical grade face mask could drastically reduce the propagation of saliva particles to 0.72 m away from the person, while without a face mask, saliva particulates could travel over 2.2 m away from the person. Pendar and Páscoa19 applied LES to investigate the transport behaviors of saliva droplets and associated hydrodynamics in a room. They concluded that increasing the safe distance to ∼4 m during a sneeze is necessary and wearing a mask and using the elbow to cover the mouth during a sneeze could reduce the contamination area to one-third and three-quarters, respectively, which served as effective protective measures. Norvihoho et al.20 used LES to examine the impact of different ventilation rates and exhaust layouts on the transmission of cough droplets in an office setting. Their results showed that the ventilation rates enhance the spreading of viral droplets in the office, but compared to the up-exhaust cases, the down-exhaust cases seemed to have performed better in protecting the healthy person.

While previous studies have provided valuable insights into the transport of respiratory droplets, there still remains a research gap concerning the impact of environmental conditions such as wind speed and relative humidity on aerosol transmission. Therefore, the main objective of this study is to address this gap by investigating the effects of different ambient wind speeds and relative humidities on the transmission behaviors of cough droplets. Moreover, the indoor environment shows unique challenges in controlling virus transmission due to its complex flow patterns21 and longer virus survival times13 compared to the outdoor environment. Thus, this study specifically focuses on the airborne transmission of cough droplets within a chamber (i.e., an indoor environment) rather than an outdoor environment. The present study is complementary to the studies of Li et al.11 and Feng et al.,12 who investigated cough droplet transmission under similar environmental conditions. It is expected that the present investigations will further enhance our understanding of respiratory droplet transmission in indoor environments.

The remainder of this paper is organized as follows: Sec. II briefly introduces the numerical methods for the indoor airflow and the cough droplets. In Sec. III, the computational domain, design cases, and their parameter settings are described. Simulation results are shown and discussed in Sec. IV. Conclusions are given in Sec. V.

In this study, the LES approach is used to simulate the indoor turbulent airflow. Within the LES framework, the large-scale eddies are directly solved by the filtered transient Navier–Stokes equations, while the small-scale eddies are modeled using a sub-grid-scale (SGS) model. The filtered equations that express the conservation of mass, momentum, and energy for a Newtonian incompressible flow are written as
u ̃ i x i = 0 ,
(1)
u ̃ i t + ( u ̃ i u ̃ j ) x j = 1 ρ p ̃ x i + 1 ρ σ ̃ i j x j 1 ρ τ i j x j ,
(2)
h ̃ s t + ( u ̃ i h ̃ s ) x i 1 ρ p ̃ t u ̃ j ρ p ̃ x i 1 ρ x i ( λ T ̃ x i ) = x j [ ρ ( u i h s ̃ u ̃ i h ̃ s ) ] ,
(3)
σ ̃ i j = μ f [ ( u ̃ i x j + u ̃ j x i ) 2 3 δ i j u ̃ k x k ] ,
(4)
τ i j = ρ ( u i u j ̃ u ̃ i u ̃ j ) ,
(5)
where u ̃ i is the filtered fluid velocity, ρ is the fluid density, p ̃ is the filtered fluid pressure, σ ̃ i j is the filtered viscous stress tensor, μf is the fluid dynamic viscosity, δij is the Kronecker symbol, h ̃ s is the filtered sensible enthalpy, λ is the thermal conductivity, T ̃ is the filtered fluid temperature, and τij is the unknown SGS stress tensor, representing the effect of the SGS motion on the resolved field, which needs to be modeled using a SGS model so that the governing equations can be solved.
In the present work, the dynamic kinetic energy SGS model is employed due to its better representation of the SGS stress. Combining with the eddy-viscosity assumption (Boussinesq hypothesis), τij can be rewritten as
τ i j = 2 3 ρ k s δ i j 2 C k ρ k s 1 2 Δ f S ̃ i j ,
(6)
S ̃ i j = 1 2 ( u ̃ i x j + u ̃ j x i ) ,
(7)
where Ck is the model constant and determined dynamically,22 Δf is the filter size, S ̃ i j is the filtered strain rate tensor, and ks is the SGS turbulence kinetic energy and can be obtained by solving the following transport equation:
ρ k s t + ρ ( u ̃ j k s ) x j = τ i j u ̃ i x j C ε ρ k s 3 2 Δ f + x j ( μ t σ k k s x j ) ,
(8)
where C ε is the model constant and determined dynamically, μt is the SGS turbulent viscosity, and σk is hardwired to 1.0. Detailed descriptions of the dynamic SGS kinetic energy model can be found in the study of Kim and Menon.23 The validation of the present LES approach can be found in  Appendix A.
Species transport and mass diffusion of water vapor in the turbulent flow are, respectively, formulated by the following equations:
( ρ Y ) t + · ( ρ Y u ) = · J + S Y ,
(9)
J = ( ρ D m + μ t S c t ) Y ,
(10)
where Y is the local mass fraction of water vapor, u is the fluid velocity, J is the mass diffusion flux of water vapor, SY is a source term representing the effect of evaporation of the dispersed phase, Dm is the mass diffusion coefficient of water vapor in the mixture, and Sct is the turbulent Schmidt number and its value is set to 0.7.22 
In the current work, the initial released cough droplets are treated as spherical pure water droplets for simplification. During transmission process, the coupled effects of evaporation, droplet breakup, and heat transfer are considered. Each droplet is tracked individually in a Lagrangian reference frame. Only the aerodynamic drag and gravitational force are considered because the droplet-to-air density ratio is of the order O(103). Droplet motion equation is written as
m p d u p d t = F d + F g = m p u u p τ p + m p g
(11)
and
τ p = ρ p d p 2 18 μ f 24 C d R e p ,
(12)
R e p = ρ d p | u u p | μ f ,
(13)
C d = a 1 + a 2 R e p + a 3 R e p 2 ,
(14)
where mp is the droplet mass, u p is the droplet velocity, F d represents the aerodynamic drag, F g represents the gravitational force, g is the gravitational acceleration and its value is 9.81 m/s2, τp is the droplet relaxation time, ρp is the droplet density, dp is the droplet diameter, Rep is the droplet Reynolds number, Cd is the drag coefficient, and a1, a2, and a3 are fitting constants that allow the drag law to be valid for a wide range of Rep, which are given by Morsi and Alexander.24 
Droplet evaporation is modeled by the following conservation equation:25 
d m p d t m ̇ p = ρ k c A p ln ( 1 + B m ) ,
(15)
where kc is the mass transfer coefficient of water vapor, which is calculated from an empirical model of the Sherwood number Sh, as shown in Eq. (16). Ap is the droplet surface area, and Bm is the Spalding mass number given by Eq. (17).
S h = k c d p D m = 2.0 + 0.6 R e p 1 2 S c 1 3 ,
(16)
B m = Y i s Y i 1 Y i s ,
(17)
where S c = μ / ( ρ D m ) is the Schmidt number, Yis is the water vapor mass fraction at the droplet surface, and Y i is the water vapor mass fraction in the bulk flow. The validation of the present evaporation model can be found in  Appendix B.
Droplet temperature is updated by solving the energy balance equation, which accounts for both convective heat transfer and latent heat transfer between the droplet and the continuous phase:22,
m p c p d T p d t = h A p ( T T p ) h f g m ̇ p ,
(18)
where cp is the droplet heat capacity, Tp is the droplet temperature, T is the temperature of continuous phase, hfg is the latent heat of phase change, and h is the convective heat transfer coefficient calculated by a modified Nusselt number Nu,
N u = h d p λ = l n ( 1 + B T ) B T ( 2 + 0.6 R e p 1 2 P r 1 3 ) ,
(19)
where Pr is the Prandtl number of the continuous phase, and BT is the Spalding heat transfer number determined by the Spalding mass number
B T = ( 1 + B m ) 1 L e · S h N u · c p v c p g 1 ,
(20)
where Le is the Lewis number, cpv is the specific heat of droplet vapor, and cpg is the specific heat of gas mixture. For more details, readers can refer to Ref. 22.

The droplet breakup dynamics are modeled by the Taylor Analogy Breakup (TAB) model,26 which takes into account droplet shape oscillation, distortion, and breakup. When the shape of a droplet changes during breakup, a dynamic drag model is used to calculate the drag force acting on the distorted (non-spherical) droplet.22 Additionally, the droplet stochastic collision and coalescence are simulated using the algorithm proposed by O'Rourke.27 This algorithm uses the concept of a collision volume to calculate the probability of collision and has second-order accuracy in spatial space.

In this study, we consider a standard single room with dimensions of x × y × z = 4 m length × 2.8 m height × 3 m width, which is nearly identical to the chamber used in the work of Yan et al.28 The chamber is ventilated via a door (2.1 m height × 0.9 m width) and a square window (1.5 m side length) on the side walls. The geometries and dimensions of the door and window are similar to those reported in the study of Pal et al.14 Other geometric parameters are shown in Fig. 1(a). We assume that a virtual person is standing at the door and coughing. The shape of the cough mouth opening is simplified as a rectangular nozzle with dimensions of 0.04 m length × 0.0048 m width, i.e., with an aspect ratio of 8.33. This simplification of mouth shape is based on a rectangular-like mouth-print, which represents the maximum human mouth opening observed and captured by Dbouk and Drikakis29 using a high-speed camera. The center of the mouth opening is located at x = 0, y = 1.63 m, and z = 0.95 m. It should be pointed out that in the present simulations, human cough is just simplified as a jet at the inlet boundary and the detailed representation of the real human body is not considered. This approach is similar to the treatments adopted by Dbouk and Drikakis29 and D'Alessandro et al.30 

FIG. 1.

(a) Schematic diagram and geometric parameters of the chamber. (b) Discretized structured grids of the computational domain.

FIG. 1.

(a) Schematic diagram and geometric parameters of the chamber. (b) Discretized structured grids of the computational domain.

Close modal

The computational domain is discretized by a structured grid system using ANSYS ICEM 19.31 The grids are well refined in the near-wall regions to capture the boundary layer flow characteristics. Near the wall, the thickness of the first grid layer is set to 0.001 m and then gradually coarsened in the outward direction with a stretching ratio of 1.1. The grids are also carefully generated in the near-mouth region. The discretized grids for the chamber are displayed in Fig. 1(b), with a total of 1.02 × 10 6 cells (see the  Appendix C for grid independence analyses).

During a human cough, the mouth can turn into an atomizer and the cough droplet size can be distributed over a wide range.12 Previous experimental measurements6 have shown that the droplet size can range over four orders of magnitude, from O( 10 1) to O(103) μm. In the present simulations, a Rosin–Rammler distribution is adopted to describe the initial size distribution of the cough droplets, with the minimum, average, and maximum diameters of 1.42, 80, and 301.91 μm, respectively. Similar initial size distribution and size values were used in the numerical work of Blocken et al.10 The total number of droplets released from the mouth is set to 2000, with a corresponding total mass of 1.004 mg based on the water density of 998.2 kg/m3. These droplets' number and mass are of the same order of magnitude as those reported in the investigation by Dbouk and Drikakis.29 

To study the natural ventilation effects on the virus spread, three different inlet wind speeds (Vin) are considered, i.e., Vin = 0.05, 1.11, and 4.17 m/s, respectively, which correspond to calm air, light air, and gentle breeze outside according to the Beaufort wind scale.32 Similar to the outdoor wind scale, we classify these values as weak, moderate, and strong natural wind-driven ventilation, respectively, for indoor environments. The corresponding Reynolds numbers of the airflow are R e = 7.19 × 10 3 , 1.59 × 10 5, and 5.99 × 10 5, respectively, based on the inlet wind speed, door height, air density of 1.225 kg/m3, and dynamic viscosity of 1.789 × 10 5 kg/(m s). These Reynolds numbers indicate that the indoor air motion is turbulent flow under natural wind-driven ventilation conditions. In addition, three different ambient relative humidities (RHs) of 0%, 45%, and 90% are taken into account to investigate the influences of indoor environmental conditions on virus transmission, which respectively represent dry air, a lower bound of the ideal humidity for comfort and health, and a highly humid environment close to saturation.12 The atmospheric pressure of 1 atm and temperature of 20  °C are assumed for the inlet air, while the wall and ground temperatures are both set to 20  °C. The initial cough droplet temperature is set to 34  °C. The aforementioned settings of the wind speeds and temperatures are roughly consistent with those adopted by Dbouk and Drikakis29 for an outdoor environment. To mimic a human cough, 2000 droplets on the mouth surface are divided into five ejections with a constant speed of 8.5 m/s over a period of 0.12 s. That is, each ejection includes 400 droplets and the ejection interval is 0.03 s. The settings of the ejection period and speed are similar to those reported in the study of Dbouk and Drikakis.29 It should be pointed out that the constant cough speed and incoming velocity direction used in the present study are somewhat different from the real human cough scenarios, further improvements are required to address different cough speeds and cough directions on aerosol transmission. Readers can refer to the supplementary material (labeled as droplet initial distribution) for the Rosin–Rammler distribution graph of droplet diameter and the detailed settings of the above five ejections.

The ANSYS Fluent 2019 R122 with an embedded discrete phase model (DPM) is used in this study to simulate two major steps of the cough droplet transmission process in the chamber. First, a fully developed indoor turbulent flow field is generated. Subsequently, the cough droplets are ejected into the indoor space to mimic a real human coughing event. One-way coupling is adopted for considering the interactions between turbulence and droplets (i.e., only the effect of turbulence on droplets is considered, but the effect of droplets on turbulence is neglected) due to the dilute droplet suspension (the ratio of the initial total droplet volume to the room volume is 2.99 × 10 11, and consequently, the mass loading is very low, 10 8).33 During the transmission process, various physical phenomena such as droplet evaporation, breakup, coalescence, and heat transfer are also taken into account.

For the continuous phase, the boundary conditions are set as follows. The door and mouth are specified as a velocity inlet with a medium turbulence intensity of 5%.34 The window is specified as a pressure outlet. Here, we focus on the effect of wind entering from the door and going out from the window on the cough droplet transmission. Thus, the boundary conditions for the door and window are set as velocity inlet and pressure outlet, respectively. These settings of boundary conditions are contrary to those mentioned in the study of Pendar and Páscoa,19 who specified the window as a velocity inlet and the door as a pressure outlet. The remaining boundaries are specified as no-slip walls. The SIMPLE algorithm is used for the pressure–velocity coupling solution, and the pressure term is discretized by the second-order scheme. The convection term and energy equation are discretized by the second-order upwind schemes. The SGS kinetic energy is discretized using the first-order upwind scheme. The transient term is discretized by the first-order implicit scheme. It should be noted that the use of first-order schemes may result in less accurate simulation results compared to those obtained using second-order schemes. Furthermore, improvements are needed to address this issue. The airflow time step is set to 0.001 s. For the discrete phase, the boundary conditions are set as follows. The door, mouth, and window are specified as “escape.” The walls at y = 0 and x = 4 planes are set as “wall-film.” The remaining walls are specified as “trap.” “Escape” and “trap” indicate that the droplet trajectory calculations are terminated. “Wall-film” consists of four impingement regimes, namely, stick, rebound, spread, and splash, based on the droplet impact energy and wall temperature.22 The DPM source terms are updated after each continuous phase iteration. The droplets' trajectories are solved using automated tracking techniques that can switch between higher-order (trapezoidal) and lower-order (implicit) tracking schemes, which improve the accuracy and stability of the simulation. The droplet heat and mass transfer equations are solved in a coupled manner. The time step for droplet evolution is also set to 0.001 s.

To accelerate the turbulence development, the indoor airflow is initialized by solving the steady RANS equations in combination with the k ω turbulent model in the shear-stress-transport formulation. Afterward, the airflow is solved by an unsteady LES in conjunction with the dynamic kinetic energy SGS model until a statistically steady state is reached. Finally, droplets are ejected into the indoor turbulent flow to simulate aerosol transmission.

1. Natural wind-driven ventilation effects

To investigate the effects of different natural wind-driven ventilation conditions on the cough droplet transmission at the early stage after ejection, the sensitivity of LES simulations with different inlet wind speeds (Vin = 0.05, 1.11, and 4.17 m/s) is performed. Figures 2(a) and 2(b) visualize the droplet velocity at the droplet solution time t = 0.191 s under Vin = 0.05 and 4.17 m/s, respectively. The probability density functions (PDFs) of the droplet velocity and temperature at t = 0.191 s under different inlet wind speeds are plotted in Figs. 3(a) and 3(b), respectively. It should be mentioned that the droplet transport phenomenon exhibits a rough similarity within a certain period of time at the early stage after ejection (see videos of droplet transmission under Vin = 0.05, 1.11, and 4.17 m/s in the supplementary material). Here, t = 0.191 s is selected as an example.

FIG. 2.

Visualizations of the droplet velocity magnitude under different inlet wind speeds: (a) Vin = 0.05 m/s, and (b) Vin = 4.17 m/s, with the same ambient relative humidity R H = 45 % and droplet solution time t = 0.191 s.

FIG. 2.

Visualizations of the droplet velocity magnitude under different inlet wind speeds: (a) Vin = 0.05 m/s, and (b) Vin = 4.17 m/s, with the same ambient relative humidity R H = 45 % and droplet solution time t = 0.191 s.

Close modal
FIG. 3.

Probability density functions (PDFs) of (a) droplet velocity magnitude and (b) droplet temperature under different inlet wind speeds with the same ambient relative humidity R H = 45 % and droplet solution time t = 0.191 s.

FIG. 3.

Probability density functions (PDFs) of (a) droplet velocity magnitude and (b) droplet temperature under different inlet wind speeds with the same ambient relative humidity R H = 45 % and droplet solution time t = 0.191 s.

Close modal

As qualitatively observed in Fig. 2(a), when Vin = 0.05 m/s, the droplet velocity decreases significantly after ejection with the initial velocity of 8.5 m/s, which is consistent with that reported in Dbouk and Drikakis's work29 with wind speed 0 m/s. This significant decrease in velocity can be explained by the droplet inertial response time, which is defined as τ p = τ p s / [ 1 + 0.15 ( ρ f W d p / μ f ) 0.687 ],4 where τ p s = ρ p d p 2 / ( 18 μ f ) is the Stokes response time and W is the droplet terminal settling velocity. Based on the initial droplet size distribution, the droplet inertial response time τp ranges from 6.3 × 10 6 s to 0.121 s, which is smaller than 0.191 s. Therefore, the droplet velocity can decrease significantly within 0.191 s. When Vin = 4.17 m/s [see Fig. 2(b)], the droplet velocity is also reduced, but its reduction is not as fast as that when Vin = 0.05 m/s. In addition, five ejections can be roughly identified when Vin = 4.17 m/s, whereas ejections agglomerate together when Vin = 0.05 m/s. This feature is related to the different natural wind-driven ventilation conditions. In the case of very weak ventilation (Vin = 0.05 m/s), the background airflow is approximately static. As a result, the velocity magnitude of the released droplets decreases rapidly and dramatically after ejection due to the small droplet inertia (in the initial stage, τp ranges from 6.3 × 10 6 to 0.121 s) and the aerodynamic drag. Meanwhile, the subsequent droplet ejections catch up with the earlier ones, leading to the occurrence of droplet agglomerations between large and small droplets. On the other hand, for strong wind-driven ventilation (Vin = 4.17 m/s), the droplets are carried by the background airflow and transported downstream, resulting in slower decreases in their velocity compared to those in weak ventilation conditions.

Figure 3(a) shows that both the minimum and maximum droplet velocities increase monotonically with increasing inlet wind speeds, and the velocity range of droplets shifts toward their initial velocity while narrowing significantly. This result confirms the qualitative visualizations in Fig. 2. From Fig. 3(b), it is seen that the maximum droplet temperature decreases dramatically with increasing inlet wind speeds, but its minimum temperature does not change much and remains close to the ambient temperature. This shrinking droplet temperature distribution pattern toward the temperature of background air can be explained by the rapid reduction in droplet temperature due to the significant enhancements of convection and evaporation heat transfer between the droplet surface and its surrounding air when the inlet wind speed increases. On the other hand, the cooling rate of droplets is not only affected by the environmental conditions and thermal properties of the droplets, but also by their size. Smaller droplets have faster thermal response time, which defined as τ T = ρ p c p d p 2 / 12 k c, where k c = 0.0244 W/m/K is thermal conductivity of the continuous phase. In our cases, the diameters of the minimum and maximum droplets change from 1.42 and 301.91 μm, respectively, in the initial stage, to 1.12 and 847.17 μm at t = 0.191 s due to evaporation, breakup, and coalescence processes. Consequently, the corresponding τT changed from 2.89 × 10 5 and 1.31 s for the minimum and maximum diameters, respectively, to 1.79 × 10 5 and 10.29 s. This results in larger droplets having significantly higher temperature than smaller droplets, as shown in Fig. 4(a).

FIG. 4.

Visualizations of the droplet temperature under different ambient relative humidities: (a) R H = 0 %, and (b) R H = 90 %, with the same inlet wind speed Vin = 1.11 m/s and droplet solution time t = 0.191 s.

FIG. 4.

Visualizations of the droplet temperature under different ambient relative humidities: (a) R H = 0 %, and (b) R H = 90 %, with the same inlet wind speed Vin = 1.11 m/s and droplet solution time t = 0.191 s.

Close modal

2. Ambient relative humidity effects

To study the effects of various ambient relative humidities on the transmission of cough droplets in the early stage after ejection, the visualizations of the droplet temperature at the droplet solution time t = 0.191 s under R H = 0 % and 90% are displayed in Figs. 4(a) and 4(b), respectively. In addition, probability density functions of the droplet temperature and diameter at t = 0.191 s under different ambient RHs are presented in Figs. 5(a) and 5(b), respectively.

FIG. 5.

Probability density functions (PDFs) of (a) droplet temperature and (b) droplet diameter under different ambient relative humidities with the same inlet wind speed Vin = 1.11 m/s and droplet solution time t = 0.191 s.

FIG. 5.

Probability density functions (PDFs) of (a) droplet temperature and (b) droplet diameter under different ambient relative humidities with the same inlet wind speed Vin = 1.11 m/s and droplet solution time t = 0.191 s.

Close modal

As qualitatively shown in Fig. 4(a), when R H = 0 %, the droplet temperature decreases fast after ejection with an initial temperature of 307.15 K due to a larger evaporation rate. When R H = 90 % [see Fig. 4(b)], the droplet temperature decreases relatively slower than that when R H = 0 %. This is because the droplet evaporates slower with less heat released when ambient RH is higher. The average droplet diameter is 180.8 μm when R H = 90 %, which is 4.7 μm larger than that when R H = 0 % ( d p ¯ = 176.1 μm). The result concerning droplet diameter is larger under higher RH is in agreement with the result of Li et al.11 In addition, the shapes of these two ejections under different RHs look very similar due to the fact that the dependence of evaporation intensity on RH is not obvious in our cases. In connection with Fig. 2, the result indicates that at the early stage after ejection, the shape of cough ejection is mainly determined by the natural wind-driven ventilation conditions (i.e., inlet wind speed) rather than the environmental conditions (e.g., RH).

Figure 5(a) shows that both the minimum and maximum droplet temperatures increase monotonically with the increased ambient RH. This finding verifies the qualitative observations in Fig. 4. Moreover, the distribution of the droplet temperature is more concentrated at a higher ambient RH and broadens at a lower RH. For dry air (RH = 0), droplets evaporate rapidly, resulting in a strong evaporative cooling that makes the droplet surface temperature lower than the ambient temperature [see Fig. 5(a)]. The PDFs of droplet diameter [see Fig. 5(b)] show that their distribution shapes are roughly similar for the three different RHs, indicating that at the early stage after ejection, the ambient relative humidity has no obvious effect on the distribution trend of droplet diameter.

To obtain a full understanding of the aerosol transmission, the front views of the evolution of cough droplet transmission at different times under different natural wind-driven ventilation conditions (Vin = 0.05, 1.11, and 4.17 m/s) are shown in Figs. 6–8, respectively. In all cases, the horizontal motions of the droplets are determined by the aerodynamic drag, while the vertical motions of the droplets are determined by the combined actions of the gravitational force and the aerodynamic drag.

FIG. 6.

Front views of the evolution of droplet transmission at different time steps: (a) t = 0.239 s, (b) t = 0.839 s, (c) t = 2 s, and (d) t = 5 s, with the inlet wind speed Vin = 0.05 m/s and the ambient relative humidity R H = 45 %.

FIG. 6.

Front views of the evolution of droplet transmission at different time steps: (a) t = 0.239 s, (b) t = 0.839 s, (c) t = 2 s, and (d) t = 5 s, with the inlet wind speed Vin = 0.05 m/s and the ambient relative humidity R H = 45 %.

Close modal
FIG. 7.

Front views of the evolution of the droplet transmission at different time steps: (a) t = 0.239 s, (b) t = 0.839 s, (c) t = 2 s, and (d) t = 5 s, with the inlet wind speed Vin = 1.11 m/s and ambient relative humidity R H = 45 %.

FIG. 7.

Front views of the evolution of the droplet transmission at different time steps: (a) t = 0.239 s, (b) t = 0.839 s, (c) t = 2 s, and (d) t = 5 s, with the inlet wind speed Vin = 1.11 m/s and ambient relative humidity R H = 45 %.

Close modal
FIG. 8.

Front views of the evolution of the droplet transmission at different time steps: (a) t = 0.239 s, (b) t = 0.537 s, (c) t = 0.839 s, and (d) t = 0.984 s, with the inlet wind speed Vin = 4.17 m/s and ambient relative humidity R H = 45 %.

FIG. 8.

Front views of the evolution of the droplet transmission at different time steps: (a) t = 0.239 s, (b) t = 0.537 s, (c) t = 0.839 s, and (d) t = 0.984 s, with the inlet wind speed Vin = 4.17 m/s and ambient relative humidity R H = 45 %.

Close modal

As observed in Fig. 6, under weak natural ventilation condition (Vin = 0.05 m/s), the droplets can travel up to 1.05 m from the mouth in the horizontal direction at t = 0.239 s, and all of them are suspended at least 1.43 m above the ground. At t = 0.839 s, the largest droplet with a diameter of 865 μm falls on the ground, and the maximum horizontal transmission distance is about 2.16 m. At this time, the larger the droplet diameter is, the farther the horizontal transmission distance is. This feature is similar to the tendency displayed by the curve Uinlet = 0 in Fig. 4 in the review paper of Pourfattah et al.4 At t = 2 s, only the droplets with a diameter smaller than 220 μm are suspended in the air. At t = 5 s, almost all droplets settle onto the ground within a horizontal distance of 2 m. This result is in consistent with the result estimated using a simplified model proposed by Pourfattah et al.4 (see the  Appendix D for details). Only a small number of droplets with a diameter smaller than 45 μm fall slowly and can remain suspended in the air for a long time.

Compared to the weak natural ventilation condition [see Fig. 6(a)], the droplets can be transported up to 1.18 m along the streamwise in the moderate natural ventilation (Vin = 1.11 m/s) at t = 0.239 s [see Fig. 7(a)]. Due to the larger background airflow speed, this distance is 0.13 m farther than that at the same time step when Vin = 0.05 m/s. Meanwhile, all droplets are suspended in the air at almost the same vertical distance of 1.4 m above the ground as that when Vin = 0.05 m/s at the same time. At t = 0.839 s, the farthest horizontal transmission distance is 2.42 m, which is 0.25 m farther than the distance at the same time when Vin = 0.05 m/s. Interestingly, no droplets reach the ground, which is different from the weak natural ventilation condition [see Fig. 6(b)]. This feature, on the one hand, is related to the relatively smaller drag force caused by the larger inlet wind speed, and on the other hand, it is likely that the higher wind speed accelerates the evaporation of droplets, resulting in the reduction of the maximum droplet size. This can be confirmed by the observed largest droplet diameters in these two different situations, which are 865 and 702 μm for Vin = 0.05 m/s and Vin = 1.11 m/s, respectively, at t = 0.839 s. Therefore, the largest droplet falls faster and then hits the ground earlier under Vin = 0.05 m/s when compared to Vin = 1.11 m/s.

At t = 2 s, the droplets with a diameter larger than 221 μm fall on the ground, and the droplets with a diameter of 40 μm propagate up to 3.25 m in the horizontal direction. Also, it is interesting to note that the shape of the suspended droplets looks like a funnel. At t = 5 s, almost all of the droplets with a diameter larger than 129 μm have settled to the ground and a few droplets suspend in the corner near the window side. This result indicates that the cough droplets can be transported over a distance larger than 4 m within 5 s after ejection under a mild natural wind-driven ventilation.

Based on the results presented in Figs. 6 and 7, it is observed that larger droplets tend to deposit at farther horizontal distance in Fig. 6, while in Fig. 7, larger droplets travel less distance. This inconsistency was also noted by Pourfattah et al.4 through a simplified mathematical model. The reasons can be explained as follows: (a) Under almost static background airflow conditions (e.g., Vin = 0.05 m/s), larger droplets tend to travel farther distance due to their larger inertial response time. (b) In the presence of a background airflow (e.g., Vin = 1.11 m/s), the sedimentation time of the droplets becomes much larger than their inertial response time. This makes the air velocity play a more significant role in the droplet propagation distance, and its effect on the propagation distance is more dominant than the initial velocity of the droplets. Consequently, smaller droplets, with their larger sedimentation time, tend to travel farther distances in moving air, which is opposite to the behaviors observed for larger droplets.

As shown in Fig. 8(a), the farthest horizontal propagation distance is 1.57 m at t = 0.239 s, which is 0.52 and 0.39 m, respectively, farther than that when Vin = 0.05 and 1.11 m/s at the same time. The vertical distance of the suspended droplets above the ground is 1.40 m, which is almost the same as those when Vin = 0.05 and 1.11 m/s at the same time. It seems that the inlet wind speed has little effect on the vertical distance of suspended droplets above the ground at the early stage after ejection. This is because when the ejection velocity of the droplets is greater than the background flow rate, the movement of the droplets at this stage is mainly controlled by their inertia, so that the droplet settling distances are not much different among all three cases. At t = 0.537 s, the droplets are carried by the turbulent flow with the farthest horizontal transmission distance of 3.1 m, and the shape of the dispersion droplets looks like a cloud floating in the air. At t = 0.839 s, some droplets with a diameter larger than 348 μm have already landed on the wall of the window side, while the rest of the droplets spread very close to the window. These phenomena do not occur at the same time for the weak and moderate natural ventilation cases due to the smaller inlet wind speeds. The droplets with a diameter smaller than 85 μm have already escaped from the window, and only a few larger droplets adhere to the sidewall. These adhesive droplets still experience the physical processes of evaporation and heat transfer with the ambient environment. Overall, within 1 s, all suspended droplets can escape from the window under the effect of the background airflow.

In Fig. 9, our simulation results are compared with the results reported in the studies of Li et al.,11 Feng et al.,12 and Dbouk and Drikakis29 with similar wind speeds. As can be seen, when the ambient wind speed is weak (e.g., V i n 0, 0.05 m/s), the horizontal travel distance is within 2.16 m. However, when the ambient wind speed is strong (e.g., Vin = 4.17 m/s), the droplets can travel more than 4 m. This indicates that the greater the ambient wind speed is, the farther horizontal distance the droplets can travel. The results of Li et al.11 imply that smaller droplets can travel farther distance than that of large droplets when Vin is not zero (e.g., Vin = 2 m/s). This result is in consistent with that observed in Fig. 7. The differences of the horizontal travel distances between Dbouk and Drikakis29 and our simulations might be mainly originated from the different settings of the boundary conditions. For example, in the study of Dbouk and Drikakis,29 an outdoor environment is considered. However, in our present study, an indoor space is simulated. In the study of Dbouk and Drikakis,29 a linear ejection velocity with the maximum velocity of 8.5 m/s is applied during the ejection period of 0.12 s. However, in our present study, a constant ejection velocity of 8.5 m/s is used during the ejection period of 0.12 s.

FIG. 9.

Comparisons of horizontal travel distances of cough droplets among Li et al.,11 Feng et al.,12 Dbouk and Drikakis's results,29 and our simulation results. D represents the horizontal travel distance.

FIG. 9.

Comparisons of horizontal travel distances of cough droplets among Li et al.,11 Feng et al.,12 Dbouk and Drikakis's results,29 and our simulation results. D represents the horizontal travel distance.

Close modal

To quantitatively investigate the evolution of the physical quantities of the suspended droplets, the maximum values of the droplet velocity and temperature as a function of time under different inlet wind speeds are plotted in Figs. 10(a) and 10(b), respectively. Figure 10(a) shows that the maximum droplet velocity decreases rapidly within the first 1.5 s after coughing in all three cases. This is because the effects of the aerodynamic drag exerted on the droplets are significant at the early stage of the droplet transmission. For the cases of Vin = 0.05 and 1.11 m/s, the maximum droplet velocity decreases very slowly after 2 s and remains almost constant over time. Similar to the evolution of the maximum velocity, the maximum droplet temperature also decreases quickly during the first 1.5 s after coughing. This is mainly because at the early stage of droplet transmission, there is a large velocity difference and temperature gradient between the droplets and the surrounding airflow, resulting in strong heat transfer between the droplets and the air, and consequently rapid cooling of the droplets. Additionally, as discussed in Sec. IV A 1, large droplets exhibit high temperatures due to their longer thermal response time compared to smaller droplets. In our cases, τT for the large droplets is about 1.3 s, indicating that the temperature of the larger droplets will drop rapidly and approach the ambient temperature in a comparable period of time. After the thermal response time of the larger droplets ( 1.3 s), the velocity difference and temperature gradient between the droplets and the surrounding airflow become smaller, resulting in a slower decrease in droplet velocity and maximum droplet temperature, which can be easily seen from the curves in Fig. 10. Eventually, the droplet velocity and temperature tend to be a constant value over time, which is typically lower than the corresponding values of the ambient airflow due to drag force (for velocity) and evaporative cooling (for temperature). As shown in Fig. 10, both cases with Vin = 0.05 and Vin = 1.11 m/s have a final droplet temperature of 287.5 K, which is about 5.6 K colder than the ambient air. The final droplet velocity is ∼0.98 m/s for Vin = 1.11 and 0.15 m/s for Vin = 0.05 m/s.

FIG. 10.

Maximum physical quantities of the suspended droplets change with time under different inlet wind speeds with the same ambient relative humidity R H = 45 %. (a) droplet velocity and (b) droplet temperature.

FIG. 10.

Maximum physical quantities of the suspended droplets change with time under different inlet wind speeds with the same ambient relative humidity R H = 45 %. (a) droplet velocity and (b) droplet temperature.

Close modal

To further study the droplet propagation behaviors, the animations of the side views of the droplet transmission process under Vin = 0.05, 1.11, and 4.17 m/s are provided in the supplementary material and labeled as Videos S1, S2, and S3, respectively. The side views of the droplet transmission at the late stage under Vin = 0.05, 1.11, and 4.17 m/s are shown in Figs. 11(a)–11(c), respectively. As observed in Video S1, when Vin = 0.05 m/s, all droplets fall to the ground almost in straight lines, without any apparent droplet dispersion phenomena in the cross section (YZ plane) during the entire transmission process. This is because when Vin = 0.05 m/s, the background airflow is nearly static and the turbulent fluctuations are very weak; thus, the indoor airflow has little effect on the droplet motions in the cross section. The final deposition width on the ground is about 0.14 m in the YZ plane [see Fig. 11(a)], which is three and a half times the length of the mouth (0.04 m).

FIG. 11.

Side views of the droplet transmission at the late stage under different inlet wind speeds with the same ambient relative humidity R H = 45 %. (a) Vin = 0.05 m/s and t = 8 s, (b) Vin = 1.11 m/s and t = 8 s, (c) Vin = 4.17 m/s and t = 1 s, and (d) streamlines of the plane x = 3.85 m for Vin = 1.11 m/s at t = 8 s.

FIG. 11.

Side views of the droplet transmission at the late stage under different inlet wind speeds with the same ambient relative humidity R H = 45 %. (a) Vin = 0.05 m/s and t = 8 s, (b) Vin = 1.11 m/s and t = 8 s, (c) Vin = 4.17 m/s and t = 1 s, and (d) streamlines of the plane x = 3.85 m for Vin = 1.11 m/s at t = 8 s.

Close modal

However, when Vin = 1.11 m/s, we can observe obvious droplet dispersion phenomena in the YZ plane during the whole transmission process (see Video S2). After ejection, the droplets move forward to the window side and disperse in the space under the effect of the local circulation of the background airflow. Finally, the droplets deposit over a wide region on the ground and a few small droplets are advected by local flow into the corner of the sidewall at Z = 3 plane [see Fig. 11(b)]. After reaching the corner area, these droplets are resuspended along the sidewall under the effect of the local upward vertical flow [see Fig. 11(d) as an example]. At t = 8 s, the droplets with a diameter of 60 μm can resuspend from the near-ground region to a height of 1.82 m, which is higher than the human height of 1.63 m. These resuspended droplets induced by the local circulation pattern will increase the inhalation risk for the nearby person. To reduce this risk, the natural ventilation of the indoor environment needs to be optimized. When Vin = 4.17 m/s, all suspended droplets have already escaped from the window at t = 1 s, while other droplets impact on the sidewall of the window side (see Video S3). The farthest distance of the impingement droplets in the YZ plane is 0.32 m away from the human mouth center [see Fig. 11(c)]. Although the residence time of the suspended droplets in the chamber is very short, the turbulent intensity is strong; hence, the droplet dispersion phenomena can be clearly observed in the YZ plane during the whole transmission process.

From all the LES simulations, it is found that both the natural wind-driven ventilation conditions and ambient environmental conditions play crucial roles in affecting the transmission process of the cough droplets. The horizontal deposition distance on the ground exhibits significant variations with the ventilation conditions and environmental relative humidities, and can be substantially farther than 2 m for larger inlet wind speed and higher relative humidity. For the weak ventilation condition (Vin = 0.05 m/s) with ambient relative humidity levels of 0%, 45%, and 90%, the farthest horizontal deposition distance on the ground is found to be 2.25 m for R H = 90 % [see Fig. 12(a)]. In all three RH cases, most droplets are deposited within 2 m under the nearly static background airflow for weak natural ventilation conditions. This finding is consistent with the simulation results reported by Feng et al.12 and Dbouk and Drikakis29 with similar settings. For the moderate natural ventilation condition (Vin = 1.11 m/s) with ambient humidity levels of 0%, 45%, and 90%, the nearest horizontal deposition distance on the ground is found to be 2.18 m for R H = 90 % [see Fig. 12(b)], which is almost the same as the farthest deposition distance at weak ventilation condition. In all three RH cases with the same moderate ventilation, all droplets are deposited at horizontal distances beyond 2 m. When the natural ventilation is strong (Vin = 4.17 m/s), all droplets are transported over a horizontal distance of 4 m within 1 s with ambient R H = 0 %, 45%, and 90%. That is to say, under strong ventilation, most of the cough droplets will be transported out of the window by the background airflow within 1 s regardless of the ambient environmental conditions. Therefore, it can be concluded that maintaining good indoor ventilation conditions can effectively remove and/or reduce the suspension time of viruses.

FIG. 12.

Front views of the droplet transmission at t = 9 s under different inlet wind speeds: (a) Vin = 0.05 m/s, and (b) Vin = 1.11 m/s, with the same ambient relative humidity R H = 90 %.

FIG. 12.

Front views of the droplet transmission at t = 9 s under different inlet wind speeds: (a) Vin = 0.05 m/s, and (b) Vin = 1.11 m/s, with the same ambient relative humidity R H = 90 %.

Close modal

We argue that the commonly recommended social distancing of 2 m in the literature is most likely based on static wind conditions and thus may be insufficient for either outdoors with a calm wind or indoors with weak/mild ventilation. For well-ventilated indoor or outdoor environments, the cough droplets will be transported quickly to several meters away within 1 2 s by the background airflow. As a result, individuals located downwind of an infected person are likely to be exposed to the virus-laden droplets exhaled by the infected person even if they are more than 4 m away. Conversely, individuals located upwind of the infected person are less likely to be exposed even if they are within 2 m. This is because the transmission of virus-laden droplets is primarily controlled by the local circulation of turbulent background airflow. Therefore, any specified safe social distance without considering the local airflow characteristics and the relative locations of individuals is not appropriate. We believe that there is no absolute safe social distancing. To be safe, we recommend wearing a face mask whenever possible, or at least covering your nose and mouth when coughing or sneezing to minimize the discharge of potentially virus-laden droplets into the air. In addition, we recommend opening doors and windows frequently to maintain well-ventilated conditions, which can remove the respiratory droplets from indoor air or shorten their suspension time in the air.

In this study, a series of numerical simulations with varying ambient wind speed and relative humidity are performed to investigate the effects of natural wind-driven ventilation and ambient environmental conditions on the cough droplet transmission in an indoor environment. The indoor airflow is solved by the LES approach, and the trajectories of the droplets are tracked by the Lagrangian method. Three different inlet wind speeds (Vin = 0.05, 1.11, and 4.17 m/s) representing weak, moderate, and strong natural wind-driven ventilation conditions, as well as three ambient relative humidities representing dry air ( R H = 0 %), lower bound of the ideal humidity for comfort and health ( R H = 45 %), and a highly humid environment close to saturation ( R H = 90 %), are considered. During the transmission process, the droplet evaporation, droplet breakup and coalescence, and heat transfer are also considered. The main findings are as follows:

  1. At the early stage of droplet transmission, the reduction in droplet velocity is more pronounced at smaller Vin. However, the droplet temperature decrease is more prominent at larger Vin as a result of the faster heat transfer and evaporative cooling under strong wind conditions. In addition, the droplet temperature reduction is more pronounced in drier environments (i.e., smaller RH) due to the faster evaporation-induced heat transfer compared to moist environments.

  2. In the case of weak natural ventilation (Vin = 0.05 m/s, R H = 45 %), the cough droplets can only travel up to a horizontal distance of 2.16 m due to the nearly static background airflow, and almost all droplets deposit within a horizontal distance of 2 m. This finding is consistent with the simulation results reported by Feng et al.12 and Dbouk and Drikakis29 with similar settings.

  3. For the moderate natural ventilation case (Vin = 1.11 m/s, R H = 45 %), all cough droplets exhibit the horizontal deposition distance exceeding 2 m due to advective transport by the background airflow. A clear dispersion phenomenon is observed in the cross-wind plane during the droplet transmission process. Some near-ground droplets with a diameter smaller than 60 μm can be resuspended to the vertical position over the human height owing to the effect of the local upward vertical flow.

  4. For the strong-ventilated case (Vin = 4.17 m/s, R H = 45 %), most suspended droplets travel over a distance of 4 m in the horizontal direction and escape from the indoor space through the window within 1 s due to the strong background airflow. Our simulations indicate that the cough droplets can be easily transported by the background airflow to a few meters downwind, and the stronger the background wind, the longer the horizontal distance of droplets spread, consistent with the findings of Pourfattah et al.4 utilizing a simplified model.

  5. The social distancing guideline of 2 m is insufficient for either outdoors with weak wind or indoors with weak/mild natural ventilation conditions, not to mention the ambient environment with a strong wind. The dispersion of cough droplets is mainly controlled by the local circulation patterns, and whether the individuals are exposed to the potentially virus-laden droplets from an infected person depends on their locations relative to the wind direction. As discussed in Sec. IV C, we emphasize that there is no absolute safe social distancing, and we strongly recommend wearing a face mask for safety, or at least covering the nose and mouth when sneezing or coughing to reduce the release of potentially virus-laden droplets into the air. Additionally, maintaining a good ventilation by opening doors and windows can effectively remove the respiratory droplets or shorten their suspension time in indoor air.

See the supplementary material for animations of the front views (X–Y planes) and side views (Y–Z planes) of the droplet transmission processes under Vin = 0.05, 1.11, and 4.17 m/s.

We gratefully acknowledge that this work is supported by the National Natural Science Foundation of China (Grant Nos. 12041601, 91852205, 11961131006, and 42075078), the NSFC Basic Science Center Program (Award No. 11988102), the Guangdong Provincial Key Laboratory of Turbulence Research and Applications (Grant No. 2023B1212060001), the Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications (Grant No. 2020B1212030001), the Science Technology and Innovation Committee of Shenzhen Municipality (Grant No. 2020–148), and the Shenzhen Science and Technology Program (Grant Nos. KQTD20180411143441009 and JCYJ20220530113005012).

The authors have no conflicts to disclose.

Liangquan Hu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Yong-Feng Ma: Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). Farzad Pourfattah: Writing – review & editing (equal). Weiwei Deng: Funding acquisition (equal); Writing – review & editing (equal). Lian-Ping Wang: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

The present LES method for simulating the airflow fields is validated using the experimental data reported in the work of Posner et al.35 As shown in Fig. 13(a), the test chamber has dimensions of 0.914 × 0.305 × 0.457 m3 in the x, y, and z directions, respectively. A square inlet and a square outlet are located in the ceiling, and the length of their sides is 0.101 m. The inlet velocity is 0.235 m/s. A partition with height of 0.15 m is located at the center of the chamber. The inlet vent is set as velocity inlet, the outlet vent is set as pressure outlet, and the other boundaries are set as no-slip conditions. The grid number used here is 120 × 40 × 60.36 The mean vertical velocity profile along the horizontal line at mid-partition height ( x = 0 0.914 m, y = 0.075 m, z = 0.2285 m) are compared with measurement data and showed in Fig. 13(b). As observed, the present LES results are in good agreement with the experimental data.35 

FIG. 13.

(a) Schematic diagram of the test chamber. (b) Comparisons of the mean vertical velocity profile along the horizontal line at mid-partition height ( x = 0 0.914 m, y = 0.075 m, z = 0.2285 m) with experimental data.35 

FIG. 13.

(a) Schematic diagram of the test chamber. (b) Comparisons of the mean vertical velocity profile along the horizontal line at mid-partition height ( x = 0 0.914 m, y = 0.075 m, z = 0.2285 m) with experimental data.35 

Close modal

The droplet evaporation model used in the current study is validated by the data reported in the study of Norvihoho et al.20 A single water droplet with an initial diameter of 100 μm and initial temperature of 37  °C is injected horizontally into quiescent air with speed of 10 m/s. The initial ambient temperature is 25  °C with relative humidity of 0%. The change of droplet diameter is compared with literature data and showed in Fig. 14. A good match between the present results and literature data can be observed.

FIG. 14.

Comparisons of present results for evaporation of 100 μm droplet with that of Norvihoho et al.20 

FIG. 14.

Comparisons of present results for evaporation of 100 μm droplet with that of Norvihoho et al.20 

Close modal

The computational domain (Fig. 1) is discretized by three different grid cells: coarse ( 6.1 × 10 5 cells), medium ( 1.02 × 10 6 cells), and fine ( 1.32 × 10 6 cells). The mean velocity magnitudes along the vertical line at the chamber center (x = 2 m, y = 1.0 1.8 m, z = 1.5 m) are compared and showed in Fig. 15 (Vin = 0.05 m/s, R H = 0 %). Similar variations are observed in the comparisons. Since the difference between the medium and fine grid cells is small, the medium grid cells of 1.02 × 10 6 is used in the present study to balance the computational cost and accuracy of the results.

FIG. 15.

Comparisons of the mean velocity magnitudes along the vertical line at the chamber center (x = 2 m, y = 1.0 1.8 m, z = 1.5 m) under three different grid cells. Vin = 1.11 m/s, R H = 0 %.

FIG. 15.

Comparisons of the mean velocity magnitudes along the vertical line at the chamber center (x = 2 m, y = 1.0 1.8 m, z = 1.5 m) under three different grid cells. Vin = 1.11 m/s, R H = 0 %.

Close modal
In the review paper of Pourfattah et al.,4 the horizontal distance D traveled by a solid particle is estimated as
D u air H W + ( u 0 u ¯ air ) τ p ,
(D1)
where uair is the horizontal mean velocity of the background air (i.e., the wind speed), H is the initial released height of the particle, W is the terminal settling velocity of the particle, u0 is the initial horizontal released velocity of the particle, and u ¯ air is the mean velocity of the background air in a room, which depends on the inlet velocity of a room (e.g., the applied velocity by the ventilation system). τp is the particle inertial response time defined as τ p = τ p s / [ 1 + 0.15 ( ρ f W d p / μ f ) 0.687 ], and τ p s = ρ p d p 2 / ( 18 μ f ) is the Stokes response time. In this study, uair = Vin, H = 1.63 m, and u 0 = 8.5 m/s.

In our simulation results displayed in Fig. 6, we find that most droplets (98%) are less than 560 μm in diameter during the whole transmission process. Therefore, we assume that the initial released droplet diameter is less than dp = 560 μm. We further assume that the droplet size will not change with time and surrounding conditions. Namely, the droplet is treated as solid particle. Based on the above assumptions and parameters, the maximum horizontal travel distance D for 39 μm d p  560 μm is estimated to be 1.98 m by Eq. (D1) (see Fig. 16), which is within 2 m and is consistent with the results shown in Fig. 6.

FIG. 16.

Horizontal travel distance of solid particles under different inlet wind speeds. This plot is based on Eq. (D1). H = 1.63 m and u 0 = 8.5 m/s.

FIG. 16.

Horizontal travel distance of solid particles under different inlet wind speeds. This plot is based on Eq. (D1). H = 1.63 m and u 0 = 8.5 m/s.

Close modal

When the air background mean velocity is increased to 1.11 and 4.17 m/s, the model results shown in Fig. 16 indicate that the minimum horizontal travel distance will exceed 2 m, and this travel distance could be larger for smaller droplet size. This observation is again consistent with our simulation results shown in Figs. 7 and 8. These comparisons demonstrate the importance of ambient mean flow on the travel distance.

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Supplementary Material