Analysis of airborne sputum droplets flow dynamic behaviors under different ambient conditions and aerosol size effects.

The coronavirus (COVID-19) is becoming more threatening with the emergence of new mutations. New virus transmission and infection processes remain challenging and re-examinations of proper protection methods are urgently needed. From fluid dynamic viewpoint, the transmission of virus-carrying droplets and aerosols is one key to understanding the virus-transmission mechanisms. This study shows virus transmission by incorporating flow-evaporation model into the Navier-Stokes equation to describe the group of airborne sputum droplets exhaled under Rosin-Rammler distribution. Solid components and humidity field evolution are incorporated in describing droplet and ambient conditions. The numerical model is solved by an inhouse code using advection-diffusion equation for the temperature field and the humidity field, discretized by applying the total-variation diminishing Runge-Kutta method. The results of this study are presented in detail to show the different trends under various ambient conditions and to reveal the major viral-transmission routes as a function of droplet size.

Introduction

Coronavirus disease (COVID-19) and its mutations, such as Alpha (B.1.1.7), Beta (B.1.351), Delta (B.1.617.2) and Omicron (B.1.1.529), with their rapid spread and superinfection, have caused fatal consequences for public health. The virus-carrying droplet-aerosol in specific indoor environments and in social behaviors has been a concern for the spread of infectious respiratory diseases (Chao et al., 2009; Kwon et al., 2012; Jayaweera et al., 2020; Dinoi et al., 2021; Issakhov et al., 2021; Bhatti et al., 2022; Shukla et al., 2022; Faleiros et al., 2022). From a fluid dynamic viewpoint, ambient conditions have been shown to affect the dynamics of droplets/aerosols, including phase change, evaporation, and heat transfer (Yang et al., 2021; Paez-Osuna et al., 2022). Exploring the transmission of virus-carrying droplets/aerosols under different ambient conditions is thus vital for preventing human-to-human viral transmission, in which multidisciplinary approach is necessary to cope with the increasing risk of public infection.

Ambient temperature and relativity humidity has been shown to affect the aerodynamics of virus-carrying droplets/aerosols (Li et al., 2018; Rosti et al., 2020, 2021; Liu et al., 2021; Wang et al., 2022). However, the details of the connection between the virus existence and the flow dynamics and polution dispersion is still not clear, due to the complicated size effects and distribution-evolution of droplets/aerosols (Srivastava, 2021; Mehmood et al., 2021; Beig et al., 2022). As early as the outbreak of the SARS-CoV virus, Casanova et al. (2010) used two potential surrogates (transmissible gastroenteritis virus and mouse hepatitis virus) in cell cultures at different temperatures (4.0, 20.0, and 40.0 °C) and humidity levels (0.2, 0.5, 0.8) and found that viruses are inactivated under higher temperature and higher relative humidity (RH). Daily confirmed cases and meteorological factors in 122 cities were collected by Xie and Zhu, (2020). Using a generalized additive model, the relationship between ambient temperature and virus infection were tested and found that at mean temperatures below 3.0 °C, each 1.0 °C rise is associated with a 4.861% increase in infection record (Xie and Zhu, 2020). Prata et al. (2020) also found that the daily number of COVID-19 cases is linear in mean temperature. Although the relationship seems quite clear, the detailed explanation from a fluid dynamic process is still less reported in literature. Necessary mechanism analysis is needed to quantify how humidity and temperature affect aerosol transport.

Droplet evaporation is largely controlled by temperature and humidity, and the vapor masses (the inhomogeneous humidity field) ejected from the respiratory tract will experience a travel route defined by those parameters. Li et al. (2020), taking Singapore as an example, revealed how airborne humidity affects the evaporation of individual cough droplets in the tropical outdoor environment by showing numerically the seven times longer evaporation time for cases at a relative humidity (RH) of 0.9 than cases at RH of 0.6. A similar conclusion was also reached by Dbouk and Drikakis (2020a), Li et al. (2018) and others (Rosti et al., 2020, 2021; Feng et al., 2020). However, the homogeneous airborne humidity differs markedly from the vapor mass exhaled from the respiratory tract, and the evaporation and transport of aerosols is subject to size factor. Li et al. (2018) treated vapor mass as an ideal mixture composed of dry air and water vapor and separately solved the continuity equations for dry air and water vapor. The results showed that the supersaturated wet air formed a “vapor plume” in front of the respiratory-track, and then the airborne droplets traveled up to 0.65 m, which is 18% higher than normal case (0.55 m). Rosti et al. (2020, 2021) introduced the advection-diffusion equation to describe the evolution of supersaturation, and high-resolution direct numerical simulations were combined with a comprehensive Lagrangian model to investigate the physical behavior of droplets. However, this model mainly concerned the evaporation of small droplets and the subsequent airborne transport, rather than the key role of humidity field evolution.

The sizes of aerosols and droplets exhaled by infected and asymptomatic individuals also play a key role in specific transmission case analysis (Morawska et al., 2009). Wang et al. (2020) applied a continuous random walk model with Lagrangian particle-tracking model to simulate the droplets transport trajectories with different sizes and found that the 100.0 μm size droplet has a much shorter lifetime (4.1s–12.3 s) compared to the 50.0 μm droplet (1.15s–43.5 s). Later, Kumar and Lee (2020), Dbouk and Drikakis (2020a, 2020b), and Pendar and Páscoa, (2020) employed the Weibull distribution of grouped droplets with specific target droplets states such as breathing and coughing. Zeng et al. (2021) proposed an evaporative flow model for droplets (with solid components) under Weibull distribution and showed that droplets (average diameter 140.0 μm) settled rapidly (t = 8.96 s) by gravity, and droplets (average diameter 80.0 μm) suspended for a longer time instead.

Along with the discussion of ambient parameter (inhouse or open public space) and the droplet size distributions, the investigation of mask then becomes one necessary part considering the respective effectiveness under those specific conditions (Elegant, 2020). Recently, Pendar and Páscoa (2020) regarded masks as fibrous microstructures and assumed that a few small particles could escape and showed that the droplet-scattering region with masks is confined to a sphere with a diameter of around 0.6 m, rather than 3.5 m in the case of a naked face. Khosronejad et al., 2020, Khosronejad et al., 2021 used a diffused interface-immersed boundary with the Navier-Stokes equations to show that saliva particulates could travel up to 0.48 m, 0.73 m, and 2.62 m for individuals with medical masks, normal masks, and without mask condition, respectively. Li et al. (2020) studied how droplets deposit on mask under constant wind condition and reported that over 65% of droplet expelled was deposited from 1.0 m distance, and the viral load on the mask reduces from 9 copies to 0.6 copies when social distancing extends from 1.0 to 2.0 m, respectively. Although these studies have analyzed the role of masks in virus prevention and control from different perspectives, the risk assessment was justified only for idealized conditions. The dynamics under different combinations of ambient temperature (summer or winter) and humidity (dry and wet) condition remains unclear.

Given that new virus mutations are ongoing worldwide, the transmission dynamics of the virus differ for different types of viruses as well as coupled ambient conditions. Thus, the study of virus-carrying droplet-aerosol flow dynamics remains an urgent task for the research community: (1) how size affects the behavior of droplets and aerosols under different distributions conditions; (2) the combinations of ambient parameters at representative locations and situations are yet to be analyzed; and (3) scenarios with and without masks for specific combinations of parameters under the cases of (2), (3). Considering those questions, the current study has been proposed with an inhouse code to deal with the coupled effects of droplet/aerosol initial sizes as well as the size-history/evolution effects, the representative ambient humidity and temperature effects on the physical mechanism of virus transmission. The following Section 2 describes the physical and numerical modeling process and development of inhouse code for the dynamic flow with airborne droplets; Section 3 then validates the proposed model for an analysis of the dynamics, and for describing in detail the effects of the parametrization of the ambient conditions (humidity-temperature combinations, inflow speeds); the comparisons for mask use under those above situations are also included in Section 3; finally, Section 4 concludes this study.

Methodology

Detailed model formulation

Cough-generated airflow model

The incompressible Navier-Stokes equations are used to model the exhaled airflow field (Zhao et al., 2005; Feng et al., 2020; Rosti et al., 2020, 2021)where u [m/s], ρ [kg/m3] and p [Pa] represent the velocity, density and pressure field, respectively, and μ [Pa·s] is dynamic viscosity coefficient of airflow.

The exhalation events, including talking, breathing, coughing and sneezing, generate droplets that are entrained into the warm and moist respiratory airflow, and the evaporative heat transfer from aerosols is strongly influenced by the temperature and humidity of the surrounding airflow. The energy equation can be used to describe the transfer of airflow temperature,

Previous studies (Li et al., 2018, 2020; Dbouk and Drikakis, 2020a, 2020b) mainly focused on the physics behaviors on the homogeneous humidity field. including droplet evaporation rate, air-flow interaction, and the process of respiratory jets. However, the inhomogeneous vapor masses ejected from the respiratory tract obviously is saturated, or close to saturation (Li et al., 2018; Rosti et al., 2021). Saturated humilities lead to smaller evaporation rates. In the present model, the advection-diffusion equation is introduced to feature the evolution of supersaturation field (Celani et al., 2005; Feng et al., 2020; Rosti et al., 2020, 2021), which is written as:where, T [°C] and φ (= RH - 1) are temperature field and supersaturation field of airflow, respectively; c [J/(kg·°C)], λ and D represent the specific heat capacity, the thermal conductivity coefficient, and the water vapor diffusion coefficient, respectively. On the one hand, the two-way coupling of mass, momentum and energy between a large number of tiny particles and the flow field increases the complexity of physical models and the difficulty of numerical calculations due to scale differences. On the other hand, it is worth noting that for tiny suspended aerosols (diameters smaller than the Kolmogorov scale (Kovasznay et al., 1975)), the collisions between droplets and force feedback on the airflow can be neglected (Casciola et al., 2010; Rosti et al., 2021). Based on that, the one-way coupling method between the tiny particles and the flow field is adopted in this study.

Droplet evaporation model

The rate of evaporation depends on the difference between droplet surface saturation vapor volume and the vapor volume of the surrounding airborne, the latter being dependent on relative humidity and ambient temperature. The expression can be written as:where m [kg], A [m2] and k [m/s] represent the droplet mass, the droplet surface area and the mass transfer coefficient, respectively; C 1 [kg/m3] and C 2 [kg/m3] denote the concentration of vapor at the droplet interface and in the airborne, respectively.

The droplet exhaled by infected and asymptomatic individuals are not pure water, which contains dissolved substances, such as salt or/and protein. As a result, the activity and saturation concentration of the water is reduced by the non-volatile components, which affects the evaporation rate of the droplets. Based on Raoult's law (Raoult, 1891), this study describes the concentration of vapor C 1 on the saliva surface as the following:where T [°C] is droplet surface temperature, x refers to the mole fraction of evaporated solvent in saliva, and [Pa] is the saturated water vapor pressure on the interface of the droplet. C 2 is closely related to relative humidity and ambient temperature,here, T [°C], RH [−] and [Pa] are ambient air temperature, relative humidity and water vapor pressure of airborne, respectively. The mass transfer coefficient k determined by a combination of the Schmidt number (Sc), Reynolds number (Re) and the diffusion coefficient (D [m2/s]) of vapor in the air (Renksizbulut and Yuen, 1983; Sazhin, 2006),where D [m] is the droplet diameter.

Convective heat exchange and evaporative heat exchange together determine the thermal balance of droplet. The heat transfer equation can be written as (Li et al., 2020):where L [J/kg] is the latent heat of the droplet. The convective heat transfer coefficient h [W/(m2∙K)] is calculated by a modified Nusselt number as:where Pr is the Prandtl number and λ is the thermal conductivity coefficient of air. The Spalding heat transfer number (B ) can be calculated by (Li et al., 2020):where [kg/s] is the droplet evaporation rate, q [J] is the heat energy transferred to the droplet and C [J/(kg·°C)] is specific heat.

Droplet tracking model

The kinematic relationship between the position of respiratory droplet and the speed of particles is controlled by a combination of the gravitational force, the Stokes drag force (Li et al., 2020; Yang et al., 2021), the buoyancy force (Kumar and Lee, 2020) and the Brownian forces (Ullersma, 1966; Zwanzig, 1973; Longest and Xi, 2007). The expressions are

The five terms in the right-hand side of Eq. (12) represents the gravitational force, the Stokes drag force, the buoyancy force, the fluctuating force and the frictional force, respectively. x [m], m [kg] and [m/s] are the position, mass and velociry of the saliva droplets, respectively. V [m3], R [m], ρ [kg/m3] and C [−] presents the volume, radius, density and the drag coefficient, respectively. The drag coefficient is a function of the droplet Reynolds number (),

The frictional force is proportional to the velocity of the Brownian particle (Kumar and Lee, 2020). Among them, the friction coefficient is given by Stokes law, λ = 6πμ R , where μ [Pa·s] is the viscosity. The fluctuating force is supposed to the accidental collision between the Brownian particle with molecules of the surrounding medium (Zwanzig, 1973; Longest and Xi, 2007), which can be summarized by giving a Gaussian random function with a first-order moment of 0 (⟨ζ(t)⟩ = 0) and a second-order moment of D (⟨ζ(t)，ζ(t’)⟩ = Dδ(t - t’)) (Zwanzig, 1973; Longest and Xi, 2007). Among them, D is the strength of the fluctuating force, which is obtained from the Einstein relation, D = K T λ, where K  = 1.38 × 10−23 J/K is the Boltzmann constant.

Particle distribution formula

Virus carriage, transport and infectivity are closely related to droplet diameter. The Rosin-Rammler distribution (Dbouk and Drikakis, 2020a, Dbouk and Drikakis, 2020b; Kumar and Lee, 2020; Zeng et al., 2021), also known as the Weibull distribution (Wells, 1934), is widely used to describe the size distribution of droplets ejected from the mouth. The formulation expressed as follows:where q and are the exponential factor and average particle diameter, respectively, which are based on the saliva injection flow rate as an input parameter for the considered seeding droplet (Kumar and Lee, 2020). Dbouk and Drikakis (2020a) obtained the fitting parameters of the Rosin-Rammler distribution (q = 8). According to that, the droplet distribution exhaled from human coughing is defined as the same value in the present numerical model.

Numerical algorithm

The current mathematical models refer to fluid flow, droplet motion, and evaporative heat transfer. The expiratory velocity is calculated from the Navier-Stokes system on Eulerian grid using the lattice Boltzmann method (Yuan et al., 2014). As the diffusion direction of the humidity and temperature fields is along the velocity field, the advection-diffusion equation (Eq. (2) and (3)) are discretized in time and space by applying the third-order total variation diminishing Runge-Kutta method and fifth-order weighted essentially non-oscillatory scheme (Shu and Osher, 1989), respectively. The physical behaviors of the droplets are controlled by the Ordinary differential equations (Eq. (4), (8), (11), and (12)) and is tracked by using the Lagrangian approach. The trilinear interpolation method enabled the one-way communication between Lagrangian-particles and the flow fields (velocity, temperature and humidity fields). The RH, which is controlled by the supersaturation field (φ) (Eq. (3)), affects the saturation vapor pressure at the surface of the droplet (Eq. (6)) and determines the evaporation rate of droplet (Eq. (4)). As for heat balance, the convective heat exchange (Eq. (8)) designed by the convective heat transfer coefficient (h) and surface-to-ambient temperature difference (Eq. (3)), while the evaporative heat exchange (Eq. (8)) is achieved by the evaporation rate of droplet (Eq. (4)). The Stokes drag force and the buoyancy force drives the transfer by coupling the relative velocities of the droplets and surrounding gas. This process involving momentum coupling, heat balance and mass transfer is therefore repeated until the droplet reaches a steady state (sedimentation, suspension or floating) after a long period of time. The detailed computational sequence is shown in Fig. S1.

Initial and boundary conditions

The current study considers large groups of virus-laden particles (i.e. droplets and aerosols) are expelled from the respiratory tract (z = 1.55 m) of an infected or asymptomatic person (height: 1.7 m) during breathing, coughing, or sneezing, and the particles deposition at the ground or on surface at z = 0 m. The various situations encompass the different seasons and ambient conditions of RH, and temperature (as listed in Table 1 ). The various ambient configurations analyzed in this study can be interpreted as representative of various locations around the world, in winter or summer, as indicated by representative cities. Therefore, to analyze the worldwide situation, we restrict ourselves to the parameters given in Table 1. For instance, to understand the situation in dry northern China (such as Changchun, Beijing) and humid southern China (such as Guangzhou, Changsha), we explore the transport of aerosols in both summer and winter. Turbulent clouds traveling at different speeds (1.0 and 5.5 m/s) exhale from the mouth, take on the humidity and temperature of the ambient air, and dissipate and merge into the ambient air in a short time and on a local scale. Therefore, shrinking the turbulent cloud cluster area (the current calculation domain is 3.2 m × 0.6 m × 0.6 m, with a grid of 800 × 150 × 150, and the total grid size up to 18 million.) improves the calculation efficiency. The inlet velocity serves as the Dirichlet boundary condition, and the velocity is only imposed at the lips. After exhaling, the inlet boundary velocity is set to zero. The Neumann outflow boundary conditions are used for the remaining boundaries. Fig. 1 shows the physical models and boundary condition used in this study. Two types of models are considered: (1) droplet or aerosol flow dynamics under different ambient conditions to study the dynamic behavior, and (2) the pair-body model to test how mask conditions affect the rate of virus transmission.

Table 1

The various situations considered for different cities with various temperature and humidity in the winter or summer.

Season	Winter
Case group	Cases 1	Cases 2	Cases 3	Cases 4	Cases 5	Cases 6
Temperature (°C)	20	20	20	5	5	5
Humidity (−)	0.4	0.9	0.1	0.1	0.9	0.4
Example city	Guangzhou	Fuzhou	Changsha	Beijing	Changchun	Urumqi
Inlet velocity (m/s)	1.0, 5.5
Season	Summer
Case group	Cases 7	Cases 8	Cases 9	Cases 10	Cases 11	Cases 12
Temperature (°C)	30	30	30	30	8	10, 40
Humidity (−)	0.9	0.8	0.1	0.4	0.8	0.4
Example city	Beijing	Guangzhou	Yinchuan	Guiyang	Lhasa	–
Inlet velocity (m/s)	1.0, 5.5

Fig. 1

Schematic model of the airborne droplet flow and transmission process. (a) Breath and/or cough model; (b) mask conditions.

For the droplet study, the initial mole fraction in expiratory solvents is set to 0.7621% (x  = 0.007621), and the volume fraction of water is set to 99.2379% (Zeng et al., 2021). To simulate the coughing jet, 1500 80- or 40-μm-diameter droplets are assumed to be released in a Rosin-Rammler distribution. Each droplet is a virus carrier, initially at rest and randomly distributed in the mouth on a 4.0 cm × 4.0 cm plane. All particles are then exhaled continuously and uniformly with the airflow exiting from the lips, mouth, and nostrils. The droplets share the same velocity and temperature (36.0 °C) as the turbulent cloud ejected from the mouth: 1.0 m/s for respiration and 5.5 m/s for coughing. The coefficient of kinematic viscosity for air is set to 1.45 × 10−5 m2/s (25 °C, 1.0 atm), and the resulting Reynolds number (based on the exhalation velocity and the average mouth diameter) is set to be 17,600.

Results and discussion

Model validation

The results from the current numerical model were validated against the published data (Redrow et al., 2011; Wei and Li, 2015) for single-droplet evaporation. Two droplet diameters (10.0 and 100.0 μm) and relative humidities (0 and 0.9, respectively) are selected to compare the evaporation time with no wind. The ambient temperature and the initial droplet temperature are 20.0 °C and 36.0 °C, respectively. Each droplet contains 1.8% initial solid volume fraction and a NaCl concentration of 0.9% (Duguid, 1946). As shown in Fig. S2, the present results are entirely consistent with previous work (Redrow et al., 2011; Wei and Li, 2015).

Effect of droplet and aerosol size

Comparison of history of droplet aerosol size distribution

In real infection processes, the history of virus-carrying droplet or aerosol flow is an important factor for understanding the transmission mechanism. Thus, useful information can be obtained by simulating the size distribution of droplets or aerosols with a specific time history. The prolonged evaporative flow of individual particles (without sedimentation) will full account of the effects different initial size of droplets or aerosols, ambient humidity and temperature conditions, and Fig. 2, Fig. 3 provided such information for the cases (i) 7, 9, and 10 and (ii) 1, 10, and 12, respectively.

Fig. 2

The evaporation rate for airborne sputum droplets with solid fraction at different diameter and realtive humidity (associated with Case group of 7, 9 and 10). The ambient air temperature, the initial droplet temperature and the expiratory velocity are 36.0 °C, 30.0 °C and 1.0 m/s, respectively.

Fig. 3

The evaporation rate for airborne sputum droplets with solid fraction at different diameter and ambient temperature (associated with Case group of 1, 10 and 12). The expiratory velocity, the initial droplet temperature, and the ambient relative humidity are 1.0 m/s, 36.0 °C and 0.4, respectively.

The evaporation rate for airborne sputum droplets with solid fraction at different diameter (20, 80, and 140 μm) and realtive humidity (0.1, 0.4, and 0.9) has compared in Fig. 2. The ambient air temperature, initial droplet temperature and expiratory velocity are 36.0 °C, 30.0 °C and 1.0 m/s, respectively. The case results are plotted for different relative humidities as an analysis of the effects of diameter for different locations and/or outdoor or indoor conditions of very high RH, moderate RH, or very low RH (dry).

Figs. 2 and 3 shows that evaporation curve is monotonic shrinkage as expected from the classical view by Wells in 1934 (Wells, 1934) and in contrast to recently published literature (Wang et al., 2021; Ng et al., 2021; Chong et al., 2021). Indeed, temperature and humid environments diminish the amount of water vapor in the airborne, which can lead to droplets evaporation and monotonic decrease. Although the droplet nonvolatile matter reduces the activity of water and its surface saturation concentration, the solid content in this study is 0.7621% instead of 3% (Wang et al., 2021).

Fig. 2 shown that larger diameter droplets or aerosols travel longer in the environment before reaching an equilibrium state (see the horizontal curve after the drop history in Fig. 2), in which the saturated vapor volume on the droplet surface (C ) is consistent with the vapor volume in the surrounding environment (C ). The range of the final diameter depends on the initial diameter of the droplets or aerosol, and the final saturated salt components define the final concentration. For example, Table S1 summarizes the final diameter of the droplets or aerosol in Fig. 2. These results show that one order of magnitude difference appears between the cases with different initial diameters for the droplets. With a larger initial diameter, the equilibrium time and the final diameter both increases. Also, higher humilities lead to smaller evaporation rates that decrease the critical droplet size. Compare the curves in Fig. 2, with increasing RH, the lifetime of droplet and critical droplet size significantly: at low RH, the droplet or aerosol size decreases rapidly, and when the RH increases, these trends slow down, and the final diameter of the droplet or aerosol (critical droplet size) becomes relatively large. Such results indicate that larger-diameter droplets or aerosol settle faster than they evaporate under the effects of inertia and gravity. Such a process of larger-diameter droplets or aerosols would contaminate surrounding surfaces/desktop on the order of minutes in the environment, whereas small droplets or aerosols evaporate faster than they sink, so they would remain a long time in the environment in the form of a dry aerosol or droplet nuclei.

Fig. 3 shows the history of the droplet or aerosol diameter for different ambient temperatures, with the expiratory velocity, initial droplet temperature, and ambient RH set to 1.0 m/s, 36.0 °C, and 0.4, respectively, which is associated with cases 1, 10, and 12 of Table 1. Fig. 3 shows that the dominant factor determining the final equilibrium diameter is the RH, because the droplets of the same initial diameter have the same final equilibrium diameter. The final equilibrium diameters are also summarized in Table S2. The ambient temperature (ambient airflow temperature) significantly modifies the diameter history of the droplets or aerosol; however, it does not determine the final diameter. In contrast, the temperature difference between the droplets or aerosol and the ambient air may change the evaporation potential across the interface [as defined in Eq. (4)] and thereby defines the interfacial rate of heat transfer [as defined in Eq. (8)].

Effect of ambient parameters

Effects of ambient parameters of different representative conditions

Cold and humid situation. The cases with different ambient parameters and parameter groups, as shown in Table 1, are simulated and presented in Fig. 4 for case 5 (winter, cold and humid climate; representative of Changchun city in winter). The outdoor ambient temperature, pressure, and RH are 5.0 °C (cold), 1.0 atm, and 0.9 (very humid). The total number of ejected droplets is 1500 at 1.0 m/s, obeying the Rosin-Rammler distribution with an average diameter of 80.0 μm. Each droplet is composed of 99.2379% water and 0.7621% salt. The initial droplet temperature is 30.0 °C.

Fig. 4

Dynamic dispersion of saliva droplet cloud with humidity field for Case group 5 (low temperature and high humidity; winter). The total number of ejected saliva is 1500 at 1.0 m/s and obeys the Rosin-Rammler distribution with averaged diameter of 80.0 μm. Each droplet is composed of 99.2379% water and 0.7621% salt. The outdoor ambient temperature, pressure, and relative humidity are 5.0 °C, 1.0 atm, and 0.9, respectively. The mouth temperature and the initial droplet temperature are both 30.0 °C. (a) t = 0.2 s; (b) t = 2.0 s; (c) t = 4.0 s; and (d) t = 10.8 s.

The droplet or aerosol diameter distribution and evolution are plotted versus time in Fig. S3 and consider the different rates of evaporation for droplets of different sizes. Fig. S3 shows that, at relatively low speed (breath speed) and under high RH, large droplets tend to fall very quickly, forming major layers of droplets after about 4.0 s. Finally, around 10.8 s, the major heavy droplets reach the bottom surface, while a small fraction of aerosols with diameters less than 10 μm remain suspended in the environment. The respective distribution histories appear in Fig. S3, which shows that the peak of the droplet distribution gradually decreases from its initial time to around 10.8 s, which indicates a quick evaporation process. However, compared with the profiles in Fig. 4, major large- and mid-sized droplets approach the bottom surface but have not yet evaporated, so these droplets may continue to exist adhered to a surface, thereby helping to obtain positive surface samples in conditions in which airborne samples gave undetectable virus concentrations.

The simulated droplet or aerosol profiles for case 5 with different initial averages diameters of 110 and 140 μm are plotted in Fig. S4 and Fig. S6, respectively. The time histories of the diameter distributions for these two initial average diameters, which also follow the Rosin-Rammler distribution, are plotted in Fig. S5 and Fig. S7, respectively. Comparing Fig. 4, Fig. S4 and Fig. S6 (1) larger droplets fall faster [e.g., around 2.6 s in Fig. S4(c) and Fig. S6(c)]: and major droplets reach the bottom surface; (2) few droplets evaporate to become dry aerosols or droplet nuclei during the dynamic process until they rest on a surface; and (3) few droplets continue to exist as aerosols suspended in the ambient environment.

In addition, comparing Fig. S3, Fig. S5 and Fig. S7, the center of the droplet size distribution moves smaller when the initial diameter is set larger. These results help explain the quicker evaporation in cold and high RH conditions. In fact, other cases show different trends as the temperature increases, but the major determining factor, as shown in Fig. 2, is the humidity field, which is more important than the temperature field (except for extreme conditions of very high temperature).

Moderate temperature and humidity. The cases with moderate temperature and humidity are plotted in Fig. S8 for case 10 (high temperature and moderate humidity; e.g., summer in Guiyang city). The total number of droplets ejected is 1500 at 1.0 m/s, and they obey the Rosin-Rammler distribution with an average diameter of 80.0 μm. The outdoor ambient temperature, pressure, and RH are 30.0 °C, 1.0 atm, and 0.4. The initial droplet temperature is set to 30.0 °C for this simulation case. Compared with the case of winter at low temperature and high humidity (case 5, Fig. 4), the increase of ambient temperature and decrease of ambient humidity clearly accelerate the evaporation. The temperature and humidity tend to create more suspended droplets or aerosol in the final stage. The layered droplet-size structure also slows, as shown in Fig. S8. The most important characteristic is that the suspended droplets of smaller diameter (for example, less than 10 μm) are significantly more numerous, which is partially due to the evaporation process. In addition, the time-dependent distribution of droplet and aerosol size in Fig. S9 deviates significantly from that of case 5 (Fig. 4, Fig. S4 and Fig. S6). This is attributed to (1) the rapid evaporation and the modification of the size distribution for case 10, and (2) the rapid reduction in major diameter from around 80 μm to 10–20 μm for case 10. These results would reflect a different trend where the ambient temperature and humidity decrease: more suspended droplets or aerosols of smaller sizes would be obtained in these conditions, which would increase the likelihood of viral transmission in human social behaviors.

Analysis of representative trends. As shown in Table 1, more cases with different temperatures, humidities, and other flow parameters are simulated in this analysis. These configurations are simulated by using our in-house code describing the basic dynamic behavior of droplets and aerosols under representative situations. The results are obtained by comparing the cases between the two cases of (i) low temperature and high humidity and (ii) high temperature and low humidity. For the normal range of parameters, the controlling factor is the general humidity. The conciliation between evaporation and gravitational force determines the equilibrium time and also the final state of the suspended droplets or aerosol. An increase in temperature and inflow speed would prolong the suspension and increase the concentration of small droplets or aerosol in the environment. More discussions as with the transmission routes should be further considered for specific situations (such as complicated inflow conditions).

Effect of humidity under different inflow conditions

As discussed in the analysis of the droplet and aerosol diameters, the ambient humidity makes the dominant contribution to the equilibrium process and thereby to the final state of the droplet or aerosol group. This section discusses the inlet speed of the droplet or aerosol group based on a simulation of normal breathing and coughing, which imposes the exhalation speeds of 1.0 and 5.5 m/s, respectively.

The profiles of the droplets or aerosol for case 10 (high temperature and moderate humidity, summer) are plotted in Fig. S10 and Fig. S11 for average diameters of 80.0 and 120.0 μm, respectively. The exhalation speed is 5.5 m/s and 1500 saliva are ejected according to Rosin-Rammler distribution. The outdoor ambient temperature, pressure, and RH are 30.0 °C, 1.0 atm, and 0.4. The mouth temperature and initial droplet temperature are both 30.0 °C. Comparing Fig. S10 and Fig. S8 shows that increasing the exhalation speed from 1.0 m/s (Fig. S8) to 5.5 m/s (Fig. S10) produces a significant difference in the flow dynamics of the droplets or aerosol: (1) In Fig. S10, the layering of different droplet or aerosol sizes happens very slowly upon increasing the inhalation speed (not till 8.3 s do we see a clear layered structure in the vertical direction). (2) Higher initial speed increases the evaporation time, so major droplets or aerosols could become suspended in final states with small size (smaller than 15 μm). (3) Around 30% of the inhaled droplets or aerosol remain suspended for up to 2.0 m, which is relatively far for normal conditions.

In addition, the case with a larger diameter of 120.0 μm, as plotted in Fig. S11, shows that increasing the initial average diameter produces a layered structure and very few droplets become suspended in the environment. Such cases follow the trends seen in Fig. 4, Fig. S4 and Fig. S6 for increasing the average diameter, which indicates different routes of transmission mechanism along with different virus types (further analysis of this issue is required). The current case 10 assumes high temperature and moderate humidity (e.g., summer in Guiyang city), so special care should be taken when dealing with long-distance droplet or aerosol routes.

Fig. 5 shows case group 10 for high ambient temperature and moderate humidity. The results show that, at relatively low inhalation speeds, the normalized droplet or aerosol number generally falls over a short distance, whereas the droplet or aerosol number increases at relatively high inhalation speed. Such a trend should be analyzed together with the effect of droplet size at equilibrium, as indicated in the discussion preceding this section: as the inhalation speed increases, the distance traveled increases, but the initial size distributions dictate whether a group of suspended droplets or aerosol exist in the final stage.

Fig. 5

Droplet transport distance distribution statistics with different exhalation velocity for Case group 10 (high temperature and moderate humidity, summer).

Masked versus unmasked

Masks have two roles: the most extensive is to protect the uninfected individual by preventing virus-carrying aerosols injected into the respiratory tract, and the second is to reduce the emission of virus-laden aerosols by the infected and asymptomatic individuals, and this section discusses the first one. Given that the previous discussion of ambient parameters shows that small-diameter droplets or aerosol may exist within the evolution of the size distribution of specific cases. The simple assumption of mask use is that it only allows droplets or aerosol to pass if they are smaller than 10 μm in diameter. Although unrealistic, it helps to explore the penetration of masks under different environmental conditions. Thus, such cases have been simulated in this section (i.e., mask versus no mask) for droplet or aerosol size distribution and filtration check.

The geometry of a mask for preventing the spread of the virus for case 10 (high temperature and moderate humidity, summer) is simulated, and the results are plotted in Fig. S12. The total number of ejected saliva is 1500 at 5.5 m/s, and the ejecta obey the Rosin-Rammler distribution with an average diameter of 80.0 μm. The outdoor ambient temperature, pressure, and RH are 30.0 °C, 1.0 atm, and 0.4. The mouth temperature and the initial droplet temperature are both 30.0 °C. Fig. 5 shows the schematic results, for example, the picture of the traveling droplets or aerosol. At 5.5 m/s inhalation speed, the major droplet or aerosol group reaches in 0.3 s the target person about 1.5 m away. It should be noted that in the current cases, the airflow and coupled transmission are considered as general field evolution under different ambient conditions, which gives a general view about how such virus-loading droplets/aerosols.

Fig. 6 shows the intake ratio for the total droplet or aerosol volume under masked conditions. With no mask, around 10%–20% would be inhaled by a person about 1.0 m away. However, according to Fig. 6, only around 0.1% is inhaled if the RH is around 0.1. Additionally, if the ambient humidity increases, this value becomes much smaller (less than 0.001% for RH = 0.9). In fact, the current results indicate that an increased humidity with smaller evaporation rates causes a rapid decay of the large droplets or aerosol at the mid-way. These results confirm the basic dynamic protection offered by wearing a mask, even if 10 μm droplets or aerosol are assumed.

Fig. 6

Droplets and virus load in-taken by the listener (with mask) with different relative humidity parameters for Case group 10 (high temperature and moderate humidity, summer). The initial droplet temperature and the expiratory velocity are 36.0 °C, 5.5 m/s, respectively.

In this study, fluid flow, droplet dispersion with force balance, droplet evaporation mass and heat transfer, and evaporative flow interactions were investigated, and explored the dynamic flow of aerosols under different environmental conditions from the perspective of droplet evaporation. However, the present study has several limitations, including the following：(1) The droplets or aerosol is idealized which both large and small droplets are assumed to be virus-carrying. Further studies are necessary to determine the viral load, virus concentration (Belosi et al., 2021) at the initial droplet diameter for each size group. (2) The risk assessment entailed in this study pertains only to idealized mask conditions. Further studies should be made on this aspect. (3) The cough process is unrealistic where droplet and airflow velocities are assumed to be same at the source. These limiting factors may affect the evaporative flow interactions for each size group in dispersion.

Conclusions

From fluid dynamics viewpoint, the transmission of virus-carrying droplets or aerosol is one key to understanding the virus-transmission mechanisms. This study is focused on the airborne transmission process by incorporating the flow-evaporation model into the Navier-Stokes equations in an inhouse code simulation. Various ambient conditions defined by RH, and temperature have been simulated, which represent major cities around the world.

The initial size distribution of virus-carrying droplets or aerosol affects the travel history of droplets. The time to reach the final equilibrium diameter depends on the initial size: 140-μm-diameter droplets require around 101 s (RH = 0.1) and 102 s (RH = 0.9), 80-μm-diameter droplets require around 5 × 10° s (RH = 0.1) and 5 × 101 s (RH = 0.9), 20-μm-diameter droplets require around 10−1 s (RH = 0.1) and 10° s (RH = 0.9). The equilibrium diameter mainly depends on the ambient RH.

For different combinations of representative city analysis, cold region and high/low humidity, moderate temperature city, and extreme conditions of (i) low temperature and high humidity and (ii) high temperature and low humidity, the controlling factor is found general for humidity. For representative ambient conditions, low temperature and high humidity (e.g., winter in Changchun) will lead to quicker rest on a surface than the conditions of high temperature and low humidity (e.g., winter in Guangzhou). The conciliation between evaporation and gravitational force determines the equilibrium time and also the final state of the suspended droplets or aerosol. An increase in temperature and inflow speed would prolong the suspension and increase the concentration of small droplets or aerosol in the environment. It is suggested that those trends with different conditions will help to identify the different transmission routes.

An ejection speed of 5.5 m/s (1.0 m/s), about 30% (5%) of the droplets or aerosol travel more than 2.0 m. Layered-separation of different droplet sizes gradually happens upon increasing the inhalation speed. Higher initial speed increases the evaporation time, so major droplets or aerosols could become suspended in final states with small size (smaller than 15 μm), while around 30% of the inhaled droplets remain suspended for up to 2.0 m, which is relatively far for normal situations and should be taken care.

Wearing a mask provides protection against viral transmission as only about 0.1% of the droplets is transmitted to a person at 1.0 m distance if the RH is around 0.1. With increased humidity, this value decreases significantly (less than 0.001% for RH = 0.9). An increase in humidity causes large droplets or aerosol to fall quickly. In fact, the current results indicate that an increased humidity causes a rapid decay of the large droplets or aerosol at the mid-way. These results confirm the basic dynamic protection offered by wearing a mask, even if 10 μm droplets or aerosol are assumed.

Credited authors

Gang Zeng: Data curation, Formal analysis, Investigation, Lin Chen: Conceptualization, Funding acquisition, Project administration, Supervision, Writing – review & editing, Haizhuan Yuan: Supervision, Funding acquisition, Writing – review & editing, Ayumi Yamamoto: Data curation, Investigation, Haisheng Chen: Supervision, Writing – review & editing, Shigenao Maruyama: Conceptualization, Supervision, Investigation, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.