Transport of exhaled droplets and aerosol suspension is a main route for the transmission of highly infectious respiratory diseases. A poorly ventilated room, where human body heat drives the flow and the pathogen motion, is one such paradigmatic situation with an elevated risk of viral transmission. Here, we report a numerical study on human body heat-driven buoyancy convection in a slender rectangular geometry with the geometric size of 12 × 1 × 3 m3. Using large-scale three-dimensional simulations, we reveal how different spacings between human body heat sources can potentially spread pathogenic species between occupants in a room. Morphological transition in airflow takes place as the distance between human heat sources is varied, which shapes distinct patterns of disease transmission: For sufficiently large distance, individual buoyant plume creates a natural barrier, forming buoyant jets that block suspension spread between occupants. Thermal plumes exhibit significant individual effects. However, for small distances, a collective effect emerges and thermal plumes condense into superstructure, facilitating long-distance suspension transport via crossing between convection rolls. In addition, we quantify the impact of morphological transition on the transport of viral particles by introducing tracer particles. The quantitative analysis shows that under certain critical distances, the infection risk becomes significantly elevated due to this transition and collective behavior. Our findings highlight the importance of reasonable social distancing to reduce indoor cross-transmission of viral particles between people and provide new insights into the hidden transitional behavior of pathogen transmission in indoor environments.

Outbreaks of respiratory diseases, such as influenza, severe acute respiratory syndrome (SARS), Middle East respiratory syndrome (MERS), and the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), have taken a heavy toll on human populations worldwide. Until vaccines are widely available, it is commonly held that the useful precautions are case isolation, wearing mask, social distancing,1–6 which relate closely on the flow physics of respiratory suspensions.7–10 Nowadays, flattening the curve is an important public health strategy to cope with the pandemic. To efficiently slow down the spread of disease, a relevant question to be answered is how people maintain a safe distance with an infected person as they interact together in an enclosed room, which is especially important as we would gradually return to the normal social interaction in post-pandemic phase. In contrast to outdoor events where dilution of contaminant can be brought about by wind, indoor social interactions between human bodies, however, could lead to a significant risk for the spread of respiratory coronaviruses, especially when the ventilation is inadequate.11–17 In indoor situations, saliva droplets or aerosols are released into the environment by asymptomatic or presymptomatic infected individuals,18–25 then suspended and mixed by the airflow for hours or even days.1,26–30 This kind of indirect transmission involves infections in large spatial and temporal scales and possibly leads to the superspreading events.31–34 

Surprisingly, little is known on long-time dispersion and transmission pattern of suspended viral particles in indoor environment. Such physical process of spreading is subtle as it involves interactions between respiratory droplets, turbulent eddies, vapor, and temperature fields1,28,35–39 and also relates to several environmental factors such as ventilation,14,40–42 ambient temperature,43,44 and relative humidity.45–49 

Respiratory viruses can spread among individuals through three primary routes.39,50 First, transmission can occur through direct contact or via contaminated surfaces (fomites) when a healthy person interacts with an infected individual. The other two routes of infection are associated with the inhalation and subsequent deposition of pathogen-laden droplets in the respiratory mucosa: droplet direct transmission and airborne transmission. The key distinguishing factor between these routes is the aerodynamic behavior of the droplets. The transmission of pathogen-laden droplets is governed by the interplay of gravitational forces, drag forces, and evaporation, which determines the transport and deposition of respiratory droplets.50 Larger droplets demonstrate ballistic behavior due to gravity [see Fig. 1(a)-I], leading to direct transmission and playing a major role in close-range spread between individuals.51 In contrast, smaller droplets exist in the form of aerosols, being controlled by aerodynamic drag and evaporation, thus enabling airborne transmission with a wider range.52–54 These tiny droplets, which are expelled from the oral cavity through respiration or sneezing, quickly evaporate and form droplet nuclei.8,55 For instance, a droplet with initial radius r0 is reached over an evaporation timescale, τ e = r 0 2 / ( θ ( 1 R H ) ), where RH is the relative humidity and θ = 4.2 × 10 10 m 2 s 1 at 25 °C.55,56 The evaporation time at 50% RH ranges from τ e = 1.2 × 10 3 s for r 0 = 0.5μm to 0.12 s at 5 μm [details can be found in Fig. 11 of  Appendix D]. Furthermore, Wang et al.39 found that the turbulent and humid air released during a sneeze leads to a longer evaporation time of droplets than predicted, especially under low temperature/high humidity conditions. They proposed modifications to the classical D2-law. The revised D2-law is capable of predicting the mean evaporation behaviors of respiratory droplets accurately.50,57 Note that we have neglected the evaporation process of liquid droplets. In this study, we consider the long-time and long-range transmission of droplet nuclei which formed due to the existence of non-evaporative substance in the droplet.

FIG. 1.

Airflow patterns driven by human body plumes. (a) Major modes of transmission of respiratory viruses during short-range and long-range transmission. During respiratory infections, individuals expel virus-laden droplets and aerosols while breathing. Large droplets settle quickly (I), but aerosols persist in the air, increasing transmission risk in poorly ventilated areas (II). Infrared cameras detect thermal emissions in enclosed spaces like classrooms, revealing elevated body surface temperatures with a distinct orange-red hue (III). (b) Illustration of the simulation set-up with the heating source of human body.

FIG. 1.

Airflow patterns driven by human body plumes. (a) Major modes of transmission of respiratory viruses during short-range and long-range transmission. During respiratory infections, individuals expel virus-laden droplets and aerosols while breathing. Large droplets settle quickly (I), but aerosols persist in the air, increasing transmission risk in poorly ventilated areas (II). Infrared cameras detect thermal emissions in enclosed spaces like classrooms, revealing elevated body surface temperatures with a distinct orange-red hue (III). (b) Illustration of the simulation set-up with the heating source of human body.

Close modal

The well-mixed suspended droplet nuclei in the room pose a risk of airborne transmission when inhaling long-term suspended small aerosol droplets. For normal breathing, the droplet nuclei radii vary between 0.1 and 5.0 μm, with a peak around 0.5 μm.58–60 Sufficiently small droplet nuclei may be suspended by the ambient airflow and mixed throughout the room until being removed by the airflow or inhaled. The aerodynamic behavior of droplet nuclei strongly depends on the local flow conditions.

Indoor environments experience two types of air flows that facilitate air mixing: buoyancy-driven convection and forced convection. Buoyancy-driven convection occurs when temperature differences create density variations, primarily due to body heat in cooler spaces. It generates turbulent thermal plumes that rise from individuals, causing localized flows. These flows are prone to hydrodynamic instabilities, promoting mixing on a larger scale. Lewis et al.61 referred to the flow field near the human body as the microenvironment, which is influenced by temperature differences between the body surface and the ambient air, the type of clothing,62,63 body geometry,64,65 body posture,66 ventilation,42,67–69 the flow of the human breath,66,70,71 and the environmental thermal stratification.72 Thermal plumes resulting from human heat sources exhibit velocities ranging from 0.1 to 0.25 ms−1 (see Refs. 72–77). These thermal plumes significantly impact air distribution72,78–81 and the dispersion of pollutants in enclosed spaces.82,83 The thermal plume's impact on contaminant intake by the respiratory system varies with the pollution source location.84 For lower sources, buoyancy flow facilitates contaminant entry.85 Conversely, a vertical upward thermal plume at the height of the human body prevents airborne particle entry in the absence of circulation. A dense and fast thermal plume poses challenges to nasal aerosol penetration.85 The flow field characteristics around the human body have been extensively studied,84,86–88 and the recent review paper by Zong et al.89 discusses the effect of the thermal plume on inhalation exposure to particulate matter.

In practice, ventilation is considered as an active control strategy to transport and exhaust those viral aerosols or droplets. For the sake of carbon dioxide removal and energy saving, the ventilating flow needs not to be too strong except in some dedicated cases. For example, the average flow speed is 4 × 10−3–8 × 10−3 ms−1 for the ventilation rate of N = 5–10 ACH.40 However, in such situations, the thermal plumes emitting from a human body becomes significant to drive the indoor flow and thus spread the droplet nuclei in enclosed spaces. The human thermal plume velocity typically ranges between 0.1 and 0.25 ms−1,72,74 based on various factors including the individual's situation and the temperature contrast between the human body and the surrounding environment, and the number of individuals in the room. Craven and Settles72 found a maximal body plume velocity of 0.24 ms−1 from their experiments of an individual heat source with a temperature difference of 5.3 °C between the human body and the surrounding environment. Murakami et al.74 showed a maximum plume velocity of 0.23 ms−1 from the simulation of the human thermal plume. However, our current research focuses primarily on the case of multiple human heat sources. Considering the body-and-ambient temperature difference of 8 °C, the speed of thermal plume generated from a person is approximately 0.89 ms−1 using the characteristic buoyancy velocity scale, giving the flow even much stronger than the normal indoor ventilating flow. Hence, the body heat source and its distribution can crucially shape the pattern of viral transmission in indoor environment [see Fig. 1(a)].

To address the question of how the body heat drives the flow pattern and affects the disease transmission, we performed large-scale simulations of indoor airflow with varying distances between human heat sources to mimic different social distancing measures. Our findings demonstrate that the spacing between occupants plays a critical role in shaping disease transmission patterns, with a morphological transition in airflow occurring as distances are altered. Our analysis reveals that under certain critical distances, the collective behavior of thermal plumes significantly elevates the risk of infection.

The remainder of the paper is organized as follows: the numerical approach is briefly introduced in Sec. II. Then, the results are shown in Sec. III. In Sec. III A, we discuss how the morphological transition in airflow takes place as the distance between human heat sources is varied. In Sec. III B, we employ the Lagrangian tracking approach to simulate the trajectories of floating aerosols, investigating the impact of heat sources on viral particle transmission through statistical analysis of their trajectories. In Sec. III C, we evaluated the risk of cross-infection at different spacings in poorly ventilated rooms. Finally, a summary is presented in Sec. IV.

We consider a room of dimensions L : W : H = 12 × 1 × 3 m 3, as sketched in Fig. 1(b), with a narrowed room being considered. Considering that the primary focus of our study is to investigate the flow pattern of convective flow and the influence of the distance between human heat sources along the x-axis on the transport and dispersion of viral particles, we simplified the problem with a small dimension in the lateral direction (y-direction). This choice was motivated by the desire to effectively capture the thermal plume dynamics while maintaining computational efficiency. We adopted a relatively small ratio of the width to the height ( W / H = 1 / 3), which is commonly used in studies of classical Rayleigh–Bénard convection.90–92 An occupant sitting in the room produces thermal plumes due to the temperature contrast between the ambient and human body. For simplicity, the heat source of the seated occupant is evenly distributed, and the ambient temperature of the room is considered to be homogeneous initially and set to the value of normal indoor temperature, i.e., T amb = 26 ° C. We simulate the buoyancy-driven effect of the body heating by setting the temperature boundary conditions. Normally, the human body exhibits temperature variations, particularly the range in the head region and other sections, such as the legs, which may have lower temperatures. Clothing plays a significant role as an insulator, where the outer surface of garments serves as the authentic boundary condition for convective heat transfer.72 The small air gap between the skin and clothing hinders heat transfer. Thus, in modeling the human thermal plume, constant surface temperature is set equal to the experimentally obtained average temperature of the region above the shoulders of human subjects. The heating temperature of the body is set to be T ocpt = 34 ° C. Here, we simplify the human body model in a sitting posture by assuming a lateral dimension of l ocpt = 0.125 H = 0.375 m with disregarding the height of the human body. It is important to note that the human considered in our study are under baseline protection such as wearing masks. When occupants of a room are wearing masks, respiratory jets are blocked; therefore, the respiratory has a slight effect on the plume dynamics. Hence, in our work, the rising plumes from human body heat present the primary pathway for the airborne pathogen, and we focus on the impact of plumes on the transport of viral particles. Consequently, we have disregarded the impact of respiratory jets in our study. The fluid is assumed to be incompressible · u = 0. The gas Prandtl number is P r ν / κ T = 0.7, where ν is the kinetic viscosity of air and κ T is the thermal conductivity. The thermal Rayleigh number defined using the occupant height h ocpt = 1.2 m is R a ocpt α g Δ h ocpt 3 / ( ν κ T ) 10 9, where Δ T ocpt T amb = 8 ° C,42  α is the thermal expansion coefficient, and g is the gravitational acceleration. A detailed list of parameters and material properties of the gas can be found in Table I. We characterize the spacing of human heat sources by λ, which equals to the length L divided by the number of human body k ( λ = L / k), which is also an important control parameter of the system.

TABLE I.

List of the material properties of the surrounding gas. We, in particular, give the values of these parameters and material properties which we employed in the numerical simulations of this study.

Properties Symbols Values
Ambient temperature  T amb  26 °C 
Thermal expansion coefficient  α  3.4 × 10 3 K 1 
Air kinematic viscosity  ν  1.562 × 10 5 m 2 s 1 
Thermal diffusivity of gas  κT  2.21 × 10 5 m 2 s 1 for air at 26 °C 
Gravitational acceleration  9.8 ms−2 
Properties Symbols Values
Ambient temperature  T amb  26 °C 
Thermal expansion coefficient  α  3.4 × 10 3 K 1 
Air kinematic viscosity  ν  1.562 × 10 5 m 2 s 1 
Thermal diffusivity of gas  κT  2.21 × 10 5 m 2 s 1 for air at 26 °C 
Gravitational acceleration  9.8 ms−2 
In this study, we conducted direct numerical simulation (DNSs) of three-dimensional (3D) turbulent convection. There are previous studies93–103 shown that the steady laminar flow of a single plume can be accurately described by similar solutions derived from boundary-layer equations. Additionally, theories exist regarding the scaling of plume ascent velocity,104 plume stem structure,105 and plume growth through the entrainment of ambient fluid.106 Here, we consider the complexity of multiple interacting heat sources within a room and applied DNS to accurately capture the interactions and complex dynamics involved in the flow within an indoor environment. Specifically, we aimed to understand the influence of strong shear resulting from the global flow structures, such as large-scale circulation, on the behavior of human thermal plumes. We consider the coupled equations of motion for the velocity field u and the temperature field T for convective flows in a poorly ventilated room with various heating sources regularly arranged below. Under the Boussinesq approximation, the governing equations are
· u = 0 ,
(1)
t u + ( u · ) u = 1 ρ p + ν 2 u + α g T e z ,
(2)
t T + ( u · ) T = κ T 2 T ,
(3)
where t is the time, u is the fluid velocity, ρ is the fluid density, T is the temperature, and e z is the vertical unit vector. All quantities studied above have been made dimensionless by the cell height H, the imposed temperature difference Δ, and the free fall velocity α g Δ H and free fall time H / α g Δ.
To numerically solve the equations, we used our finite difference solver AFiD with high performance message passing interface (MPI).107–109 The governing equations are numerically solved by DNS using the multi-scalar second-order finite-difference method with a fractional third-order Runge–Kutta scheme, which has been validated many times in our previous studies.28,38,110–113 For boundary conditions, at all solid boundaries, no-slip boundary conditions are applied for the velocity. We apply a distributed heating to the bottom plate as follows:
T bot ( x ) = { T ocpt , for ( n 1 / 2 ) λ x ( n 1 / 2 ) λ + l ocpt , T amb , for others ,
(4)
where n = 1 , 2 , , k , l ocpt = H / 8 is the width of occupants. A constant temperature T top = T amb holds on the top plate, and an insulating condition is adopted on the sidewalls. About the initial conditions, the temperature field is constant set to ambient temperature. The velocity, the systems are motionless initially, without any perturbation. For each run, the uniform grid spacing is adopted horizontally while stretched grid spacing is adopted vertically to guarantee more grid points near the top and bottom walls for resolving small scales inside boundary layer. We employ a grid resolution of 2048 × 170 × 512 to adequately resolve both the Kolmogorov length scale for the velocity field u and the Batchelor length scale for the temperature field T. The Kolmogorov scale is estimated by the global criterion η = ( ν / ε ) 1 / 4 and the Batchelor scale η B = η P r 1 / 2.114 Concretely, for each simulation, more than 15 grid points are nested within thermal BL. The maximal grid spacing Δ g max satisfies Δ g max 3 η and Δ g max 3 η B. Additionally, we have verified grid independence; details can be found in Fig. 8 of  Appendix A. The time step adopted is small enough to guarantee numerical stability and resolve the Kolmogorov timescale of turbulent fluctuations. The simulations were performed for at least 900 free fall time units, and we sample the last 300 free fall units for our statistical analysis. To ensure that the steady state is reached, we compared the last 100 free fall units to the full sampling duration, and the variation is 1% difference. We have compared our numerical results of the body plume velocity with the experimental results of Craven and Settles,72 which shows good agreement [see Fig. 7 in  Appendix A].

For the droplet nuclei in the room, we assume a well-mixed room. The assumption of well mixedness is widely applied in the theoretical modeling of indoor airborne transmission.115–117 The fate of ejected droplets in a well-mixed environment depends on the relative magnitudes of two speeds: the settling speed of the droplets in still air (vs) and the ambient air circulation speed within the room (va).55 Drops with a radius of r and density ρd descend through still air with a density ρa and dynamic viscosity νa at the Stokes settling speed v s ( r ) = 2 Δ ρ g r 2 / ( 9 ν a ). This speed is determined by the balance between gravity and viscous drag,55,118 where g represents the gravitational acceleration and Δ ρ = ρ d ρ a. Consideration is given to a room of area A, depth H, and volume V = HA, with a ventilation outflow rate Q and an outdoor air change rate (commonly expressed as air changes per hour, or ACH) of λ a = Q / V. The mean air velocity v a = Q / A determines the air mixing time τ a = H / v a, which operates on the room scale. The timescale for droplet settling in a well-mixed environment corresponds to that in a still environment. Equating the characteristic times for droplet settling, τ s = H / v s, and removal, τa, indicates a critical droplet radius r c = 9 λ a H ν a / ( 2 g Δ ρ ).55 For a room with a height of H = 3 m and a ventilation rate of λa = 5ACH, corresponding to r c = 6.22 μ m. The “airborne” droplets of interest in our study, those with a radius of r < 5 μm, thus constitute a significant fraction of those emitted during most respiratory events.58,59,119 Furthermore, following Wang et al.,39 we assume a uniform viral load across all droplets at the time of their ejection. Additionally, we do not consider any decay in the viral load.

For aerosols, we apply the spherical point-particle model and consider the conservation of momentum120–122 as follows:
d u i , n d t = ( β + 1 ) D u i , g , n D t + ( β + 1 ) 3 ν ( u i , g , n u i , n ) r n 2 f d + g β e ̂ z .
(5)
The notations we used in equations are as follows: u i , n and u i , g , n are the velocities of particles and air at the location of the droplets, respectively. β is a dimensionless measure of the particle density relative to the fluid density and is defined as β 3 ρ / ( ρ + 2 ρ l ) 1 with ρl the density of particles. fd is the prefactor for the drag corrections defined as f d = 1 +  0.169 R e p 2 / 3 with Rep the particle Reynolds number. Because the volume fraction considered is dilute, we chose not to couple the particles to the momentum field, as such a coupling would have a negligible effect. Considering that the majority of exhaled aerosols are smaller than 5 μm,53,119 even a large fraction is <1 μm for most respiratory activities, including those produced during breathing, talking, and coughing, we investigate the effect of heat sources on virus particle transmission from the Lagrangian viewpoint by tracking fluid particles and calculating particle statistics. Our focus here is on transmission via aerosols, which are small enough (and noninertial) that they can be regarded as faithful tracers of the fluid flow, i.e., u i , n = u i , g , n.17,123,124 We record particle information every ten-time steps, at approximate time intervals of 0.1, so that the Lagrangian statistics produced are also well resolved in time. At each time step, the particle velocities are interpolated from the instantaneous Eulerian velocity field using a tri-cubic polynomial interpolation scheme.125,126 The consequence is that the Lagrangian statistics that we produce in this work are well resolved in space.127 

The flow structure driven by regularly arranged occupants in a poorly ventilated room is first examined. Figures 2(a)–2(f) show the typical instantaneous flow structure at different distances λ of body heating sources, visualized by the volume rendering of temperature field. It is clearly seen that the fluctuating thermal plume is emitted from the heating source and rises into the ambient air. For large distance λ, i.e., with small number of occupants as shown in Figs. 2(a)–2(c), those ejected plumes self-organize into the circulating flow (Multimedia view). The interior spaces of room are then separated by the airflow pattern formed by several individual circulations. For small distance λ, i.e., with large number of occupants as shown in Figs. 2(d)–2(f), the collective buoyancy effect is found, where the large-scale flow induces strong shear effect to favor the merging of thermal plumes (Multimedia view). The plume merger connects several heating sources, resulting in the superstructure in the enclosed room. This phenomenon is also observed in geometries with smaller aspect ratios, where the width extends across a wider range [see Fig. 9 in  Appendix B]. In the large-scale upwelling region, although near the source each plume will behave independently, they do interact by drawing together due to the entrainment of ambient fluid in the region between them. After the two plumes touch, they will interact in a region where the self-similar solutions for a single plume do not hold. At an even larger distance from the sources, the two plumes will have merged into a single plume.128 This merging phenomenon has been extensively discussed by Kaye and Linden.101 We further note that the merging of buoyant plume has also been observed for the convective dissolution of multiple sessile droplets, where these droplets are composed of long-chain alcohol with the initial size of hundred micrometers.129 

FIG. 2.

The snapshot of the temperature and velocity (arrows) field in the steady state for the different distancing of people in the room: λ = L in (a), λ = L / 2 in (b), λ = L / 4 in (c), λ = L / 8 in (d), λ = L / 16 in (e), and λ = L / 32 in (f). For all cases, R a ocpt is fixed at 109, and Pr is set to 0.7. Multimedia available online.

FIG. 2.

The snapshot of the temperature and velocity (arrows) field in the steady state for the different distancing of people in the room: λ = L in (a), λ = L / 2 in (b), λ = L / 4 in (c), λ = L / 8 in (d), λ = L / 16 in (e), and λ = L / 32 in (f). For all cases, R a ocpt is fixed at 109, and Pr is set to 0.7. Multimedia available online.

Close modal

The overall airflow pattern undergoes a morphological transition from the individual to the collective behavior of thermal plumes as the distance between occupants decreases as shown in Figs. 3(a)–3(c). At λ = L, the motion of plume exhibits strong individual effect, i.e., thermal plume from heating source directly moves upward without interaction with the adjacent plumes. With decreasing the distance λ (e.g., λ = L / 4 and λ = L / 16), body plumes start to merge with each other as favored by the shearing of the large-scale mean flow, and eventually, the strong collective behavior appears with all thermal plumes contribute energy to a single large-scale flow.

FIG. 3.

Individual and collective effects of body plume motions. Behavior of body plumes for different distances between occupants: λ = L in (a), λ = L / 4 in (b), and λ = L / 16 in (c). (d) and (e) The Reynolds number ratio R e z / R e as a function of the horizontal position x obtained from the slice of the velocity field at mid-width. Here, R e z = w 2 z H / ν is the averaged vertical Reynolds number, R e = u 2 + v 2 + w 2 z H / ν, where · z denotes the averaging over vertical direction. The right axis (marked in red) represents the actual values of Rez, while the dashed line represents the spatially averaged Re. As shown in (d) at λ = L, the plume emission at the peak position leads to local flow enhancement, showing significant individual effects. However, at λ = L / 16 in (e), the location of the plume-rising area does not entirely depend on the individual heat source, the behavior of the plume shifts to a collective effect. (f) The horizontal Reynolds number Rex as a function of the ratio between the length and x distancing L / λ. Here, R e x = u 2 V , t H ν is the spatially and temporally averaged horizontal Reynolds number. For all cases, R a ocpt is fixed at 109, and Pr is set to 0.7.

FIG. 3.

Individual and collective effects of body plume motions. Behavior of body plumes for different distances between occupants: λ = L in (a), λ = L / 4 in (b), and λ = L / 16 in (c). (d) and (e) The Reynolds number ratio R e z / R e as a function of the horizontal position x obtained from the slice of the velocity field at mid-width. Here, R e z = w 2 z H / ν is the averaged vertical Reynolds number, R e = u 2 + v 2 + w 2 z H / ν, where · z denotes the averaging over vertical direction. The right axis (marked in red) represents the actual values of Rez, while the dashed line represents the spatially averaged Re. As shown in (d) at λ = L, the plume emission at the peak position leads to local flow enhancement, showing significant individual effects. However, at λ = L / 16 in (e), the location of the plume-rising area does not entirely depend on the individual heat source, the behavior of the plume shifts to a collective effect. (f) The horizontal Reynolds number Rex as a function of the ratio between the length and x distancing L / λ. Here, R e x = u 2 V , t H ν is the spatially and temporally averaged horizontal Reynolds number. For all cases, R a ocpt is fixed at 109, and Pr is set to 0.7.

Close modal

To quantitatively distinguish the individual and collective effects, we plot in Figs. 3(d) and 3(e) the profiles of R e z / R e along the horizontal direction, which indicates the strength of buoyancy-driven jet from the heat source. Here, Rez is the Reynolds number defined by the vertical velocity magnitude, and Re is the overall Reynolds number defined by flow speed considering all velocity components. At λ = L in the individual regime [see Fig. 3(d)], the location of the peak coincides with the location of the heat source, and the value of the local Rez is approximately twice of the overall Reynolds number Re. This indicates that the airflow for small number of occupants (in individual regime) is mainly contributed by the buoyancy-driven jet with the dominance of the vertical momentum. At λ = L / 16 under the collective regime [see Fig. 3(e)], the location of the plume-rising area does not have one-to-one correspondence with the individual heat sources due to merging of plumes. In this case, the horizontal momentum becomes dominant with the emerged superstructure as shown in Fig. 3(f). The consequence of the enhanced horizontal momentum is the potential transport of viral aerosols over a long lateral distance.

It is highly desirable to understand how the airflow transition affects the aerosol transportation, which is important to disease transmission. As most of exhaled aerosols are smaller than 5 μm,53,119 and a large fraction is < 1μm for most respiratory activities, including those produced during breathing, talking, and coughing, we employ the Lagrangian tracking approach to simulate the trajectories of the floating aerosols. The effect of heat sources on viral particle transmission is examined through statistical analysis on the trajectory of aerosols. As the size of aerosols is small enough, they can be regarded as faithful tracers of the fluid flow.17,123,124

The positions of thousands of Lagrangian tracer particles are initialized randomly over the domain at a time when the turbulent flow has attained a statistically stationary state [see Fig. 10 in  Appendix C]. To track the trajectories of particles, we initially marked a subset of particles for tracking purposes, as shown in Fig. 4(a). After a certain period of time, we observed that a majority of these particles were predominantly distributed within a large-scale circulation, while only a small fraction of the particles were transported to more distant regions [see Fig. 4(a)]. We plotted the trajectories of these particles over this period, with the color of the trajectories representing the corresponding time [see Fig. 4(b)]. Through analyzing these trajectories, we can clearly observe two distinct stages of particle transport. With the obtained trajectories, the statistical behavior of particles can be characterized by their mean-squared displacement (MSD). Figure 4(c) shows the MSD vs time t for various distance λ. The trends of MSDs for different λ are seen to exhibit similar behavior, namely, at short time, the particle motion is ballistic ( r 2 ( t ) t 2) since the aerosols are advected by the circulation [see Figs. 4(a) and 4(b)]; after a certain time tc, a short range of diffusive motion ( r 2 ( t ) t 1) is found and eventually the MSD becomes independent of t due to the finite-size effect.

FIG. 4.

Transport characteristics of aerosol transport. (a) Ballistic and diffusive motion of the particles. Typical spatial distribution of aerosols in the ballistic and the diffusive regime. (b) Examples of particle trajectories for both ballistic and diffusive motion. (c) The mean-squared displacement (MSD) r x 2 ( t ) of particles as a function of time. The straight dotted lines indicate the theoretical scaling laws corresponding to ballistic and diffusive regimes. The measured MSD as a function of L / λ at tt0 = 1 in (d) and tt0 = 100 in (e), where t0 is the time of particle release.

FIG. 4.

Transport characteristics of aerosol transport. (a) Ballistic and diffusive motion of the particles. Typical spatial distribution of aerosols in the ballistic and the diffusive regime. (b) Examples of particle trajectories for both ballistic and diffusive motion. (c) The mean-squared displacement (MSD) r x 2 ( t ) of particles as a function of time. The straight dotted lines indicate the theoretical scaling laws corresponding to ballistic and diffusive regimes. The measured MSD as a function of L / λ at tt0 = 1 in (d) and tt0 = 100 in (e), where t0 is the time of particle release.

Close modal

To quantify how rapidly the aerosols spread inside the room, we plot in Figs. 4(d) and 4(e) the magnitude of MSD as a function of L / λ at two time instants. At tt0 = 1 where the motion of aerosols is still within the ballistic stage, the trend of displacement in horizontal direction r x 2 ( t ) is similar to that of Rex [see Fig. 3(f)] because the amount of displacement is set by the strength of the flow. At tt0 = 100 in diffusive stage [see Fig. 4(e)], the transition from individual to collective regime can be clearly distinguished: the horizontal displacement decreases with increasing L / λ in the individual regime, and then increases with L / λ in the collective regime.

This above trend can be understood from the following visualizations. Figures 5(a)–5(d) present the snapshots of the spatial distribution of particles overlapped with the contour of temperature field. In the individual regime, as illustrated in Figs. 5(a)–5(c), the thermal plumes from heating source behave as buoyant jets. Consequently, the movement of particles is restricted within the circulation in between buoyant jets and thus particle horizontal displacement decreases with heat source density. However, the effect of buoyant jets has been broken down once the collective regime enters with the merging of plumes taking place. With plume merging, the airflow in the room becomes dominated by the superstructure, enabling a much further horizontal transport. Since the flow intensity increases with larger number of heat source, there is also opportunity for particles to travel longer distances by crossing several superstructure [see Fig. 5(d)].

FIG. 5.

Instantaneous spatial distributions of particles overlaid with the thermal plume: λ = L in (a), λ = L / 2 in (b), λ = L / 4 in (c), and λ = L / 16 in (d). In the individual regime [as shown in (a)–(b)], where distancing between occupants is large, thermal plumes act as buoyant jets and exhibit significant individual effects, limiting the spread of virus particles. However, in the collective regime [as shown in (d)], where the number of occupants is increased, thermal plumes condense and merge into the superstructure, enhancing the intensity of airflow and facilitating long-distance transport of aerosols.

FIG. 5.

Instantaneous spatial distributions of particles overlaid with the thermal plume: λ = L in (a), λ = L / 2 in (b), λ = L / 4 in (c), and λ = L / 16 in (d). In the individual regime [as shown in (a)–(b)], where distancing between occupants is large, thermal plumes act as buoyant jets and exhibit significant individual effects, limiting the spread of virus particles. However, in the collective regime [as shown in (d)], where the number of occupants is increased, thermal plumes condense and merge into the superstructure, enhancing the intensity of airflow and facilitating long-distance transport of aerosols.

Close modal

As mentioned above, the crossing of particle in between superstructure provides a possible route for long-distance air transmission between the two occupants [see Fig. 6(a)] and, thus, leads to the enhanced infection for respiratory diseases. To quantify this process, we calculate the frequency for particles to crossover multiple neighbor occupants, i.e., f = 1 t total 1 N A i ( C ), where N is the total number of particles, t total is the total evaluation time, and A i ( C ) is the counts of crossing over multiple heat sources of number C for ith particle. Once the horizontal displacement of the particles exceeds the distance between the heat sources, we count once on A i ( C ), i.e., we execute the operation A i ( C ) = A i ( C ) + 1 once | X i , t + τ X i , t | > C · λ. Figure 6(b) shows the frequency of particles crossing as a function of L / λ for various C. It is seen that when the distance λ exceeds a critical value ( L / λ c = 4), there is a significant enhancement in the frequency compared to the cases in regime of individual effect. This finding suggests that the reduction of the distance between occupants results in a vast increase in particle crossing, which in turn induces a high infection risk. By taking advantage of the morphological transition of the flow structure, we can minimize the risk of disease transmission by adjusting the spacing between heat-carrying bodies. This can help to inform the development of effective strategies for controlling the spread of infectious diseases in indoor environments.

FIG. 6.

Infection risk assessment. (a) Cross infection path: the flows established through the large-scale circulation provide a path for air transmission between the two occupants, and hence, a possible infection route. (b) The frequency of viral particles crossing over multiple heat sources as a function of L / λ. Here, the frequency is calculated by f = 1 t total 1 N A i ( C ), where A i ( C ) is the counts of viral particles released from a certain occupant crossing over multiple neighbor occupants of number C, and N is the total number of particles during a large duration t total. Once the horizontal displacement of the particles exceeds the distance between the heat sources, we count once on A i ( C ), i.e., we execute the operation A i ( C ) = A i ( C ) + 1 once | X i , t + τ X i , t | > C · λ.

FIG. 6.

Infection risk assessment. (a) Cross infection path: the flows established through the large-scale circulation provide a path for air transmission between the two occupants, and hence, a possible infection route. (b) The frequency of viral particles crossing over multiple heat sources as a function of L / λ. Here, the frequency is calculated by f = 1 t total 1 N A i ( C ), where A i ( C ) is the counts of viral particles released from a certain occupant crossing over multiple neighbor occupants of number C, and N is the total number of particles during a large duration t total. Once the horizontal displacement of the particles exceeds the distance between the heat sources, we count once on A i ( C ), i.e., we execute the operation A i ( C ) = A i ( C ) + 1 once | X i , t + τ X i , t | > C · λ.

Close modal

In summary, we have performed a series of large-scale direct numerical simulation (DNS) to examine the aerosol transport with different spacings between the heat source, mimicking different social distancing. As the distance between occupants changes, the overall airflow pattern undergoes a morphological transition. We identified two regimes of thermal plume behavior from heating sources at different distancing: the individual regime and collective regime. In the individual regime, where distancing between occupants is large, thermal plumes act as buoyant jets and exhibit significant individual effects. These individual buoyant jet plumes act as a natural barrier, trapping viral particles in a localized region and limiting their spread. It is especially important when normal indoor activities are resumed, and one relies on the natural way of precautions based on the flow physics of the indoor flow. In contrast, in the collective regime, where the number of occupants is increased, thermal plumes condense and merge into the superstructure, enhancing the intensity of airflow and facilitating long-distance transport of aerosols. We evaluated the risk of cross-infection at different spacings in poorly ventilated rooms. When the distance λ exceeds a critical value ( L / λ c = 4), there is a significant enhancement in the frequency compared to the cases in regime of individual effect.

Our study uncovers a critical behavior in the spread of aerosols that is dependent on the distance between heat-carrying bodies. Beyond certain critical distances, we observe a transition from the individual to the collective regime, which facilitates the long-distance tunneling of viral particles and increases the risk of infection. This finding has important implications for understanding and mitigating the transmission of respiratory diseases in indoor environments. This provides decision-makers with guidelines for determining the maximum number of occupants based on the underlying physics of viral transmission in enclosed spaces. This indicates that localized air patterns contribute significantly to a potentially wider range of infection risks, underscoring the importance of both physical distancing and mask precautions. The establishment of effective physical distancing measures must not only account for the inherent trajectory and lifespan of respiratory droplets but also consider the global airflow dynamics within a room. This becomes particularly crucial when addressing the long-time and long-distance transmission of viral particles. Moreover, this emphasizes the significance of well-designed heating, ventilation, and air conditioning (HVAC) systems as valuable references in mitigating transmission risks.

Our direct numerical simulations of airflow patterns and aerosol transport in poorly ventilated rooms reveal that arranging occupants at a reasonable distance is crucial to minimize the transmission of potentially infectious aerosols. Accommodating a minimal number of people by utilizing the morphological transition of airflow in a room is the most effective way to reduce cross-contamination. However, we oversimplified the problem by using a simplified boundary temperature distribution, limiting the generalizability of our conclusions to specific scenarios. Factors such as variations in body temperature, clothing insulation, and activity level can all influence the characteristics of the human body plume and, consequently, the transport of respiratory droplets. Understanding the flow patterns generated by these factors will be a key focus of future research. Nevertheless, our findings is especially important when normal indoor activities are resumed and one relies on the natural way of precautions based on the flow physics of the indoor flow. It also provides guidance on coping with the potential outbreak of respiratory diseases other than SARS-CoV-2 in the future.

This work was sponsored by the Natural Science Foundation of China under Grant Nos. 11988102, 92052201, 91852202, 11825204, 12102246, 12032016, 11972220, and 12372219; the Program of Shanghai Academic Research Leader under Grant No. 19XD1421400; and the Shanghai Science and Technology Program under Project Nos. 19JC1412802 and 20ZR1419800.

The authors have no conflicts to disclose.

Chao-Ben Zhao: Conceptualization (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing—original draft (equal); Writing—review & editing (equal). Jianzhao Wu: Writing—review & editing (equal). Bofu Wang: Writing—review & editing (equal). Tienchong Chang: Writing—review & editing (equal). Quan Zhou: Writing—review & editing (equal). Kai Leong Chong: Conceptualization (equal); Supervision (equal); Writing—original draft (equal); Writing—review & editing (equal).

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

We have compared our numerical results of the body plume velocity with the experimental results of Craven and Settles,72 which shows good agreement (see Fig. 7). additionally, we have observed that our results are slightly lower than the experimental values, especially near the heat source location. This discrepancy may be attributed to the assumptions made in our model. Our model has simplified or not fully accurately captured the flow behavior around the heat source.

FIG. 7.

Plume centerline velocity distribution with height above head. It shows good agreement with the experiment by Craven and Settles.72 

FIG. 7.

Plume centerline velocity distribution with height above head. It shows good agreement with the experiment by Craven and Settles.72 

Close modal

To validate the consistency of our results across different grids, we compared temperature profiles averaged over time from the center position of the thermal plume for three sets of different grids. We can see that they exhibit the same consistency, as illustrated in Fig. 8.

FIG. 8.

The temperature profile along the central axis of the plume as a function of height. The temperature profile along the central axis of the plume as a function of height, for all the cases fixed R a ocpt = 109, Pr = 0.7, λ = L.

FIG. 8.

The temperature profile along the central axis of the plume as a function of height. The temperature profile along the central axis of the plume as a function of height, for all the cases fixed R a ocpt = 109, Pr = 0.7, λ = L.

Close modal

Figure 9 illustrates a snapshot of the temperature field of a human thermal plume in a room with a central aisle. We introduce a corridor in the middle of the room to disrupt the continuity of heat sources in the y-direction. The results reveal the enduring presence of self-organizing global flow structures (large-scale circulation) within the system, exerting substantial shearing effects on certain thermal plumes.

FIG. 9.

Snapshot of temperature field of human thermal plume in a room with a central aisle.

FIG. 9.

Snapshot of temperature field of human thermal plume in a room with a central aisle.

Close modal
Figure 10 shows the initial spatial distribution of particles. We explicitly mention that the particles are released randomly within a defined region of the computational domain, which spans ( 0.05 H , L 0.05 H ) × ( 0.05 H , W 0.05 H ) × ( 0.05 H , 0.95 H ) of the confined indoor space. The exact position of X = ( X 1 , X 2 , X 3 , ) are given by
X 1 = 0.05 H + ( L 0.1 H ) · ζ ,
(C1)
X 2 = 0.05 H + ( W 0.1 H ) · ζ ,
(C2)
X 3 = 0.05 H + ( H 0.1 H ) · ζ ,
(C3)
where the function “ζ” generates a random number between zero and one.
FIG. 10.

Initial spatial distribution of particles. The black wireframe represents the entire computational domain, while the red wireframe indicates the area where particles are initially randomly distributed.

FIG. 10.

Initial spatial distribution of particles. The black wireframe represents the entire computational domain, while the red wireframe indicates the area where particles are initially randomly distributed.

Close modal

Evaporation times τe for the different humidity (Fig. 11).

FIG. 11.

Evaporation times τe for the different humidity.

FIG. 11.

Evaporation times τe for the different humidity.

Close modal
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