# Human body heat shapes the pattern of indoor disease transmission

Chao-Ben Zhao1, Jian-Zhao Wu1, Bo-Fu Wang1,2, Tienchong Chang1,2, Quan Zhou1,2
and Kai Leong Chong1,2∗

1Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai
Institute of Applied Mathematics and Mechanics, School of Mechanics and
Engineering Science, Shanghai University,
Shanghai 200072, PR China
2Shanghai Institute of Aircraft Mechanics and Control,
Zhangwu Road, Shanghai 200092, China

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>

A one-sentence summary of the paper: Our study uncovered two thermal plume
regimes—individual and collective—and a critical distancing threshold that
triggers a transition between them, revealing novel disease spread patterns
through the resulting morphological phase transition of airflow.

> Exhaled droplet and aerosol-mediated transmission of respiratory diseases,
> including SARS-CoV-2, is exacerbated in poorly ventilated environments where
> body heat-driven airflow prevails. Employing large-scale simulations, we
> reveal how the human body heat can potentially spread pathogenic species
> between occupants in a room. Morphological phase transition in airflow takes
> place as the distance between human heat sources is varied which shapes
> novel patterns of disease transmission: For sufficiently large distance,
> individual buoyant plume creates a natural barrier, forming a “thermal
> armour” that blocks suspension spread between occupants. However, for small
> distances, collective effect emerges and thermal plumes condense into super-
> structure, facilitating long-distance suspension transport via crossing
> between convection rolls. Our quantitative analysis demonstrates that
> infection risk increases significantly at critical distances due to
> collective behavior and phase transition. This highlights the importance of
> maintaining reasonable social distancing indoors to minimize viral particle
> transmission and offers new insights into the critical behavior of pathogen
> spread.

## Introduction

Outbreaks of respiratory diseases, such as influenza, severe acute respiratory
syndrome (SARS), middle East respiratory syndrome, and now the 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 (?, ?, ?, ?, ?, ?), which relate
closely on the flow physics of respiratory suspensions (?, ?, ?). 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 SARS-CoV-2,
especially when the ventilation is inadequate (?, ?, ?, ?, ?, ?, ?, ?). In
indoor situations, saliva droplets or aerosols are released into the
environment by asymptomatic or presymptomatic infected individuals (?, ?, ?,
?, ?, ?, ?, ?), then suspended and mixed by the airflow for hours or even days
(?, ?, ?, ?, ?, ?). This kind of indirect transmission involves infections in
large spatial and temporal scales, and possibly leads to the superspreading
events (?, ?, ?, ?).

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 fields (?, ?, ?, ?, ?, ?, ?), and also
relates to several environmental factors such as ventilation (?, ?, ?, ?),
ambient temperature (?, ?) and relative humidity (?, ?, ?, ?, ?). 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
$4-8$mm/s for the ventilation rate of $N=5-10$ACH (?). 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. Considering the body-and-ambient temperture difference of $8^{o}$C,
the speed of thermal plume generated from a person is approximately 800mm/s
using the characteristic buoyancy veloctiy 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.

To address the question of how the body heat drives the flow pattern and
affects the disease transmission, we carry out a series of large-scale
simulations of indoor flow to examine the aerosol transport with different
spacings between the heat source, mimicking different social distancing. We
discover a transition in flow structure from individual to collective and
condensed thermal plumes as heat-carrying body density increased, shaping
different disease transmission patterns. In individual regime, the so-called
thermal armour is formed which traps the aerosols in localized circulation. On
the contrary, in collective regime, merging of plumes enables the long-
distance aerosol transport. We evaluated the risk of cross-infection at
different spacings in poorly ventilated rooms.

Figure 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. C-H, The snapshot of the temperature (color) and
velocity (arrows) field in the steady state for the different distancings of
people in the room: $\lambda=L$ in C, $\lambda=L/2$ in D, $\lambda=L/4$ in E,
$\lambda=L/8$ in F, $\lambda=L/16$ in G, $\lambda=L/32$ in H.

## Results

### Overall airflow patterns

We consider a room of dimensions $L:W:H=12m\times 1m\times 3m$, as sketched in
Fig. 1B, with a narrowed room being considered. 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. The heating
temperature of the body is set to be $T_{ocpt}$=34∘C and the width of the heat
source to be 0.375m. The fluid is assumed to be incompressible
$\nabla\cdot\boldsymbol{u}=0$. The gas Prandtl number is
$Pr\equiv\nu/\kappa_{T}=0.7$, where $\nu$ is the kinetic viscosity of air and
$\kappa_{T}$ is the thermal conductivity. The thermal Rayleigh number defined
using the occupant height $h_{ocpt}=1.2m$ is $Ra_{ocpt}\equiv\alpha g\Delta
h^{3}_{ocpt}/(\nu\kappa_{T})\approx 10^{9}$, where $\Delta\equiv
T_{ocpt}-T_{amb}=8^{\circ}C$ (?), and $g$ is the gravitational acceleration.
We characterize the spacing of human heat sources by $\lambda$, which equals
to the length $L$ divided by the number of human body $k$ ($\lambda=L/k$),
which is also an important control parameter of the system. Further details of
control parameters are given in the Supplementary Materials.

The flow structure driven by regularly arranged occupants in a poorly
ventilated room is first examined. Figure 1C-H shows the typical instantaneous
flow structure at different distances $\lambda$ 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 $\lambda$, i.e. with small
number of occupants as shown in Fig. 1C-E, those ejected plumes self-organize
into the circulating flow (see supplementary Movie. S1). The interior spaces
of room are then separated by the airflow pattern formed by several individual
circulations. For small distance $\lambda$, i.e. with large number of
occupants as shown in Fig. 1F-H, the collective buoyancy effect is found,
where the large-scale flow induces strong shear effect to favour the merging
of thermal plumes (see supplementary Movie. S2). The condensation of plumes
connects a number of heating sources, resulting in the super-structure in the
enclosed room.

Figure 2: Individual and collective effects of body plume motions. A-C,
Behavior of body plumes for different distances between occupants: $\lambda=L$
in A, $\lambda=L/4$ in B, $\lambda=L/16$ in C. D, E, The Reynolds number ratio
$Re_{z}/Re$ as a function of the horizontal position $x$ obtained from the
slice of the velocity field at mid-width. Here $Re_{z}=\sqrt{\langle
w^{2}\rangle_{z}}H/\nu$ is the averaged vertical Reynolds number,
$Re=\sqrt{\langle u^{2}+v^{2}+w^{2}\rangle_{z}}H/\nu$ where
$\langle\cdot\rangle_{z}$ denoting the averaging over vertical direction. As
shown in D at $\lambda=L$, the plume emission at the peak position leads to
local flow enhancement, showing significant individual effects. However, at
$\lambda=L/16$ 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 $Re_{x}$ as a function of
the ratio between the length andx distancing $L/\lambda$. Here
$Re_{x}=\sqrt{\langle u^{2}\rangle_{V,t}}H\nu$ is the spatially and temporally
averaged horizontal Reynolds number.

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 Fig. 2A-C. At $\lambda=L$, 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 $\lambda$ (e.g. $\lambda=L/4$ and
$\lambda=L/16$), body plumes start to merge with each other as favoured by the
shearing of the large-scale mean flow, and eventually the strong collective
behaviour appears with all thermal plumes contribute energy to a single large-
scale flow.

To quantitatively distinguish the individual and collective effects, we plot
in Fig. 2D,E the profiles of $Re_{z}/Re$ along the horizontal direction, which
indicates the strength of buoyancy-driven jet from the heat source. Here,
$Re_{z}$ is the Reynolds number defined by the vertical velocity magnitude,
$Re$ is the overall Reynolds number defined by flow speed considering all
velocity components. At $\lambda=H$ in the individual regime (Fig. 2D), the
location of the peak coincides with the location of the heat source, and the
value of the local $Re_{z}$ 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 $\lambda=L/16$ under the collective
regime (Fig. 2E), 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 super-
structure as shown in Fig. 2F. The consequence of the enhanced horizontal
momentum is the potential transport of viral aerosols over a long lateral
distance.

Figure 3: 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) $\langle{r_{x}}^{2}(t)\rangle$ of particles as a function of time. The
straight dotted lines indicate the theoretical scaling laws corresponding to
ballistic and diffusive regimes. D,E, The measured MSD as a function of
$L/\lambda$ at $t=1$ in D and $t=100$ in E. F-I, Instantaneous spatial
distributions of particles overlaid with the thermal plume.

### Viral transmission analysis

It is highly desirable to understand how the airflow transition affect the
aerosol transportation, which is important to disease transmission. As most of
exhaled aerosols are smaller than $5\mu m$ (?, ?), and a large fraction is
$<1\mu m$ 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 (?, ?, ?).

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. With the obtained trajectories, the
statistical behavior of particles can be characterized by their mean-squared
displacement (MSD). Figure 3C shows the MSD versus time $t$ for various
distance $\lambda$. The trend of MSDs for different $\lambda$ are seen to
exhibit similar behavior, namely, at short time, the particle motion is
ballistic ($\langle{\vec{r}}^{2}(t)\rangle\sim t^{2}$) since the aerosols are
advected by the circulation (Fig. 3A,B); after a certain time $t_{c}$, a short
range of diffusive motion ($\langle{\vec{r}}^{2}(t)\rangle\sim t^{1}$) is
found and eventually the MSD becomes independent of $t$ due to the finite-size
effect.

To quantify how rapidly the aerosols spread inside the room, we plot in Fig.
3D, E the magnitude of MSD as a function of $1/\lambda$ at two time instants
$t=1$ and $t=100$ (i.e. about $3$ secs and $6$ mins in real time). At $t=1$
where the motion of aerosols is still within the ballistic stage, the trend of
displacement in horizontal direction $\langle r_{x}^{2}(t)\rangle$ is similar
to that of $Re_{x}$ (see Fig. 2F) because the amount of displacement is set by
the strength of the flow. At $t=100$ in diffusive stage (see Fig. 3D), the
transition from individual to collective regime can be clearly distinguished:
the horizontal displacement decreases with increasing $\lambda$ in the
individual regime, and then increases with $\lambda$ in the collective regime.

This above trend can be understood from the following visualizations. Figure
3F-I presents the snapshots of the spatial distribution of particles
overlapped with the contour of temperature field. In the individual regime, as
illustrated in Fig. 3F-H, the thermal plumes from heating source behave as
“thermal armour”. Consequently, the movement of particles is restricted within
the circulation in between “thermal armour” and thus particle horizontal
displacement decreases with heat source density. However, the effect of
“thermal armour” 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 super-structure, 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 super-structure (Fig. 3I).

Figure 4: 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 $L/\lambda$. Here, the frequency is calculated by
$f=\frac{1}{T}\frac{1}{N}\sum\limits_{i}A_{i}(C)$, where $A_{i}(C)$ is the
counts of viral particles released from a certain occupant crossing over
multiple neighbor occupants of number $C$, $N$ the total number of particles
during a large duration $T$. 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
$|X_{i,t+\tau}-X_{i,t}|>C\cdot\lambda$.

### Risk assessment

As mentioned above, the crossing of particle in between super-structure
provides a possible route for long-distance air transmission between the two
occupants (see Fig. 4A), and thus leads to the enhanced infection for
respiratory diseases. To quantify this process, we calculate the frequency for
particles to cross the large-scale airflow, i.e.,
$f=\frac{1}{T}\frac{1}{N}\sum\limits_{i}A_{i}(C)$, where $N$ is the total
number of particles, $T$ the total evalution time and $A_{i}(C)$ the counts of
crossing over multiple heat sources of number $C$ for i-th particle. Figure 4B
shows the frequency of particles crossing as a function of $L/\lambda$ for
various $C$. It is seen that when the distance $\lambda$ exceeds a critical
value ($L/\lambda_{c}=4$), 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 of 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.

## Discussion

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 ultizing the
morphological phase transition of airflow in a room is the most effective way
to reduce cross-contamination.

Our study 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 “thermal armour” and exhibit significant individual effects,
limiting the spread of virus particles. In contrast, in the collective regime,
where the number of occupants is increased, thermal plumes condense and merge
into the super-structure, enhancing the intensity of airflow and facilitating
long-distance transport of aerosols. Our study provides decision-makers with
guidelines for determining the maximum number of occupants based on the
underlying physics of viral transmission in enclosed spaces.

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 guidances on coping with the potential
outbreak of respiratory diseases other than SARS-CoV-2 in the future.

## Materials and Methods

In the current study, we conducted direct numerical simulations (DNS) of
three-dimensional (3D) turbulent convection with spatially harmonic heating.
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

$\displaystyle\nabla\cdot\boldsymbol{u}=0,$ (1)
$\displaystyle\partial_{t}\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}=-\frac{1}{\rho}\nabla
p+\nu\nabla^{2}\boldsymbol{u}+\alpha gT\boldsymbol{e}_{z},$ (2)
$\displaystyle\partial_{t}T+(\boldsymbol{u}\cdot\nabla)T=\kappa_{T}\nabla^{2}T,$
(3)

where $t$ is the time, $\boldsymbol{u}$ is the fluid velocity, $\rho$ is the
fluid density, $T$ is the temperature, and $\boldsymbol{e}_{z}$ is the
vertical unit vector.

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 the literature (?,
?). At all solid boundaries, no-slip boundary conditions are applied for the
velocity. For boundary conditions, we apply a distributed heating to the
bottom plate:

$T_{\mathrm{bot}}(x)=\begin{cases}T_{ocpt},&\mbox{for $(n-1/2)\lambda\leq
x\leq(n-1/2)\lambda+l_{ocpt}$ }\\\ T_{amb},&\mbox{for others}.\par\end{cases}$
(4)

where $n=1,2,\cdots,k$, $l_{ocpt}=H/8$ is the width of occupants. A constant
temperature $T_{\mathrm{top}}=T_{amb}$ 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. And
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 to adequately resolve both the Kolmogorov length scale for the
velocity field $\boldsymbol{u}$ and the Batchelor length scale for the
temperature field $\theta$. For the temporal resolution, the time step adopted
is set to be small enough to guarantee the numerical stability and resolve the
Kolmogorov time scale of turbulent fluctuations.

For aerosols, we apply the spherical point-particle model. Considering that
the majority of exhaled aerosols are smaller than $5\mu m$ (?, ?), even a
large fraction is $<1\mu m$ 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 (?, ?, ?). Further
details of numerical parameters are given in the Supplementary Materials.

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## Acknowledgments

Funding: This work was supported by the Natural Science Foundation of China
under Grant Nos. 11988102, 92052201, 91852202, 11825204, 12032016, 12102246,
91952102 and 11972220, the Shanghai Science and Technology Program under
Project No. 20ZR1419800, the Shanghai Pujiang Program under grant No.
21PJ1404400. Author contributions: K. L. C. conceived and designed research.
C. -B. Z. and J.-Z. W. conducted the numerical simulations. K. L. C., B.-F.
W., T. C., Q. Z. supervised the project. C.-B.Zhao, K. L. Chong and J.-Z. Wu
wrote the manuscript. All authors contributed to the analysis of the data.
Competing interests: The authors declare that they have no competing
interests. Data and materials availability: All data needed to evaluate the
conclusions in the paper are present in the paper. Additional data related to
this paper may be requested from the authors.