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*Corresponding author. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Virus transmission by aerosol transport during short conversations

Rohit Singhal1,† S. Ravichandran2,∗,† Rama Govindarajan  Sourabh S. Diwan1,∗,†
Department of Aerospace Engineering, Indian Institute of Science, Bengaluru,
KA 560012, India  Nordic Institute for Theoretical Physics, KTH Royal
Institute of Technology and Stockholm University, Stockholm, 10691-SE, Sweden
International Centre for Theoretical Sciences, Tata Institute of Fundamental
Research, Bengaluru, KA 560089, India  These authors contributed equally to
the work.

###### Abstract

Pathogens like the SARS-CoV-2 are transmitted not only through violent
expiratory events like coughing, but also through routine activities like
breathing/speaking/singing. We perform direct numerical simulations of the
turbulent transport of potentially infectious aerosols in short conversations.
It is shown that a two-way conversation significantly reduces the aerosol
exposure compared to a relative monologue by one person and relative silence
of the other. This is because the interaction of the jets ejected from the
mouth of each speaker produce a “canceling” effect. Unequal conversation is
shown to significantly increase the risk of infection to the person who talks
less. Interestingly, a small height difference is worse for infection spread,
due to reduced interference between the two speech jets, than two faces at the
same level! For small axial separation, speech jets show large oscillations
and reach the other person intermittently. We suggest a range of lateral
separations between two people to minimize transmission risk. A realistic
estimate of the infection probability is provided by including exposure
through eyes and mouth, in addition to the more common method of using inhaled
virions alone. We expect that our results will provide useful inputs to
epidemiological models and to disease management.

###### keywords:

Jets; turbulence simulations; disease transmission; two-people conversations;
infection probability

††articletype: RESEARCH ARTICLE

Impact Statement Asymptomatic transmission, through activities like speaking,
is believed to be an important route of COVID-19 spread across the world. The
evolving turbulent jet that emerges from the mouth of a speaker can carry
aerosols to large distances. While direct numerical simulations of turbulent
speech flow are becoming common, very little is understood about two-way
conversations. We simulate these to obtain spatio-temporal distributions of
aerosols and calculate infection probabilities. We show that the interaction
between the two speech jets plays a key role in determining this probability.
The risk of infection is reduced considerably during a “dialogue” as compared
to a “monologue” and a small height difference is found to be more dangerous
than the two speakers of equal heights. The present results can inform public
health guidelines for minimizing risk of transmission, such as introducing a
lateral separation between conversing people.

##  1 Introduction

SARS-CoV-2, responsible for the COVID-19 pandemic, is known to be transmitted
through more than one mode. Around the beginning of the pandemic (in early
2020), the primary mode of infection was believed to be through droplet
transmission by an infected symptomatic person, through violent expiratory
events like coughing and sneezing. Based on this, WHO recommended maintaining
a physical distance of $1$m between people to minimize the spread of
infection, whereas the recommendation of CDC-USA is a separation of 6ft. By
mid-2020, it became clear that asymptomatic transmission of the virus was
equally likely, through the virus-laden droplets released by an infected
person while talking, singing, breathing etc. (Asadi et al., 2020; Morawska &
Cao, 2020) and that this could potentially cause a rapid spread of the
disease. It was then realized that public health measures such as masking and
physical distancing would be necessary in ordinary situations since an
infected person (symptomatic or asymptomatic) who merely engaged in a
conversation could spread the disease (Asadi et al., 2020). More recently,
infection through the airborne virus has been much in the news. These findings
highlight the need to better understand the different transmission modes of
SARS-CoV-2, in which the fluid dynamics of droplet/aerosol transport plays a
key role (Bourouiba, 2021). In particular, scientific attention has been
focused on physical distancing guidelines as set out in public health
advisories, which were based at first on a study done in the 1930s (Wells,
1934). According to that study, large droplets (>100$\mu$m) settle faster than
they evaporate but can cover $\approx$1m horizontally while doing so, causing
direct transmission. It was estimated that smaller droplets cannot directly
transmit the disease due to their fast evaporation. Recent research has shown
that smaller droplets can cause disease as well. There is an entire spectrum
of droplet sizes released during different expiratory events, and a
significant part of this size spectrum can be transported over long distances
by the turbulent jet/puff generated during such events (Bourouiba, 2021). The
turbulent jet further contains water vapour which slows down evaporation. The
complexity of the transmission dynamics of SARS-CoV-2, coupled with its
asymptomatic transmissibility, could thus have played a large role in the
ongoing pandemic afflicting the world (Prather et al., 2020), with several
countries experiencing second or third waves of the contagion at present.

Bourouiba (2021) discusses the break-up of mucosalivary fluid bubbles and the
formation of droplets ranging from 1$\mu$m to 500$\mu$m (Duguid, 1946; Johnson
et al., 2011) during different expiratory events. Among this range, the large
virus-laden droplets predominantly cause direct transmission or contaminate
surfaces close to their source (Basu et al., 2020; Bhardwaj & Agrawal, 2020)
turning them into “fomites”. Small droplets can stay airborne for longer
times, their longevity being a function of their composition as well as
prevalent ambient conditions like the relative humidity and temperature.
Furthermore, small droplets can evaporate completely while airborne and turn
into what are called droplet nuclei. The SARS-CoV-2 has been found to survive
in aerosols (typically consisting of micro-droplets and droplet nuclei of
sizes less than 5$\mu$m) with a half-life of about 15 minutes for typical
indoor conditions (Marr et al., 2019; US dept of Homeland Security, 2020;
Schuit et al., 2020); another study (Van Doremalen et al., 2020) found the
virus half-life in aerosols to be 1.1 hour (Greenhalgh et al., 2021). The
long-range transport of disease-causing viruses through such droplets and
droplet nuclei, and the impact of the ambient conditions on virus survival,
are open questions of fluid dynamical interest (Bourouiba, 2020), with some
studies suggesting that droplets with diameters in the range of
(2.5$\mu$m-19$\mu$m) have the greatest potential for causing the initial
nasopharyngeal infection (Basu, 2021; Smith et al., 2020). Droplets exhaled
while speaking are typically smaller than 10$\mu$m (Asadi et al., 2019) and
lie in the described range for higher nasopharyngeal infection.

Most of the fluid dynamical studies on human expiratory flows have focused on
droplet transport during coughing and sneezing, due to their direct relevance
to symptomatic transmission. Several experimental (Bourouiba et al., 2014;
Clark & De Calcina-Goff, 2009; Gupta et al., 2009; Nielsen et al., 2012; Tang
et al.,2009; Wei & Li, 2017) and numerical (Dbouk & Drikakis, 2020; Fabregat
et al., 2021; Liu et al.,2017; Qian & Li, 2010) studies of cough and sneeze
flows have been reported and their outcomes are being incorporated into
epidemiological models (Chaudhuri et al., 2020; Dbouk & Drikakis, 2021). On
the other hand, the fluid dynamics of speech and breathing has received much
less attention until very recently. Shao et al., 2021 provide a risk
assessment of virions exhaled due to normal breathing in elevators,
classrooms, and supermarket settings. Chong et al., 2021 carried out a direct
numerical simulation (DNS) of a turbulent vapour puff and found that droplets
can last O(100) times longer in cold humid air than predicted by classical
models (Wells, 1934); they used an inlet flow-rate profile typical of a cough
but the results are equally applicable to other expiratory flows including
speech. Recent experiments (Giovanni et al., 2021) on the effects of airflow
velocity on droplet trajectory in speech and vocal exercises have been used to
construct models of the behavior of different droplets. Abkarian et al.(2020)
carried out a large eddy simulation of speech flows generated by the
repetition of certain phrases. They showed that beyond a certain distance from
the mouth of the speaker, speech flow behaves like a steady jet that spreads
at a typical half-angle of $10^{\circ}$. These results were incorporated by
Yang et al. (2020) into a simplified model for the transport of aerosol
particles away from a speaker’s mouth, using known properties of steady jets
like the $1/x$ variation of the velocity and scalar concentration with the
axial distance $x$. They calculated the probability of infection of a silent
listener based on conservative estimates of the minimum number of virions
$(N_{inf}\approx 100)$ that must be inhaled to cause infection (Basu, 2021;
Kolinski & Schneider, 2020) and presented space-time maps of the risk of
infection. This study highlighted the fact that the disease transmission by
speech involves not just distances but also exposure times (see also Tan et
al., 2021) and that these should also be incorporated in the public health
guidelines.

The idealized scenario studied so far, where a single person engages in an
extended monologue, is of limited applicability, and typical public
interactions are dialogues of short time-spans, e.g., over- the-counter
conversations at a supermarket. The present study is devoted to gauging
infection probability from such a short, unmasked conversation between two
people, by performing a DNS of speech flow. We compute turbulent transport of
speech aerosols, which play a key role in the transmission of virus. We
estimate the total viral ingestion by a listener by including exposure through
the eyes and mouth, in addition to the aerosols inhaled through the nose (more
commonly only the last is used to determine viral dose). We show that the
active participation in the conversation of the second person significantly
alters the evolution of the jet from the first person, dramatically mitigating
infection probability. Any temporal asymmetry in speech enhances the risk of
infection to the person who speaks less. Secondly and rather interestingly, a
small vertical or lateral separation between the mouths of the speakers
actually increases the infection probability, due to a less effective
interference of the two jets. At large vertical or lateral separation,
infection probability is lower, as would be expected. We discuss the
implications of these results for improving physical distancing guidelines and
for epidemiological modelling. Also, our results offer interesting
experimental test cases and can be used to validate flow-modelling approaches
such as the Reynolds-averaged Navier Stokes equations.

The paper is organized as follows. First, we describe the computational
geometry and set-up (Section 2), followed by the parameters needed to estimate
the infection risk. We present two cases: one in which one of the people is a
passive listener, and the other in which both people converse. Next, we
describe our method of determining the total exposure and ingestion of
virions, which is used to calculate the probability of infection for the above
two cases (Section 3). Lastly, a case of conversation with temporal asymmetry
in speech duration is analyzed before a discussion (Section 4) and summary
(Section 5) of the results.

##  2 Numerical Experiments

Figure 1: (a) Side view $(z=0)$ of the computational domain in Case I showing
the location $(L_{s},y_{s})$ of the silent (and susceptible) listener,
represented by a circular region of interest (ROI) as shown in the figure. (b)
Three-dimensional representation of the computational domain in Case II, where
both Person 1 and Person 2 speak, and $L_{s}$ and $y_{s}$ are treated as
parameters. The orifices are displaced symmetrically about $y=0$. (c) Inlet
flow rate (in $l/s$) from the orifice ‘O’ in Case I. (d) Inlet flow rates from
the two orifices in Case II.

We use two types of computational domain for the direct numerical simulation
of speech flows. The first type of simulation, Case I, is done using a domain
whose side view is shown in Figure 1a (the lateral size of the domain is the
same as that shown in Figure 1b); the co-ordinate system used is shown in the
figure. Here, a silent susceptible listener is prescribed at
$(L_{s},y_{s},z=0)$ with respect to the origin ’O’ where the speaker (Person
1) is placed. A circular region of interest (ROI) of diameter 17.2cm (which is
approximately the diameter of a human face) is used to represent the face of
Person 2 (Figure 1a). To examine the 6ft rule, the length of the computational
domain is chosen to be $L_{x}=1.96$m, and the vertical and lateral extents are
chosen to be $L_{y}=L_{z}=0.98$m which are sufficient for the study due to the
small spread angle $(\approx 20^{\circ}-28^{\circ})$ of speech flows (Yang et
al., 2020). The grid resolution used is $256\times 128\times 128$ in
$L_{x}\times L_{y}\times L_{z}$, which is equivalent to $256^{3}$ for a
cubical domain with size $L_{x}$; this resolution has been tested for grid
convergence by using higher resolutions of $512^{3}$ and $1024^{3}$(see Fig.
S1). As done in previous studies (Ravichandran & Narasimha, 2020; Singhal et
al., 2021), open boundary conditions are imposed on the four lateral
boundaries at $y=0,L_{y}$ and $z=0,L_{z}$ and the boundary at $x=L_{x}$; this
enables advection of the turbulent structures out of the domain, without
affecting the upstream flow. A free-slip condition is used for the boundary at
$x=0$, except for the orifice at ‘O’ (see below).

Case II is a set of simulations where the two people are in conversation
(Figure 1b), and the flows thus generated interact with each other. We account
for a possible height difference between the speakers by a vertical separation
$y_{s}$. The orifice for Person 1 is located at $(0,0.5*y_{s},0)$ and that for
Person 2 (whose susceptibility to infection is being estimated) at
$(L_{s}+\Delta,-0.5*y_{s},0)$, as shown schematically in Figure 1b. To account
for the obstacle created by Person 2 to the incoming speech flow for this
case, the boundary at $x=L_{x}=(L_{s}+\Delta)$ , excluding the orifice, is
prescribed to be a free-slip wall (instead of an outflow boundary) similar to
that at $x=0$. In order to avoid wall effects, we choose an axial location
that is at a distance $\Delta(\approx 0.15m)$ upstream of $L_{x}$ to place the
ROI. The axial length of domain is varied among three values,
$L_{s}\in\\{0.6\rm{m},1.2\rm{m},1.8\rm{m}\\}$ , keeping the lateral lengths
$L_{y}=L_{z}=0.98$m fixed. We also wish to understand the rate of infection
when the conversation is temporally asymmetric, i.e., Person 1 speaks more
than Person 2. We simulate such situations with $L_{s}=1.2$m and $y_{s}=0$ and
name it “Case IIt”.

The speaker’s mouth is modelled as an elliptical orifice with half-axis
dimensions of $a=1.5\rm{cm},b=1.0\rm{cm}$ , following Abkarian et al.(2020).
We use the characteristic diameter $d=2\sqrt{ab}=2.45cm$ of the ellipse as the
length-scale for non-dimensionalisation; the non-dimensional size of the
domain is therefore $(80d\times 40d\times 40d)$ for $L_{s}=1.8$m. Prescribing
speech flow at the orifice is not straightforward as it involves complex
linguistic expression, which can continue for an extended period of time. Here
we follow Abkarian et al.(2020) in choosing a simple repetitive phrase for
simulating speech flow. Abkarian et al. (2020) show that speech phrases with
plosive sounds (e.g., words containing the letter “P” - oshtavya in Sanskrit)
induce puffs that travel further than those having fricative sounds (e.g., the
letter “S”). Furthermore the speech jet produced by plosives was found to be
directed primarily in the axial direction (Abkarian et al., 2020), obviating
the need to introduce flow directionality at the orifice. We therefore use a
repeated utterance of the phrase "Peter Piper picked a peck" and prescribe the
inlet velocity at the orifice (in the axial direction) using laboratory
measurements of flow velocity associated with this phrase (see Figure 4c of
Abkarian et al., 2020). We do not include the inhalation part of the cycle
used in Abkarian et al. (2020) as it does not affect the transport of
aerosols, except for a region very close to the orifice (Abkarian et al.,
2020). In the present study, the exhalation phase of the phrase has a volume
of 0.5 litre per cycle, as in Abkarian et al. (2020), which lasts for 2.8s,
followed by a halt in speaking of 4.2s, which makes one complete cycle of
speech (Figure 1c and d). The present simulations are validated against the
results of Abkarian et al. (2020), and show a good overall agreement. The
cycle-averaged centreline velocity from our simulations reproduces the $1/x$
variation after $x\approx 0.4$m (see Fig. S1) and the “flow length” exhibits
the $t^{1/4}$ variation for small $t$ (typical of a “puff”), followed by the
$t^{1/2}$ variation at large $t$ (typical of a “starting jet”) as obtained in
Abkarian et al. (2020) (Fig. S2).

For Case I, we have used 10 speech cycles for Person 1 and zero flow rate for
Person 2 (who acts like a silent listener, though not physically present in
the domain); see Figure 1a and c. For the simulations in Case II, flow rates
at both orifices (corresponding to Persons 1 and 2) consist of 10 staggered
cycles shown in Figure 1d; this represents each person taking turns to speak
the same phrase. In Case IIt, Person 1 and Person 2 have 10 cycles of
conversation, following which, Person 2 stops speaking, and Person 1 continues
for another 10 cycles (see Figure 7). For this case, the total speech time
(including the silent intervals) is 140s for Person 1 and 70s for Person 2.
The maximum flow velocity at the orifice, $u_{c}=1.167m/s$, is used as the
velocity scale for all the cases. Since these flow speeds are small compared
to the speed of sound in air and also because the change in density within the
flow is much smaller than the ambient density
$(|\Delta\rho|\textfractionsolidus\rho\approx 0.05$ for $\Delta T=14^{\circ}C$
at the orifice; see appendix A) we may make the Boussinesq approximation. This
involves treating the velocity field as effectively incompressible with the
density differences relevant only for the buoyancy term in the momentum
equation. The buoyancy module of our code has been validated for a steady
plume (Singhal et al., 2021) wherein the self-similar decay of the centreline
temperature has been accurately captured. For the speech flows considered
here, it turns out that the buoyancy forces are much smaller than the inertia
forces. As a result, the flow does not show any perceptible deflection in the
vertical direction on average (Fig. S3; see also Figure 4b) and is
qualitatively similar to that reported in Abkarian et al. (2020) who did not
solve for temperature . This is borne out by the small value of
$Fr^{-2}(=0.00842)$ obtained for the present simulations based on the orifice
conditions, where $Fr$ is the Froude number (appendix A). Furthermore, the
temperature drops rapidly as the speech jet issues from the orifice (by about
$10^{\circ}$ C over 0.5m from the orifice; see Fig. S4) and once the steady
“self-similar” regime is established for $x>0.45$m Fig.( S5b), the Froude
number based on local parameters is expected to remain constant (Turner,1986).
Thus, buoyancy effects are estimated to be negligible over the entire flow
evolution.

In order to model the dynamics of virus laden droplets generated during
speech, we make two simplifications. First, that most droplets
($O(10\mu\rm{m})$ or less) produced during speech are small enough that they
follow fluid streamlines. The smallness can be quantified in terms of the
Stokes number of the droplets, defined as the ratio $\tau_{p}/\tau_{f}$, where
$\tau_{p}$ is the timescale on which the velocity of a droplet adjusts itself
to the local fluid velocity, and $\tau_{f}=d/u_{c}$ is the flow timescale. For
droplet sizes typical of speech flows, Stokes numbers are much smaller than 1
(Yang et al., 2020), which means that within a fraction of the flow timescale,
a droplet attains the same velocity as the flow. In other words, it behaves as
a passive scalar. In the present work, $\tau_{p}\sim 0.5$ms (for $10\mu\rm{m}$
droplets) and $\tau_{f}\sim 20$ms, giving $\tau_{p}/\tau_{f}\sim 0.025$ which
is much less than 1. Since the flow velocity rapidly drops away from the
orifice, the passive-scalar approximation is justified in the entire domain.
This approach has been used before for studying the interaction of droplets
and turbulence in a cloud flow (Ravichandran & Narasimha, 2020) and allows us
here to represent small droplets in speech flow by scalar fields of
concentration $C_{s1}$ and $C_{s2}$ emanating from Persons 1 and 2
respectively (for Case I, $C_{s2}=0$). We prescribe unit concentration at the
orifices during speaking and 0 otherwise. Secondly, we note (a) that the total
liquid content in these flows is quite small (Asadi et al.,2019), as are the
temperature changes resulting from droplet evaporation; and (b) that the half-
life of SARS-CoV-2 in speech droplets/nuclei ($\approx$ 15 mins to 1 hr) is
much longer than the time-scales of evaporation (a few seconds; Chaudhuri et
al., 2020; Yang et al., 2020). Thus, even if the liquid part of a droplet has
evaporated, the nucleus containing the infection survives. We therefore ignore
the thermodynamics of evaporation in this study, and thus treat small droplets
and droplet nuclei as interchangeable, consistent with Abkarian et al. (2020)
and Yang et al. (2020). We refer to both as “aerosols”. The resulting
governing equations, namely the Boussinesq Navier-Stokes equations, the
continuity equation and the scalar transport equations, are solved using the
finite difference DNS solver, Megha-5 (for further details, see appendix A).

##  3 Results

###  3.1 Case I: Person 1 speaking and Person 2 being a silent listener

Figure 2: Aerosol flux for Case I. (a) Variation in time of the aerosol flux
$f_{I}$ through the circular ROI centred at $(L_{s},y_{s}=0)$. Close to the
orifice, the flow is resembles a series of puffs rather than a jet. (b) The
total scalar exposure $F_{I}(L_{s})$ over the simulation time (140s), for
different height differences $y_{s}$. Interestingly, $F_{I}$ varies non-
monotonically with $L_{s}$ for non-zero $y_{s}$. This is because only a part
of the speech-flow jet intersects the ROI for $L_{s}\lesssim 10y_{s}$. (c)
Vertical variation of normalized $F_{I}(L_{s},y_{s})$ and its comparison with
a Gaussian curve. The normalized $F_{I}$ profiles show best collapse with a
virtual origin at 0.32m upstream of the orifice.

Studies (Chaudhuri et al., 2020; Yang et al., 2020) have provided spatial and
temporal separation guidelines using calculations based on the total viral
load on a susceptible person present at a specific location. The infection
probability is then calculated from the number of virions inhaled by the
person. However, several studies have found that infections may also be caused
by viral contact through other exposed areas like the eyes (Chen et al., 2020;
Coroneo & Collignon, 2021; Sun et al., 2020; Xie et al., 2020) or the mouth
and lips (WHO scientific brief 27 March, 2020). In order to include these
possibilities, we estimate the exposure of a listener to virus-laden aerosols
over an ROI positioned in front of their face. Towards this, we determine
aerosol concentration $(\phi)$ which is related to the passive scalar
concentration as $\phi=\phi_{o}$ ($C_{s}/C_{so}$), where $\phi_{o}$ represents
the volume of droplets (i.e., liquid volume) per unit volume of air at the
orifice and $C_{so}=1$ (see appendix A). We specify $\phi_{o}=6\times 10^{-9}$
(Yang et al., 2020) as a representative value. As mentioned earlier, the
speech droplets evaporate fast and turn into droplet nuclei but the total
number of virions carried by the speech aerosols remains the same; see Yang et
al. (2020). The flux of aerosols through an ROI centered at
$(L_{s},y_{s},z=0)$ is given as

$f(L_{s},y_{s},t)=\int\int\phi_{1}\ u\ dy\ dz,\quad\quad
2\sqrt{{(y-y_{s})}^{2}+z^{2}}<D_{ROI}\ $ (1)

where $\phi_{1}$ is the aerosol concentration corresponding to the speech flow
of Person 1, $u$ is the axial velocity and $D_{ROI}=17.2$cm is the diameter of
the ROI. The aerosol flux through the ROI is taken as the viral exposure to
the face of the listener. Since the deflection of the flow by buoyancy is
negligible, the $y$ and $z$ directions are equivalent. We therefore only need
to vary $L_{s}$ and $y_{s}$. Figure 2a shows the time series of the aerosol
flux across an ROI for Case I, $f_{I}$ $(L_{s},y_{s}=0,\ t)$, using the data
obtained from the simulations. The transition of the flow from a puff-like
behavior to jet-like behavior can be seen from the decrease in the
oscillations of the scalar flux with increase in $L_{s}$. The total aerosol
exposure to the listener $F(L_{s},y_{s},t_{c})$ over time $t_{c}$ is given by
the time integral of the aerosol flux through the ROI,

$F(L_{s},y_{s},t_{c})=\int_{0}^{t_{c}}f(L_{s},y_{s},t)dt,\ $ (2)

where $t_{c}$ is the duration of exposure and $F$ is units of volume (here
ml). Here we show results for $t_{c}=140$s (although the total speech duration
is 70s) as this represents the time until which the listener at $L_{s}=1.8$m
continues to receive aerosol flux from the speaker (Figure 2a and b). The
aerosol exposure $F_{I}(L_{s},y_{s},140s)$ for Case I is plotted in Figure 2b
as a function of axial location $L_{s}$ at five different vertical locations
$y_{s}$. For $y_{s}=0$, the total exposure is practically constant for
$L_{s}<0.5$m, because the area of the ROI is larger than the cross-section of
the jet till this axial location. Once the jet cross-section area exceeds the
area of the ROI, the total exposure starts decreasing (Figure 2b). When the
listener is not aligned with the speaker face to face $(y_{s}\neq 0)$, $F_{I}$
increases with $L_{s}$ for $L_{s}\lesssim 10y_{s}$, before decaying at large
$L_{s}$.

The vertical profiles of $F_{I}(L_{s},y_{s},140\rm{s})$ are found to be bell-
shaped curves with the peak in $F_{I}$ decreasing with increase in $L_{s}$.
Suitably scaled versions of the profiles of $F_{I}$ are plotted in Figure 2c.
The $F_{I}$ profiles at $L_{s}=0.25\rm{m},0.40\rm{m}$, and $0.50\rm{m}$, show
an evolution in shape, and show an approach towards a Gaussian distribution
(represented by a dashed curve); this is consistent with the observation that
flow is in a transition state between a puff and steady jet for these
distances. For $L_{s}>0.50$m, the curves collapse well onto the Gaussian curve
(Figure 2c). Note that the Gaussian distribution for the scalar flux profiles
is typical of a self-similar jet (Turner, 1986). A more commonly used
indicator for the start of the self-similar regime is the $1/x$ variation of
the centreline scalar concentration. Such a variation starts from
$L_{s}\approx 0.45$m (Fig. S1 and S5b) and we find it convenient to associate
this location with the beginning of steady, self-similar speech flow. These
results are useful for determining viral exposure to the listener’s eyes and
mouth, as will be seen.

Figure 3: Comparison of the different characterizations of aerosol
concentration. The axial variation of locally averaged aerosol concentration
$(\Phi_{new})$ calculated by equation (4) and its comparison with aerosol
concentration based on the “top-hat” formulation $(\Phi_{H})$. Steady self-
similar behavior is established only for $L_{s}\gtrapprox 0.45m$.

We first obtain a conservative estimate of the aerosol flux inhaled by a
listener by using the fact that the domain of inhalation is much more
localized than has been considered in some of the previous studies. In
particular, Yang et al. (2020) have used a measure of aerosol concentration
($\Phi_{H}$) that is averaged over an area with radius $r_{H}=xtan\alpha$,
where the half-cone angle $\alpha$ encompasses 90% of scalar within it. These
effectively represent “top-hat” quantities and therefore have been denoted by
a subscript “H”. Using this formulation, Yang et al. estimated the number of
inhaled virions as

$N(\Phi_{H})\ =\
c_{v}\Phi_{H}(x)Q_{r}t=c_{v}\phi_{o}a_{H}Q_{r}t/(xtan\alpha),\ $ (3)

where $c_{v}$ is the number of virions per unit droplet volume (or the total
saliva volume) in the exhaled air during speech, $a_{H}$ is the orifice radius
and $Q_{r}$ is the rate of inhalation by the listener taken as $0.1l/s$.
However, the actual distributions of axial velocity and scalar concentration
in a steady self-similar jet are Gaussian, and it is therefore of interest to
relate $r_{H}$ and $\Phi_{H}$ with their Gaussian counterparts. The relevant
length scale characterizing a Gaussian distribution is the “$1/e$ width”
$(b_{e})$, defined as a radial distance where a quantity reaches $1/e$ times
its centreline value - for axial velocity we denote it as $b_{ue}(x)$ and for
aerosol concentration as $b_{\phi e}(x)$. The centreline values of the mean
axial velocity and aerosol concentration are denoted as $U_{c}(x)$ and
$\phi_{c}(x)$ respectively. We get the relation between the top-hat width of
Yang et al. (2020) and the Gaussian widths as $r_{H}=1.516b_{\phi e}$
$(x)=1.819b_{ue}(x)$ (see SI Text, section 1). Thus, the top-hat radius
considerably overestimates the lateral spread of the aerosol distribution. As
a result of this, the top-hat velocity $(v_{H}(x))$ and aerosol concentration
$(\Phi_{H}(x))$ are underestimated in relation to their Gaussian counterparts
as $v_{H}$ $(x)=0.39U_{c}(x)$ and $\Phi_{H}(x)=0.462\phi_{c}(x)$; (see Fig. S5
for a graphical comparison). Thus, a susceptible listener can be expected to
get exposed to a lot more number of virions than estimated in Yang et
al.(2020). To make a realistic estimate of the aerosol flux inhaled by a
listener, we consider inhalation to be a “sink” flow (with nose at its
centre), drawing in air from a hemispherical domain with a radius of $6.2$cm
(Abkarian et al., 2020; Haselton & Sperandio, 1988). An average aerosol
concentration is then calculated over this localized circular region centered
at the jet axis (rather than over the much larger area, $\pi r_{H}^{2}$) given
as

$\Phi_{new}=(b_{\phi e}/6.2)^{2}[1-exp(-(6.2/b_{\phi_{e}})^{2})]\phi_{c},$ (4)

which represents the average scalar concentration a listener is likely to
ingest through the inhalation process; here $b_{\phi e}$ is measured in cm
(for derivation, see SI Text, section 2). Using $\Phi_{H}(x)=0.462\phi_{c}(x)$
and $b_{\phi e}=0.137x$ for a steady self-similar jet (Singhal et al., 2020),
we obtain a relation between $\Phi_{new}$ and $\Phi_{H}$ as a function of $x$.
This is plotted in Figure 3, which shows that $\Phi_{new}$ > $\Phi_{H}$ for
$x>0.45m$; the region up to this axial location is shown shaded as the
Gaussian profile does not apply in this region. The figure also shows that for
$x>1\rm{m},\Phi_{H}(x)\approx 0.5\Phi_{new}(x)$; the use of the top-hat
profile thus underestimates the risk of infection by direct inhalation of
virions by about $50\%$, supporting our expectation mentioned earlier.

###  3.2 Case II: Persons 1 and 2 engaged in conversation

Figure 4: Aerosol flux distribution and concentration contours for Case II
(two-way conversation) compared with Case I (Person 2 silent). (a) Aerosol
flux $f_{II}$ for $L_{s}=1.2m$ for different vertical separations $(y_{s})$
for Case II. (b-e): Contours of aerosol concentration for $L_{s}=1.2m$ at
$t\approx 35s$ for $y_{s}$ of (b) $0d$ in Case I, (c) $0d$ in Case II, (d)
$4d$ in Case II and (e) $8d$ in Case II. In contrast to Case I shown in (b),
it is evident in (c-e) that the passage of aerosol from one person to another
is inhibited by the existence of two speech jets. The outlines of the jets are
shown in white for Person 1 and black for Person 2. Filled colour contours are
shown only for $\phi_{1}$. A side view of the circular region of interest
(ROI) in front of Person 2 is represented by a grey rectangle. Here
$\phi_{1}=\phi_{o}$ $(C_{s1}/C_{so})$ and $\phi_{2}=\phi_{o}(C_{s2}/C_{so})$,
(see supplementary movie S1 for the time evolution of the interaction between
two speech jets for c-e).

Thus far, the listener was entirely passive. In this set of simulations (Case
II), Person 2 is present at a location $L_{x}=L_{s}+\Delta$ and $-y_{s}$ with
respect to Person 1, i.e., Person 2 is of the same height or shorter than
Person 1 (Figure 1b). We calculate the flux of aerosols emanating from the
mouth of Person 1 $(\phi_{1})$ at the ROI located in front of Person 2 (who is
a susceptible individual) and denote it as
$f_{II}(L_{s},-y_{s}\textfractionsolidus 2)$ (1). Simulations are performed
for three different values of $L_{s}\in\\{0.6\rm{m},1.2\rm{m},1.8\rm{m}\\}$
with $y_{s}$ taking one of the seven values among $0d,2d,4d,6d,8d,12d$ and
$16d$ for each $L_{s}$; a couple of other values of $y_{s}$ are also chosen
where needed. As described earlier, each person speaks for 10 speech cycles in
a staggered manner to have a total speech time of 70s (including silent
intervals; Figure 1d). The simulations are run longer than this duration,
until almost all of the aerosols expelled by Person 1 pass through the ROI.
Figure 4a shows the time variation of $f_{II}$ for $L_{s}=1.2$m as $y_{s}$ is
varied, which exhibits a non-monotonic behaviour. As $y_{s}$ increases from
$0$, $f_{II}$ increases many folds until $y_{s}=8d$ and the time at which
$f_{II}$ peaks (at a given $y_{s}$) goes on decreasing. As $y_{s}$ increases
beyond $8d$, $f_{II}$ shows a decrease whereas the time at which it peaks
increases (Figure 4a).

Figure 4(c to e) show the contour plots of the instantaneous aerosol
concentration for Case II, for $y_{s}=0d,4d,8d$ respectively, with the color
contours representing different values of $\phi_{1}$ and the line contours
representing an iso-line for $\phi_{1}$ and $\phi_{2}$ equal to $1.2\times
10^{-10}$; Figure 4b shows the aerosol contours for Case I for comparison. For
$y_{s}=0d$ (Case II; Figure 4c), the jets issuing out of the two orifices
impinge on each other and “cancel” out each other’s effect to form a cloud in
the middle. As a result, the flux of $\phi_{1}$ at Person 2 is much smaller
for $y_{s}=0d$ than what it would be if Person 2 was silent as in Case I (see
Figure 4b). For $y_{s}=4d$, the jets are found to be “sliding over” each other
with a region of overlap in the aerosol distributions (Figure 4d). Although
the head of flow from Person 1 could be seen deviating away from the Person 2,
the overlapped region of $\phi_{1}$ manages to project a part of the aerosol
flux on the latter’s face (grey rectangle). For $y_{s}=8d$ in Figure 4e, the
interference between the two jets is significantly reduced and therefore the
aerosols from Person 1 find it easier to reach the ROI positioned in front of
Person 2. For higher $y_{s}$, only a small fraction of the speech jet from
Person 1 can be expected to intersect the ROI, due to the large vertical
separation. Thus, the competing effects of the jet interference and vertical
separation lead to the observed maximum in the aerosol flux for an
intermediate $y_{s}$ (Figure 4a). The time evolution of the interaction of
speech jets from Persons 1 and 2 for the cases shown in Figure 4c-e has been
presented as supplementary movie S1 (see SI). The values for
$F_{II}(L_{s},y_{s},t_{c})$ (2) are calculated for all the simulations of Case
II and used to calculate the virion exposure, as explained in next section.

###  3.3 Viral exposure and infection probability for Cases I and II

Figure 5:  (a) The number of virions and (b) the probability of infection for
Case I (curves) and Case II (symbols). Each symbol for Case II corresponds to
a different simulation. The probability of infection in Case II is always much
lower than in Case I, which means that a dialogue is always better than a
monologue. The red-dashed curve, showing viral exposure through inhalation
alone (Yang et al.,2020), is always lower by about 50% than the solid black
curve which includes exposure to the eyes and the mouth. The locations of the
symbols correspond to the axial separation $L_{s}$ between the speakers and
each symbol is accompanied by a horizontal bar whose length is proportional to
$y_{s}$, for $y_{s}\in\\{0d,2d,4d,6d,8d,12d,16d\\}$. In Case II, the infection
probability for a given $L_{s}$ varies non-monotonically with $y_{s}$.

In this section, we determine the virion ingestion by a susceptible individual
due to the aerosol transport from speech flows. First, the total virion
exposure to a passive listener in Case I is calculated using inhaled virions
as well as the ones projected onto the eyes and mouth area. The number of
inhaled virions in Case I is calculated using the same expression as in (3)
but by replacing $\Phi_{H}$ by $\Phi_{new}$ (4), as

$N(\Phi_{new})=\ c_{v}\Phi_{new}Q_{r}\ t$ (5)

We use $c_{v}=7\times 10^{6}\rm{ml}^{-1}$ (Yang et al., 2020). To calculate
the infection through eyes and mouth, we first obtain the total number of
virions projected onto the listener’s face (or passing through the ROI in
front of their face) as $N_{I,f}=c_{v}F_{I}(L_{s},y_{s},140\rm{s})$, i.e.,
integrated over 140s. Towards this, we use $F_{I}$ for $y_{s}=0$ (blue curve
in Figure 2b) which corresponds to the maximum aerosol exposure for Case I. We
assume that viral exposure to the eyes and mouth is the fraction $\kappa$ of
$N_{I,f}$, where $\kappa=A_{em}/A_{ROI}$; here $A_{em}$ is the total area of
eyes and mouth, and $A_{ROI}$ is the area of the ROI (i.e., the projected face
area). The total number of virions that are ingested by the listener
(potentially causing infection) for Case I is, therefore,

$N_{I,i}\ =\ N(\Phi_{new})\ +\ \kappa\cdot N_{I,f}\ .$ (6)

Reasonable estimates for the areas of the eyes (radius $\approx 1.2\rm{cm}$)
and of the mouth and lips area ($\approx 15.55\rm{cm}^{2}$), suggest that
these amount to about $10.6\%$ of the area of the face $(\kappa=0.106)$.

For Case II the total exposure to Person 2 over 140s from the aerosols
expelled by Person 1 is denoted by $F_{II}(L_{s},-0.5*y_{s},140\rm{s})$ (1 and
2). The total virion exposure to the face of Person 2 is therefore given as
$N_{II,f}\ =\ c_{v}F_{II}(L_{s},-0.5*y_{s},140\rm{s})$. To calculate the
virion exposure causing infection we assume that

$N_{II,i}/N_{II,f}\ =\ N_{I,i}/N_{I,f},$ (7)

where $N_{II,i}$ represents the number of virions ingested by Person 2 through
inhalation and exposure to eyes and mouth for a given simulation of Case II.
Note that $N_{I,i}$ and $N_{II,i}$ represent conservative estimates of
ingested virions, assuming that the entire virion exposure to eyes and mouth
leads to infection.

Figure 5a presents a summary plot of the virion exposure to Person 2 (for
Cases I and II) obtained using different measures. The number of inhaled
virions using the method of Yang et al. (2020), $N(\Phi_{H})$(3), is plotted
as a red dashed curve, whereas $N(\Phi_{new})$ obtained by averaging over a
localized region (5) is plotted as a black dashed curve. Note that the region
$x<0.45$m is not considered as self-similarity of speech flow does not hold
and calculations using steady state parameters are expected to be unrealistic
in this region (Fig. S5b). As can be seen from Figure 5a, $N(\Phi_{new})$ is
nearly twice that of $N(\Phi_{H})$ for the entire range of separation
distances, $L_{s}$, consistent with Figure 3 . Note that both $N(\Phi_{H})$
and $N(\Phi_{new})$ have been calculated in the context of Case I, i.e., with
Person 2 as a passive listener. The total number of ingested virions for Case
I, $N_{I,i}$, shows even higher values as compared to $N(\Phi_{new})$ as
expected (6). We propose that $N_{I,i}$ is a more realistic (although
conservative) measure for the viral load for determining the risk of
infection, as compared to the estimate based on inhaled virions alone. The
number of ingested virions for Case II, $N_{II,i}$, is shown as symbols for
three different values of $L_{s}$. The maximum number of ingested virions for
this case at each $L_{s}$ is seen to occur at an intermediate value of $y_{s}$
consistent with Figure 4a. Figure 5a clearly shows that the viral exposure to
Person 2 when they are actively engaged in a conversation (Case II) is lower
as compared to that when Person 2 is a passive listener (Case I) by a factor
more than two. This is due to the interference of the speech jets from the two
persons during a conversation as discussed earlier.

Figure 6:  (a) The number of virions $N_{II,i}$ ingested by Person 2 as a
function of the vertical separation $y_{s}/d$ for different axial separations
$L_{s}$. (b) Variation of normalized $N_{II,i}$ with $y_{s}/L_{s}$. Risk of
infection varies non-monotonically with vertical separation.

In Figure 5b we plot the probability of infection corresponding to the viral
load $N$ calculated as

$p(N)=1-exp(-N/N_{inf}),$ (8)

where $N_{inf}\approx 100$ is the characteristic number of virions causing
infection (Yang et. al., 2020). For Case I, $p(N_{I,i})$ is found to be as
high as $0.5$ when $L_{s}=0.6$m. Thus, there is a real risk of inflection to a
passive listener for a 2ft separation distance, even when the other person
speaks for a very short time (speech time =70s and exposure time = 140s). As
the separation distance increases the infection probability decreases
monotonically, with $p(N_{I,i})<0.2$ for $L_{s}=1.8$m (Figure 5b). Thus, it is
much safer to adhere to the 6ft rule even while listening to someone speak for
a short span of time, in order to minimize the risk of infection. For Case II
involving conversations, $p(N_{II,i}$ ) is much less than $p(N_{I,i})$ as
expected and the risk of infection is therefore low. The maximum $p(N_{II,i})$
among all the conversation cases is about $0.2$ for $L_{s}=0.6$m for a
conversation of a little over a minute. For a longer conversation or for
asymmetric speech times (see the next section) between the people, the risk of
infection can be expected to be higher than this. Increasing separation
between people during conversation is again a sure-shot way of reducing the
infection probability (Figure 5b).

Figure 6a provides a graphical representation of the dependence of $N_{II,i}$
on $L_{s}$ and $y_{s}$. As follows from the discussion above, the viral dose
to Person 2 decreases considerably with increase in $L_{s}$ and $N_{II,i}$
peaks at an intermediate vertical separation $y_{s}$ (see also Figure 5).
Figure 6b plots the variation of the normalized viral dose (by its peak value
for a given $L_{s}$) with $y_{s}/L_{s}$. We find that maximum of the
normalized $N_{II,i}$ occurs at $y_{s}/L_{s}$ of $0.12-0.15$ which can be a
useful result from the scaling point of view. Furthermore,
$N_{II,i}/max(N_{II,i})$ is significantly reduced for
$y_{s}\textfractionsolidus L_{s}>0.3$, implying that for a given separation
distance between two people, the viral load during a short conversation can be
expected to be low if the condition $y_{s}\gtrsim L_{s}\textfractionsolidus 3$
is satisfied.

###  3.4 Case IIt: Temporal asymmetry in speech during conversation

Figure 7: The role of temporal asymmetry in determining viral exposure, shown
by plotting (bottom panel) the number of virions ingested as a function of
time in Case IIt compared to Case I. The panels at the top show the inlet flow
rates. For both cases $L_{s}=1.2m$ and $y_{s}=0$. For Case IIt, the
conversation does not stop incoming aerosols from reaching Person 2 but merely
delays it by about 100s (see supplementary movie S2 for time evolution of the
flow for this case). The time variation of $N_{IIt,i}$ shown here can in
principle be extrapolated to estimate the viral load for conversations longer
than used here.

So far, we have presented conversation cases where the number of speech cycles
and the total speech time were the same for Persons 1 and 2 (Figure 1d). This
corresponds to a scenario wherein the exchange of short phrases during a
conversation is more or less “symmetric” (although staggered in time). The
objective of this exercise was to quantify the difference between one-way and
two-way conversations. However, in reality, conversations can take place in a
variety of different ways and in most of the cases there is likely to be an
asymmetry of speech between two people. Here we present one instance of an
“asymmetric” conversation (Case IIt) wherein both the persons engage in a
symmetric conversation for 70s, after which Person 2 stops talking while
Person 1 continues for another 70s; see Figure 7. Both the persons are
considered to be of equal height $(y_{s}=0)$ with a separation distance,
$L_{s}=1.2$m, and the simulation is run for about 250s. Figure 7 shows the
variation of the number of virions ingested, $N_{IIt,i}$, with time interval,
$t_{c}$, obtained by varying $t_{c}$ in the calculation of
$F(L_{s},y_{s},t_{c})$ in (2) for Case IIt. Also shown for comparison is
$N_{I,i}(t_{c})$ for Case I representing a one-way conversation. It is seen
that the number of ingested virions for Case IIt is negligibly small until
about $100$s (cf. Figure 4a) after which $N_{IIt,i}$ starts increasing as the
aerosols from Person 1 (who continues to speak after 70s) reach Person 2. The
increase in $N_{IIt,i}$ is initially rapid but tapers off after Person 1 stops
speaking after 140s. Overall, the shapes of curves and the number of virions
ingested by Person 2 at the end of conversation are similar between
$N_{IIt,i}$ and $N_{I,i}$ (Figure 7).

The probability of infection for Person 2 at $t_{c}=250$s for Case IIt comes
out to be $p(N_{IIt,i})=0.32$ (8), which is slightly higher than that obtained
for Case I (Figure 5b). This suggests that whenever there is an asymmetry in
speech between people engaged in a conversation, the person who talks less is
at an enhanced risk of catching infection than the person who talks more. This
is true even when both the persons are of nearly the same height. The time
evolution of the speech jets for Case IIt is presented as supplementary movie
S2 (see SI). Note that the asymmetry in time need not occur at the end of a
conversation but could be embedded within it, in which case the curve for
$N_{IIt,i}$ in Figure 7 will show an “up and down” variation, providing
temporally varying and non-monotonic probability of infection.

##  4 Discussion

###  4.1 Lateral spread and temporal evolution of the interacting speech jets

Figure 8: (a) Lateral width ($r_{s}$ from (9)) normalised by the corresponding
domain length between the two persons for Case II. (b) Lateral width of the
flow on a log-log scale for Cases I and II, with $L_{s}=1.8m$. For Case II,
$y_{s}=0$.

Although the collision of the speech jets during conversations can restrict
the axial transport of aerosols (especially for people of comparable heights;
Figure 4), it can result in considerable lateral spread of the flow. This
aspect is relevant to how the interaction of two people generates a “cloud” of
infected aerosols that might affect disease spread in poorly ventilated rooms.
We define the lateral width ($r_{s}$) of the flow using

$\int_{r=0}^{r_{s}}\int_{\theta}\int_{x}\phi rdrd\theta
dx=0.9\int_{r=0}^{\infty}\int_{\theta}\int_{x}\phi rdrd\theta dx.$ (9)

The lateral widths of the flow in Case II (two-speaker) simulations are
compared for different axial separations in Figure 8a, and with Case I
(single-speaker case) in Figure 8b. The axial length of the domain,
$L_{s}+\Delta$ ($\equiv L_{x}$; see Figure 1b), acts as a rough scaling length
for the lateral spread (Figure 8a); $r_{s}/(L_{s}+\Delta)$ reaches a value of
0.3 after $120$s, which is about $0.55$m for $L_{s}=1.8$m. Figure 8b shows
that the collision of the speech jets from the two persons initially increases
the rate of lateral spread, ($r_{s}\propto t^{1.0}$), which is significantly
higher than that in Case I where lateral spread is $\propto t^{0.6}$. This
provides an estimate of the rate of evolution of the aerosol cloud that might
affect a third person positioned nearby; it should also be possible to
extrapolate the rate of growth for longer conversation times using the power
laws in Figure 8b. At later times, i.e. after the conversation ends, the rate
of spread in Case II becomes slower, but the lateral spread continues to
remain much larger than that for Case I.

Figure 9: (a) Flow oscillation due to the collision of alternate speech jets
from two people, represented by the stagnation point ($x_{s}$) in the
centreline velocity profile. The location $x_{s}$ is normalized such that
Person 1 is at $x=-1$ and Person 2 is at $x=1$. (b) A measure of average
velocity, based on the domain-averaged kinetic energy, as a function of time
for Cases I and II. For Case II, $y_{s}=0$.

In Figure 9a, we plot the location of the stagnation point, $x_{s}$, between
the two speech jets for Case II. For small axial separation as $L_{s}=0.6$m,
we see that the location of the stagnation point nearly reaches either speaker
at $(2x_{s}/(L_{s}+\Delta))-1=\pm 1$. This wide oscillation is responsible for
the non-zero aerosol flux even in a face-to-face ($y_{s}=0$) conversation for
$L_{s}=0.6$m (see Figure 6). This is not true, however, for larger axial
separations $L_{s}$. The oscillation of the stagnation point is a feature of
the pulsatile nature of the flow and is not captured if the flows are taken to
be steady jets (for which the stagnation point would remain fixed in place at
mid-domain). The oscillation of the stagnation point also suggests that the
“safe” axial separation between two persons facing each other depends on the
duration of the speech pulses in Figure 9a. Following Abkarian et al., (2020),
the time $(t^{*})$ taken by a jet to reach the separation distance $L_{s}$ is
given as $t^{*}=L_{s}^{2}tan\alpha/2\bar{v}_{o}d$, which gives $t^{*}$=
(3.34s, 13.35s, and 30.04s) for $L_{s}$= (0.6m, 1.2m ,and 1.8m) respectively.
For $L_{s}=0.6$m, $t^{*}=3.34$s is comparable to the half of the speech-cycle
duration, which is $3.5$s. Note that the average orifice velocity
$\bar{v}_{o}=0.39m/s$.

Figure 9b, plots the time variation of a measure of velocity related to the
averaged kinetic energy as $\sqrt{2\overline{K.E.}}/3$, with $\overline{K.E.}$
defined as follows.

$\overline{K.E.}=\frac{1/2\int(u^{2}+v^{2}+w^{2})dV|_{\phi>0.01\phi_{o}}}{\int
dV|_{\phi>0.01\phi_{o}}},$ (10)

where $V$ represents the domain volume. The decay of the domain-averaged
kinetic energy is nearly identical for Cases I and II until the end of the
speech duration (i.e., 70s), after which the kinetic energy decays at
different cases for different cases. The decay rate is larger for the two
lower values of $L_{s}$ (Case II), presumably because of the enhanced
dissipation resulting from more intense gradients generated by the collision
of speech jets; Figure 9b. The residual velocity in the domain reaches fairly
low values after the conversation ends and can soon become comparable to the
speed of the background air motion found in typical indoor environments. For
example, a threshold background current in an air-conditioned room is of the
order of 15cm/s (Matthews et al. 1989), shown by dashed horizontal line in
Figure 9b. Under such conditions, the transport of virus-laden aerosols can be
expected to switch to the airborne transmission mode.

###  4.2 Relevance towards public health guidelines and epidemiological
modelling

Our results provide guidance, in addition to physical distancing, for safe
interaction among people engaged in short conversations (up to about a minute
or two). This situation arises routinely in day-to-day life, for example in
shopping malls or stores, where over-the-counter conversations take place for
short durations. We show the effect of buoyancy to be negligible, since the
rate of spread of the speech flow in the vertical and horizontal planes is
practically identical (see Fig. S3). So our results on the height difference
(vertical separation $y_{s}$) between two people also apply to lateral
separation (denoted by $z_{s}$; Figure 10). This is an important result since
the lateral separation between people can be controlled but the height
difference cannot! We find surprisingly that the probability of infection from
a two-way conversation is maximum, not when the two people face each other
directly, but when $y_{s}$ (or $z_{s}$) is about 12-15% of $L_{s}$ (Figure
6b); for an axial separation of 4ft, this amounts to about 15cm of vertical or
lateral separation. However, infection probability is considerably lower when
$y_{s}\textfractionsolidus L_{s}$ (or $z_{s}\textfractionsolidus L_{s}$ ) is
greater than 0.3 (Figure 6b) and this helps in devising the following general
rule for minimizing the risk of infection. When people in a conversation turn
their faces away from each other very slightly, it can be worse than facing
each other directly, but an angle greater than $tan^{-1}(0.18/1.2)\approx
9^{\circ}$ (Figure 10b) can be very helpful. This implies a total “angular
separation” of $18^{\circ}$ or higher between two people for a safe
conversation which still enables eye contact. This angular separation matches
well with the angle of spread of the speech jet ($20^{\circ}$; Fig. S3), which
means that the speech jet from one person effectively misses the face of the
other person, thereby minimizing viral exposure. As expected, for the three
axial separations considered, the probability of infection is the least for
$L_{s}=6$ft (Figure 5).

Note that the above results are based on the “directed jet” behaviour of
speech flow (which is the main cause of the sustained aerosol transport away
from the speaker’s mouth) generated by the repetition of the phrase “Peter
Piper picked a peck” (Abkarian et al., 2020). Abkarian et al. (2020) also
considered other phrases containing fricatives and plosives, and found that
the flow associated with the utterance of isolated syllables could be directed
upward or downward from the axis at angles as high as $40-50^{\circ}$.
However, these flow patterns are contained within a distance of 0.5m from the
speaker and die out within 100ms after they are expelled from the mouth
(Abkarian et al., 2020). For two people separated by more than 0.6m and for
time scales of interaction of the order of 1-2mins as considered here, the
transient aerosol transport associated with individual syllables is likely to
be less relevant. It will of course be better if the listener can stay
entirely outside the zone of influence of the speech flow as suggested by
Abkarian et al. (2020), although it may not be always possible.

Based on our simulations (and taking cues from previous studies), we formulate
the following additional guidelines (apart from physical distancing) for short
unmasked conversations.

  * •

Let the other person speak! Any two-way conversation is far better than a
monologue. The one who talks less is at a higher risk of infection.

  * •

In close-up conversations, a slight tilt of the head or a small lateral
separation, even while maintaining eye contact, will go a long way in reducing
risk.

  * •

As expected, a greater separation is better, not only while coughing/sneezing
but even during short conversations.

Figure 10: Lateral/angular separation to minimize transmission risk. (a)
Schematic of two persons separated laterally by a distance $z_{s}$ such that
the risk of infection is minimized. (b) Schematic of two people conversing
with their heads turned away from each other by an “angular separation” to
achieve the same effect as in (a). For an axial separation up to 6ft, a tilt
angle $(\alpha)$ of $9^{\circ}$ or more is recommended.

The present results can also provide useful inputs to epidemiological models
to improve their prediction accuracy. In this connection, Chaudhury et al.
(2020) have presented a framework for determining the infection rates for
different expiratory events such as coughing, sneezing, breathing, talking
etc. and incorporating them in an SEIRD (susceptible-exposed-infected-
recovered-deceased) model (see also Dbouk and Drikakis, 2021). They have
highlighted the need for accurate data on viral exposure based on fluid
dynamical simulation of these expiratory events. Since the present simulations
provide a spatio-temporal distribution of the aerosol field for speech flows,
the quantities of interest can be easily calculated, e.g., the temporal
variation of the viral load presented in Figure 7 can be used for determining
time-dependent infection probabilities for speech flows, which is a crucial
input to the model used in Chaudhuri et al. (2020) . For realistic
conversation scenarios these probabilities could not have been accurately
inferred from the idealized cases of “one-way” conversation (which would
overestimate the risk) or “symmetric” conversation (which would underestimate
the risk).

The limitations of the present work should be noted. Parameters like $c_{v}$
could have different magnitudes than those used here (Wölfel et al., 2020).
Also, the number of droplets exhaled is a function of the loudness of speech
(Asadi et al., 2019) and can introduce variability in the number of virions
exhaled by an infected person. The results presented here are relevant for
poorly ventilated spaces whereas a cross ventilation can make a difference.
Finally, the speech cycles used are based on utterance of simple repetitive
phrases and therefore can only serve as an approximation to the real speech
patterns. Notwithstanding these, our simulations provide general scientific
principles on which public health guidelines can be based. However, in view of
the large number of variables involved, mask mandates are crucial in curbing
disease transmission (Chu et al., 2020).

##  5 Summary of key results

We have performed direct numerical simulations of the turbulent transport of
aerosols by the flows generated in human speech (involving repetitive
utterance of certain phrases) during short conversations. Our main results
are:

  1. 1.

We have computed the total exposure to aerosol on a listener due to speech
flow from a speaker not only through inhalation but also through the mouth and
eyes. A conservative estimate of the probability of infection is calculated
based on the total number of ingested virions and is shown to be higher by a
factor of 2 or more compared to previous estimates.

  2. 2.

In conversations, the active involvement of both people significantly lowers
the aerosol exposure (and the associated risk of infection) with respect to
the case where one person is a passive listener. This is because of the
interference between the two speech jets, which effectively shuts off onward
transmission of infected aerosols. On the other hand, asymmetry in the speech
pattern of two people, invariably present in real conversations, reduces the
jet interference and enhances the risk of infection to the person who talks
less.

  3. 3.

For identical speech phrases from both speakers, the probability of infection
peaks when there is a small vertical/lateral separation between them, due to a
less effective interaction between the two jets. When this separation is
greater than 0.3 times the axial (perpendicular) distance between them, the
risk of infection is seen to be minimal.

  4. 4.

The collision of the speech jets results in a considerable lateral spread of
the aerosol cloud potentially infecting surrounding air in an indoor
environment. For small axial separations between two people, the speech jets
oscillate considerably in time and can penetrate to reach the other person’s
face in an intermittent fashion.

We believe simulations such as the ones presented here can provide realistic
estimates of infection probabilities for any speech pattern of interest and
can provide useful inputs to epidemiological models. Lastly, the sensitive
dependence of viral exposure during short conversations on several parameters
studied here provides insights into the complexity of deciding foolproof
public health guidelines to curb the spread of SARS-CoV-2.

##  Appendix A Governing equations and numerical method

The Boussinesq Navier-Stokes equations, with scalar transport equations for
temperature and passive scalar, are given as

$\vec{\tilde{\nabla}}\ \cdot\ \vec{\tilde{u}}\ =0,$ (11)
$D{\vec{\tilde{u}}}/D\tilde{t}=-\vec{\tilde{\nabla}}\tilde{p}+\frac{1}{Re}\vec{\tilde{\nabla}}^{2}\vec{\tilde{u}}+\frac{1}{Fr^{2}}\tilde{\theta}e_{y},$
(12) $D\tilde{\theta}/D\tilde{t}=\frac{1}{Re\cdot
Pr}\tilde{\nabla}^{2}\tilde{\theta},$ (13)
$D\tilde{C_{s}}/D\tilde{t}=\frac{1}{Re\cdot
Sc}\tilde{\nabla}^{2}\tilde{C_{s}},$ (14)

where $\tilde{u}$ is the fluid velocity, $\tilde{p}$ is the pressure,
$\tilde{\theta}$ is the temperature difference between the flow and the
ambient, $\tilde{C_{s}}$ is the passive scalar concentration, $D/Dt$ is the
material derivative, $Re=u_{c}d/\nu$ is the Reynolds number,
$Fr^{2}=u_{c}^{2}/\beta dg\Delta T_{o}$ is the Froude-number squared, $\nu$ is
kinematic viscosity of air, $g$ is acceleration due to gravity, $\beta$ is the
coefficient of thermal expansion for air and non-dimensional quantities are
represented by a tilde $\textasciitilde$. The density difference between the
speech fluid and the ambient is given by
$\Delta\rho\textfractionsolidus\rho=-\beta\Delta T$; with $\beta\approx
3.4\times 10^{-3}$ 1/K and a temperature difference at the orifice, $\Delta
T_{o}=14^{\circ}$C, we get $\Delta\rho\textfractionsolidus\rho\approx-0.05$,
supporting the Boussinesq approximation. The passive scalar at the orifice is
taken as unity, $C_{so}=1$. The governing parameters are the Prandtl number,
$Pr$, and the Schmidt number $Sc$ governing the diffusion of the temperature
and scalar respectively, which are both assigned unit values. For the length
and velocity scales mentioned above characteristic of respiratory flows,
$Re=1906$ and $Fr^{-2}=0.00842$. For the conversation cases (with both the
persons speaking) two equations are solved for the passive scalar – one each
for $\tilde{C_{s1}}$ and $\tilde{C_{s2}}$.

The DNS solver Megha-5 which is used previously for canonical jet , plumes
(Singhal et al., 2021) and cumulus & mammatus clouds (Ravichandran &
Narasimha, 2020; Ravichandran et al., 2020), is employed here. The code
discretizes the governing equations in a Cartesian geometry using a second-
order scheme in space and uses a second-order accurate Adams-Bashforth scheme
for time-stepping. For more details see Singhal et al. (2020) and Ravichandran
et al. (2020).

#### Acknowledgements

The simulations were carried out at the Supercomputer Education and Research
Centre at Indian Institute of Science, Bengaluru.

#### Funding Statement

SR gratefully acknowledges support through the Swedish Research Council grant
no. 638-2013-9243. SSD acknowledges support from Indian Institute of Science,
Bengaluru in the form of the start-up grant no. 1205010620.

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