# Towards realistic simulations of human cough:
effect of droplet emission duration and spread angle

Mogeng Li  Kai Leong Chong  Chong Shen Ng  Prateek Bahl  Charitha M. de Silva
Roberto Verzicco  Con Doolan  C. Raina MacIntyre  Detlef Lohse
<EMAIL_ADDRESS>Physics of Fluids Group, Max Planck Center for Complex
Fluid Dynamics, J. M. Burgers Center for Fluid Dynamics and MESA+ Research
Institute, Department of Science and Technology, University of Twente, 7500AE
Enschede, The Netherlands  Max Planck Institute for Dynamics and Self-
Organisation, 37077 Göttingen, Germany  Shanghai 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  School of Mechanical and Manufacturing
Engineering, UNSW Sydney, Kensington, NSW, 2052, Australia  Dipartimento di
Ingegneria Industriale, University of Rome ‘Tor Vergata’, Roma 00133, Italy
Gran Sasso Science Institute - Viale F. Crispi, 7 67100 L’Aquila, Italy
Biosecurity Program, The Kirby Institute, UNSW Sydney, Kensington, NSW, 2052,
Australia  College of Public Service & Community Solutions and College of
Health Solutions, Arizona State University, Phoenix, Arizona, USA

###### Abstract

Human respiratory events, such as coughing and sneezing, play an important
role in the host-to-host airborne transmission of diseases. Thus, there has
been a substantial effort in understanding these processes: various analytical
or numerical models have been developed to describe them, but their validity
has not been fully assessed due to the difficulty of a direct comparison with
real human exhalations. In this study, we report a unique comparison between
datasets that have both detailed measurements of a real human cough using
spirometer and particle tracking velocimetry, and direct numerical simulation
at similar conditions. By examining the experimental data, we find that the
injection velocity at the mouth is not uni-directional. Instead, the droplets
are injected into various directions, with their trajectories forming a cone
shape in space. Furthermore, we find that the period of droplet emissions is
much shorter than that of the cough: experimental results indicate that the
droplets with an initial diameter $\gtrsim\,10\upmu$m are emitted within the
first 0.05 s, whereas the cough duration is closer to 1 s. These two features
(the spread in the direction of injection velocity and the short duration of
droplet emission) are incorporated into our direct numerical simulation,
leading to an improved agreement with the experimental measurements. Thus, to
have accurate representations of human expulsions in respiratory models, it is
imperative to include parametrisation of these two features.

###### keywords:

COVID-19, pathogen transmission, respiratory droplets

###### MSC:

[2010] 76T10 , 76F65

††journal: International Journal of Multiphase Flow

##  1 Introduction

Since the outbreaks of SARS-CoV in 2003 and SARS-CoV-2 in 2019, the role
played by the turbulent multiphase flow in the transmission of infectious
diseases via the airborne route has received increasing attention. The host-
to-host transmission of respiratory disease is a complicated process with
multiple stages including the exhalation, dispersion and inhalation of the
pathogen [Bourouiba, 2020, Zhou and Zou, 2021, Bourouiba, 2021, Chong et al.,
2021, Smith et al., 2020].In particular, a central piece to the puzzle is the
dispersion of pathogen-carrying droplets in turbulent flows.

A classic approach to mathematically predict the transmission of an infectious
disease among populations is the compartmental models [Tolles and Luong,
2020]. Developed in 1927, the most basic compartmental model is an SIR model
[Kermack and McKendrick, 1927], where the population is separated into three
so-called compartments, namely ‘Susceptible’, ‘Infected’ and ‘Removed’.
Individuals can transfer between compartments based on ordinary differential
equations while the total population remains constant. The infection rate of
susceptible individuals can be improved with more physical insight via dose-
response models [Brouwer et al., 2017], or the Wells–Riley model in particular
for diseases transmitted via the airborne route [Noakes et al., 2006]. The
application of these two groups of models and a detailed comparison are
presented in the review paper by Sze To and Chao [2010]. In essence, the
Wells–Riley model centers on the concept of ‘quantum’, which is the dose
required to start an infection, while the dose-response model uses the amount
of pathogen taken in by the susceptible individual. In both approaches, the
complicated transmission process that involves a series of stages (exhalation,
dispersion, ventilation, inhalation, etc) is parametrised based on some
physical assumptions or empirical observations [Lelieveld et al., 2020, Bazant
and Bush, 2021, Jones et al., 2021, Nordsiek et al., 2021]. As a typical
example, the room where the infectious disease is spread from is assumed to be
‘well-mixed’, i.e. the pathogen distributes uniformly in the room. More
recently, some efforts have been made to incorporate the temporal and spatial
inhomogeneity in the model [Mittal et al., 2020a, Yang et al., 2020]. A
sophisticated software tool has been developed to estimate the infection risk
under a number of scenarios, including various modes of ventilation and
different levels of activities [Mikszewski et al., 2020]. These models are
relatively easy to use and hence highly appealing when developing social
guidelines based on the risk of transmission at a wide range of real-life
scenarios. However, the derivation of these models, especially the selection
of the parameters hinges on the detailed understanding of the governing
physics in each step of the transmission.

Figure 1:  (a) Schematic of the setup used to capture high-speed frames of
droplets expelled during coughing. (b) Schematic of the spirometry test
performed by the subject for coughing.

Alongside the effort in the epidemiology community, there has also been an
ongoing endeavour in understanding and modelling the flow physics in the
transmission of respiratory diseases. In the pioneering work of Wells [1934],
the lifespan of a droplet is illustrated through a simple ‘evaporation-falling
curve’, where the fate of a small droplet is considered to be full
evaporation, while that of a large droplet is ground deposition. This
simplistic model was then improved by Xie et al. [2007], where the salinity of
the droplet, the ambient temperature and relative humidity conditions and the
buoyant respiratory jet were taken into consideration. The effect of ambient
conditions on the droplet evaporation is then incorporated into one parameter,
the effective evaporation diffusivity proposed by Balachandar et al. [2020].
The coupling between the carrying fluid and the disperse phase is further
incorporated into the puff model by Bourouiba et al. [2014], and a good
agreement with the laboratory measurement of a multiphase puff is observed,
but a direct comparison with clinical data, such as the speed and the
trajectory of the puff is still lacking.

There is now a consensus that the flow physics in the transmission of
respiratory diseases involve a multiphase flow [Bourouiba, 2020]. Indeed,
various human respiratory events, such as breathing, speaking, coughing and
sneezing, can be simplified as transient turbulent jets with liquid droplets
as the second phase [Balachandar et al., 2020, Mittal et al., 2020b, Bourouiba
et al., 2014, Bourouiba, 2020, Ng et al., 2021, Chong et al., 2021, Abkarian
et al., 2020, Stadnytskyi et al., 2020]. The flow field associated with a
single respiratory event have been investigated through both experiments with
human subjects [VanSciver et al., 2011, Bahl et al., 2020] and laboratory or
numerical simulations [Bourouiba et al., 2014, Wei and Li, 2017, Chang et al.,
2020, Abkarian et al., 2020, Ng et al., 2021, Chong et al., 2021, Fabregat et
al., 2021]. The general picture of a cough or sneeze is that the droplet-
laden, warm and moist air is injected at a high speed over a short duration
($\lesssim 1$ s), which propagates under the effect of both the initial
momentum and buoyancy [Bourouiba et al., 2014, Ng et al., 2021], while in
continuous events such as speaking, a ‘puff train’ assimilates to a turbulent
jet like structure in the far field [Abkarian et al., 2020]. The fate of the
droplets varies largely depending on their size [Ng et al., 2021]: large
droplets with a diameter $\sim\mathcal{O}(1000\upmu\text{m})$ deposit on the
ground in a near ballistic motion, while small droplets with a diameter
$\sim\mathcal{O}(10\upmu\text{m})$ are trapped in the humid puff and have
prolonged lifetimes compared to the prediction of Wells [1934]. Condensation
is observed at low ambient temperature and high relative humidity settings
[Chong et al., 2021]. Similar to the need for judicious assessment of the
accuracy of theoretical models with clinical data, a direct comparison of
numerical simulations with real respiratory events is imperative to test and
elevate the accuracy of these models.

Accordingly, in this study, we report a unique pair of datasets of human
coughs with both experimental and numerical components with similar
conditions. The numerical dataset is obtained using direct numerical
simulations (DNS) based on the methods employed by Chong et al. [2021] and Ng
et al. [2021], and the experimental component is the extension of the method
used by Bahl et al. [2020] to coughs. We reveal that both the droplet emission
duration and the initial spread angle have strong influence on the far-field
behaviour of the flow field. These two refinements to the flow inlet
conditions are validated through the experimental data collected from a human
subject in §2. Two numerical simulation cases are formulated based on these
refined conditions in §3, and their detailed effect on the simulated flow is
discussed in §4.

##  2 Experimental observations

To capture the flow dynamics of both the airflow and the droplets expelled
during a cough from human exhalations, we utilise two sets of experiments.
Namely, a spirometer is employed in the subject’s mouth to capture the
volumetric flow rate expelled during a cough (Fig. 1(b)), and Particle
Tracking Velocimetry (PTV) experiments quantify the motion of the expelled
droplets. For the latter, a volume illumination configuration is employed
using a high powered pulsed LED light source (GsVitec L5), positioned at
approximately $45^{\circ}$ from the image plane, to capture high-speed imaging
data of the scattered light from the droplets expelled during a cough. The
high-speed camera was placed $0.6$ m away from the image plane and was
equipped with a $35$ mm lens with an f-stop of F4 to capture a field-of-view
of approximately $0.6\times 0.35$ m2. This arrangement provided a resolution
of 312.5 $\mu$m/pixel and a depth of field of 40 mm. Respiratory droplets
expelled by the subject were used as tracer particles, and no additional
seeding was introduced. The basic setup is depicted in Fig. 1(a). We note that
all the experiments were performed at an ambient temperature of 22 ∘C.

To perform precise particle tracking, the high-speed image sequences are first
pre-processed to minimise background/sensor noise and isolate head movement.
After that, PTV was performed on the image sequence using Lavision® Davis 8.4.
Here an initial Particle Image Velocimetry (PIV) pass on the image sequence
was employed to obtain a velocity field estimate and then individual droplets
were tracked using PTV algorithm [Cowen and Monismith, 1997]. Further details
on the experimental setup and processing can be found in Bahl et al. [2020].

(a)(b)

Figure 2:  (a) Flow rate and (b) bulk injection velocity profile computed from
the flow rates as a function of time. In the present study and Gupta et al.
[2009], the flow rate is measured experimentally by a spirometer, whereas the
inlet velocity profile used by Chong et al. [2021] is converted to flow rate
by assuming a circular mouth shape with a constant diameter $D=23$ mm. The
scatter of the peak location and magnitude from 25 subjects of Gupta et al.
[2009] is shown by the horizontal and vertical error bars at the peak flow
rate, and the variation of the cough duration is shown by the horizontal error
bar centred around $t=0.5$ s. The three black curves (rep1, rep2 and rep3) are
three repeating measurements of the same subject.

In order to draw a direct comparison of the flow rates in our experiments,
Fig. 2 shows the flow rate of coughs measured by the spirometer compared to
the representative profile obtained using the same method in the experiments
of Gupta et al. [2009], and the inlet velocity condition assumed in our
previous work [Chong et al., 2021]. Although the flow rate of the experiments
peaks within a shorter time after the start of the cough, the overall duration
and intensity of the coughs are very comparable with previous works, i.e. the
cough duration is $\mathcal{O}(1)$ s. Three repeating measurements are
conducted on the same subject, and an average of these measurements is used as
the injection flow rate in the simulation to be detailed in §3.

Fig. 3 shows the droplet trajectories of a single cough produced by the same
subject and captured by PTV for a duration of approximately 0.4 s. Note that
as limited by the intrusive nature of the spirometer measurement, the
trajectories presented here are not obtained simultaneously with the flow rate
date shown in Fig. 2. However, the close agreement between the three repeating
spirometer curves in Fig. 2 indicates that measurements performed at different
times by the two methods can still be reliably compared. The droplets are
emitted into various directions, and the ballistic trajectories form a cone
shape in the space with an apex angle around $90^{\circ}$, as delineated by
the red dashed lines. Such spreading phenomena can also be observed in the
droplet trajectories captured by Bourouiba et al. [2014] and the inline
holography in Shao et al. [2021]. In the atomisation process of a high-speed
gas jet shearing a liquid film, which is considered as a major mechanism of
droplet generation in respiratory tracts [Johnson and Morawska, 2009], there
is also experimental evidence that droplets are ejected at an angle of
$20^{\circ}-40^{\circ}$ relative to the jet flow direction [Descamps et al.,
2008]. Note that the spreading angle of the initial velocity field discussed
here is a near field quantity ($\mathcal{O}(D)$ where $D$ is the mouth
diameter), which is different from the far field ($>\mathcal{O}(5D)$)
spreading angle of the respiratory jet typically reported to be
$10^{\circ}-15^{\circ}$ with respect to the centre-line [Gupta et al., 2009,
Abkarian et al., 2020, Dudalski et al., 2020].

Figure 3:  Droplet trajectories of a single cough captured by PTV. The flow
direction is from left to right, and the mouth is located at $x=0$ and at
height $z\approx 0.18$ m. The region marked by a blue shade has a streamwise
extent of $0<x<2D$, where $D$ is the diameter of the mouth. Figure 4:  Number
of droplets within $x<2D$ from the mouth in each frame, normalised by
$N_{peak}$, magnitude of the first peak in the $N|_{x<2D}$ versus $t$ curve.
The red curve is from the present experimental (PTV) dataset, while the blue
curve is from the DNS results of Chong et al. [2021]. The black curve is from
the same simulation as the blue curve, but with droplet emission suppressed
after $t_{\text{cutoff}}=0.05$ s, which is marked by the vertical dashed line
in the figure.

By visually inspecting the raw PTV images, the droplet emission rate is found
to be the highest at the beginning of a cough, then it quickly diminishes long
before the flow injection velocity at the mouth reaches zero [de Silva et al.,
2021]. In other words, the droplet emission is concentrated at the onset of
the cough for a much shorter duration than that of the flow injection. To
provide further quantitative evidence regarding this observation, we utilise
the numerical dataset in Chong et al. [2021], and compare the number of
droplets in a region close to the mouth (as marked by the blue shaded box in
Fig. 3) before and after removing all droplets emitted after
$t_{\text{cutoff}}=0.05$ s. We choose to focus on this region mostly due to
the consideration that here the droplet number is predominantly dependent on
the emission and streamwise transportation, with negligible contribution from
other factors such as evaporation, deposition or droplets leaving the
measurement volume. Fig. 4 shows the number of droplets within $2D$ from the
mouth in the streamwise direction in each frame, where $D$ is the mouth
diameter. Each curve is normalised by the magnitude of the first peak emerging
around $t\approx 0.03$ s, thus eliminating the effect of the total number of
droplets on the comparison between cases. In the PTV dataset, the number of
droplets captured in a frame reaches its maximum between 0.02 s and 0.04 s,
before it approaches 0 at 0.15 s after the start of the cough, when the
injecting flow rate is still more than 60% of its peak value (see Fig. 2). The
DNS case of Chong et al. [2021] shows a similar trend as in the PTV data
between 0 and 0.1 s, but instead of reducing to 0, the droplet number then
quickly increases and the value almost triples at 0.4 s, indicating a very
dense droplet population close to the mouth. In the DNS data of Chong et al.
[2021], droplets were emitted at a constant rate over a duration of 1 s. At
the later stage of the cough when the injection velocity was low, the droplets
still entered the flow field at a constant rate, leading to the accumulation
of droplets close to the mouth. After removing all the droplets emitted after
$t_{\text{cutoff}}=0.05$ s (black curve in Fig. 4), better agreement with the
PTV result is observed.

Note that the PTV method in this study can confidently detect droplets with
$d\gtrsim 10\upmu$m although smaller droplets are also likely to be detected.
Therefore, the emission cutoff time discussed above is only strictly
applicable to droplets in this size range, and it is likely that smaller
aerosol particles formed deep in the lung [Johnson and Morawska, 2009] are
emitted throughout the entire cough when the air flow is present.

To summarise, two major features of a real human cough can be concluded from
the experiments: first, the initial velocities of droplets are not uni-
directional. Instead, droplets with a relatively high momentum spread into
various directions after exiting the mouth, as evidenced by their trajectories
which form a cone shape with an apex around $90^{\circ}$. Second, the majority
of droplets with $d\gtrsim 10\upmu$m are emitted between 0-0.05 s, which is
less than 10% of the duration of a cough based on a nonzero injection flow
rate. Incorporating these features in numerical simulations are therefore
mandatory in order to achieve a better representation of realistic human
respiratory events.

Figure 5:  Schematics of the flow inlet conditions in DNS. The left figure is
for the $\theta=0^{\circ}$ (uni-direction) case, and the right one is for
$\theta=45^{\circ}$.

##  3 Numerical investigations

In this section, we provide details of the numerical cases designed to study
the effect of the spread angle and droplet emission cutoff time.

We consider an incompressible fluid of gas phase, with both temperature and
vapour concentrations coupled to the velocity field by employing the
Boussinesq approximation. The gas phase is solved using DNS by a staggered
second-order accurate finite difference scheme and marched in time using a
fractional-step third-order Runge–Kutta approach [Verzicco and Orlandi, 1996,
van der Poel et al., 2015]. Buoyancy effect of the puff as a result of the
temperature and vapour mass fraction difference between the exhalation and the
ambient environment is considered in the simulation. For droplets, we apply
the spherical point-particle model, and consider the conservation of momentum
(Maxey-Riley equation), energy, and mass. During the evaporation of droplets,
the temperature and vapour mass fraction of the surrounding air are modified,
which leads to a change in the density of the local gas phase. The momentum
equation of the gas phase is not affected by the presence of droplets except
through the modification of density. We use the measurement of Xie et al.
[2009] for the initial droplet size distribution. For the domain size and
other details on the implementation, we refer readers to Chong et al. [2021]
and Ng et al. [2021].

The spread angle, $\theta$, is defined as half of the cone apex angle.
Therefore the $90^{\circ}$ apex angle observed in Fig. 3 is equivalent to
$\theta=45^{\circ}$. Two configurations are simulated in the present numerical
study, with $\theta=0^{\circ}$ and $\theta=45^{\circ}$, respectively.
Schematics of the inlet conditions can be found in Fig. 5. The initial
velocity vectors at the inlet cross section have the same magnitude within
each case, and they spread radially in the $\theta=45^{\circ}$ case. The
velocity distribution is then smoothed with an axis-symmetric Gaussian
function.

The mouth shape in both cases is assumed to be a circle with diameter $D=23$
mm, which remains constant in time. By integrating the streamwise velocity
across the mouth area, the flow rate of the exhalation can then be expressed
as

$\dot{m}_{0^{\circ}}=\rho\int_{0}^{D/2}2\pi rG(r)U_{0^{\circ}}\mathrm{d}r$
(1a) and $\dot{m}_{45^{\circ}}=\rho\int_{0}^{D/2}2\pi
rG(r)U_{45^{\circ}}\cos{\phi(r)}\mathrm{d}r,$ (1b)

where $\rho$ is the air density, $G(r)$ is the Gaussian function and $r$ is
the variable of integration with $r=0$ at the centre-line of the mouth.
$\phi(r)$ is the angle between the injection velocity vector and the
horizontal axis in the $\theta=45^{\circ}$ case, which can be related to $r$
as

$\tan\phi(r)=\frac{r}{D/2}.$ (2)

The injected flow rate and forward momentum govern the propagation of the
respiratory jet. For a given flow rate, the injection velocity magnitudes,
$U_{0^{\circ}}$ and $U_{45^{\circ}}$, can then be determined using (1b). Here
we assume that both cases have the same flow rate profile as the spirometer
result in the present study (see Fig. 2). The resulting centreline velocity of
the $\theta=45^{\circ}$ case is approximately 16% higher compared to the
$\theta=0^{\circ}$ case. The injected forward momentum of the former is then
7% higher than the latter.

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Exp.DNS$\theta=0^{\circ}$DNS$\theta=45^{\circ}$$t=0.01$s$t=0.02$s$t=0.05$s$t=0.20$s

Figure 6:  Droplet distribution of (a-d) the experiment, (e-h) DNS with a
spread angle of $\theta=0^{\circ}$ and (i-l) $\theta=45^{\circ}$. From left to
right, the four columns correspond to $t=0.01$ s, 0.02 s, 0.05 s, and 0.20 s.
The experimental case is rotated by $10^{\circ}$ counter-clockwise to align
the cough with the streamwise direction for better comparison, and the
original field-of-view is marked by the solid lines. Only droplets with an
initial diameter greater than $10\,\upmu$m are shown in the DNS results, and
only every the second droplet is shown in the figure. The grey dots in (h) and
(l) are droplets emitted between 0.05 s to 0.20 s.

A total number of 5000 droplets with the initial size distribution following
Xie et al. [2009] are introduced to the simulation domain at a constant rate
over 0.4 s. The total number of droplets is also chosen based on the same
study. Since the droplet distribution in the flow field is relatively sparse,
the droplet-to-droplet interaction is negligible. For the same reason the
modification to evaporation rates due to droplet clustering is also
insignificant. Therefore, it is possible to remove droplets emitted after a
certain time in post processing to study the effect of the droplet emission
cutoff time.

##  4 Results and discussion

The spatial distribution of the droplets in the DNS cases at a range of time
instances from 0.01 s to 0.2 s are shown in Fig. 6. Overall, the $45^{\circ}$
spread angle case qualitatively agrees with the experimental results: droplets
form a ‘cloud’ after entering the flow field, instead of concentrating along
the centre-line as in the $0^{\circ}$ case. Both the experiment and the
$\theta=45^{\circ}$ case have a large population of droplets above the centre-
line of the mouth ($z_{m}=0.184$ m), which is absent in the $\theta=0^{\circ}$
case. Furthermore, the droplet density near the mouth is much lower in Fig.
6(d) compared to earlier time instances, and such a decrease is also observed
in the simulation in Fig. 6(l) after excluding all the grey dots, which are
droplets emitted after 0.05 s. This observation confirms our previous
hypothesis that the rate of droplet emission during a cough varies with time,
and large droplets are released only at the early stage of the cough. In the
literature, the common practice is to assume that the droplets are uniformly
distributed in the discharging fluid [Bourouiba et al., 2014, Wei and Li,
2017] or emitted at a constant rate over the entire duration of the cough
[Chong et al., 2021, Ng et al., 2021, Fabregat et al., 2021]. Incorporating
the short droplet emission time in the model can significantly improve the
agreement between the simulation and the experiment.

(a)(b)(c)Exp.DNS $\theta=0^{\circ}$DNS $\theta=45^{\circ}$

Figure 7:  Histogram of droplet distribution at $t=0.20$ s of (a) the
experiment, (b) DNS with a spread angle of $\theta=0^{\circ}$, and (c)
$\theta=45^{\circ}$. Only droplets emitted between 0 and 0.05 s are included
in the histogram.

The contribution of the spread angle is further analysed in Fig. 7, a 2D
histogram of the droplet spatial distribution at $t=0.20$ s for the experiment
and the $\theta=0^{\circ}$ and $\theta=45^{\circ}$ DNS cases. The short
droplet emission time is taken into account, thus only droplets emitted before
$t=0.05$ s are included when computing the histogram. The histogram peaks at
$x\approx 0.1$ m and $z\approx 0.19$ m in the experiment (Fig. 7a), with
droplets scattered both above and below $z_{m}$. These major features are
successfully replicated in the $\theta=45^{\circ}$ case (Fig. 7c). In the
$\theta=0^{\circ}$ case (Fig. 7b), however, very few droplets can be found
above the mouth location $z_{m}$. The build-up of droplets near the mouth is
absent, and the droplet distribution is almost uniform in the streamwise
direction.

(a)(b)(c)(d)(e)(f)LargeMediumSmall

Figure 8:  Droplet trajectories at $t=0-0.2$ s for DNS cases
$\theta=0^{\circ}$ (left column) and $\theta=45^{\circ}$ (right column). The
top, middle and bottom row correspond to large, medium and small droplets,
with an initial diameter of $d_{0}>500\,\upmu$m, $d_{0}\approx 90\,\upmu$m and
$d_{0}<40\,\upmu$m. All the trajectories shown in the figure are from droplets
emitted within $t=0-0.05$ s.

Droplet trajectories of cases $\theta=0^{\circ}$ and $\theta=45^{\circ}$ are
compared side-by-side in Fig. 8. As expected, droplets with an initial
diameter $d_{0}>500\,\upmu$m exhibit predominately ballistic behaviour, and a
non-zero spread angle introduces a larger scatter in their trajectories. For
the medium-sized droplets, the streamwise deceleration is complete before they
reach the floor. Comparing Fig. 8(c) and (d), one would notice that droplets
in the $\theta=45^{\circ}$ case settle closer to the mouth. Instead of carried
downstream by the jet with a similar velocity, droplets emitted with a large
spread angle penetrate deeper sideways into the still ambient fluid, so they
experience larger relative velocity (therefore higher aerodynamic drag and
stronger deceleration) compared to the $\theta=0^{\circ}$ case. The reduction
in the settling distance is not limited to droplets emitted with a high wall-
normal velocity component: it is also seen in those with small or zero wall-
normal component, as a result of the lower streamwise velocity in the widened
jet.

(a)(b)

Figure 9:  Top-view of the droplet deposition location on the ground for (a)
$\theta=0^{\circ}$ and (b) $\theta=45^{\circ}$ cases. The colour of each dot
is determined by the droplet initial diameter $d_{0}$, which changes from blue
to yellow with increasing $d_{0}$. The solid lines are the convex hull of the
deposition locations of droplets with the same $d_{0}$.

The settling behaviour of droplets is further explored by examining the
distribution of deposition locations in the ground plane of $z=0$ in Fig. 9.
The location where a droplet reaches the ground is represented by a dot in the
$x-y$ plane, with the colour determined by its initial diameter $d_{0}$. The
contaminated region on the ground for each size class is shown by the convex
hull. Although the mouth is at $z\approx 0.18$ m in the simulation domain, for
smaller droplets ($d_{0}\lesssim 100\,\upmu$m), the streamwise velocity has
already decreased to 0 before reaching the $z=0$ plane. Therefore, the
deposition pattern reported in Fig. 9 would still be applicable when the mouth
is at $z\approx 1.5$ m above the ground, which is the case of a standing
person.

Figure 10:  Number percentages of deposited droplets in four spanwise bins at
$x\approx 0.2$ m. The $x$-axis labels indicate the edges of each bin. The
light blue bars are data from Xie et al. [2009], while the blue and black
lines represent cases $\theta=0^{\circ}$ and $\theta=45^{\circ}$,
respectively.

Comparing Fig. 9(a) and (b), it is apparent that the deposition location
scatters over a wider range in the spanwise direction for $\theta=45^{\circ}$,
while it is mostly concentrated along the centre-line in the
$\theta=0^{\circ}$ case. This is a progress towards a more realistic cough, as
evidenced by the improved agreement with the data of human subjects [Xie et
al., 2009] (Fig. 10). In Fig. 9(b), the spanwise width of the convex hull
increases with increasing $d_{0}$, presumably because larger droplets have a
higher initial momentum and they can continue for a longer distance along the
direction of their initial velocity. The streamwise extent of the convex hull
is shorter in the $\theta=45^{\circ}$ case compared to the $\theta=0^{\circ}$
case, indicating that droplets (especially those with $d_{0}\lesssim
100\,\upmu$m) have a strong tendency to deposit closer to the mouth.

We note that the initial size distribution of droplets exhaled during
respiratory events is presently actively investigated. The simulation results
can be affected by the size distribution: for instance, if we use the
distribution of Duguid [1946], which has more small ($\mathcal{O}(10)$
$\upmu$m) droplets, a higher number of trajectories can be expected in Fig.
8(e-f). However, here our focus is on capturing the dominant effects of an
actual cough.

The scalar fields of relative humidity ($RH$) are shown in Fig. 11. At
$t_{1}=0.15$ s, despite the higher injected streamwise momentum, the
$\theta=45^{\circ}$ case has a shorter and wider turbulent jet compared to the
$\theta=0^{\circ}$ case. The flow also appears more turbulent close to the
mouth ($x\lesssim 0.2$ m) in the former. The cone angle $\alpha$ of the jet,
which can be determined from $\tan\alpha=R_{\text{jet}}(x)/x$, where
$R_{\text{jet}}(x)$ is the jet radius at the streamwise location $x$, is very
different from the spread angle $\theta$. Comparing the white reference lines
in the figure with the scalar field of the jet, it is shown that the jet cone
angle is approximately $10^{\circ}$ in the $\theta=45^{\circ}$ case, while it
is only slightly smaller in the $\theta=0^{\circ}$ case. Therefore, a
$45^{\circ}$ change in the spread angle $\theta$ only results in a less than
$5^{\circ}$ change in the cone angle $\alpha$.

(a)(b)(c)(d) Figure 11:  Relative humidity in the spanwise midplane ($y=0$)
for (a,c) $\theta=0^{\circ}$ and (b,d) $\theta=45^{\circ}$ cases. The top row
is at $t_{1}=0.15$ s and the bottom row is $t_{2}=0.6$ s. The white lines in
each plot form a cone angle of $\alpha=10^{\circ}$.

At $t_{2}=0.6$ s (Fig. 11c and d), a vortex ring separated from the main jet
is observed in both cases. These vortices are formed at the front of the jet,
which can be seen attached to the jet at $t=t_{1}$. As the injection velocity
decreases and the jet front decelerates, the self-induced velocity of the
vortex ring eventually exceeds the velocity of the jet front, then the vortex
ring detaches from the jet. Such vortex rings have also been observed in the
large-eddy simulation results by Liu et al. [2021], and their formation,
strength and velocity are shown to be very sensitive to the initial conditions
of the simulation.

##  5 Conclusions and outlook

In this study, we present a unique comparison of both experimental and
numerical datasets on a human cough with matched conditions. Specifically, we
implemented two important features of the human cough with insight gained from
the comparison with the experimental data: the injection of the coughing flow
which is shown to be better described with a spread angle of $45^{\circ}$, and
the emission of large droplets only occurs for a short period of time
($\approx 0.05$ s) at the beginning of a cough, whose length is much longer
($\approx 1$ s). The contribution of these features to the simulation result
is examined by comparing two DNS cases with the conventional and with the
modified inlet conditions based on the aforementioned experimental
observations from a human cough. Overall, a significant improvement in the
agreement with the experimental data is observed after implementing the
modifications. We further demonstrate that by considering the $45^{\circ}$
initial spread angle, the simulated jet appears thicker and shorter. Droplets
deposit on the ground closer to the mouth in the streamwise direction, but
scatter over a wider range in the spanwise direction, which is consistent with
the experimental observations. The aim of future work must be to extend the
present comparative approach between experiments and numerical simulations to
other respiratory events such as singing and in particular speaking, which
accumulatively plays an even more important role in the aerosol release
[Abkarian et al., 2020, Yang et al., 2020].

## Acknowledgements

This work was funded by the Netherlands Organisation for Health Research and
Development (ZonMW), project number 10430012010022: “Measuring, understanding
& reducing respiratory droplet spreading”, the ERC Advanced Grant DDD, Number
740479, Foundation for Fundamental Research on Matter with Project No.
16DDS001, which is financially supported by the Netherlands Organisation for
Scientific Research (NWO). C de Silva acknowledges ARC LIEF funding Grant,
Number LE200100042, which supported the experimental work. The use of the
national computer facilities in this research was subsidized by NWO Domain
Science. We also acknowledge PRACE for awarding us access to MareNostrum in
Spain at the Barcelona Computing Center (BSC) under the project 2020235589 and
Irene at Très Grand Centre de calcul du CEA (TGCC) under PRACE project
2019215098. KL Chong is also supported by Shanghai Science and Technology
Program under the project no. 19JC1412802.

## Data availability

As part of the ZonMW funding agreement, data of the DNS cases are available as
open access, which can be accessed via the link
https://doi.org/10.4121/17099468. Larger files such as the scalar fields will
be available upon request. The experimental data cannot be directly shared due
to ethics limitations.

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