eﬀect of droplet emission duration and spread angle Mogeng Liab Kai Leong Chongcab Chong Shen Ngab Prateek Bahld

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arXiv:2210.00534v1 [physics.flu-dyn] 2 Oct 2022

Towards realistic simulations of human cough:

eﬀect of droplet emission duration and spread angle

Mogeng Lia,b, Kai Leong Chongc,a,b, Chong Shen Nga,b, Prateek Bahld,

Charitha M. de Silvad, Roberto Verziccoa,b,e,f, Con Dooland, C. Raina

MacIntyreg,h, Detlef Lohsea,b,∗

aPhysics 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

bMax Planck Institute for Dynamics and Self-Organisation, 37077 G¨ottingen, Germany

cShanghai 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

dSchool of Mechanical and Manufacturing Engineering, UNSW Sydney, Kensington, NSW,

2052, Australia

eDipartimento di Ingegneria Industriale, University of Rome ‘Tor Vergata’, Roma 00133,

Italy

fGran Sasso Science Institute - Viale F. Crispi, 7 67100 L’Aquila, Italy

gBiosecurity Program, The Kirby Institute, UNSW Sydney, Kensington, NSW, 2052,

Australia

hCollege 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 nu-

merical 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.

∗Corresponding author

Email address: d.lohse@utwente.nl (Detlef Lohse)

Preprint submitted to International Journal of Multiphase Flow October 4, 2022

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 ini-

tial diameter &10µm 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

2010 MSC: 76T10, 76F65

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 dis-

eases 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 infec-

tious 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’. Indi-

viduals can transfer between compartments based on ordinary diﬀerential equa-

tions while the total population remains constant. The infection rate of suscep-

tible individuals can be improved with more physical insight via dose-response

models (Brouwer et al., 2017), or the Wells–Riley model in particular for dis-

eases 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 in-

fection, 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, in-

halation, 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 dis-

ease 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 incorpo-

rate the temporal and spatial inhomogeneity in the model (Mittal et al., 2020a;

Yang et al., 2020). A sophisticated software tool has been developed to esti-

mate 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 measure-

ment 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 res-

piratory 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 inves-

tigated 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 (.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 tur-

bulent 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 ∼ O(1000µm) deposit on the ground in a near ballistic motion,

while small droplets with a diameter ∼ O(10µm) are trapped in the humid puﬀ

and have prolonged lifetimes compared to the prediction of Wells (1934). Con-

densation 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 nu-

merical 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 condi-

tions. 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 nu-

merical 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 vol-

umetric 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◦from the image plane, to capture high-speed imaging data of the scattered

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arXiv:2210.00534v1[physics.flu-dyn]2Oct2022Towardsrealisticsimulationsofhumancough:eﬀectofdropletemissiondurationandspreadangleMogengLia,b,KaiLeongChongc,a,b,ChongShenNga,b,PrateekBahld,CharithaM.deSilvad,RobertoVerziccoa,b,e,f,ConDooland,C.RainaMacIntyreg,h,DetlefLohsea,b,∗aPhysicsofFluidsGroup,Max...

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eﬀect of droplet emission duration and spread angle Mogeng Liab Kai Leong Chongcab Chong Shen Ngab Prateek Bahld

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