The pioneering work of Taylor on the turbulent dispersion of aerosols is one century old and provides an interesting way to introduce both diffusive processes and turbulence at an undergraduate level. Low mass particles transported by a turbulent flow exhibit a Brownian-like motion over time scales larger than the velocity correlation time. Aerosols and gases are, therefore, subjected to an effective turbulent diffusion at large length scales. However, the case of a source of pollutant much smaller than the integral scale is not completely understood. Here, we present experimental results obtained by undergraduate students in the context of the COVID-19 pandemic. The dispersion of a fog of oil droplets by a turbulent flow is studied in a wind tunnel designed for pedagogical purposes. It shows a ballistic-like regime at short distance, followed by Taylor's diffusive-like regime, suggesting that scale-free diffusion by the turbulent cascade process is bypassed. Measurements show that the dispersion of CO_{2} emitted when breathing in a natural, indoor air flow is not isotropic but rather along the flow axis. The transverse spread is ballistic-like, leading to the concentration decaying as the inverse-squared distance to the mouth. The experiment helps students understand the role of fluctuations in diffusive processes and in turbulence. A Langevin equation governing aerosol dispersion based on a single correlation time allows us to model the airborne transmission risk of pathogens, indoors and outdoors. The results obtained in this study have been used to provide public health policy recommendations to prevent transmission in shopping malls.

## I. INTRODUCTION

The challenge of offering experimental physics courses online during the COVID-19 pandemic motivated the following question: can students learn experimental techniques during lockdowns while simultaneously working to reduce viral transmission risk indoors? A group of students studying experimental physics at our university worked on several projects to reduce the risk of airborne transmission of SARS-CoV-2, including the filtration efficiency of face masks, the calibration of CO_{2} sensors using candles or dry ice in a closed transparent box, and the analysis of epidemic data. We report here the work on a project to understand the dispersion of viral particles both indoors and outdoors. This project has required a detailed understanding of both diffusion processes and turbulent flows.

However, turbulence is not an easy subject to teach. At the undergraduate level, topics in fluid mechanics mainly include hydrostatics, potential laminar flows, and viscous flows. The concept of turbulence appears when teaching the drag force exerted on a moving spherical solid as a function of the Reynolds number (see, e.g., Ref. 1). In this approach, one of the fundamental aspects of turbulence, namely, space and time fluctuations, is bypassed. Here, we show that turbulence can be taught by considering the dispersion of particles in a regime where concentration is low enough that it acts as a passive scalar field; that is, the concentration has no dynamical effect on the flow. We treat this dispersion theoretically in Sec. II and also present very simple experiments. In Sec. III, we make use of controlled experiments performed both in a small wind tunnel used for pedagogical purposes, using a fog of oil droplets, and in the large corridors of two shopping malls under various ventilation conditions using CO_{2}. We show that the concentration decays faster than expected by the turbulent dispersion theory of Taylor,^{2} following instead a ballistic law. Indeed, introducing a single velocity decorrelation time provides a perfect fit to the data. This suggests, as shown previously by Villermaux *et al.*,^{3,4} that concentration fluctuations bypass the turbulent cascade: diffusion does not follow a continuous process starting from the injection scale and down to the dissipation scale.^{5} The Kolmogorov scaling law is not observed. The work presented here can be used to introduce students to the turbulent dispersion not only of pollutant molecules but also of aerosols. We finally discuss the relevance of the results to the airborne transmission of SARS-CoV-2, which has motivated this research. SARS-CoV-2 is mainly transmitted through the airborne route:^{6} infection happens when a susceptible person inhales viral particles emitted by another infected person. In between their respective respiratory tracts, the air carrying viral particles is gradually diluted by turbulent dispersion. Near an emitter, the concentration of viral particles is, therefore, higher than far away, which has implications on the airborne transmission risk.

## II. TURBULENT DISPERSION OF PARTICLES

### A. Brownian motion

*d*, mass

*m*, and density

*ρ*in a gas of viscosity

_{p}*η*without turbulence. The force exerted by the gas on the particle can be decomposed into two parts: the average hydrodynamic Stokes

^{1}force $ \u2212 3 \pi \eta d v$ and a noise

^{7}$ \Gamma ( t )$, which describes the thermal fluctuations around this average. The particle dynamics is described by the Langevin equation

*τ*, which is the particle velocity response time to a force. $ \Gamma ( t )$ is only known statistically as it results from the random collisions of gas molecules with the particle, which are uncorrelated. Therefore, we define the ensemble average $ \u2009 \xb7 \u2009 \xaf$ as the average over many realizations of the motion with different values of the noise. The noise is Gaussian, has zero mean, and is

_{S}*δ*-correlated in time

^{7}

*δ*stands for the Dirac delta function.

*τ*. Particle inertia, which drives the homogeneous solution, can, therefore, be neglected in the $ t \u226b \tau S$ limit. In this overdamped case, the particle velocity is directly proportional to the noise force: $ v ( t ) = \tau S \Gamma ( t ) / m$. The noise does not induce any average motion: $ v ( t ) \xaf = 0$. However, as the velocities are

_{S}*δ*-correlated in time, the particle diffuses around its initial position $ x ( t = 0 ) = 0$ with a mean squared displacement $ \sigma R 2 = x 2 ( t ) \xaf$ given by Taylor's theorem

^{2,8,9}

*σ*typically increases as $ t$, which is a key characteristic of diffusion.

_{R}### B. Turbulent diffusion

*x*axis in cylindrical coordinates $ ( r , \theta , x )$.

*L*denotes the typical large-scale length, which is in our case set by the transverse dimensions of the space. The Reynolds number $ Re$ compares inertia with viscous effects

*ν*being the kinematic viscosity of air. When $ Re$ is large enough, typically above 10

^{3}, the fluid motion given by the Eulerian velocity field $ u ( r , t )$ becomes unsteady, chaotic, and irregular. We, therefore, treat this field statistically by using the Reynolds decomposition: we split

**u**into its ensemble average $ u \xaf$, which is a drift velocity, and its fluctuating part with zero mean $ u \u2032$, i.e., $ u = u \xaf + u \u2032$.

^{10}Likewise, we decompose the particle velocities $v$ into $ v \xaf$ and $ v \u2032$. The typical particle fluctuating velocity

*σ*is the root mean squared fluctuating velocity $ \sigma V 2 \u2261 v \u2032 2 \xaf$.

_{V}^{2}published one century ago that turbulent dispersion, induced by the random velocity fluctuations, is diffusive. The particle dynamical equation reads

*τ*, which is the case if the particles are light enough. Particles, therefore, move with the fluid: $ v \xaf = u \xaf$ and $ v \u2032 = u \u2032$ so that their velocity correlation functions are the same: $ v \u2032 ( t ) v \u2032 ( t + \tau ) \xaf = u \u2032 ( t ) u \u2032 ( t + \tau ) \xaf$. As the fluctuating velocity $ u \u2032$ acts as a noise, Taylor's picture of turbulent diffusion is a Brownian-like motion analogous to molecular diffusion. The turbulent kinetic energy plays the role of temperature.

_{S}*σ*with zero mean. As the individual particle probability distributions are identically distributed, the probability of finding a particle at transverse radial position

_{R}*r*will be Gaussian, according to the central limit theorem, i.e., proportional to $ exp \u2009 ( \u2212 r 2 / ( 2 \sigma R 2 ) )$. The average concentration $ C \xaf$ is proportional to this probability. The prefactor is set by conservation of mass as the total concentration along the transverse section must be equal to the injection rate $ m \u0307 / v \xaf$

^{5,11,12}The velocity of a transported particle is, therefore, correlated as well, which leads to an anomalous diffusion at short distances. Because the particles have negligible inertia, their velocities are the fluid velocities: velocity correlation results from the underlying flow structure driving their motion rather than from the particle dynamics. On a phenomenological ground, the simplest hypothesis would be to assume that the velocity fluctuation is a low-pass filtered noise associated with a typical relaxation time

*T*. We assume that an elementary volume of fluid follows a Langevin equation of the type of Eq. (1) driven by a white noise with a linear friction term and including inertia. The particle velocity correlation, which is equal to the fluid velocity correlation, is not a Dirac

*δ*as previously but decays exponentially

^{7–9,13}

*T*is called the Lagrangian integral time scale, defined in the general case as

^{8,9}

*r*=

*0) is, therefore, expected to decay as $ 1 / x$. Conversely, at short distances ( $ x \u226a v \xaf T$), the formula predicts a ballistic-like regime of the form*

## III. EXPERIMENTS

### A. Field experimental setup

We performed “field” experiments in the large corridors of two commercial malls and of our university during lockdown, while they were empty. The experimental setup is schematized in Fig. 1. A controlled CO_{2} source of constant mass rate $ m \u0307$ was obtained by sublimating dry ice sticks in an open cylindrical container of diameter 20 cm heated by a power controlled hot plate. The source was positioned at height 1.1 m above the ground. The imposed sublimation rate, measured using a scale, was equal to $ m \u0307 = 1.5 \u2009 g / s$. This rate was chosen to measure CO_{2} concentrations with high relative accuracy without saturating the sensors.

Our initial motivation was to characterize the dispersion of breath in public spaces. The source used generated CO_{2} at about 150 times the exhalation rate of an adult at rest. This increased generation rate does not change the dispersion equations as the measured CO_{2} concentration is still very small in the $ 10 \u2212 5$− $ 10 \u2212 4$ volume fraction range. The experiment had been designed believing that the turbulent dispersion would be statistically isotropic, as expected for an effective diffusion caused by large scale incoherent turbulent motion.^{14} Instead, our preliminary observations using a source of micron-sized glycerol/water droplets showed that in most large public spaces, there are air drafts causing bulk horizontal transport and biased dispersion. After identification of the mean air flow direction, non-dispersive infrared (NDIR) CO_{2} sensors were placed downstream from the source. They recorded the CO_{2} concentration $ C \xaf$ over the duration of each experiment around 30 min. The initial and final values of $ C \xaf$ were used to determine the background CO_{2} level *C _{e}*.

After an initial short transient time, a concentration field in a statistically steady state was established. CO_{2} concentration in excess was measured in eight different locations. The draft wind velocity was measured using a hot-wire anemometer. It ranged from 0.1 to $ 2 \u2009 m / s$ depending on the location. The flow Reynolds number $ Re = u \xaf L / \nu $ was between 10^{5} and 10^{6} with $ L \u2243 1 \u2009 m$, the typical corridor scale. Measurements were done with different ventilation flow rates and recycled air fractions; the use of fire safety ventilation, namely, mechanical smoke extractors and smoke vents, was tested when available. Measurements were performed with entrance doors both open and closed as open doors create large drafts in some locations. Control sensors (not shown in Fig. 1) placed immediately upstream to the left and to the right of the source showed no concentration increase: convection dominated over turbulent diffusion.

### B. Field results

_{2}. In Fig. 2, the CO

_{2}concentration is rescaled by $ m \u0307 / \rho CO 2 v \xaf$ and plotted as a function of space

*x*. The reasonable collapse between data shows that the CO

_{2}concentration is proportional to the inverse wind velocity $ v \xaf$. The fraction of fresh air injected in the ventilation system had no effect on the measured concentrations: in practice, only the local airflow disperses CO

_{2}. In other words, the airflow induced to renew air is negligible in comparison to natural air drafts.

*α*is the dispersion cone angle, determined by the turbulent fluctuation rate $ \sigma V / v \xaf$. We find $ a = 0.35 \u2009 m$, which is close to the actual diameter of the dry ice container and $ \alpha = 0.10$. We have also included in Fig. 2 the CO

_{2}concentration in the wake of a volunteer breathing through the mouth in a fan-induced wind of velocity $ v \xaf = 0.3 \u2009 m / s$. We find $ a \u2248 0.27 \u2009 m$ and a slightly higher fluctuation rate $ \alpha = 0.14$ in that case. The fluctuation rate depends on the processes creating the mean flow (fans, natural indoor air drafts, wind outdoors) and the geometry, as obstacles upstream destabilize the flow and create larger fluctuations.

### C. Wind tunnel setup

In order to better characterize the turbulent transport at play, we studied after lockdown the dispersion of a fog composed of micrometer-sized oil droplets in a small wind tunnel used for experimental physics student projects, schematized in Fig. 3. The $ 120 \xd7 23 \xd7 23 \u2009 cm 3$ test section is illuminated from above by a LED array with a diffuser. A turbulence generating grid (3 × 3 grid of $ 7 \xd7 7 \u2009 cm 2$ separated by $ 1 \u2009 cm$ wide bars) is inserted immediately after the air inlet contraction section. $ u \xaf$ is measured with a hot-wire anemometer (testo 405i). We estimate the flow Reynolds number $ u \xaf L / \nu $ with *L* being the grid mesh size to be between 10^{5} and 10^{6}—as in the field experiments. Oil vapor is injected at a controlled rate through a $ 6 \u2009 mm$ nozzle heated by a resistor at a controlled power. A fog of droplets nucleates at few millimeters downstream. The fog is made dilute enough so that double scattering is negligible, and the light intensity scattered is proportional to the local drop concentration. We qualitatively check the dilution by putting the light behind the tunnel rather than above. High resolution pictures with a 1 or $ 2 \u2009 s$ exposure time are taken with a Digital Single-Lens Reflex camera (Nikon D800). A series of ten pictures under the same conditions is averaged to achieve a satisfying statistical convergence. The mean green channel from the pictures in raw format is used to measure the light intensity. The image of the tunnel without fog is subtracted to remove residual background light. The light intensity profile $ I 0 ( x )$ is then calibrated by placing a diffusive black sheet inside the tunnel. $ I 0 ( x )$ is flat along the tunnel axis, except at the start and the end of the tunnel where it falls off due to greater distance to the illumination source.

*z*-axis of the scattered intensity. The concentration profile is expected to depend on the distance to the central axis in a Gaussian way: $ \u223c \u2009 exp \u2009 ( \u2212 ( ( y \u2212 y 0 ) 2 + ( z \u2212 z 0 ) 2 ) / 2 \sigma R 2 )$. Integrating it over

*z*, the image intensity field $ I ( x , y )$ remains Gaussian along

*y*

### D. Wind tunnel results

Transverse intensity profiles at regularly spaced distances from the fog injection nozzle are shown in Fig. 4. As the droplets are formed by vaporizing oil, the fog is hot and slightly buoyant so the centerline $ y 0 ( x )$ gets slightly shifted upwards. The centerline slope, which is the transverse-to-longitudinal mean particle velocity ratio, is at most $ 10 \u2212 2$, allowing any additional effects of this heating to be ignored. The profiles are nicely fitted by a Gaussian, which leads to the measurement of the three parameters of Eq. (15): the concentration on the axis, $C$, the radius *σ _{R}*, and the centerline $ y 0 ( x )$. Assuming that $ v \xaf$ is constant along the flow axis, mass conservation imposes that $ C ( x ) \sigma R 2 ( x ) v \xaf$ must be constant proportional to the rate of emission by the source. Figure 5 shows typical profiles $ C ( x ) \sigma R 2 ( x )$, which are indeed flat outside the injection zone. It means that the droplets, once nucleated, do not change much and keep the same light scattering properties; this validates the use of this measurement technique.

Figure 6 shows various profiles of the dispersion radius $ \sigma R ( x )$. The dispersion of aerosols is significantly larger with a grid generating turbulence than without. In all cases, Eq. (10) provides an excellent fit to the data within the scatter. It constitutes a very striking result: a model based on a single correlation timescale is sufficient to adequately represent all the data. At short distance $ x < v \xaf T$ from the source, the dispersion takes place in a cone: *σ _{R}* is linear in

*x*. This controversial regime has been explored in a series of recent papers.

^{3,4,15–20}At large distance $ x > v \xaf T$, conversely, the diffusion regime predicted by Taylor with a constant diffusivity is recovered, in which the radius increases as the square root of time, i.e.,

*x*here.

From each profile $ \sigma R ( x )$, two quantities are, therefore, extracted: the turbulent fluctuation rate $ \sigma V / v \xaf$ and the cross-over length $ v \xaf T$, which is the cross-over distance between the ballistic and diffusive regimes. These quantities are plotted in Fig. 7 as a function of the mean flow velocity $ u \xaf$. The fluctuation rate $ \sigma V / v \xaf$ is independent of the wind velocity $ u \xaf$, as expected, and equals to $ \u223c 7.5 %$. The cross-over length $ v \xaf T$ is very small when no grid is introduced at the entrance of the wind tunnel. It is much larger with a turbulence generating grid. It tends to a nonzero value as $ u \xaf \u2192 0$ and increases with the fluid velocity $ u \xaf$. The Lagrangian correlation length $ \sigma V T$ increases from $ 0.7$ to $ 2.5 \u2009 cm$ in the range of velocities explored, which is significantly smaller than the grid mesh size: correlations come from intermediate scale flow structures. The slope of the curve gives a characteristic time around $ 9 \u2009 ms$, which is much larger than the Stokes time of oil droplets, around $ 0.3 \u2009 ms$. As shown in the discussion of Eq. (6), this is consistent with the dynamics of particles with negligible inertia. To the best of our knowledge, there is no simple mechanistic explanation to the observed dependence of *T* with $ u \xaf$. This dependence is a near-field effect of the grid, which is why we have used a single decorrelation time, independent of $ u \xaf$, to fit the field data of Fig. 2.

## IV. RELEVANCE FOR THE ASSESSMENT OF THE AIRBORNE TRANSMISSION RISK OF SARS-CoV-2

SARS-CoV-2 is transmitted in the airborne route by respiratory aerosols: droplets of mucus, ranging between 200 nm and $ 200 \u2009 \mu m$, which can contain viral particles.^{6,21,22} These droplets are produced in the respiratory tract during normal respiratory activity by various fluid instabilities.^{23–25} The air exhaled by an infected person contains active SARS-CoV-2 viral particles, which may start an infection when one of the virions is deposited on an ACE2 receptor of the epithelium and successfully bypasses the immune response. Infection may also start if virions deposit on the eyes. The probability of infection increases with the intake viral dose, defined as the amount of virus particles inhaled by a person, cumulated over time. It, therefore, increases with the exposure time and with the concentration of viral particles in the air.^{26,27}

The viral transmission risk by a given infected patient is, therefore, determined by the concentration in viral particles in the air inhaled by another susceptible person. This concentration is given by the rate of emission of viral particles in exhaled air and by the dilution of the air between exhalation and inhalation. In closed rooms, viral particles accumulate, and their concentration eventually reaches a steady state between exhalation and ventilation, which is the replacement of contaminated air by fresh air. This explains the large transmission clusters in poorly ventilated indoor spaces.^{6} The average transmission risk, uniform in space, can be determined by assuming that viral particles are well-mixed by turbulence. In that case, dilution is due to the natural or forced air exchanges with outside air.^{26,27} Near an infected person, however, the concentration is much higher and decays with distance, and the dilution from the exhalation concentration of viral particles to the average concentration of viral particles is controlled by the local airflow. The total transmission risk is, therefore, the sum of an average risk, controlled by ventilation, and an additional risk in the dispersion cone of exhalations. Outdoors, in the absence of homogenizing flow, this short-range risk is the only transmission risk. Respiratory activity, such as unmasked coughs and sneezes, can create large airflows that dominate over the pre-existing flow patterns. In that case, air is exhaled in a buoyant jet that gets gradually mixed at its boundaries as it travels.^{23,25} Face masks strongly alter the airflow by blocking the jet; small jets can remain along the nose and the cheeks if the mask is not properly fitted.^{28,29} The stopping distance of a masked respiratory puff is around^{29} $ 20 \u2009 cm$; the more filtering masks make the puff stop earlier. Beyond this very near field distance, dilution is fully controlled by the ambient airflow, which is what we have extensively developed here.

Our analysis applies to any public space where either face mask wearing is mandatory and there is a horizontal air draft or the air velocity is much larger than the exhalation velocity. We have found that in large indoor corridors, the airflow is a horizontal draft with $ u \xaf = 0.1 \u2212 1 \u2009 m / s$. Outdoors, the wind is also horizontal with larger velocities. Understanding the dilution in these public spaces allows us to assess their transmission risk, given biological data on pathogen contagiousness.^{26,27} This is needed to ground infection prevention policies on a solid physical basis, as some (such as the “6 ft rule”) are not (see Refs. 22 and 30 for a historical perspective and critical discussion). Exhaled CO_{2} is a commonly used risk proxy in well-mixed rooms;^{26,27} its spatial decay allows for a simple translation of the short-range risk into an equivalent long-range risk. For the gentle draft of shopping mall corridors $ u \xaf \u2243 0.2 \u2009 m / s$, the increase in risk at $ 6 \u2009 ft$ corresponds to a CO_{2} concentration excess of $ 50 \u2009 ppm$.

## V. CONCLUSIONS

In this article, we have investigated experimentally and theoretically the dispersion of a passive scalar (the concentration of particles) by a turbulent flow. This constitutes an accessible problem to teach turbulence at an undergraduate level, combining possible experiments and theory. Using any source of fog, a fan and a camera, the Reynolds decomposition between average and fluctuating quantities can be illustrated qualitatively and quantitatively. The derivations based on physical reasoning and dimensional analysis remain linear and much simpler than attempting to solve the Navier–Stokes equation. While the Reynolds number dependence of the drag force exerted on a solid is useful to introduce dimensionless numbers, asymptotic regimes, viscosity, and inertial effects, as it is usually done in fluid dynamics textbooks (see, e.g., Ref. 1), the diffusion of a passive scalar provides a way to understand the role of fluctuations to enhance turbulent mixing. The applications to the dispersion of odors and animal olfactory search,^{31} the dispersion of pollutants, or airborne viral transmission are immediately appealing to students.

## ACKNOWLEDGMENTS

Unibail-Rodamco-Westfield has funded this work under CNRS Contract No. 217977 and provided access to Forum des Halles and Carré Sénart as well as technical assistance. This article involves seven undergraduate students who have worked on the problem in the context of their final-year experimental physics courses (“Phy Exp ”) at the Université de Paris Cité. The authors thank the “Phy Exp ” team and, in particular, the lathe-mill operator, Wladimir Toutain, and the technician, Thibaut Fraval De Coatparquet, for their assistance. The authors thank Alessandra Lanotte and Luca Biferale for fruitful discussions. The authors thank the referees for their insightful comments, which have helped them to improve the manuscript.

## AUTHOR DECLARATIONS

### Conflict of Interest

This work was funded by Unibail-Rodamco-Westfield on behalf of Conseil National des Centres Commerciaux (CNCC), who asked the authors to make recommendations for a health protocol aiming to reduce and quantify the transmission risk in shopping centers. The conclusions of the present article are, therefore, of direct interest for the funding company. The authors declare no financial competing interest. The funding company had no such involvement in study design, in the collection, analysis, and interpretation of data, nor in the writing of the article. The authors had the full responsibility in the decision to submit it for publication.

## REFERENCES

*An Introduction to Fluid Dynamics*

*The Fokker-Planck Equation: Methods of Solution and Applications*

*Turbulence in Fluids*

*illusio*, and playing the scientific game: A Bourdieusian analysis of infection control science in the COVID-19 pandemic