2. Large-eddy simulations of respiratory flows

In order to provide the reader with an overview of the model and method used in this paper, the numerical simulations are briefly presented in this section. A thorough description is available in the Supplementary Materials.

2.2. Computational domain and meshing

The computational domain is a three-dimensional parallelepiped of dimensions 4 m (width) x 4 m (height) x 5 m (length) in the directions $x$ , $y$ and $z$ , respectively. A human manikin with mouth and nose openings is included in the domain. The face of the manikin is symmetric along the $x$ -axis and directed toward the positive values of the horizontal $z$ -axis. It is positioned in a way such that the jets or puffs may freely develop without interacting with the limits of the domain. Here, $y$ is the vertical direction, pointing upward. In simulations involving the table, the table is sufficiently large to ensure that the flow does not interact with its borders. The horizontal extension of the table is not considered here.

For the numerical grid, a first coarse tetrahedral mesh is generated in the three-dimensional parallelepiped computational domain, with elements of edge size around 10 cm. Then, conformal meshing of the surface is obtained inside the in-house solver, which uses a grid cutting algorithm and a subsequent remeshing with the MMG3D mesh adaptation library (Dapogny et al., Reference Dapogny, Dobrzynski and Frey2014) to generate the initial grid (Grenouilloux et al., Reference Grenouilloux, Leparoux, Moureau, Balarac, Berthelon, Mercier, Bernard, Bénard, Lartigue and Métais2023), which faithfully describes the boundaries. That grid has a grid size of 1 mm close to the face of the manikin and 5 mm and the remaining surfaces, but remains coarse (grid size approximately 10 cm) in the volume, including in the regions where the jets will develop. Due to its coarse grid level in the volume, that initial mesh is too coarse to be used for the flow predictions. It is refined on the fly, during the calculation, in regions where the vorticity norm is higher than a small threshold and where the air composition differs from that of the environment. The MMG3D library is used to adapt the grid (Grenouilloux et al., Reference Grenouilloux, Leparoux, Moureau, Balarac, Berthelon, Mercier, Bernard, Bénard, Lartigue and Métais2023). In addition, the mesh is never coarsened during the simulation and a maximum gradient of mesh size is imposed to prevent grid discontinuities. The smallest grid size, approximately 0.8 mm, is imposed at the exhalation location. The target grid size then varies spatially to follow the growth of the jet. In addition, the grid is refined near the table’s surface (grid size 1 mm) to resolve the boundary layer that develops along it. For the breathing cases for instance, the size of the first element at the table is of the order of 1–3 wall units, depending on the case. This results in final grids ranging from 110 to 170 million elements for breathing simulations, and from 105 to 216 million elements for laughing cases. Snapshots of the grid during the simulations are shown in figure S1 of the Supplementary Materials. To assess mesh convergence, additional simulations for all configurations were conducted using different target grid sizes. The comparisons between results obtained on different grid sizes are presented in figures S3 and S5–S8 of the Supplementary Materials.

Figure 2. Flow rate signals used in the simulations (solid lines: exhalation; dashed lines: inhalation): (a) first four cycles of the signal for the nasal breathing flow with an exhaled volume of 1 l cycle⁻¹. The total duration of the simulations is 80 s, which corresponds to 20 cycles of 4 s, identical to the first four cycles displayed. Half of the flow rate is imposed at each nostril. (b) First 6 s of the flow rate imposed for the laughing flow simulations. The exhalation signal is injected over 2 cycles of 1.5 s each, then the emission is set to zero for the rest of the simulation (between $t=\textit{3}$ and $t=\textit{100}$ s). The exhaled volume is 1 l cycle⁻¹.

2.3. Characteristics of the simulations performed

In this study, we will successively consider nose breathing in § 3 and laughing in § 4. Their characteristics will be discussed in detail in each section, but it proves useful to comment about their common features and differences in the present paragraph. In nose breathing, air flow is exhaled and inhaled periodically at the two nostrils of the manikin. The flow rate signal for the case at 1 litre per breath is displayed in figure 2a. For the laughing case, the flow is exhaled (and inhaled) at the mouth of the manikin. It consists in two series of intense exhalations, followed by zero emission for the rest of the simulation, as shown in figure 2b. Although the exhaled volume per cycle is the same for the two signals displayed in figure 2, the peaks in the flow rate for laughing are approximately ten times higher than the maximum flow rate in the nose breathing case. Both the nose breathing and the laughing cases generate downward flows whose interaction with a table will be the subject of §§ 3 and 4.

No-slip (zero velocity) Dirichlet boundary conditions are imposed on the table and the manikin’s body. The solid surfaces are modelled as adiabatic and impermeable boundaries, thus there is no mass or heat transfer from/to the air flow and ambient environment. In particular, this applies to the body of the emitter, so that thermal boundary layer effects are neglected, as often performed in computational fluid dynamics (CFD) studies for transport of pathogens. In an unperturbed thermal boundary layer, the velocities are of the order of 10 cm s⁻¹, which is much less than the order of magnitude of the velocities in the jets considered (a few metres per second). Note also that experiments on manikins have shown that the velocities in the thermal convective boundary layer are decreased when the manikin is seated at a table (Licina et al., Reference Licina, Pantelic, Melikov, Sekhar and Tham2014), the table screening the flow associated with the lower part of the body.

In the simulation, the temperature and relative humidity of the exhaled air has to be imposed. Data from the literature allow us to define the typical ranges for these parameters (Bourouiba et al., Reference Bourouiba, Dehandschoewercker and Bush2014): $30$ to $35^\circ$ C and $85\,\%$ – $100\,\%$ . For the nose breathing, a temperature of $32^\circ$ C has been used for the exhaled air, to render the heat exchange role of the nose. For the laugh flow, we have used a temperature of $34^\circ$ C at the mouth exit, which is often chosen in simulations (Chong et al., Reference Chong, Ng, Hori, Yang, Verzicco and Lohse2021). However, the absolute values are only indicative, the important figures being obviously the non-dimensional numbers that quantify the relative effects of the different physics involved which are discussed next. Relative humidity has been set to $90\,\%$ in the exhaled air and $50\,\%$ in the ambient air, whose temperature is fixed to $25^\circ$ C. The compositions of exhaled air and ambient air are provided in terms of mass fraction in Table 1.

Table 1. Characteristics of the exhaled and ambient air: $T_{exh}$ : exhaled air temperature; $T_{amb}$ : Ambient air temperature; RH: relative humidity. Mass fraction of oxygen ( $Y_{\rm O_2}$ ), nitrogen ( $Y_{\rm N_2}$ ), argon ( $Y_{\rm Ar}$ ), water vapor ( $Y_{H_2O}$ ) and carbon dioxide ( $Y_{\rm CO_2}$ )

The different flow configurations considered (breathing or laughing) are both situations in which a warmer flow is exhaled into an ambient flow at rest. In the study of buoyant jets, the inertial and thermal effects can be quantified by the Reynolds number $ Re=U_0 D_0/\nu$ and the densimetric Froude number $Fr=U_0/\sqrt {g' D_0}$ , where $U_0$ is the jet-exit velocity, $D_0$ is the hydraulic diameter of a nostril (with $D_0=\sqrt {4S/\pi }$ , and $S$ representing its surface area), $\nu$ the kinematic viscosity of air and $g'$ the reduced gravity due to the density difference between the jet and the ambient fluid. Additionally, it is useful to introduce a length scale $l_m$ , which represents the transition from jet-like to plume-like motion some distance away from the jet exit

(1) \begin{align} l_m = M_0^{3/4}/B_0^{1/2}&=D_0 Fr, & Q_{0} &= \frac {\pi D^2}{4}{U_0}, & M_0 &= Q_0 U_0 \nonumber \\ B_0&=g'Q_0, & g'&=g\frac {|{\rho _{amb}-\rho _{jet}}|}{\rho _{amb}}, & Fr&=\frac {U_0}{\sqrt {g'D}}, \end{align}

where $M_0$ , $B_0$ are the specific fluxes of momentum and buoyancy at the jet exit, with $Q_0$ as the volume flux (Fischer et al., Reference Fischer, List, Koh, Imberger, Koh and Brooks1979). The densities of the jet and ambient air are denoted by $\rho _{jet}$ and $\rho _{amb}$ . In our simulations, the inflow velocity is not constant, so that the time-averaged velocity during injection will be used (Abkarian et al., Reference Abkarian, Mendez, Xue, Yang and Stone2020) as a reference velocity $U_0$ .

3. Breathing flow

Nose breathing is first considered. Characteristics of human nasal respiration, respiration frequency, exhaled volume and air velocities and direction vary with various factors, such as anatomy, age, posture and physical activity, etc. (Gupta et al., Reference Gupta, Lin and Chen2010). Here, we do not aim to examine all possible flow conditions, we rather use values that are typical of breathing and explore the effect of some parameters, focusing on the interaction with a horizontal table. We define a 4 s cycle, with an exhalation time of 2.4 s (figure 2a), consistently with the longer exhalation times reported by Gupta et al. (Reference Gupta, Lin and Chen2010). While the flow orientation varies according to the shape of the nose and nasal opening, here, we set the jets to be inclined by ${45}^{\circ }$ downwards with respect to the streamwise axis $z$ and by ${25}^{\circ }$ with respect to the symmetry plane (see figure 1a, b). The breathing volume is varied from 0.66 to 2 l per breath, the latter corresponding to a heavy breathing. The flow rates are obtained by multiplying the signal shown in figure 2a of 1 l per breath, by a constant. The hydraulic diameter of each nostril is $D_0={\sqrt {4S_0/\pi }}=1.23$ cm, with a surface area of $S_0={1.18}\,\textrm {cm}^{2}$ . Twenty cycles are computed, for a total physical time of 80s simulated.

The exhaled air is at a temperature of $32^\circ$ C and the relative humidity has been set to $90\,\%$ . The ambient air is at $25^\circ$ C with a relative humidity of $50\,\%$ .

With a total volume of 1 l per breath from the two nostrils, this yields ${Re}\approx 1450$ and ${Fr}\approx 25$ at each nostril exit, using the time-averaged bulk velocity. The relevant characteristics of the breathing cases discussed in this paper, obtained by varying the flow rate or the table position, are reported in table 2.

Table 2. Characteristics of numerical simulations for breathing flow. The inflow signals are presented in figure 2a. Here, ${U}_0$ denotes the mean velocity of the inflow over one cycle of exhalation only; $Re$ and $Fr$ : the mean Reynolds and mean Froude numbers over the exhalation time of one cycle; $h_t$ : the vertical distance of the table to the emitter’s nose; $l_m$ : the characteristic length of the buoyant jet; $y_{min}/l_m$ and $z_{@y_{min}}/l_m$ : the maximum vertical penetration of the breathing jets and the associated horizontal location, respectively

During nose breathing, emitted particles are less than a few microns in diameter and originate from the lower respiratory tract (Bagheri et al., Reference Bagheri, Schlenczek, Turco, Thiede, Stieger, Kosub, Clauberg, Pöhlker, Pöhlker, Moláček, Scheithauer and Bodenschatz2023). Their small size makes them readily transported by the exhaled flow and suspended for long times in the surrounding environment (Morawska & Cao, Reference Morawska and Cao2020; Netz & Eaton, Reference Netz and Eaton2020). The impact of evaporation on their trajectory is negligible. As a consequence, the trajectory of the exhaled particles is determined by the flow itself, which justifies focusing on the jet dynamics. The jet is thus seeded at constant concentration with massless tracer particles, whose dynamics is essentially the same as the small droplets exhaled during nose breathing, allowing us to characterise the cloud generated by the nose breathing over time. A snapshot of the emitted cloud at the end of the 5th cycle is shown in figure 1a,b.

As observed for mouth breathing by Abkarian et al. (Reference Abkarian, Mendez, Xue, Yang and Stone2020) in cases of periodic exhalations, each breath feeds one exhaled flow whose characteristics are similar to those of constant turbulent jets. In nose breathing, two main negatively buoyant jets (Turner, Reference Turner1966; Fischer et al., Reference Fischer, List, Koh, Imberger, Koh and Brooks1979) are formed (one at each nostril), which travel first downwards then lift due to buoyancy effects. Using the residence time of tracers $t_r$ , which denotes the time since their injection, an average trajectory $\boldsymbol{\vec {X}}(t_r)$ of the jets can be calculated as

(2) \begin{equation} \boldsymbol{\vec {X}}(t_r)=\frac {1}{2N}\sum _{j=1}^{2} \sum _{i}^{N} \vec {x}_{i,j}(t_r), \end{equation}

where $\vec {x}_{i,j}(t_r)=(x,y,z,t_r)$ is the coordinate vector of the $i$ th tracer emitted from nostril $j$ and $N$ is the number of tracers injected at each nostril. The symmetry of the geometry with respect to $x$ yields an average trajectory in the ( $z$ , $y$ ) plane (see figure 1a). We consider particles with injection times $t_i$ ranging from 60 to 80 s to obtain well-converged trajectories from $t_r=0$ to 25 s, discarding the initial transient phase, which may be particularly long.

The trajectory of the unbounded jet (no table), non-dimensionalised by the length scale $l_m=D_0 Fr$ , is shown in figure 3a. The inclined, unbounded jet is initially momentum driven and its trajectory first follows a straight line. As the jet gradually expands in the cross-stream direction, it decelerates due to both the entrainment of the ambient fluid and the opposing buoyancy force. Eventually, the buoyancy overcomes the vertical component of the momentum flux, causing the jet to reach a terminal depth at $y\approx 1.3 l_m$ before going upwards, along a skewed trajectory that is best characterised by a quartic polynomial (Papakonstantis and Christodoulou, Reference Papakonstantis and Christodoulou2020, see also figure S2 in the Supplementary Materials). The length scale $l_m$ indeed characterises this transition from jet-like to plume-like motion. For the buoyant jet considered here, the mean value for $l_m$ is found to be approximately 30.4 cm.

Figure 3. Average trajectories of breathing jets, with different origins and normalisations. (a) Cases with and without the table, for a flow rate of 1 l breath⁻¹ ( $Fr\approx$ 25, ${Re}\approx$ 1450), normalised by $l_m=D_0 Fr$ . Horizontal dashed grey lines show the positions of the tables from the nose of the emitter, denoted by a blue $\star$ , at the origin; (b) same trajectories as in (a) with the origin shifted at the location of minimum height at the jets ( $z_{@y_{min}}$ ; $y_{min}$ ); (c) trajectories for cases with the table at $h_t=$ 32 cm below the emission, normalised by $h_t$ , for different flow rates; (d) same trajectories as in (c), but normalised by $l_m$ and with the origin shifted at the location of minimum height at the jets ( $z_{@y_{min}}$ ; $y_{min}$ ), as in (b); comparison with the case without table at 1 l per breath.

We have seen that flows exhaled from nostrils exhibit buoyant jet-like features and may transport particles to a potential interlocutor seated opposite to the emitter. This raises two questions: (i) How do these flows interact with a surface placed beneath the emitter? (ii) What are the consequences of this interaction on forward propagation of the jet and associated droplet transport?

A key point to recall from the rare relevant impinging jet studies is that, depending on the operating conditions – such as jet–wall separation, inclination of the flow and Reynolds number – jets are observed to reach and attach to the surface. Clinging results from a ‘Coanda-like effect’ in the lower part of the jet, where the presence of the wall restricts the entrainment of ambient air, thereby creating a vertical pressure gradient that maintains the jet along the surface. Sharp & Vyas (Reference Sharp and Vyas1977) explored the conditions that lead to the clinging of a jet (refer to figure 5 in the cited work). In the case just discussed, clinging is estimated to occur for tables placed less than 1.6 $l_m$ (49 cm) lower than the nose. Simulations of the interaction between breathing jets and a horizontal table are presented with three nose–wall separation heights: $h_{t}=22$ , $32$ and $42$ cm (which correspond to 0.73 $l_m$ , 1.06 $l_m$ and 1.39 $l_m$ , respectively). Clinging is thus expected for the three cases (Sharp & Vyas, Reference Sharp and Vyas1977).

In figure 3a, the average trajectories of the ‘table jets’ are compared with that of the free jet, at $Fr=24.87$ . Initially, the table jets develop as unbounded jets. Then, the interaction with the table results in the formation of buoyant wall jets (see figure 1a and Supplementary Movie S4). When the table is higher than the terminal height of the free jet trajectory (cases 22 and 32 cm), the jet development is strongly constrained, yielding a shorter penetration in the vertical and the horizontal directions. For lower table heights, the jet–wall interaction is less pronounced. First, the jet trajectory barely differs from the free jet in the descending region. However, due to clinging, the wall prevents the jet from lifting as in the free jet due to buoyancy forces (Sharp & Vyas, Reference Sharp and Vyas1977). However, the difference in forward penetration is small.

In order to compare the trajectories of the wall jets, they are plotted with their origin shifted to the location of the minimum vertical position ( $z_{@y_{min}}$ ; $y_{min}$ ) in figure 3b. The figure clearly shows that the interaction with the table has modified the shape of the trajectory by bending it along the wall. In addition, figure 3b shows that the three wall jet trajectories actually superimpose on the ascending portion, even though the jets have interacted with the table at different distances from the source, and thus different velocities, diameters, Froude numbers: in particular, friction forces are different, as higher velocities are obtained closer to the wall when $h_t$ is small: the decrease in the forward motion is thus not related to viscous friction. We thus observe that the interaction with the table has not modified the balance between forward transport and buoyancy. The table thus limits the forward penetration by making the exhaled jets form a wall jet earlier in their development as the table is moved closer to the injection.

To probe further the effect of exhaled volume on the propagation of breathing clouds, we use simulations for three different exhaled volumes (0.66, 1.0 and 2.0 l per breath, corresponding to approximately $Fr=17$ , $25$ and $50$ , respectively) at constant $h_{t}=32$ cm (figure 3c). Contrary to the jets at 1 and 2 l breath⁻¹, which have the same descending portion, the jet with the lowest breathing volume goes up before reaching the table and does not cling. The trajectories of the clinging jets are initially similar, until $z\approx 1.4 h_t$ , then they strongly depart in the ascending portion. For the largest breathed volume (2 l cycle⁻¹), a notable increase in the near-wall horizontal extent of the jet is observed. In figure 3d, the trajectories are shifted to the location of the minimum vertical position and non-dimensionalised by $l_m$ , as in figure 3b. It is first found that the free jet at 1 l breath⁻¹ behaves similarly as the case at 0.66 l breath⁻¹ and the table at 32 cm. As $h_t/l_m=1.59$ for that case, our results are consistent with the experimental findings of Sharp & Vyas (Reference Sharp and Vyas1977) that clinging should not occur for jets inclined at ${45}^{\circ }$ when the emission-to-surface distance is larger than $h_t/l_m=1.6$ . However, for the clinging jets, the ascending portions of the trajectories do not match, contrary to what was observed in figure 3b. This is consistent with the nonlinear behaviour of cling length with $Fr$ predicted by Sharp & Vyas (Reference Sharp and Vyas1977): contrary to the free jet case, the interaction with the table makes the shape of the trajectory of the table jets dependent on the Froude number.

The nose breathing simulations have thus shown that periodic breathing yields the formation of two negatively buoyant turbulent jets which interact with the table if placed sufficiently close to the emitter. This interaction may limit the downward penetration of the jet, and also modifies its forward penetration. Let us define as $z_0$ the forward penetration, the distance in the $z$ direction at which the jets come back to the altitude of the emission $y=0$ . For the free jet, the trajectory in figure 3a intersects $y=0$ at $z_0\approx 2.7\,l_m$ , for instance. The results in terms of forward penetration $z_0$ as a function of the table distance are displayed in figure 4. The results can be interpreted in the following way: for small values of $h_t/l_m$ , the jet is constrained in its descending motion. As it interacts with the table, it forms a wall jet, bent along the table, but with a limited forward penetration compared with the free jet. However, when the table is lower (here, around $1.3$ to $1.4\,l_m$ ), the table may cling to the table after an unperturbed descending phase, which yields the characteristic bent trajectory without a large effect on the forward penetration.

Figure 4. Horizontal penetration of the jets $z_0$ , measured as the distance at which the average trajectory reaches the emission height $y=$ 0, normalised by $l_m$ . The result for each case is reported as a function of the reduced table distance $h_t/l_m$ . The series with blue squares correspond to the results at constant $l_m$ plotted in figures 3a and 3b. The series with red diamonds correspond to the results at constant $h_t$ plotted in figures 3c and 3d. The horizontal dashed line represents the terminal height of the free jet (‘no table – 1 l per breath’).

We now examine how the table may modify the virus intake by comparing the results for $Fr \approx 25$ with different table positions. We aim to quantify the worst-case scenario, in which a susceptible individual is positioned at the location of highest viral exposure. We will assume that virus intake may be modelled by calculating the number of particles passing through a volume located around the receiver’s nose, represented as a sphere of radius 5 cm (a volume of 0.5 l approximately). We will assume that the height of the emitter’s nose ( $y=0$ ) is the height at which infectious particles may be inhaled by an interlocutor.

In turbulent jets, the concentration of the discharged material decreases as the inverse of the distance from the emission $1/s$ . Figure 3a shows that ‘table jets’ at $h_{t}=22$ and $32$ cm return to $y=0$ by travelling a shorter distance than the free jet. Consequently, if these jets reach a susceptible individual, they would deliver a higher concentration of infectious particles.

We will first consider for each of the two jets of each case at $Fr \approx 25$ (shown in Figure 3a) one sphere placed at the locations where the trajectory of the jet gets back to $y=0$ , as if evaluating the most adverse placement for each configuration. In order to smooth the results, we add 4 other spheres positioned by displacing the first one by 10 cm (twice the radius) in the $-x$ , $+x$ , $-z$ and $+z$ directions and average the data. For each of the ten spheres (5 spheres per jet and 2 jets), the number of particles entering that sphere is calculated over time as a proxy for virus intake, ten results are averaged, yielding a typical measure of virus intake in the core of the jet. While the quantitative results depend on the spheres’ locations and sizes, all of our tests estimating viral intake produce the same overall trends and conclusions. Results are non-dimensionalised by the total number (for the two nostrils) of particles injected per cycle and plotted in figure 5 for all cases at $Fr \approx 25$ , without a table and with the three different table positions. First, differences between the free jet and the ‘42 cm’ table jet are small. On the contrary, transmission risks are increased in the ‘22 cm’ and ‘32 cm’ table jets by two effects: first, the time at which the first particles reach the interlocutor’s face is smaller than for the free jet; then, the slope of the curves, which informs us about the particle flux, indicates that the local particle concentration opposite to the emitter tends to increase as the table distance is decreased.

In addition, we now discuss the cases of a susceptible person located this time in front of the emitter ( $x=0$ ), at different distances in the $z$ direction. In the same way, we place fictive spherical probes of 5 cm radius at different positions, right opposite to the emitter: ( $x=0$ ; $y=0$ ; $z=55$ − $95$ cm). The number of particles entering the spheres is now calculated for each case, and plotted in figure 6. At a given position, the different cases are not ordered the same in terms of virus intake: at $z=0.55$ and $z=0.65\, \text{m}$ , the virus intake is significantly larger for the 22-cm case, while at $z=0.95$ m, the virus intake is much larger for the 42-cm case. The virus intake indeed depends on the forward penetration of the jet, so that the position of the maximum virus intake differs for the different cases: the closer the table, the closer the position at which maximum virus intake is found. In addition, the closer the table, the higher the maximum virus intake: the maximum virus intake is obtained for the 22-cm case (figure 6a), then the 32-cm case (figure 6c), then the 42-cm case (figure 6c). It is also seen that the free jet behaves differently from the others, with lower values on the centreline, which is consistent with the orientation of the jet at the nose exit. Figure 6 also illustrates the time required for the jets to reach a steady state in terms of particle behaviour. In particular, the trajectories of the particle clouds from the first few cycles resemble those of isolated puffs rather than a continuous jet, resulting in reduced forward penetration. This transient behaviour accounts for the different slopes observed in the virus intake over time before the system reaches a steady state.

Interestingly, although the virus intake is lower on the centreline than along the jet direction, the orders of magnitude reported in figures 5 and 6 are similar, which show the robustness of the results with respect to the exact position of the receiver.

Figure 5. Virus intake over time for the cases at $Fr \approx$ 25, with and without a table, for a position of a receiver in the core of the jet, where the trajectory of the jet returns to $y=$ 0. Virus intake is modelled as the number of tracers entering a spherical probe of radius 5 cm at the position of the receiver’s nose. The plots are normalised by the number of particles emitted per breath ( $N_t$ ), which is the same for the presented cases.

Figure 6. Virus intake over time for the cases at $Fr \approx$ 25, with and without a table, for different positions of a receiver in front of the emitter, modelled as the number of tracers entering a spherical probe of radius 5 cm at the position of the receiver’s nose: $x=$ 0; $y=$ 0; $z=$ 55 cm (a), $z=$ 65 cm (b), $z=$ 75 cm (c), $z=$ 85 cm (d), $z=$ 95 cm (e). Results are normalised by the number of particles emitted per breath ( $N_t$ ).

4. Laughing flow

While breathing flow is characterised by periodic exhalations at moderate Reynolds numbers, some of the human exhalation types are shorter and faster: shout, cough, laughter, sometimes speech, not to mention sneeze. We consider here a signal mimicking laughter, composed of 2 series several short and high-speed exhalations directed downwards (see figure 1c). Laughter indeed generates high-speed downward flows, as illustrated by Bhagat et al. (Reference Bhagat, Wykes, Dalziel and Linden2020), and by an experiment shown in their movie S3 (and figure S10).

An inflow signal of 1.5 s per cycle is considered, with a flow rate profile that captures the intermittent feature of laughing flows with high-frequency bursts generating coalescent puffs: the signal consists of an exhalation of 1 s with 5 peaks at 2 to 3 l s⁻¹ and an inhalation of 0.5 s (in practice, a cycle of a signal previously used for speech (Abkarian et al., Reference Abkarian, Mendez, Xue, Yang and Stone2020) is compressed in time and increased in magnitude so that the exhalation lasts 1 s and the exhaled volume per cycle is 1 litre (see figure 2b). Two consecutive cycles are computed. After the 2 cycles of injection, a large puff is formed. The simulation is continued until 100 s without any flow rate at the mouth to isolate the dynamics of the puff.

The flow exits from an elliptic mouth surface of hydraulic diameter $D_0={2.46}\textrm {cm}$ and is directed downwards, with a ${45}^{\circ }$ angle (figure 1c). The exhaled air has a temperature of $T_{exh}= {34}^{\circ }\textrm {C}$ with relative humidity (RH) of 90 $\,\%$ . The flow is released into a still environment at $T_{exh}= {25}^{\circ }\textrm {C}$ and 50 $\,\%$ RH.

The peak velocity during the injection is approximately 7.1 m s⁻¹ and the average value during injection is approximately 2.34 m s⁻¹. This yields a maximum (respectively average) Reynolds number during the emission of approximately $Re_{max}=11750$ (respectively $Re=3900$ ). The densimetric Froude number, based on the maximum (respectively average) velocity is $Fr_{max}=69$ (respectively $Fr=23$ ). The characteristic length of the free flow can be calculated based on the average Froude number: $l_m=D_0 {Fr}\approx$ 57 cm. Particles with initial diameters ranging from ${1}$ to ${60}\,{\unicode{x03BC} \textrm {m}}$ (as will be apparent, the dynamics of larger particles can be extrapolated from the simulations) with a uniform distribution are injected at the inflow with an initial velocity equal to that of the gas. These particles are used as a proxy for expiratory droplets and evaporate down to a minimum diameter. As already detailed in § 2.1, thanks to the one-way coupling, the dynamics of the particles is computed without an effect on the flow. Thanks to this assumption, a large number of particles can be injected, and a uniform distribution is used. The objective is to document the fate of droplets as a function of their size. The evolution of the particle cloud is studied in the presence of a horizontal table located 30 cm lower than the person’s mouth (figure 1c,d), or without the table. The conditions selected yield $h_t/l_m\approx 0.53$ . Supplementary Movie S5 illustrates the dynamics of cloud in the presence of a table.

Figure 7. Snapshots of particle clouds emitted during the laughing flow at 6 different time instants: configurations with the table (1st and 3rd column; a1-6) and without the table (2nd and 4th column; b1-6). Each column shows the same configuration. Droplets are coloured by their initial diameter. The $\star$ symbol indicates the mouth of the emitter. Three spherical probes used for evaluating the particle fluxes at the emitter/receiver face height are displayed by $\circ$ . They are located at 50, 100 and 150 cm from the mouth exit and mimic different separation distances between two people seating at a table.

Figure 7 shows the evolution in time of the particle cloud in both configurations. In the ‘no-table’ case (b series), the cloud is first dominated by inertia and moves along the direction imposed at the mouth (see figure 7b1). When buoyancy overwhelms the vertical momentum, the cloud is deflected upwards, as shown by the dynamics of the smallest particles. The largest droplets leave the cloud early in the descent stage and are mostly settled out of the cloud during its ascent. As in the breathing flow cases, when a table is placed above the location of the terminal descent of the free jet (a series), the exhaled flow impacts the table (figure 7a1), which cancels its vertical momentum and limits its forward motion. The maximum streamwise distance reached in the ‘no-table’ configuration is more than 2.5 m, but less than 2 m in the case with the table. A number of particles deposit on the table, which will be analysed further.

To quantify the impact of the table on the particles that may be inhaled by a person located in front of the emitter at different distances, three spherical probes with a radius of 5 cm are placed opposite the emitter at 50, 100 and 150 cm. Their location is displayed in figure 7. Figure 8 shows the histograms of diameters for the particles having entered those spheres during the simulation. At short distances (50 cm, figure 8a), the puff–table interaction does not change the results significantly. The puff passes lower than the location of the sphere. At the distance of 1 m (figure 8b), the differences are pronounced. While the results without the table are similar to those at the previous location, the number of particles having passed through the sphere is an order of magnitude larger. This corresponds to the location where the table puff lifts off the table. In that table case, few particles reach the probe at $z=150$ cm (figure 8c). On the contrary, for the free jet, the maximum number of particles gathered is for the 150 cm location. In addition, the free jet carries droplets from all the injected sizes: flow motions counteract sedimentation and transport some of the largest droplets simulated to a height where they may be inhaled. This is not the case for the table jet, for which the number of droplets collected decreases with the initial diameter. In particular, droplets with an initial diameter larger than ${50}\,{\unicode{x03BC} \textrm {m}}$ are never found to reach the probes. The deposition of particles on the table is now examined to understand this difference.

Figure 8. Airborne particles in laughing flow: (a–c) fraction of particles passing through spherical probes centred 50 (a), 100 (b), 150 cm (c) away from the emitter (red circles in figure 7) during the $\textit{100}$ s simulated, as a function of their initial diameter. Here, $N_t$ denotes the total number of particles per bin.

Figure 9. Time evolution of particle deposition on the table for the puffs released during laughter: (a) first cycle; (b) second cycle. Droplets of initial diameter ranging from $\textit{1}$ to $\textit{60}\,{\mu \textrm {m}}$ are divided into 4 groups, as indicated in the legend. Note that the size distribution of emitted particles is uniform. Here, $t$ denotes the physical time with $t_i$ representing the time of the end of emission of cycle $i$ .

Figure 10. Histogram of particles deposited on the table as a function of the streamwise coordinate $z$ for laughing flow. Approximately $N_t=$ 33 000 particles are injected, with a uniform distribution between 1 and $\textit{60}\ {\mu {\textrm {m}}}$ . Particles are grouped into 4 bins based on their initial diameter, i.e. (a) 1–15 $\mu {\textrm {m}}$ , (b) 15–30 $\mu {\textrm {m}}$ , (c) 30–45 $\mu {\textrm {m}}$ , (d) 45–60 $\mu {\textrm {m}}$ . The vertical dashed line represents the streamwise position of the geometric impinging point of the puffs.

Figure 9 shows the proportion of the particles deposited on the table over time for the table case. The particles from the two cycles are separated (first cycle, figure 9a and second cycle, figure 9b) and the time is shifted to the end of each cycle for comparison. The particles are grouped into 4 bins according to their initial diameter. First, the proportion of deposited particles increases with their size. More than 90 $\,\%$ of the emitted particles of initial diameter larger than ${45}\ {\unicode{x03BC} \textrm {m}}$ are deposited on the table at the end of the simulation, but this ratio drops only to 20 $\,\%$ for the particles of initial diameter smaller than ${30}\,{\unicode{x03BC} \textrm {m}}$ . In addition, the dynamics is very different: particles within the ${30}$ – ${60}\,{\unicode{x03BC} \textrm {m}}$ range rapidly deposit after the emission, consistently with a rapid deposition at the puff’s impact on the table, while the deposition of the smallest particles is more progressive and characteristic of a slow sedimentation. The differences between the particles of the two cycles is small, but in general, the deposition rate is higher for the second cycle. We interpret this difference as an effect of the second cycle in maintaining the momentum of the particles of the first cycle, slightly limiting their sedimentation compared with those of the second cycle.

In figure 10, we present the density in the $z$ direction of particles deposited over the table, integrated over the spanwise direction $x$ . Particles are grouped into the same 4 bins as in figure 9 and results are plotted at 5 s, soon after the injection is completed, and at 100 s, at the end of the simulation. The maximum deposition of the largest droplets occurs near the geometric stagnation point, which is consistent with the deposition profile in impinging jets with large nozzle-to-surface distances (Burwash et al., Reference Burwash, Finlay and Matida2006). It confirms that the deposition is due to the impact of those particles due to their initial momentum. In contrast, the deposition of smaller particles is more progressive in time and spans over a larger region: it is associated with their sedimentation from the cloud. A peak in the density of small deposited particles is observed below the particle cloud. Figure 10 gathers results from the two cycles simulated, but they are separated in figure S9. Differences between the two cycles are mainly visible for the smallest particles: the particles emitted during the first cycle are pushed further downstream and contribute more significantly to the deposition for $z \gtrapprox 0.8$ m.

Overall, the table acts as a filter that collects the largest particles simulated at impact. This has a tremendous effect on the distribution of airborne particles and explains the absence of large airborne particles observed in figure 8 in the table case.