Speech can produce jet-like transport relevant to asymptomatic spreading of virus

Orders of Magnitude.

The typical human adult has a head with approximate radius 7 cm. We may define the characteristic length scale of the mouth, whose shape is approximately elliptical, with the radius a of a circle having the same surface area. Measurements show that the average mouth opening areas are approximately 1.2 cm2${\mathrm{c}\mathrm{m}}^2$ for breathing and 1.8 cm2${\mathrm{c}\mathrm{m}}^2$ (with peak values of the order of 5.0 cm2${\mathrm{c}\mathrm{m}}^2$) for speaking (22). For an order-of-magnitude estimate of the Reynolds numbers, a=1 cm$a=1\hspace{0.17em}\text{cm}$ is chosen. It is perhaps surprising to many that typical airflow speeds are u≈0.5 to 2 m/s$u\approx 0.5\hspace{0.17em}\text{to}\hspace{0.17em}2\hspace{0.17em}\text{m/s}$ (volumetric flow rates ≈0.2 to 0.7 L/s$\approx 0.2\hspace{0.17em}\text{to}\hspace{0.17em}0.7\hspace{0.17em}\text{L/s}$) when breathing and u≈1 to 5 m/s$u\approx 1\hspace{0.17em}\text{to}\hspace{0.17em}5\hspace{0.17em}\text{m/s}$ (volumetric flow rates ≈0.3 to 1.6 L/s$\approx 0.3\hspace{0.17em}\text{to}\hspace{0.17em}1.6\hspace{0.17em}\text{L/s}$) when speaking (Table 1). When breathing, exhalation and inhalation occur approximately evenly over a cycle with period about 3 to 5 s (22, 29), while during speaking the exhalation period is generally lengthened so that two-thirds or even greater than four-fifths of the time may be spent in exhalation.

The local fluid mechanics of exhaled and inhaled flows of speed u are characterized by Reynolds numbers Re=2ua/ν$\text{Re}=2ua/\nu$ (the kinematic viscosity of air ν≈1.5×10−5 m2$\nu \approx 1.5\times 10^{-5}\hspace{0.17em}{\mathrm{m}}^2$/s), which have typical magnitudes Re = O(7×102 to 3×103)$=\hspace{0.17em}O\left.7\times 10^2\hspace{0.17em}\text{to}\hspace{0.17em}3\times 10^3\right)$ when breathing and Re =O(1×103 to 7×103)$=O\left.1\times 10^3\hspace{0.17em}\text{to}\hspace{0.17em}7\times 10^3\right)$ when speaking. Inertial effects are expected to dominate these flows, which will also generally be time dependent and turbulent, as discussed below.

Breathing and Blowing as Jet-Like Flows.

We characterize first the nature of breathing and blowing flows (Fig. 1). We set up a laboratory experiment with a laser sheet (1 × 2 m × 3 mm), where no light hits the speaking subject, who sits adjacent to the sheet. A fog machine generates a mist of microscopic aqueous droplets whose large-scale motions are observed with a high-speed camera oriented perpendicular to the sheet. We obtain the velocity field of exhalation (during both breathing and speaking) by observing how the air stream drags and deforms the cloud in the sheet of light using correlation image velocimetry (see typical images in Fig. 1 A and C, with details in Materials and Methods).

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Fig. 1.

Flow visualization snapshot of exhalation in a laboratory-generated fog and parallel to a laser sheet in two different breathing situations. (A–D) Calm breathing (A) with the corresponding flow speeds shown with the color code and arrows (B) and a case of strong blowing (C) with the corresponding velocity field (D). Note the much higher velocities associated with blowing. However, the flows in the two cases are qualitatively similar over a sufficiently long period of a few seconds and exhibit jet-like features. The field of view in all of the images is 1 m. (E) Sketch of blowing out a candle (or not). (F) Sketch of the qualitative contrast between exhalation and inhalation for breathing and speaking.

The flows are qualitatively similar during breathing or strong blowing (Fig. 1 A and C), although the velocity magnitudes can be quite different (Fig. 1 B and D). For instance, typical velocities observed in the airflow while breathing with a slightly open mouth (∼1 ×2 cm$\sim 1\hspace{0.17em}\times 2\hspace{0.17em}\text{cm}$) remain of the order of 0.3 to 1 m/s as visible in Fig. 1B (Movie S1), while velocities can be as high as a few meters per second in the blowing stream (Fig. 1D) (Movie S2). Most significantly, a jet-like, conical structure is visible for the two different situations as depicted by the white lines in Fig. 1 A and C, with a cone angle 2α≈20○$2\alpha \approx 20^{○}$. We can expect stronger propagation when breathing after exercising, as the volumetric flow rates are increased, which could make breathing in such a case closer to blowing. These observations call for comparison for the more complex situation relevant for pathogen transport, which is the case of speaking, where aerosols are produced during speech (13, 14). Next, though, we comment on a fundamental asymmetry of exhalation and inhalation.

Asymmetry of Exhalation and Inhalation.

At these Reynolds numbers, we expect exhalation and inhalation to be asymmetric. A reader may be aware that one extinguishes a candle by blowing, but it is not possible to do so by inhalation (Fig. 1E), which is a characteristic of the flows for breathing and speaking. Long exhalation should produce starting jet-like flows propagating away from the individual over a significant distance of the order of 1 m (e.g., Fig. 1 A–D), while inhalation is more uniform and draws the air inward from all around the mouth (Fig. 1F); it is this asymmetry that explains the phenomenon related to extinguishing a candle (Fig. 1E). These outflows are in fact responsible for transporting large droplets and aerosols away from the speaker.

For such inertially dominated flows, a continuous or long outflow should be similar to an ordinary jet (30), and during the initial instants over a time T the propagation distance, while smaller than the naive estimate L=uT=O(1) m$L=uT=O\left(1\right)\hspace{0.17em}\mathrm{m}$ (see below), is still larger than the typical size of the head (e.g., Fig. 1). Moreover, since L≫a$L\gg a$, it follows that, in ordinary circumstances, one breathes in little of what is breathed out. Wearing a mask (as recommended as a mitigation strategy for COVID-19) should be expected to produce more symmetric flow patterns during exhalation and inhalation, localizing airflow around the face.

Speaking, Plosive Sounds, and Jet-Like Flows.

Flows exiting from an orifice are well known to produce vortices, even in the absence of coughing, and these drive the transport about the head, as evident in Fig. 1. Speaking introduces two further differences: 1) The typical time of inhalation is about one-quarter to one-half of the exhalation time (29) and 2) language includes rapid pressure and flow rate variations associated with sound productions (plosives, fricatives, etc.), as previously characterized acoustically by linguists (26). We also note that the stop consonants, or what are referred by linguists as plosives consonants, such as (P, B, K, …), have been demonstrated recently to produce more droplets (14). In these cases, the vocal tract is blocked temporarily either with the lips (P, B) or with the tongue tip (T, D) or body (K, G), so that the pressure builds up slightly and then is released rapidly, producing the characteristic burst of air of these sounds; in contrast, fricatives are produced by partial occlusion impeding but not blocking airflow from the vocal tract (31).

We now visualize flow during speaking, which seems different from breathing as, for instance, when saying a sentence like “We will beat the corona virus,” as shown in Fig. 2A (and visible in Movie S3). A color code illustrates the average speeds (averaged over the time to say the phrase), but note that these are not representative of the true instantaneous velocities, which in the remainder of this section were estimated from Movies S1–S5. Over the approximately 2.5 s to say the sentence, the airflow is more jerky and changes direction, depending on the sound emitted. In this particular case, the sentence contains starting vowels (in “We” and “will”) and pulmonic consonants as fricatives (as V and S in “virus”) and plosives (like B and K in “beat” and “corona”). Three different directions are revealed when averaging the velocity field over the time to say the sentence in Fig. 2A: “We will beat” being slightly up and to the front with a typical velocity of about 5 to 8 cm/s, “the corona” being directed downward between −40○$-40^{○}$ and −50○$-50^{○}$ with higher velocities of almost 8 to 12 cm/s while saying the two syllables “coro.” Finally, the short air puff associated to “virus” is directed upward at about 50○$50^{○}$ with speeds of 5 to 7 cm/s. We believe that an interlocutor and potential receiver of the exhaled material will be most exposed after a few seconds by the horizontally directed part of the flow, whose velocity reaches, in this case, the ambient circulation speed at about 0.5 m at most.

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Fig. 2.

Mean velocity field produced when speaking three different sentences. A color code illustrates the average speeds but note that single images of the magnitude of speeds are not representative of the true instantaneous velocities, which were estimated from Movies S1–S5. (A) “We will beat the corona virus,” which is a mixture of vowels, fricatives and plosives. (B) “Sing a song of six pence” (SSSP) (25), mainly composed of the fricative “S” except the last word that starts with “P.” (C) The distance traveled by the extremity of the air puff as a function of time when saying “pence” at the end of SSSP for three different runs. (D) “Peter Piper picked a peck” (PPPP) (25), which is mainly composed of many plosives P.

Next, we illustrate a sentence of the same time lapse of about 2.5 s containing many times the same starting fricative S as in “Sing a song of six pence” (25) with only one starting bilabial plosive sound P in the last word: Most of the air puffs produced are emitted downward at an estimated angle of −50○$-50^{○}$ from the horizontal (and become visible in this sequence only when the airflow hits a nearby table and crosses the laser sheet; Fig. 2B and Movie S4). However, a distinct, directed air puff appears in front of the speaker when “pence” is pronounced (Fig. 2B), which propagates forward at initially high speeds of about 1.4 m/s as visible in Movie S4, but decelerates rapidly to ≈1$\approx 1$ m/s at 0.5 m distance from the mouth; the puff has a speed of 30 cm/s at about 0.8 m (Movie S4).

These images of typical speech raise the question of the dynamics of individual puffs. In Fig. 2C we report the distance L traveled by the air puff as a function of time t when pronouncing pence. The data demonstrate that the starting plosive sounds like P induce a starting jet flow, which grows initially for very short timescales of under 10 to 100 ms as t1/2$t^{1/2}$, but rapidly transitions to a slower movement characterized by a t1/4$t^{1/4}$ response, typical of puffs (32) and vortex rings (33). In fact, when looking at the flow, a vortex ring stabilizes the transport over a distance of almost 1 m. This transition between two different dynamics, ending with the dynamics of an isolated puff, is also measured in coughs (10).

In contrast, when we speak a sentence with many P sounds, such as “Peter Piper picked a peck” (PPPP) (25), as illustrated in Fig. 2D, the distribution of the average velocity field approaches that of a conical jet with average velocities of tens of centimeters per second and over long distances of about 1 m. Peak velocities are seen at the emission of the sound P with values close to 1.2 to 1.5 m/s (Movie S5). This more directed flow situation shares features of breathing and blowing and thus material will be transported faster and farther than individual puffs. But, unlike breathing, we believe that this distinct feature of language is more likely to be important for virus transmission since droplet production has been linked to the types of sounds (14).

It should be evident that language is complicated (Fig. 2 A and B). Given the possibility of asymptomatic transmission of virus by aerosols during speech, we have focused on the phrases in language, those usually containing plosives, that produce directed transport in the form of approximately conical turbulent jets (Fig. 2D and also see Figs. 4 and 6).

In addition, to see that thermal effects are small until the jet speeds are reduced to closer to ambient speeds, consider the Richardson number Ri=gdρdzρ(du/dz)2$Ri=\frac{g\frac{d\rho }{dz}}{\rho {\left.du/dz\right)}^2}$. So approximately Ri≈ΔρρgΔz(Δu)2$Ri\approx \frac{\mathrm{\Delta }\rho }{\rho }\frac{g\mathrm{\Delta }z}{{\left.\mathrm{\Delta }u\right)}^2}$. For a 15 °C$15\hspace{0.17em}°\mathrm{C}$ temperature change in air, Δρρ≈0.05$\frac{\mathrm{\Delta }\rho }{\rho }\approx 0.05$, so with Δu≈0.5$\mathrm{\Delta }u\approx 0.5$ m/s and a length scale say Δz≈0.1$\mathrm{\Delta }z\approx 0.1$ m (which is relatively large), we find Ri≈0.2<1$Ri\approx 0.2<1$. The thermal effects should be expected to be important at longer distances where the jet speed is reduced (usually where the ventilation may also matter) or if a mask is used which decreases the flow speed substantially.

We document the distinct role of the individual plosives in the phrase “Peter Piper picked a peck” (PPPP) with the time-lapse images displayed in Fig. 3A (also Movie S5). By performing correlation image velocimetry to calculate the vorticity field ω=∇∧u$\omega =\nabla \wedge \mathbf{u}$, where u is the in-plane velocity field, as shown in Fig. 3B, we could follow the vortical structures created by the pronunciation of Ps in PPPP. Vortices shed from the mouth are clearly visible, interact, and survive downstream where they easily reach the meter scale. The transition from puff-like dynamics associated to single plosives to the development of turbulent jet-like flow during longer sentences seems to be associated with the sequential accumulation of “puff packets” pushing air exhaled from the mouth. We explore this transition in more detail using the numerical simulations below.

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Fig. 3.

Time evolution of the flow from a phrase, “Peter Piper picked a peck” (PPPP), with many plosives, spoken parallel to a laser sheet. The speaker is indicated by the dotted curve to the left. (A) Flow visualization with the individual plosives identified. (B) Vorticity field with individual vortices clearly visible for each plosive P pronounced in the sentence. Note the interactions between the first vortices, as well as the different upward angle of the vortex produced when the syllable “Pi” is pronounced in “picked.”

Characteristic Features of a Steady Turbulent Jet.

In a high-Reynolds-number steady turbulent jet, it is of interest to characterize the volume flux, linear momentum transport, and kinetic energy transported by the jet, as well as the entrainment of the surrounding air that dilutes the jet (34). These properties also help to understand the fluid dynamics of breathing and speaking. There are at least three significant conclusions that characterize the flow: 1) Denoting the direction of the jet as x, the typical axial speed of the jet as v(x)$v\left(x\right)$, and its cross-sectional area as A(x)$A\left(x\right)$, in a steady jet issuing into an environment at a constant pressure, the flux of linear momentum is constant, or v2A=constant$v^{2}A=\text{constant}$. If the exit flow near the mouth is characterized by a speed v0$v_0$, volumetric flow rate Q0$Q_0$, and area A0$A_0$, we conclude that v(x)/v0=(A0/A(x))1/2<1$v\left(x\right)/v_0={\left.A_0/A\left(x\right)\right)}^{1/2}<1$. For a conical jet-like configuration of angle α (Fig. 1), then beyond the mouth A(x)∝(αx)2$A\left(x\right)\propto {\left.\alpha x\right)}^2$. 2) The corresponding volume flux Q=vA$Q=vA$, so that the outflow leads to a volume flux Q/Q0=(A(x)/A0)1/2>1$Q/Q_0={\left.A\left(x\right)/A_0\right)}^{1/2}>1$; i.e., there is entrainment of the surrounding air into the jet, which is an important feature of mixing of the surroundings. 3) Any material expelled from the mouth with concentration c0$c_0$ is reduced in concentration as the jet evolves, with c(x)/c0=Q0/Q(x)$c\left(x\right)/c_0=Q_0/Q\left(x\right)$. Since the jets are approximately conical, then the above results predict that the characteristic quantities vary with distance as v(x)∝(αx)−1$v\left(x\right)\propto {\left.\alpha x\right)}^{-1}$, A(x)∝(αx)2$A\left(x\right)\propto {\left.\alpha x\right)}^2$, Q∝αx$Q\propto \alpha x$, and c∝(αx)−1$c\propto {\left.\alpha x\right)}^{-1}$. Although these arguments are based on the assumption of a steady jet, we shall now see that they apply approximately to the unsteady features of speaking on the timescale of many cycles and far enough from the mouth or exit of an orifice.

Starting Jets and Puffs.

A jet formed by the sudden injection of momentum out of an orifice is referred to as a starting jet. Such flows reach a self-preserving behavior some distance downstream of the source, where the penetration distance grows over time like L∝t1/2$L\propto t^{1/2}$ (32, 35); see also Eq. 2.

On the other hand, a rapid release of air, or puff, injects a finite linear momentum into the fluid, e.g., Fig. 2. For the inertially dominated flows of interest here, the linear momentum of the puff is conserved, so that the distance traveled is L∝t1/4$L\propto t^{1/4}$ (32, 35), similar to interrupted jets, i.e., starting jets when the flow is suddenly stopped.

However, during breathing or speaking, the interrupted jet and the puffs are released one after the other and interact with each other in front of the source, as illustrated by Fig. 3. The jet is neither continuous like in starting jets nor isolated like in classical puffs. What are then the dynamics of such a “train of puffs”? In the next section, we use numerical simulations to investigate the dynamics of puff trains and quantify their growth in space and time.

Contrasting Four Situations of Exhalation.

We contrast four situations with comparable period and given volumes exhaled and inhaled, with zero net outflow over one cycle (Fig. 4 A–D): 1) normal breathing with a 4-s period split into intervals of exhalation (2.4 s) and inhalation (1.6 s); 2) a breathing-like signal but with a (slow) speaking-like distribution of exhalation (2.8 s) and inhalation (1.2 s); 3) a spoken phrase, “Sing a song of six pence” (25); and 4) a phrase with many plosive sounds, “Peter Piper picked a peck” (25). We ran either one-cycle simulations using the flow rate profiles over a single period, followed by no further outflow, to quantify a single “atom” of breathing and speaking, or simulations for many periods (or cycles) to understand how the local environment around an individual is established and changes in time. Different volumes of exhalation typical of speaking, from 0.5 to 1 L per breath, were studied (see the full table of runs in SI Appendix, Table S1).

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Fig. 4.

Numerical simulations of periodic breathing versus speaking signals for cycles of 4.0 s. The jets issue from a sphere with an open elliptic orifice of semiaxes 1.0 and 1.5 cm. (A–D) volumetric flow rate signals for cases with 0.5 L per breath (hence the “50” in the labels), where the vertical dashed lines mark the separation between exhalation and inhalation. (A) Case B50 is a breathing-like signal with 2.4 s exhalation and 1.6 s inhalation. (C) Case P50 and (D) case S50 are speaking signals sampled from ref. 25 and recorded during articulation of “Peter Piper picked a peck” and “Sing a song of six pence,” respectively, with speaking time of 2.8 s and a 1.2-s inhalation. The P50 and S50 signals have been adjusted to 0.5 L per breath. (B) Case C50 is a calm signal of the same macroscopic characteristics as P50 and S50, but with a smooth signal similar to B50. Three series of simulations have been performed at different volumes per breath, i.e., 0.5, 0.75, and 1.0 L per breath. For the simulation of “Peter Piper picked a peck” for instance, the simulations at 0.75 and 1.0 L per breath are referred to as P75 and P100 and are obtained by multiplying the input flow rate signal of P50 by 1.5 and 2.0, respectively. (E–H) Examples of jets obtained for cases B75, C75, P75, and S75 after nine cycles (36 s), as visualized by perfect tracers issued from the mouth and color coded by their residence time in the computational domain (dark blue tracers were exhaled during the last cycle). The scale is the same for all plots. For each case, a point marked by a “+” is positioned to indicate the axial and radial extent of the jet. The x and y coordinates of the point are reported as (x;y$x;y$) in E–H. The sphere representing the head is shown to the left in E–H.

The results of simulations of these different flow rate profiles are shown in Fig. 4 E–H for an exhaled volume of 0.75 L per breath. To visualize the flow, tracers injected at the inflow are shown, color coded by the residence time of the tracers. For every case, a conical jet flow is produced, with similar cone angles as well, which is reminiscent of typical features of turbulent jets studied in laboratory experiments and many applications (e.g., refs. 34 and 37 and Figs. 1 and 2). Qualitatively, we observe that breathing produces a jet with an axial flow comparable to speaking, which some may find surprising. Jet lengths in particular are very similar, despite a factor of 2.6 in the peak flow rate of cases P75 and C75 for instance (SI Appendix, Table S1). The phrase with plosives produces qualitatively a rougher jet (Fig. 4G) due to the ejection of vortex rings away from the main jet and vortex interactions. Speaking jets (P75 and S75) yield the largest cone angles and consequently an axial extent somewhat reduced compared to breathing (B75 and C75). Short high-speed puffs associated with speaking thus seem to increase the jet entrainment, but do not enhance the long-range transport in the axial direction.

For all cases, even those with complex phonetic characteristics, we observe that the resulting jets display many of the features of a turbulent jet, which leads to transverse spreading and mixing of the exhaled contents with the environment. These features actually build up over the continual cycles of exhalation and inhalation in both breathing and speaking. Particle residence time (Fig. 4 E–H) notably shows the progressive formation of the jet. However, a striking feature is the absence of obvious signature of the flow pulsation in the far field. From the global point of view, all computed jets, whatever the details of the inflow signal, are similar to steady turbulent jets away from the immediate vicinity of the mouth.

Quantifying the Jets.

We ran simulations for three different flow rate signals and exhaled volumes (0.5, 0.75, and 1 L per breath) to understand the characteristics of the flows generated by multicycle breathing and speaking. Because the flows are time varying and turbulent, we quantified the cone half-angle α by determining the angle inside of which reside 90% of the exhaled tracer particles (Fig. 5A). The included angles differed from case to case, but were of the order of 10 to 14○$14^{○}$ (SI Appendix, Table S1). The typical jet lengths, L(t)$L\left(t\right)$, were also calculated based on the criterion that 90% of the tracers are located upstream of x=L$x=L$ at time t. Raw data of L(t)$L\left(t\right)$ are presented in SI Appendix, Fig. S2, and the jet angles are reported in SI Appendix, Table S1. First, higher mean flow rates (exit speeds) produce longer lengths, as expected. For a given exhaled volume per cycle, different types of exhalation produce comparable jet lengths, as suggested by the qualitative analysis of Fig. 4 E–H. Modulation of the inflow signal (cases P and S) systematically tends to increase the lateral growth of the jet, increasing the jet angle and decreasing the jet length.

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Fig. 5.

Jet characteristics in the simulation of the breathing and speaking signals shown in Fig. 4 A–D. (A) Example of calculation of jet length L and angle α for S75, based on the emitted tracer particles color coded here by residence time, as described in Fig. 4 E–H. L(t)$L\left(t\right)$ is such that 90% of the particles are located upstream of x=L$x=L$ at time t. The cone angle α is calculated to enclose 90% of the particles and is a cone passing through the mouth exit (radius is 1 cm at x=0$x=0$). This angle is verified to remain stable with time after the initial cycles. (B) Principle of the calculation of the cycle-averaged velocity fields presented in C and D. The velocity is time averaged independently over each cycle. (C and D) Progressive formation, along increasing cycles of exhalation and inhalation, of a turbulent jet-like velocity profile v(x)$v\left(x\right)$ in the far field. Examples of cases C50 (C) and S50 (D): cycle-averaged axial velocity along the x axis from the mouth exit to 2.0 m downstream, for different cycles, extending to 14 cycles or 56 s. The black dashed line is the v(x)∝1/x$v\left(x\right)\propto 1/x$ scaling, plotted here as a guide for the eye, which is suggested by a model of a steady turbulent jet. (E) Evolution of the nondimensional jet length L/a$L/a$ as a function of 2v0t/(aα)$2v_{0}t/\left(a\alpha \right)$ (Eq. 2), with a≈1.22$a\approx 1.22$ cm the equivalent radius of the mouth exit and v0$v_0$ the average axial speed during exhalation for the different simulations. Two power laws are plotted as a guide for the eye to assess the evolution of L with time. Raw data for L(t)$L\left(t\right)$ are plotted in SI Appendix, Fig. S2.

A Train of Puffs.

We ran the multicycle simulations over many periods to quantify the development of the transient velocity field. To filter the turbulent fluctuations that prevent direct comparisons of the velocity fields as a function of time, we performed time averages over each period (Fig. 5B) to produce an approximate profile for the distribution of axial speeds in the exhaled jet. In the far field, although time varying, breathing and speaking may be viewed as periodic processes where the timescales are much longer than an individual period. Moreover, we have already explained that inhalation has little effect on exhalation because of the differences expected of high-Reynolds-number motions. Indeed, when we plot the axial speed as a function of axial distance, we find that for each period of exhalation, the axial velocity falls along the curve v(x)∝x−1$v\left(x\right)\propto x^{-1}$ for both speech and breathing, shown, respectively, in Fig. 5 C and D. Not only does the head of the jet evolve as that of a starting jet, but the whole flow downstream of a certain distance from the mouth behaves similarly to a steady turbulent starting jet. This is particularly striking as the near-mouth flow is laminar and completely different from a steady jet (Fig. 5 C and D). Thus, at the Reynolds numbers characteristic of breathing and speaking, a train of puffs transitions to a turbulent, jet-like flow that dominates the transport associated with breathing and speaking.

A Diffusive-Like, Directed Cloud of Exhaled Air.

For growing jets at constant angle, we can estimate the spreading of the cloud with time. The time t it takes to reach an axial distance from the orifice, or the mouth, is estimated by

The theoretical prediction for the length of the exhaled air column for a starting jet (Eq. 2) is then compared to the nondimensional numerical data in Fig. 5E. The scaling captures quantitatively the trends provided that v0$v_0$ is defined as the average speed at the orifice exit (mouth) during exhalation. The peak velocity is not relevant: Strikingly, the details of the flow rate signal do not impact the scaling, but influence only the spreading angle of the jet. In addition, for one-cycle simulations (1P75 and 1S50), we recover that the whole exhaled material acts as a unique large puff (32), and L∝t1/4$L\propto t^{1/4}$ is obtained, which is consistent with the experiments (Fig. 2C).

These results allow the quantification of concentration of exhaled material in the far field. From the previous results, we expect the concentration field of the exhaled cloud is quasi-steady and falls off with distance, c(x)/c0∝a/(αx)$c\left(x\right)/c_0\propto a/\left.\alpha x\right)$. Note that for a=1 cm$a=1\hspace{0.17em}\text{cm}$, α=10○$\alpha =10^{○}$, and L=2 m$L=2\hspace{0.17em}\mathrm{m}$ (the 6-foot rule), then for directed jets the concentration of any exhaled material has fallen off by a factor of (a/(αL))≈0.03$\left.a/\left.\alpha L\right)\right)\approx 0.03$. Typical dilution levels of 0.04 to 0.05 have been found in the different simulations at 1.5 m, which is consistent with this estimate. It is evident that this result is not an especially large dilution and the concentration is much larger than might be estimated based on a model of diffusion from a sphere.