Growth of respiratory droplets in cold and humid air

The ambient conditions surrounding liquid droplets determine their growth or shrinkage. However, the precise fate of a liquid droplet expelled from a respiratory puff as dictated by its surroundings and the puff itself has not yet been fully quantified. From the view of airborne disease transmission, such as SARS-CoV-2, knowledge of such dependencies is critical. Here, we employ direct numerical simulations (DNS) of a turbulent respiratory vapor puff and account for the mass and temperature exchange with respiratory droplets and aerosols. In particular, we investigate how droplets respond to different ambient temperatures and relative humidity ($\mathrm{RH}$) by tracking their Lagrangian statistics. We reveal and quantify that in cold and humid environments, as there the respiratory puff is supersaturated, expelled droplets can first experience significant growth, and only later followed by shrinkage, in contrast to the monotonic shrinkage of droplets as expected from the classical view by Wells in 1934. Indeed, cold and humid environments diminish the ability of air to hold water vapor, thus causing the respiratory vapor puff to supersaturate. Consequently, the supersaturated vapor field drives the growth of droplets that are caught and transported within the humid puff. To analytically predict the likelihood for droplet growth, we propose a model for the axial RH based on the assumption of a quasistationary jet. Our model correctly predicts supersaturated RH conditions and is in good quantitative agreement with our DNS. Our results culminate in a temperature-RH map that can be employed as an indicator for droplet growth or shrinkage.

Figure 1

Comparison of our employed initial droplet size distribution to that from Ref. [13]. Here, 5000 droplets are included in total. Note that a one-to-one comparison is not possible because the experimental measurements are taken after some time after expulsion, whereas our droplet distributions are prescribed as initial conditions for our DNS.

Figure 2

Flow visualization snapshots from our direct numerical simulations of water droplets in a warm humid puff in ambient air at (a) ${\theta }_{\text{amb}}=30{}^{\circ }\mathrm{C}$ and (b) ${\theta }_{\text{amb}}=10{}^{\circ }\mathrm{C}$, both at ${\text{RH}}_{\text{amb}}=90\%$. Corresponding movies can be seen in Movies S1 and S2 [48]. The snapshots show (i) vertical 2D planes of the local $\text{RH}$ fields, (ii) the instantaneous droplets spatial distribution, (iii) the heavy large droplets which already fell on the ground, and (iv) the instantaneous droplet size histograms vs distance. The local $\text{RH}$ planes are taken from the vertical midplane of the puff and are plotted on the background for clarity. Droplets are color coded by their instantaneous sizes. Initial droplet sizes are prescribed with a distribution similar to Ref. [13] (plotted in Fig. 1) and are injected evenly in time with the same local inflow velocity. The initial temperature of the droplets and puff is $34{}^{\circ }\mathrm{C}$. Both snapshots are taken at 0.6 s corresponding to the cutoff time of the puff. In the colder conditions of (b), the expelled humid puff oversaturates (seen as the lighter colored $\text{RH}$ field and visible in Movie S2), which in turn dictates the growth of smaller droplets caught within the puff. Correspondingly, the droplet counts are confined within a narrower range of sizes in (b) as compared to (a) [see Fig. 3].

Figure 3

Histogram of droplet sizes at (a) $t=0.6\mathrm{s}$ and (b) 1.2 s for ${\theta }_{\text{amb}}=10,20,30{}^{\circ }\mathrm{C}$. As a point of reference, the solid lines denote the initial droplet size distributions based on the experimental measurement by Ref. [13] (see Appendix). The bin size of the histogram is $2\mu \mathrm{m}$. At $t=0.6\mathrm{s}$ (a), the peak of the histogram for ${\theta }_{\text{amb}}=30{}^{\circ }\mathrm{C}$ marginally shifts down, whereas the overall histogram for ${\theta }_{\text{amb}}=10{}^{\circ }\mathrm{C}$ remains roughly similar to the initial distribution. The gap is more pronounced at $t=1.2\mathrm{s}$ (b): For ${\theta }_{\text{amb}}=30{}^{\circ }\mathrm{C}$, the peak of the histogram is halved from the initial peak, whereas in contrast, for ${\theta }_{\text{amb}}=10{}^{\circ }\mathrm{C}$, the histogram still remains roughly unchanged. The relative invariance of the histogram for ${\theta }_{\text{amb}}=10{}^{\circ }\mathrm{C}$ is due to the droplet growth dictated by the supersaturated vapor puff, which is in turn determined by cold and humid ambient conditions, as described in the main text.

Figure 4

Lagrangian statistics of droplets with initial droplet diameters $d_{\text{init}}\approx 15\mu \mathrm{m}$ for ${\theta }_{\text{amb}}=30$ and $10{}^{\circ }\mathrm{C}$. (a), (b) Change in droplet surface area $d^2/d_{\text{init}}^2$ vs time. (c), (d) Temperature difference between droplets and surrounding fluid, ${\theta }_{\text{drop}}-{\theta }_{\text{fluid}}$, vs time. (e), (f) Local $\text{RH}$ vs time experienced by droplets. The time is shifted to the initial expelled time of each droplet. The ensemble average value is shown by the solid curve. The droplet counts are indicated by the colors (see inset color bars). For ${\theta }_{\text{amb}}=30{}^{\circ }\mathrm{C}$ (a), droplets undergo pure evaporation after being expelled. Conversely, for ${\theta }_{\text{amb}}=10{}^{\circ }\mathrm{C}$ (b), most droplets undergo growth in the initial stages and then evaporate. This growth in (b) is dictated by the supersaturated local $\text{RH}$ values [see (f) and visible in Movie S2], which exceed 100%.

Figure 5

Comparison of the results from our present DNS results (light curves) with those from the model for axial $\text{RH}$ derived from Eq. (9) (smoother darker curves) at $t=0.2\mathrm{s}$ and ${\text{RH}}_{\text{amb}}=90\%$. The data are plotted vs distance normalized by the mouth diameter, $x/d_{\text{mouth}}$. The different colored curves represent data for ${\theta }_{\text{amb}}$ of $10{}^{\circ }\mathrm{C}$ (red), $20{}^{\circ }\mathrm{C}$ (green), and $30{}^{\circ }\mathrm{C}$ (blue), respectively. The model $\text{RH}$ curves all extend beyond $x/d_{\text{mouth}}\approx 25$ since the model assumes a fully developed jet, which is not the case for the present simulated case of a puff. However, the near-field axial model accurately captures the puff, implying quasistationary self-similar jet characteristics.

Figure 6

${\theta }_{\text{amb}}$ vs distance map for the $\text{RH}$ model at $t=0.2\mathrm{s}$. The contours are computed for ${\text{RH}}_{\text{amb}}=90\%$ with the color coded by the local $\text{RH}$. The boundary for $\text{RH}$ oversaturation (local $\text{RH}>100\%$) is shown by the dashed line. With decreasing ${\theta }_{\text{amb}}$, the range with local $\text{RH}>100\%$ (width of the reddish area under the dashed curve) increases, which promotes growth of the smaller droplets because of the longer exposure to supersaturated regions. The most serious supersaturation conditions in this plot are for ${\theta }_{\text{amb}}=5{}^{\circ }\mathrm{C}$ and $x/d_{\text{mouth}}\approx 10–15$, namely beyond a local $\text{RH}$ of $130\%$. For a given ${\theta }_{\text{amb}}$, the location of the maximum is marked by a dashed red line.

Figure 7

Map of maximum local $\text{RH}$ supersaturation values (see Fig. 6) for different ${\theta }_{\text{amb}}$ and ${\text{RH}}_{\text{amb}}$ based on the $\text{RH}$ model. For the case of $R{H}_{\text{amb}}=90\%$ these are the values of the maximal local $\text{RH}$ in Fig. 6 (red dashed curve). The values can be used as a gauge to indicate the tendency of droplets to experience supersaturation, given that the droplets are caught within the turbulent puff. The color is coded by the upper bounds for supersaturation. The plot indicates that in cool and humid environments, small droplets within a coughing puff will tend to grow once expelled. To assess the performance of our model, we also plot our DNS results for droplets with an initial size of $\approx 15\mu \mathrm{m}$ and indicate growth or shrinkage using circle and cross symbols, respectively. (The detailed breakdown is plotted in Fig. 8.) A good agreement between our DNS and the $\text{RH}$ model is seen.

Figure 8

Change of droplet surface area $d^2/d_{\text{init}}^2$ vs time for different ${\text{RH}}_{\text{amb}}$ and ${\theta }_{\text{amb}}$ for initial droplet diameters $d_{\text{init}}\approx 15\mu \mathrm{m}$. The ensemble average value is shown by the solid curve. For humid and cold ambient conditions, droplets tend to grow in the initial stages.