Fluid dynamics simulations show that facial masks can suppress the spread of COVID-19 in indoor environments

The Coronavirus disease outbreak of 2019 has been causing significant loss of life and unprecedented economical loss throughout the world. Social distancing and face masks are widely recommended around the globe in order to protect others and prevent the spread of the virus through breathing, coughing, and sneezing. To expand the scientific underpinnings of such recommendations, we carry out high-fidelity computational fluid dynamics simulations of unprecedented resolution and realism to elucidate the underlying physics of saliva particulate transport during human cough with and without facial masks. Our simulations: (a) are carried out under both a stagnant ambient flow (indoor) and a mild unidirectional breeze (outdoor); (b) incorporate the effect of human anatomy on the flow; (c) account for both medical and non-medical grade masks; and (d) consider a wide spectrum of particulate sizes, ranging from 10 micro m to 300 micro m. We show that during indoor coughing some saliva particulates could travel up to 0.48 m, 0.73 m, and 2.62 m for the cases with medical-grade, non-medical grade, and without facial masks, respectively. Thus, in indoor environments either medical or non-medical grade facial masks can successfully limit the spreading of saliva particulates to others. Under outdoor conditions with a unidirectional mild breeze, however, leakage flow through the mask can cause saliva particulates to be entrained into the energetic shear layers around the body and transported very fast at large distances by the turbulent flow, thus, limiting the effectiveness of facial masks.


Introduction
As the Coronavirus Disease of 2019 (COVID-19) pandemic, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is spreading around the globe, it is causing devasting loss of life, threatens to topple national health care systems, and wreaks havoc on the global economy. The World Health Organization (WHO), the United States Centers for Disease Control (CDC), and other public health organizations are recommending social distancing and the use of facial coverings as two of the most important weapons to curb the spread of the pandemic [1][2][3][4][5] . However, recommended social distancing guidelines vary considerably with the WHO recommending at least 1 m while the CDC recommending at least 1.8 m of separation 5 . There is now mounting evidence that such guidelines are based on outdated scientific studies carried out decades ago 6 , which, among others, have overlooked the physics of respiratory emissions, the range of droplet size distribution such emissions produce, the turbulent flow processes that transport respiratory droplets and the potential effects of ambient wind conditions on such transport [6][7][8][9][10][11][12][13][14] . Several experimental and computational studies are now beginning to emerge, seeking to fill this major knowledge gap and yield improved science-based physical distancing guidelines (see Refs. 5 and 7) for excellent recent reviews).
Experimental studies have suggested, for instance, that under certain conditions, pathogen-bearing droplets emitted during violent expiratory events (coughing and sneezing) can travel up to 7-8 m [8][9][10][11] . Dbouk and Drikakis 12,13 employed a statistically stationary Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) model to simulate droplet transport during coughing and showed that while saliva droplets from coughing travel less than 2 m under stagnant ambient air, they could travel very fast and for several meters downwind when a mild ambient breeze is present as a 3 result of turbulence transport. In another recent study, Cummins et al. 14 investigated the dynamics of various size individual droplet dispersion in a source-sink pair flow filed using a simplified numerical model and showed that unlike the large droplets, the intermediatesized droplets have a minimal dispersion range. Studies are also emerging, seeking to quantify the effect of facial masks as a means to limit the transport of pathogen-bearing droplets. Fischer et al. 9 designed a low-cost experimental method to probe the respiratory droplet transmission of regular speech through different facial masks. The results showed that N95 respirators, surgical, and cotton masks strongly reduce respiratory droplet count, whereas the bandana did not show the same effectiveness. Verma et al. 10 emulated coughs using the vaporized liquid mixture to visualize the effectiveness of different mask materials and designs. Their study demonstrated that uncovered coughs could spread farther than the recommended 1.8 m social distance, while homemade and commercial facial masks can significantly curtail the speed and spreading range of the saliva particulates, even though with some leakage around mask edges.
Coughing and sneezing give rise to jet-like flows whose Reynolds number is approximately 10,000 11 . At this Reynolds number, the flow becomes fully threedimensional and turbulent and is dominated by energetic coherent structures spanning a range of scales. While the critical role of turbulence in transporting expiratory droplets is well recognized [11][12][13] , studies resolving the scales of turbulence generated during an expiratory event as it interacts with facial coverings and/or ambient turbulence have yet to report in the literature. Moreover, no studies have been reported that take into account the presence of the body of the human that generates the droplets. While for expiration in stagnant air such simplification may be adequate, when the ambient wind is present, neglecting the presence of the human is a gross over-simplification as the body generates additional turbulence and sheds energetic coherent structures that could significantly impact the transport of respiratory droplets by the flow. In this work, we employ a highfidelity large-eddy simulation (LES) CFD model to study the transport of droplets of various sizes generated during coughing with and without facial masks and under varying ambient flow conditions. Our work is significant because we: 1) employ unprecedented numerical resolution to carry out the LES, using over three orders of magnitude finer grids that previous studies (e.g., see Refs. 12 and 13) and resolving the energy of turbulent eddies 4 down to 0.5 mm in size; 2) incorporate into the simulation the full three-dimensional geometry of facial coverings; 3) take into account the entire anatomy of the human that generates the cough; and 4) report simulations both for stagnant ambient air and a mild breeze.
We employ the open-source Virtual Flow Simulator massively parallel CFD code, which has been developed by our group and applied in the past extensive to a wide range of biological, environmental, and engineering turbulent flows [15][16][17][18][19][20][21][22][23]  (c) indoor cough with a non-medical grade facial mask; (d) outdoor cough without a facial mask; and (e) outdoor cough with a non-medical grade facial mask.

Governing equations
We employ a fully coupled aerodynamics and saliva transport model capable of carrying out LES of stratified, turbulent flows with the presence of arbitrarily complex geometries, such as a face, mouth, or facial masks 15 . We solve the spatially-filtered continuity and Navier-Stokes equations with the Boussinesq assumption to simulate the incompressible, stratified, turbulent flow of dilute air-saliva mixture using a second-order accurate finite-5 difference fractional step approach, which satisfies the discrete continuity equation to machine zero at each time step. The Navier-Stokes equations with the Boussinesq assumption in non-orthogonal, generalized, curvilinear coordinates { }, and compact tensor notation (repeated indices imply summation) are as follows ( = 1,2, or 3 and = 1, 2, and 3): where are the transformation metrics, is the Jacobian of the transformation, is the contravariant volume flux, is the Cartesian velocity component, is the pressure, is the Reynolds stress tensor for the LES model, is the Kronecker delta, is the dynamic viscosity, is the gravitational acceleration, 0 is the background density (the density of air in our case), ̅ is the density of the air-saliva mixture, and ( = 1, 2, and 3) is the body force introduced by the facial mask computed using the mask model, as described below.
Our method employs second-order finite differencing numerics, the dynamic Smagorinsky model 25 for modeling the unresolved subgrid scales of turbulence in the LES. The pressure Poisson equation is solved using an algebraic multigrid acceleration along with generalized minimal residual method (GMRES) solver and the discretized momentum equation is solved using a matrix-free Newton-Krylov method. The density of the air-saliva mixture is modeled as where is the volume fraction of the saliva particulates and is the density of saliva particulates, which is considered to be equal to that of the water (=1000 / 3 ). For dilute air-saliva mixture in which the volumetric saliva concentration barely exceeds O(0.01), the saliva particulates concentration is modeled as an active scalar whose transport is computed using the following convection-diffusion equation 17 : is the contravariant volume flux of saliva particulates in the vertical direction due to the settling velocity ( ) of the particulates, is the laminar Schmidt number (=1) (21), is the turbulent Schmidt number (=0.75) 21 , and is the eddy viscosity obtained from the LES model. Finally, it should be noted that in this study, we do not consider the evaporation of saliva during and after coughing, as described in Bourouiba et al. 6 . Although evaporation can play a role in the transport of saliva particulates 6 , dropping its effects should not affect the generality of our findings as they pertain to the relative effects of indoor and outdoor transport with and without facial masks.
In fact, we could reasonably argue that our simulations may provide more conservative estimates of transport as droplets do not evaporate.

Numerical model of the human anatomy and facial masks
We employ the sharp interface curvilinear immersed boundary (CURVIB) method with wall model reconstruction 15 to simulate the anatomy of the human body (Fig. 1a) while a diffused interface immersed boundary method is used to incorporate the presence of facial masks and their different porosities. Namely, the facial mask is modeled by applying a drag force on the unstructured grid nodes used to discretize the three-dimensional geometry of the mask (see Fig. 1d). The so prescribed force is then distributed to the cartesian grid nodes of the background grid via a smoothed discrete delta function, which is used as the kernel for transferring information between the two grids. That is, the facial mask momentum drag force term (i.e., in Eqn. 2) is given by: where is the drag coefficient, is the projected area of the facial mask, is the local cartesian velocity vector, and Δ is the smoothed discrete delta function. The approach we employ for implementing the drag force over the mask geometry is essentially identical to that presented in Yang and Sotiropoulos 26 for prescribing forces over wind turbine structures. The geometry of the two facial masks, with an average thickness of 2.2 mm, were created using an open-source software, Blender (www.blender.org) (Fig. 1b,c). The details of the numerical method and validation studies for jet-like flows, which resemble the cough, are documented extensively elsewhere (e.g., see Refs. 8-16).

Results
We report a series of high-fidelity numerical simulation results of the aforementioned five

Cough saliva particulates transport under indoor conditions without a facial mask
In Figs. 2 and 3, we illustrate the instantaneous simulation results of a cough without a facial mask and in a room with stagnant air conditions for saliva particulate sizes of 10 µm 8 (Fig. 2), 40 µm (Fig. 3), and 80 µm (Fig. 3). For the sake of brevity, the results obtained for the rest of the particulate sizes are not presented here but can be made available upon request. As seen in Fig. 3e-h, the 80 µm saliva particulates settle down soon after they are emitted, reaching a maximum distance of 1.13 m from the person. The finer particulates, however, travel farther with the 10 µm particulates traveling the farthest away from the person (Fig. 2). The 10 µm saliva particulates constitute less than 5% of the saliva emitted during coughing [11][12][13] , however, they are large enough to carry SARS-CoV-2 viruses. Thus, to be conservative, in the rest of the study, we focus our attention on the simulation results of the 10 µm saliva particulates. We continued the numerical simulations until the forward momentum of the 10 µm saliva particulates reached machine zero (i.e., 10 8 ) (Fig. 2).

Cough saliva particulates transport under indoor conditions with facial masks
In Fig. 4, we plot the instantaneous simulation results of the spreading of 10 µm saliva particulates after coughing with the medical and non-medical grade facial masks. As seen in this figure, both mask types act as effective sinks of forward momentum and limit the spreading of the saliva particulates by both re-directing portions of the cough in the vertical direction and dissipating its forward momentum. Vortices "B" and "T" in Fig. 4m,n mark the vertical re-direction of the cough towards the mask openings at the bottom (anterior 9 neck) (see vortices "B" in Fig. 4m,n) and top (over the top of the nose), respectively. The portion of the cough flow that passes directly through the fabric of the facial masks transporting some of the particulates forward is marked by vortices "P1," "P2," and "P3" in Fig. 4m,n. Since the medical-grade facial mask (Fig. 4a-f,m) has a greater drag coefficient than the non-medical grade mask (Fig. 4g-l,n), the cough flow rate passing through the medical-grade mask seems to be about 24% smaller than that for the non-

Saliva transport during coughing under outdoor conditions
In this section, we simulate the same cough processes without a facial mask and with the non-medical grade facial mask by replacing the background flow condition of the indoor stagnant air with a turbulent background flow condition, which corresponds to a unidirectional mild breeze with mean-flow velocity of 4.5 m/s. These simulations also incorporate and resolve the full anatomical details of the person. Before initiating the cough in the simulation, we first simulate the mild breeze that flows from behind the person and into the computational domain to obtain the fully developed turbulent flow past the human body. As the incoming flow is diverted around and over the body, a complex turbulent flow arises dominated by energetic three-dimensional shear layers shed from the body and a three-dimensional separation zone in the wake of the person (i.e., the front side of the body) (see Fig. 6). Once the turbulent flow around the person reaches statistical equilibrium, we initiate in the simulation the cough process to simulate saliva transport with and without facial masks in an ambient environment with a mild turbulent breeze flow. A snapshots of the so computed instantaneous flow fields are shown in Fig. 6.
As seen in Fig. 7a- Furthermore, as seen in Fig. 7f-j, the saliva transport process for the case with the facial mask is almost identical to that of the case without a facial mask. This is because, and as we discussed in the previous section, facial masks exhibit leakage and divert some of the saliva particulates upward toward the top of the head where particulates can escape the recirculation zone as they get entrained by the fast traveling outer turbulent flow. Once this happens and saliva particulates have escaped into the ambient flow, they can travel forward very rapidly, also reaching approximately 2.2 m ahead of the person in less than a second. As seen in Fig. 7j, the transport of the saliva particulates in the case with the facial mask occurs very rapidly and as fast as that of the case without the facial mask. Therefore, our results show that, unlike indoor conditions, the use of facial masks in outdoor conditions (with a unidirectional and forward blowing mild breeze) cannot limit the transport of the fine saliva particulates during the cough as a result of the entrainment of mask leakage flow by the fast traveling ambient turbulent flow.

Conclusion
We carried out high-fidelity LES on a computational grid that is three orders of magnitude finer than previously reported simulations to study saliva transport during coughing. We Finally, we should note that because of the high computational cost of the simulations, we limited our scenarios for the outdoor conditions to a unidirectional mild breeze. Our numerical method, however, can be used to computationally investigate a variety of ambient flow conditions as well as other mask types and designs.

Acknowledgments
This work was supported by grants from the National Science Foundation (EAR-0120914) and a sub-award from the National Institute of Health (2R44ES025070-02). The first author would like to thank Aram Khosronejad for the proofreading and her contributions to enhance the readability of the paper.

Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.   (19) with the medical and non-medical facial masks, respectively. The "P1" to "P3" vortices in (m) and (n) mark the portion of the cough flow that passes through the fabric of the facial masks, while the "T" and "B" vortices illustrate the cough flow that is re-directed toward the top and bottom of the facial masks, respectively.  particulates in these cases, it is clear that they will continue propagating for many meters away from the source before they begin to settle to the ground.