# A Multi-Scale Approach for Modelling Airborne Transport of Mucosalivary Fluid
## 3 Numerical Approximation
### Initial and Boundary Conditions  
In the present work, a 3D computational domain, represented in Figure 1, was employed; it consists of an open air volume starting from the mouth print of a standing coughing person. A length \(L=4\) m, a width \(W=1\) m, and a height \(H=3\) m were adopted for the entire domain, in accordance with [PERSON] and [PERSON] [27]. The reference frame origin, \(O=(0,0,0)\), is placed in the same plane where the mouth print itself is inserted. It is worth noting that \(x\)-axis is aligned with the streamwise direction, i.e., \(0\leq x\leq 4\); \(y\)-axis is the transverse direction: \(-0.5\leq y\leq 0.5\). Lastly, \(z\)-axis is employed for vertical direction, thus \(0\leq z\leq 3\). Moreover, the mouth print shape was approximated to be rectangular with an area of \(2\times 10^{-4}\) m\({}^{2}\) and its center, \(P_{m}\), was placed in the same position selected by [PERSON] and [PERSON]: \(P_{m}=(0,0,1.63)\)[27]. The reason behind this choice is due to the fact a regular shape allows to use a fully orthogonal hexahedral cells. Furthermore, the number of grid cells is fixed to \(5.83\times 10^{6}\) which is the result of our former investigations about the solution space-time convergence [33]. For the sake of discussing the employed boundary conditions (BCs), it is noteworthy to mention that, for all the simulations presented below, both the Eulerian and Lagrangian phases are introduced from the mouth print boundary at \(x=0\) m. The remaining part of the \(y\)-\(z\) plane at \(x=0\) m is such that all the variables have a null gradient through it. The ground is handled as a standard viscous wall. Differently, symmetry conditions are used on lateral boundaries. Zero gradient condition is also set for all the variables at the domain top with the exception of the pressure. Specifically, the pressure is reduced to its hydrostatic level. The \(y\)-\(z\) plane at \(x=4\) m is handled as a physical outflow. By contrast, the pressure is imposed to decrease linearly, starting from the atmospheric pressure level at \(z=0\) m.  
The initial temperature of the carrier fluid is 20 \({}^{\circ}\)C with relative humidity fixed at 50%. The ground is at 25 \({}^{\circ}\)C, while the air and droplets ejected by human mouth are at 34 \({}^{\circ}\)C. No background flow is included in the presented computations. However, initial fields consistent with atmospheric conditions were adopted. These fields are obtained through a preliminary computation which does not provide Lagrangian particles into the domain. It should be noted that cloud evolution is strongly influenced by this conditions [33]. In order to mimic the mass injection from human mouth two different strategies were adopted, see Figure 2. A first approach consists in imposing a stepped velocity inlet at the mouth boundary. A stream-wise velocity inlet step, having an amplitude equal to 8.5 m/s, was applied over 0.12 s. These values were deduced on the basis of measurements carried out by [PERSON] et al. [34]. Figure 1: Computational domain discretization. Secondly, following [PERSON] et al. [35], a sinusoidal approximation of breathing, hereinafter coughing/breathing configuration, is adopted to simulate a real-life condition. In particular, the transient sinusoidal velocity profile presents an amplitude of 1 m/s and a frequency of 0.2 Hz. A velocity peak of 5 m/s was also mounted on the sinusoidal velocity profile in order to reproduce coughing effect as showed by [PERSON] et al. [36]. In both the configurations, carrier fluid and injected particles were subjected to the same velocity condition. Note also that, for the second strategy, particles are injected only during the velocity peak. Moreover, turbulence intensity, Tu, is fixed at 15% and the mixing length is equal to \(7\times 10^{-3}\) at the mouth boundary. As regards saliva droplets, it is important to put in evidence that total mass laden into the domain for a single cough event is 7.7 mg, according to the experimental measurements performed by [PERSON] et al.