## Appendix B Staggered-Grid Finite-Volume Approach  
As described by [PERSON] et al. [34], the quasi-3D model uses a staggered grid with two different basic elements: volumes and connectors. Volumes have associated scalar information such as pressure, density or temperature, as well as the cell volume. Connectors contain vector information, like flow velocity, momentum or the orientation of the connector in space, together with its area. With this configuration, a volume might have attached as many connectors as needed, but a connector will always connect only two volumes. In Figure 14 two volumes and a connector are shown schematically (volumes and connectors actually have an undefined shape). The method is based on the solution of the 3D Euler conservation equations without source term:  
\[\partial\rho/\partial t+\
abla\cdot(\rho\mathbf{u})=0 \tag{10}\]  
\[\partial(\rho\mathbf{u})/\partial t+\
abla\cdot(\rho\mathbf{u}\mathbf{u})=-\
abla p \tag{11}\]  
\[  
velocity from its left and right, \(u_{Ln}\) and \(u_{Rn}\), are shown. As a result, the momentum in the connector follows a one-dimensional momentum equation, whose discretization gives:  
\[(\rho_{c}u_{c}A_{c})^{n+1}=(\rho_{c}u_{c}A_{c})^{n}+(\Delta t/\Delta L)\Big{[} \Big{(}\rho u_{n}^{2}+p\Big{)}_{L}+\Big{(}\rho u_{n}^{2}+p\Big{)}_{R}\Big{]}A_{c} \tag{10}\]  
Here, \(u_{n}\) is the velocity projection onto the direction orthogonal to the connector surface and subscripts \(R\) and \(L\) denote the variables taken from the volumes at the right and left of the connector, respectively. With this simplification, a single equation is solved for each connector instead of three coupled equation for each volume. While usually the number of connectors is higher than the number of volumes in a 3D mesh, the fact that the momentum equations for each direction are not coupled reduces drastically the computation time when compared to a regular 3D method. Once the momentum is calculated, its value is used to compute the mass and energy conservation equations in the next time step. It is worth noticing that some terms in the momentum and energy equations, like density or pressure, must be evaluated in the connectors, but are only calculated in the volumes. To solve this, an upwind approach is adopted, the required values being taken from the right or left volumes, depending on the flow direction. Finally, the momentum in the volumes is computed by distributing the momentum calculated in the connectors among the volumes they connect, taking into account their size. In a uniform 1D mesh, half the momentum is thus assigned to each volume. As the orientation of the connectors in space is known, the resulting momentum vector of a volume can be calculated from the vector sum of the momentum in the connectors:  
\[(\rho_{c}\mathbf{u}V)_{v}^{n+1}=\frac{1}{2}\sum_{c=1}^{N_{c}}(\rho\mathbf{u} _{c}A_{c}\Delta L)_{c}^{n+1} \tag{11}\]  
The resulting method is a second-order accuracy method based on an explicit scheme with a staggered grid, as shown in Figure 12.  
The fact that the resulting scheme offers second-order accuracy, together with the simplifications adopted in the momentum equation, results in nonphysical oscillations, especially in the vicinity of significant pressure gradients. In order to mitigate such overshoots, two different flux limiters have been used here: the momentum diffusion term (MDT) proposed in [35] and the flux-corrected transport (FCT) methodology proposed in [34]. In the case of the MDT, the main goal is to add a diffusion term to the momentum equation so that the mass flux computed at that connector is conveniently limited.