# Smoke Back-Layering Phenomenon under the Combined External Wind and Stack Effects in a Staircase
## 2 Numerical Modeling  
FDS (fire dynamics simulator software) is regarded as a practical tool for simulating fire-induced thermodynamics and fire science. The model has been subjected to numerous validations and calibrations in high-rise building fire simulations [17; 18; 19]. FDS (version 6.7.3) is employed in the current work. FDS solves the Navier-Stokes equations with a low-Mach number formulation for thermally driven flow. It includes both a LES (Large Eddy Simulation) model and a DNS (Direct Numerical Simulation) model. LES is adopted as the turbulence model. Deardorff turbulent viscosity is used in FDS 6 by default,  
\[\mu_{LES}/\rho=C_{v}\Delta\sqrt{k_{\text{sgs}}} \tag{1}\]  
where \(C_{v}=0.1\). Based on scale similarity, the subgrid scale (sgs) kinetic energy is taken from an algebraic relationship [20]. The LES filter width is taken as the geometric mean of the local mesh spacing in each direction, \(\Delta=\left(\delta x\delta y\delta z\right)^{1/3}\). In an LES calculation, the convective heat flux can be expressed as  
\[q_{c}{}^{\prime\prime}=h(T_{g}-T_{w}) \tag{2}\]where \(h\) is the convective heat transfer coefficient, which is based on a combination of forced and natural convection correlations:  
\[h=\max\biggl{[}C(T_{g}-T_{w})^{1/3},\frac{k}{L}Nu,\frac{k}{\delta n/2}\biggr{]} \tag{3}\]  
where \(C\) is an empirical coefficient for natural convection, \(L\) is a characteristic length related to the size of the physical obstruction, \(k\) is the thermal conductivity of the gas, \(N_{u}\) is the Nusselt number, and \(\delta n\) is the gas phase cell size. The last term of the \(h\) formula aims to ensure that the LES convective heat flux converges to the DNS heat flux at fine resolutions. The Radiative Transfer Equation (RTE) is solved using the Finite Volume Method (FVM). A radiation fraction of 0.35 is prescribed as a lower bound in order to limit the uncertainties in the radiation calculation induced by uncertainties in the temperature field. Heat losses to the walls are calculated by solving the 1-D Fourier's equation for conduction [18; 20]. More details can be seen in [20]. The simulation model is a high-rise building with 12 stories, as shown in Figure 1, consisting of the staircase, atrium, and compartment. The building is 36.6 m high. The ground floor is 3.6 m high, and the other floors are 3.0 m high. The cross-sections of staircase, atrium and compartment are, respectively, 4.5 m (L) \(\times\) 3 m (W), 2.4 m (L) \(\times\) 2.4 m (W), and 2.4 m (L) \(\times\) 2.4 m (W). Three doors (D1-D3) connect the staircase, atrium, compartment, and surroundings. The sizes of the doors are all 1.8 (H) \(\times\) 1.2 m (W). The window located on the back of the staircase is 3 m high by 2 m wide. The doors (D1-D3) on the eighth floor (8F) and window are always open. The doors (D1-D3) on the first (1F), third (3F), fifth (5F), and seventh (7F) floors remain open, respectively, and other doors remain closed. The internal lining of the building is specified as \"CONCRETE\", and its density, specific heat, and conductivity are 2200 kg/m\({}^{3}\), 0.88 kJ/(kg K), and 1.2 W/(m K) [20]. The ambient temperature was 20 \({}^{\circ}\)C. The external wind is provided by a wind screen and blows into the staircase from the window on the top floor. Wind velocity ranges from 0 to 5.5 m/s. The effects of open stair doors below the fire floor, heat release rate of fires, and the external wind velocity are considered. A vertical column of 23 thermocouples are located at the centerline of the staircase, as shown in Figure 1. It is mainly used to determine the smoke back-layering time and distance. The fire locates on 8F and at the center of the compartment. The reaction was specified as propane gas fire. The HRRs are designed as 2, 3, and 4 MW.