# Analysis of Flow Field Characteristics in the Three-Phase Jet Fire Monitor Head
## 2 Materials and Methods
### Numerical Simulation Methods  
The study of the fire monitor jet and part of the IFF flow involves the two-phase flow of air and water, so it is necessary to use the VOF model under a fixed Eulerian mesh to track the air-water interface, and the VOF transport equations are as follows:  
\[\frac{\partial\gamma}{\partial t}+(\mathbf{u}\cdot\
abla)\gamma=0 \tag{1}\]  
where \(\gamma\) is the fractional function, \(\gamma=0\) indicates that the flow field does not contain air, \(\gamma=1\) indicates that there is only air in the flow field, \(0<\gamma<1\) indicates that the flow field contains both air and water phases, and there is a gas-liquid interface in the grid. The VOF model solves a single momentum equation over the entire solution domain, and the velocity field of the solution is shared by two phases, air and water, with the following mass and momentum equations:  
\[\frac{\partial\overline{\rho}}{\partial t}+\
abla\cdot(\overline{\rho}\mathbf{u} )=0 \tag{2}\]  
\[\frac{\partial}{\partial t}(\overline{\rho}\mathbf{u})+\
abla\cdot(\overline{\rho }\mathbf{u}\mathbf{u})=-\
abla p+\
abla\cdot\left[\overline{\mu}\Big{(}\
abla\mathbf{u}+ \
abla\mathbf{u}^{T}\Big{)}\right]+\overline{\rho}g+F_{s} \tag{3}\]  
where \(g\) is the gravitational acceleration, \(\mathrm{m}/\mathrm{s}^{2}\); \(p\) is the pressure, \(\mathrm{Pa}\); \(\overline{\rho}\) is the equivalent density, \(\mathrm{kg}/\mathrm{m}^{3}\); \(\overline{\mu}\) is the equivalent viscosity, \(\mathrm{Pa}\cdot\mathrm{s}\). \(F_{s}\) is the interfacial force at the gas-liquid interface generated by the interfacial tension, \(\mathrm{N}/\mathrm{m}^{3}\). \(\overline{\rho}\) and \(\overline{\mu}\) are denoted as follows:  
\[\overline{\rho}=\gamma\rho_{a}+(1-\gamma)\rho_{w} \tag{4}\]  
\[\overline{\mu}=\gamma\mu_{a}+(1-\gamma)\mu_{w} \tag{5}\]  
where \(\rho_{a}\) is the density of air, \(\mathrm{kg}/\mathrm{m}^{3}\), \(\mu_{a}\) is the viscosity of air, \(\mathrm{Pa}\cdot\mathrm{s}\); and \(\mu_{w}\) is the viscosity of water, \(\mathrm{Pa}\cdot\mathrm{s}\). \(F_{S}\) is denoted as the following:  
\[F_{S}=\sigma_{f}\kappa\mathbf{n} \tag{6}\]  
Figure 1: Diagram of TPJFM head dimensions: (**a**) three-phase jet fire-fighting vehicle; (**b**) three-phase jet fire monitor; (**c**) three-phase jet fire monitor head model; (**d**) diagram of three-phase jet fire monitor head dimensions. where \(\sigma_{f}\) denotes the two-phase interfacial tension, N/m; \(\kappa\) denotes the interfacial curvature, m\({}^{-1}\); and \(n\) denotes the unit vector perpendicular to the two-phase interface. The internal water flow field of the TPJFM head is extracted as the computational domain. The head's water inlet is selected as the computational domain's inlet, and the nozzle outlet is the outlet. All other surfaces of the computational domain are considered wall surfaces. The polyhexcore mesh division method is used, with local densification applied to regions such as the nozzle outer wall, powder nozzle, and supporting blade fillets. The grid division of the IFF of the TPJFM head is shown in Figure 2.  
In Fluent 2020R2, boundary conditions are set as follows: velocity inlet at 12.8 m/s, pressure outlet at atmospheric pressure, and wall surfaces set as no-slip walls. A pressure-based steady-state calculation model is used, with the realizable k-epsilon model and enhanced wall functions. The VOF model is employed, setting air and water as the two phases. Convergence residuals for each variable are set at 1 \(\times\) 10\({}^{-5}\), and iterations stop when all parameters converge below the set residual values.