## 2 Problem Formulation
### Computational Approach and Modeling  
Reynolds-averaged Navier-Stokes code for steady-state reacting computations was used to investigate the flow and combustion characteristics of the four-point LDI combustor [40; 41; 42]. The conservation equations of mass, momentum, energy and species mass are expressed for steady flow as:  
\[div(\rho\overline{u})=S_{m} \tag{1}\]  
\[\frac{\partial}{\partial x_{j}}(\rho\overline{u_{i}u_{j}})=-\frac{\partial P} {\partial x_{i}}+\frac{\partial}{\partial x_{j}}(\mu\frac{\partial u}{\partial x }-\rho\overline{u_{i}u_{j}})+\overline{F_{i}} \tag{2}\]  
\[\
abla\overline{u}[(\rho E+p)]=\
abla(k_{eff}\
abla T-\sum_{j}h_{j}j_{j}+ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline}}}}}}}}}}}}}} }\overline{u})+S_{h} \tag{3}\]  
\[\
abla(\rho\overline{u}Y_{i})=-\overline{\
abla J_{i}}+R_{i}+S_{i} \tag{4}\]  
where,\(\rho\), \(\overline{u}\) and \(S_{m}\) represent density, fluid velocity vector and flow mass through control volume, \(P\) and \(\overline{F_{i}}\) represent static pressure and all the power on the elemental volume; \(k_{eff}\), \(j_{j}\) and \(S_{h}\) represent effective coefficients of heat conduction, diffusion mass of species \(j\) and reaction heat and other volume heat source; \(Y_{i}\), \(J_{i}\), \(R_{i}\) and \(S_{i}\) represent mass fraction, diffusion flux, net generation rate and extra generation rate by source defined by dispersed phase and user for species \(i\), respectively.  
In this paper, the realizable \(\kappa-\varepsilon\) closure turbulence model is considered to solve the turbulence problem of the four-point LDI combustor. The equations of turbulent kinetic energy \(K\) and turbulent dissipation \(\varepsilon\) could be written as  
\[\frac{\partial}{\partial x_{i}}(\rho ku_{i})=\frac{\partial}{\partial x_{j}}[( \mu+\frac{\mu_{\tau}}{\sigma_{k}})\frac{\partial k}{\partial x_{j}}]+G_{k}+G_{ b}-\rho\varepsilon-Y_{M}+S_{k} \tag{5}\]  
\[\frac{\partial}{\partial x_{i}}(\rho\varepsilon u_{i})=\frac{\partial}{ \partial x_{j}}[(\mu+\frac{\mu_{\tau}}{\sigma_{\varepsilon}})\frac{\partial \varepsilon}{\partial x_{j}}]+\rho C_{1}E\varepsilon-\rho C_{2}\frac{ \varepsilon^{2}}{k+\sqrt{\
u\varepsilon}}+G_{1\varepsilon}\frac{\varepsilon} {k}C_{3\varepsilon}G_{b}+S_{\varepsilon} \tag{6}\]  
where, \(G_{k}\) is turbulent kinetic energy generated by the average velocity gradient, \(G_{b}\) is turbulent kinetic energy generated by buoyancy. The \(G_{k}\) and \(G_{b}\) could be written as  
\[G_{k}=\mu_{\tau}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}})\frac{\partial u_{i}}{\partial x_{j}} \tag{7}\]  
\[G_{b}=\beta g_{i}\frac{\mu_{\tau}}{\Pr_{\tau}}\frac{\partial T}{\partial x_{i}} \tag{8}\]  
where, \(C_{1\varepsilon}\), \(C_{2}\), \(\sigma_{k}\) and \(\sigma_{\varepsilon}\)are the constants of 1.44, 1.9, 1.0 and 1.2, respectively.  
By solving the RANS equations coupled with the chemistry, the numerical simulations of the combustion process in the four-point LDI combustor induced by the flame for the H\({}_{2}\)/Air mixture were performed. The eddy dissipation concept (EDC) combustion model [40; 43] was used to solve the interaction of the turbulence and chemistry. The reaction consists of two processes. One of these processes is that the reaction is induced by collision in turbulence micro-scale structure which is controlled by chemical kinetics. The other process is that the mixing time is more than the reaction time in the vortex clouds and the reaction rate is governed by the mixing rate. The H\({}_{2}\)/Air reaction mechanism is a reduced mechanism which consists of 11 species (H\({}_{2}\), O\({}_{2}\), OH, H\({}_{2}\)O\({}_{2}\), HO\({}_{2}\), H\({}_{2}\)O, H, N, NO, O, N\({}_{2}\)) and 23 lumped reaction steps [40; 44; 45; 46; 47]. A splitting operator method was used to separately treat the aerodynamic process and the chemical process in the LDI combustion simulation.