## Appendix A  
The turbulent kinetic energy transport equation and turbulent dissipation rate transport equation of the rke model are  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial\big{(}\rho ku_{j}\big{)}}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{ \mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+G_{k}+G _{b}-\rho\varepsilon, \tag{1}\]  
\[\frac{\partial\big{(}\rho\varepsilon\big{)}}{\partial t}+\frac{\partial \big{(}\rho\varepsilon u_{j}\big{)}}{\partial x_{j}}=\frac{\partial}{ \partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}} \bigg{)}\frac{\partial\varepsilon}{\partial x_{j}}\Bigg{]}+\rho C_{1}S \varepsilon-\frac{\rho C_{2}\varepsilon^{2}}{k+\sqrt{\varepsilon v}}+C_{1 \varepsilon}\frac{\varepsilon}{k}C_{3\varepsilon}G_{b}, \tag{2}\]  
where the turbulent Prandtl number for \(k\) is \(\sigma_{k}=1.0\), the turbulent Prandtl number for \(\varepsilon\) is \(\sigma_{\varepsilon}=1.2\), \(C_{1}=\max\Big{[}0.43,\frac{\eta}{\eta+5}\Big{]}\), \(\eta=S_{\varepsilon}^{k}\), \(S=\sqrt{2S_{ij}S_{ij}}\), mean strain-rate tensor \(S_{ij}=\frac{1}{2}\Big{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\Big{)}\), \(C_{2}=1.9\), \(C_{1\varepsilon}=1.44\). The turbulent kinetic energy \(k\) generated by the mean velocity gradients \(G_{k}\) and the turbulent kinetic energy \(k\) generated by the buoyancy \(G_{b}\) are calculated as follows:  
\[G_{k}=\mu_{t}S^{2}, \tag{3}\]  
\[G_{b}=\gamma g_{i}\frac{\mu_{t}\partial T}{\text{Pr}_{t}\partial x_{i}}, \tag{4}\]  
where the turbulent Prandtl number of energy is \(\text{Pr}_{t}=0.85\) and thermal expansion coefficient is \(\gamma=-\frac{1}{\rho}\Big{(}\frac{\partial\rho}{\partial T}\Big{)}_{p}\). \(C_{3\varepsilon}\) represents the degree to which turbulent dissipation rate \(\varepsilon\) is affected by buoyancy, which is calculated as follows:  
\[C_{3\varepsilon}=\tanh\Bigl{(}\Big{|}\frac{w}{u}\Big{|}\Bigr{)}, \tag{5}\]  
where \(w\) is the component of the velocity parallel to the gravity vector and \(u\) is the component of the velocity perpendicular to the gravity vector. The turbulent viscosity \(\mu_{t}\) is calculated as follows:  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon}, \tag{6}\]where \(C_{\mu}\) is not a constant, \(C_{\mu}=\left(A_{0}+A_{5}\frac{\mu_{t}k}{\varepsilon}\right)^{-1}\), \(U^{*}=\sqrt{S_{ij}S_{ij}+\Omega_{ij}\Omega_{ij}}\), mean rotation rate tensor \(\Omega_{ij}=\frac{1}{2}\Big{(}\frac{\partial u_{ij}}{\partial x_{j}}-\frac{ \partial u_{ij}}{\partial x_{i}}\Big{)}\), \(A_{0}=4.04\), \(A_{5}=\sqrt{6}\cos(\phi)\), \(\phi=\frac{1}{3}\arccos\Big{(}\sqrt{6}\omega\Big{)}\), \(\omega=\frac{S_{ij}S_{ij}}{(\varepsilon/\sqrt{2})^{3}}\). The effective thermal conductivity \(\kappa_{\varepsilon}\) required for the energy Equation (3) is calculated as  
\[\kappa_{\varepsilon}=\kappa+\frac{C_{\mu}\mu_{t}}{\text{Pr}_{t}},\] (A7)  
where \(\kappa\) is thermal conductivity and \(\text{Pr}_{t}=0.85\). The turbulent kinetic energy transport equation and turbulent dissipation rate transport equation of SST model are  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial\big{(}\rho ku_{j}\big{)}} {\partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{ \mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+P_{k}- \beta^{*}\rho k\omega,\] (A8)  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial\big{(}\rho\omega u_{j} \big{)}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+ \frac{\mu_{t}}{\sigma_{\omega}}\bigg{)}\frac{\partial\omega}{\partial x_{j}} \Bigg{]}+P_{\omega}-\beta\rho\omega^{2}+2(1-F_{1})\frac{\rho\sigma_{\omega 2}}{\omega}\frac{\partial k}{ \partial x_{j}}\frac{\partial\omega}{\partial x_{j}},\] (A9)  
where \(\beta^{*}\), \(\sigma_{k}\), \(\beta\), \(\sigma_{\omega}\) and \(\sigma_{\omega 2}\) are the model constants; \(\beta^{*}=0.09\), \(\sigma_{k}=1.0\), \(\beta=0.075\), \(\sigma_{\omega}=0.5\), \(\sigma_{\omega 2}=0.856\).