## 2 Computational Model
It was found that all models needed to be calculated for at least 70 cycles to meet this condition (approximately 35 h of computational time). The flow solutions presented in this paper rely on the RANS equations used for turbulence modelling, while the laminar model (see e.g., [21]) is solved for comparisons. The four RANS models tested included: k-\(\varepsilon\), k-\(\omega\) and shear-stress-transport (SST)-k-\(\omega\) and transition-SST provided by ANSYS FLUENT 13.0 [23]. The RANS equations (shown in index notation) are:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}(\rho u_{i})=0, \tag{1}\]  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial}{\partial x_{j}}(\rho u _{i}u_{j})=-\frac{\partial p}{\partial x_{i}}+F_{i}+\frac{\partial}{\partial x _{j}}\big{[}\tau_{ij}\big{]}+\frac{\partial}{\partial x_{j}}\Big{(}-\rho \overline{u_{i}^{\prime}u_{j}^{\prime}}\Big{)}+\frac{\partial}{\partial x_{j }}\Big{(}-\rho\overline{u_{i}^{\prime 2}}\Big{)}, \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho E)+\frac{\partial}{\partial x_{i}}[u_{i}(\rho E +p)]=\frac{\partial}{\partial x_{j}}\Bigg{(}(k)_{eff}\frac{\partial T}{ \partial x_{j}}+u_{i}\big{(}\tau_{ij}\big{)}_{eff}\Bigg{)}, \tag{3}\]  
where  
\[\big{(}\tau_{ij}\big{)}_{eff}=\mu\Bigg{(}\frac{\partial u_{j}}{\partial x_{i} }+\frac{\partial u_{i}}{\partial x_{j}}\Bigg{)}-\frac{2}{3}\mu_{eff}\frac{ \partial u_{k}}{\partial x_{k}}\delta_{ij}. \tag{4}\]  
These have a form similar to the laminar model of Navier-Stokes equations, but the solution variables now represent the ensemble-averaged (or time-averaged) values. The new terms at the end of Equation (2) are the Reynolds stresses representing the turbulence effect. For compressible flow, the fluctuation of density may affect the turbulence. In this case, the transport equations can be interpreted as Favre-averaged Navier-Stokes equations, with velocities representing the mass-averaged values [20,22]. The effective stress tensor \(\big{(}\tau_{ij}\big{)}_{eff}\) represents the stress tensor under the influence of turbulence with effective viscosity, \(\mu_{eff}=\mu+\mu_{t}\), defined as a sum of laminar viscosity, \(\mu\), and turbulent viscosity, \(\mu_{t}\). Similarly, the effective thermal conductivity, \(k_{eff}=k+k_{t}\), is the sum of mean thermal conductivity, \(k\), and turbulent conductivity, \(k_{t}\). The turbulent thermal conductivity is calculated as, \(k_{t}=\mu_{t}c_{p}/Pr_{t}\) and the turbulent Prandtl number, \(Pr_{t}\), has a constant value of 0.85. The eddy viscosity, \(\mu_{t}\), is calculated using the turbulence model. The Reynolds stresses are solved through additional equations provided by the turbulence model. In this study, turbulence is assumed isotropic so that the Boussinesq hypothesis is applicable to relate the Reynolds stresses to the mean velocity gradient as follows [23]:\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\left(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}\left( \rho k_{e}+\mu_{t}\frac{\partial u_{k}}{\partial x_{k}}\right)\delta_{ij}. \tag{5}\]  
The subscripts \(i\), \(j\) and \(k\) refer to the coordinates \(x\), \(y\) and \(z\), respectively. The term, \(k_{e}=\left({u_{1}^{\prime}}^{2}+{u_{2}^{\prime}}^{2}+{u_{3}^{\prime}}^{2} \right)/2\), is known as the turbulent kinetic energy. The Kronecker delta (\(\delta_{ij}=1\) if \(i=j\) and \(\delta_{ij}=0\) if \(i\
eq j\)) is introduced to correctly model the normal component of the Reynolds stress [20]. The RANS models used require additional equations to solve for turbulent viscosity, \(\mu_{t}\), and turbulent kinetic energy, \(k_{e}\), to obtain the Reynolds stresses using Equation (5). The Reynolds stresses are then used to correctly model the momentum equation for the turbulence-affected flow through Equation (2).