# Research on Computational Method of Supersonic Inlet/Isolator Internal Flow
## 2 RANS Computation Method  
### Governing Equations  
This paper uses the finite volume method to solve the compressible RANS equations in integral form:  
\[\frac{\partial}{\partial t}\iint\limits_{V}\mathbf{W}dV+\iint\limits_{ \partial V}\mathbf{F}dS=\iint\limits_{V}\mathbf{H}dV, \tag{1}\]  
where \(\mathbf{W}\) is the vector of the conservative variables, \(\mathbf{F}\) is the normal flux at \(\partial V\), and \(\mathbf{H}\) is the source term. The field variables like density and pressure are decomposed by Reynolds averaging, and considering the compressibility of high \(Re\) number flow, other variables like velocities and temperature are decomposed into Favre-mean and fluctuating components by Favre averaging. To make the equations look more concise, the symbols indicating the above averaging methods are omitted in this paper. Since the compressible RANS introduces a new unknown term in the governing equations: Reynolds stress, so it is necessary to add the corresponding turbulence model to close the equations. RANS contains a variety of turbulence models, which are roughly divided into two categories: the Reynolds stress model and the eddy viscosity model. The latter is widely used due to its simplicity, economy, robustness, easy implementation, and excellent performance. Eddy viscosity models can be further classified as one-equation models (such as the Spalart-Allmaras model [14]), two-equation models (such as the \(k-\varepsilon\)[15] and \(k-\omega\)[16] models) and so on. The \(k-\omega\) SST model [17], which is adopted in this paper, has the best evaluation performance among the current eddy viscosity models. It can adapt to the pressure gradient changes and accurately simulate the flow near boundary layers through the wall function. The model adopts the Wilcox \(k-\omega\) model near the wall and the \(k-\varepsilon\) model at both the edge of the boundary layer and the free shear layer, which combines the advantages of these two turbulence models and is widely used in industry. With the \(k-\omega\) SST turbulence model, the vector \(\mathbf{W}\) is presented as:  
\[\mathbf{W}=\left[\rho,\rho\mathbf{U},\rho E,\rho k,\rho\omega,\right] \tag{2}\]  
where \(\mathbf{U}=\left[u,v,w\right]\) is the velocity vector, \(\rho\) is the air density, \(E\) is the energy, \(k\) is the turbulent kinetic energy, and \(\omega\) is the specific dissipation rate. \(E\) is calculated based on the following equation:  
\[E=c_{v}T+\frac{1}{2}\left(u^{2}+v^{2}+w^{2}\right), \tag{3}\]  
where the \(c_{v}\) is specific heat at constant volume. The flux term \(\mathbf{F}\) of viscous flow consists of two parts:  
\[\mathbf{F}=\mathbf{F}_{c}+\mathbf{F}_{v}, \tag{4}\]where \(\mathbf{F}_{c}\) is the inviscid flux, and \(\mathbf{F}_{v}\) is the viscous flux. Each is defined as follows [18; 19]:  
\[\mathbf{F}_{c}=\left[ \begin{array}{c}\rho(\mathbf{U}-\mathbf{U_{s}})\cdot \mathbf{n}\\ \rho\mathbf{U}(\mathbf{U}-\mathbf{U_{s}})\cdot\mathbf{n}+p\mathbf{n}\\ \rho E(\mathbf{U}-\mathbf{U_{s}})\cdot\mathbf{n}+p\mathbf{U}\cdot\mathbf{n}\\ \rho k(\mathbf{U}-\mathbf{U_{s}})\cdot\mathbf{n}\\ \rho\omega(\mathbf{U}-\mathbf{U_{s}})\cdot\mathbf{n},\end{array} \right] \tag{5}\]  
\[\mathbf{F}_{v}=\left[ \begin{array}{c}0\\ \tau\mathbf{n}\\ \tau\mathbf{U}\cdot\mathbf{n}+(\mu+\mu_{T}/\sigma_{k})\forall k\cdot\mathbf{n} +K_{T}\forall T\cdot\mathbf{n}\\ (\mu+\mu_{T}/\sigma_{k})\forall k\cdot\mathbf{n}\\ (\mu+\mu_{T}/\sigma_{\omega})\,\forall\omega\cdot\mathbf{n},\end{array} \right] \tag{6}\]  
where the \(\mathbf{U_{s}}\) is the velocity vector of control volume's boundary, \(p\) is the pressure, the \(\triangledown\) operator is used to calculate the gradient, \(\mu_{T}\) is the eddy viscosity, \(K_{T}\) is the thermal conductivity, and the positive direction of \(\mathbf{n}\) is defined as pointing to the outside of the control volume. Moreover, the \(\sigma_{k}\) and \(\sigma_{\omega}\) are constants determined by the standard \(k-\omega\) SST turbulence model [17].