## 2 Computational Schemes  
Scale adaptive simulation (SAS) is an improved URANS formulation based on introducing the von Karman scale into the turbulence scale equation [20]. The information provided by the von Karman length-scale allows SAS models to dynamically adjust to resolved structures in a URANS simulation, which results in a LES-like behavior in unsteady regions of the flow field. At the same time, the model provides standard RANS capabilities in stable flow regions. The computational resources consumed by scale adaptive simulation are only half that of the large eddy simulation [20], which is very suitable for engineering applications. However, under the condition of low Mach number flow, SAS cannot trigger the LES-like behavior in the flow region far from the wall due to the flow instabilities loss. In the flow region near the wall, it cannot show the scale resolution ability due to the defects of the RANS equation [21]. Given the advantages and disadvantages of the SAS turbulence model, combined with the widely used standard \(k-\varepsilon\) turbulence model in engineering, according to the literature [22], the SAS equation can be rewritten as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho U_{i}k)}{\partial x_{ i}}=G_{k}-\rho\varepsilon+\frac{\partial}{\partial y}\Bigg{[}\frac{\mu_{t}}{ \delta_{k}}\frac{\partial k}{\partial x_{j}}\Bigg{]}+Y_{M} \tag{1}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho U_{i} \varepsilon)}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{(}\mu_{t} \frac{\partial\varepsilon}{\partial x_{j}}\Bigg{)}+c_{1}\varepsilon\frac{ \varepsilon}{k}G_{k}-c_{2t}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{2}\]where \(k\) is the turbulence kinetic energy, \(\epsilon\) is the dissipation rate, \(\mu_{t}\) is the turbulent viscosity, \(G_{k}\) represents the generation of turbulence kinetic energy, \(G_{k}=-\rho\overline{u_{i}u_{j}}\frac{\partial u_{j}}{\partial x_{i}}\), \(Y_{M}=2\rho eM_{T}^{2}\) is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, \(M_{T}=\sqrt{k/a^{2}}\) is the turbulent Mach number, and \(a\) is the speed of sound. This compressibility modification is available when the compressible form of the ideal gas law is used. The model constants \(c_{1\varepsilon}\), \(\delta_{k}\), and \(\sigma_{\varepsilon}\) have the following default values:  
\(c_{1\varepsilon}=1.44\), \(c_{u}=0.09\), \(\sigma_{k}=1.0\), \(\sigma_{\varepsilon}=1.3\). In Equation (2), the user-defined source term is:  
\[S_{\varepsilon}=0.47\frac{\sqrt{k}}{L_{\
u\kappa}}G_{k}-2\frac{u_{t}}{ \varepsilon}\left(\frac{\partial\varepsilon}{\partial x_{j}}\right)^{2}+5 \frac{u_{t}}{k}\left(\frac{\partial k}{\partial x_{j}}\right)\left(\frac{ \partial\varepsilon}{\partial x_{j}}\right)-\frac{\rho}{3}\left(\frac{ \partial k}{\partial x_{j}}\right)^{2} \tag{3}\]  
The von Karman length-scale is redefined as:  
\[L_{\
u\kappa}=\kappa\frac{\|S\|}{\|U^{\
u}\|} \tag{4}\]  
where \(\kappa=0.41\) is the von Karman constant. The first velocity derivative \(\frac{\partial U}{\partial y}\) is represented in \(L_{\
u\kappa}\) by \(U^{\prime}\), which is equal to \(S_{ij}\), a scalar invariant of the strain rate tensor \(S\): \(U^{\prime}=S=\sqrt{2\cdot S_{ij}S_{ij}}\), \(S_{ij}=\frac{1}{2}(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j} }{\partial x_{i}})\). The second velocity derivative, \(U^{\prime\prime}\), is generalized to 3D using the magnitude of the velocity Laplacian:  
\[U^{\prime\prime}=\sqrt{\frac{\partial^{2}U_{i}}{\partial x_{k}^{2}}\frac{ \partial^{2}U_{i}}{\partial x_{j}^{2}}} \tag{5}\]  
The model also provides direct control of the high wave number damping.