## 2 Numerical Methods  
### Governing Equations  
This work solves the compressible Navier-Stokes equations of an ideal gas in 3D, with Laplacian artificial viscosity and zero bulk viscosity, which are written in Favre filtered form as follows:\[\frac{\partial\overline{\rho}}{\partial t}+\
abla\cdot\left( \overline{\rho}\tilde{\mathbf{u}}\right)=\
abla\cdot\left(\mu_{av}\
abla\overline{ \rho}\right), \tag{1}\] \[\frac{\partial(\overline{\rho}\tilde{\mathbf{u}})}{\partial t}+\
abla \cdot\left(\overline{\rho}\tilde{\mathbf{u}}\tilde{\mathbf{u}}\right)+\
abla\overline {p}-\
abla\cdot\hat{\mathbf{\sigma}}=\
abla\cdot\left(\mu_{av}\
abla(\overline{ \rho}\tilde{\mathbf{u}})\right)-\
abla\cdot\mathbf{\tau}^{gss}+\
abla\cdot\left( \overline{\mathbf{\sigma}}-\hat{\mathbf{\sigma}}\right),\] (2) \[\frac{\partial\overline{\rho}\overline{e}}{\partial t}+\
abla \cdot\left(\overline{\rho}\tilde{\mathbf{u}}\tilde{\mathbf{v}}\right)+\
abla\cdot \left(\overline{\rho}\tilde{\mathbf{u}}\right)-\
abla\cdot\left(\hat{\mathbf{\sigma} }\cdot\tilde{\mathbf{u}}\right)+\
abla\cdot\hat{\mathbf{q}}\] \[\qquad=\
abla\cdot\left(\mu_{av}\
abla(\overline{\rho}\overline{ e})\right)-\
abla\cdot\mathbf{q}^{gss}+\
abla\cdot\left(\overline{\mathbf{\sigma}\cdot \mathbf{u}}-\hat{\mathbf{\sigma}}\cdot\tilde{\mathbf{u}}\right)-\
abla\cdot\left(\overline {\mathbf{q}}-\hat{\mathbf{q}}\right), \tag{3}\]  
with  
\[\overline{p} =\left(\gamma-1\right)\left(\overline{\rho}\overline{e}-\frac{1} {2}\overline{\rho}\tilde{\mathbf{u}}\cdot\tilde{\mathbf{u}}\right),\] \[\hat{\mathbf{\sigma}} =2\tilde{\mu}\tilde{\mathbf{S}}-\frac{2}{2}\tilde{\mu}\delta(\
abla \cdot\tilde{\mathbf{u}})\,,\] \[\tilde{\mathbf{S}} =\frac{1}{2}\Big{(}\
abla\tilde{\mathbf{u}}+(\
abla\tilde{\mathbf{u}})^{ T}\Big{)}\,, \tag{4}\] \[\tilde{\mu}(\tilde{T}) =\mu_{ref}\left(\frac{\tilde{T}}{T_{ref}}\right)^{3/2}\left( \frac{T_{ref}+T_{s}}{\tilde{T}+T_{s}}\right),\] \[\tilde{q} =-\tilde{k}\
abla\tilde{T}\,,\quad\tilde{T}=\frac{\overline{p}}{ \overline{p}R}\,,\quad\tilde{k}=\frac{\tilde{\mu}C_{p}}{\Pr}\,,\]  
where \(\tilde{(\cdot)}\), \(\tilde{(\cdot)}\) stand for a low-pass filtering operator and the Favre-filter operator (\(\tilde{\varphi}=\overline{\rho}\overline{\sigma}/\bar{\rho}\), for any quantity \(\varphi\)), respectively, \(\rho\) is density, \(\mathbf{u}\) the velocity vector, \(e\) the total energy per unit mass, the filtered pressure \(\overline{p}\) is computed by the usual ideal gas equation of state, \(\gamma=C_{p}/C_{V}\) is the ratio between specific heat capacities at constant pressure and constant volume, \(\hat{\mathbf{\sigma}}\) is the resolved viscous stress tensor with \(\tilde{\mathbf{S}}\) being the Favre-filtered strain-rate tensor, \(\delta\) is the unit tensor, \(\hat{\mathbf{q}}\) is the resolved heat fluxes dominated by [PERSON]'s law, \(\tilde{\mu}\) and \(\tilde{k}\) are the corresponding dynamic viscosity and thermal conductivity of the fluid at the filtered temperature \(\tilde{T}\), \(\Pr\) is the Prandtl number and \(R\) is the gas constant, in this paper, \(\gamma=1.4\) and \(P_{r}=0.72\). The first terms in the right-hand side of Equations (1)-(3) are the Laplacian artificial viscosity terms and are presented in detail in Section 2.4. In the right-hand side of momentum equation, \(\mathbf{\tau}^{gss}=\overline{\rho}(\overline{\tilde{\mathbf{u}}}-\tilde{\mathbf{u}} \tilde{\mathbf{u}})\) is the Favre-averaged subgrid scale stress. Note that the difference between \(\overline{\sigma}\) and \(\hat{\mathbf{\sigma}}\) is usually neglected [43]. In the right-hand side of energy equation, \(\mathbf{q}^{gss}\) is the subgrid total energy flux vector defined as  
\[\mathbf{q}^{gss}=\overline{(\rho e+p)\mathbf{u}}-(\overline{\rho}\overline{e}+ \overline{p})\tilde{\mathbf{u}}\,. \tag{5}\]  
The third and fourth terms in the right-hand side of the energy Equation (3) are viscous dissipation and heat diffusive subgrid contributions which can be neglected [44]. The explicit SGS modeling of \(\mathbf{\tau}^{gss}\) and \(\mathbf{q}^{gss}\) is presented in detail in Section 2.3.