## 4 Results and Discussion
### Compressible Channel Flow  
In this section, the compressible turbulent channel flow [61] is considered. The channel flow is a canonical configuration for wall-bounded turbulence which consists of simple boundary conditions and geometry. In this paper, we consider the computation of the turbulent channel flow at \(Re_{b}=\)15,334 and \(M_{b}=1.5\), where \(M_{b}\) and \(Re_{b}\) stand for the bulk Mach number and bulk Reynolds number, respectively, defined by  
\[Re_{b} =2\rho_{b}u_{b}h/\mu_{w} \tag{57}\] \[M_{b} =u_{b}/c_{w} \tag{58}\]  
where \(h\) is half-height of the channel, and  
\[\rho_{b} =\frac{1}{V}\int_{V}\rho\,\mathrm{d}V \tag{59}\] \[u_{b} =\frac{1}{\rho_{b}V}\int_{V}\rho u\,\mathrm{d}V \tag{60}\]  
are the bulk density and bulk velocity in the channel, respectively. \(\mu_{w}\) and \(c_{w}\) are the dynamic viscosity and speed of sound at the wall temperature, respectively. The simulation is carried out in a rectangular box whose size is \(2\pi h\times 2h\times\pi h\) in the x, y, z directions. Periodicity conditions are set in streamwise (x) and spanwise (z) directions and isothermal and no-slip walls are set at \(y=\pm h\). The initial field for this testcase consists a laminar velocity profile, which is perturbed in spanwise direction. The flow is developed statistically stationary by a uniform pressure gradient, which ensures that the mass flux remains nearly constant. The detailed procedure follows that reported in [19]. The detail of spatial resolutions is given in Table 2, where the \(\Delta^{+}\) values are estimated as element size divided by the number of solution points per direction. Note that the grid spacing \(y_{1}^{+}\) is controlled by the distance of first solution point from the wall, which is set in order to capture the small structures developing in the near-wall regions [62], as suggested by some rules-of-thumb for wall-resolving LES [20]. The friction Reynolds number is defined as \(Re_{\tau}=h/\delta_{\
u}\), with \(\delta_{\
u}=v_{w}/u_{\tau}\) being the viscous length scale, \(\
u\) the kinetic viscosity and \(u_{\tau}\) the friction velocity. In this study, the \(Re_{\tau}\) computed from wall stress and average density near the wall is 404, which is lower than the target \(Re_{\tau}=500\) reported by [61]. This gap is  
Figure 17: Local Mach number along \(y,z=\pi\) line and time evolution of max Mach number of compressible TGV simulation, \(Ma=1.25\), \(Re=1600\), \(120^{3}\) DOFs, \(p3\). (a) local Mach number along \(y,z=\pi\) line; (b) max Mach number. resulted from the insufficient resolution [17], which is not as fine as DNS which produce the target value. Figure 18 shows the mean velocity profile simulated by LES using \(p3\) FR-DG method with the original DB artificial viscosity. DNS results in [61] are taken as reference. These quantities are presented in semi-local coordinates [63]. It can be observed in Figure 18 that the viscous sublayer is well resolved compared with the DNS result. The resolved buffer layer (about \(10<y^{+}<40\)) deviates from DNS results, indicating that the turbulent transport is damped in this near-wall region. The bias prediction of the buffer layer leads to the mismatch of the log layer (about \(y^{+}>40\)) to DNS results, while the logarithmic dependence is well predicted. We note that in semi-local coordinates the near-wall velocity is used for non-dimensionalization, which is computed from the wall friction in the simulation. As addressed before, the wall stress is underestimated taken the DNS results as reference. This underestimation also partly contributes to the bias of the log layer, and the bulk velocity \(u_{b}\) may be a more proper choice for non-dimensionalization for a fair comparison, as reported in [17], which is however not available for this test case. Figure 19 show the Reynolds stresses. A general agreement of trends can be observed.