## 4 Results and Discussion
### Compressible Taylor-Green Vortex  
In this section, for a further assessment of the DB method in solving compressible problems, the TGV problem is extended to Mach number 1.25, while the other parameters of simulation follow those in Section 4.1. DNS results on \(512^{2}\) meshes [42] is taken as reference. Note that in the reference work, the Ducros sensor [60] is applied for adaptive dissipation control of the scheme. Therefore a combination of the DB method and the Ducros sensor, referred to as DBs, is applied in this testcase for a relatively fair comparison, which takes the following form:  
\[\mu_{\beta,DBs}=s_{D}\cdot c_{\beta}|\
abla\cdot\bar{\mathbf{u}}|(h/P)^{2}\,, \tag{55}\]  
where \(s_{D}\) is defined by  
\[s_{D}=H(-(\
abla\cdot\bar{\mathbf{u}}))\frac{(\
abla\cdot\bar{\mathbf{u}})^{2}}{(\
abla \cdot\bar{\mathbf{u}})^{2}+\bar{\mathbf{w}}\cdot\bar{\mathbf{w}}+\epsilon}\,, \tag{56}\]  
where \(H\) is the Heaviside function, \(\bar{\mathbf{w}}\) is the resolved vorticity vector, \(\epsilon=10^{-32}\) is a small constant to avoid zero in denominator. Note that \(s_{D}=1\) leads to the original DB method defined by Equation (28). Figure 13 displays the time evolution of the turbulent kinetic energy \(E_{k}\) of the compressible Taylor-Green vortex simulated by the original DB method (marked as DB) and the Ducros-sensor-involved DB method (marked as DBs). A different feature of the compressible Taylor-Green vortex problem from its incompressible counterpart is that the turbulent kinetic energy does not decrease from the initial moment but increases slightly before \(t=4\) when internal energy is converted into kinetic energy. The increase in kinetic energy at the early stage can be observed for DBs method, while the kinetic energy predicted by the original DB method exhibits no significant change at the same time period. Both methods generally agree well with the reference, while the DBs produces more satisfying results because the further reduced dissipation. Figure 13b plots the total viscous dissipation rate, which follows the definition in reference [42] by \(e_{c}+e_{s}\). It can be observed that the DB method recovers the peak value of the total viscous dissipation rate well when working with the Ducros sensor. Figure 13c plots the sum of total viscous dissipation rate and the extra dissipation brought by the artificial viscous terms, namely, \(e_{c}+e_{s}+e_{a}\) to show influence of the DB models on the dissipative process. While in incompressible TGV simulation the DB model at \(p3\) provides reasonable compensation when the flow becomes under-resolved, as plotted in Figure 5b, in this compressible case, however, the DB models behave over dissipative at the early stage of the flow. It can be observed that the Ducros sensor helps control the over dissipation. Figures 14-16 plot contours of dilatation rate \(\
abla\cdot\mathbf{u}\), the artificial viscosity \(\mu_{av}\) defined by Equation (30) and local Mach number respectively on surfaces of the computational domain via DB and DBs methods at \(t=2\). Taking previous study [42] as reference, multiple shock waves are well captured despite the lower resolution than reference. The profiles of shockwaves by DB and DBs methods are similar since that the DBs method mainly reduces artificial viscosity in smooth regions, as shown in Figure 15. Figure 17a shows the line profiles on center line of the computational domain defined by \(y,z=\pi\) at \(t=2.5\). It can be observed that both DB models are capable of capturing the shockwave, which is represented by the sudden drop of Mach number. Time evolution of the max Mach number in the computational domain is plotted in Figure 17b. In spite of the early stage before \(t=4\), two major peaks of the max Mach number is observed at about \(t=6\) and \(t=9\), respectively. The peak values are more or less smoothed by the original DB method and are recovered well when the Ducros sensor is applied.