## 4 Results and Discussion
### Incompressible Taylor-Green Vortex
#### 4.1.1 Effects of Polynomial Orders  
First, the effects of the solution polynomial order on dissipative property are discussed. Figures 1 and 2 illustrate a general overview of the dissipative evolution of the TGV testcase. It can be observed that in the \(p1\) results, all solutions deviate from the reference due to larger overall dissipation. The results of overall dissipation rate at \(p1\) are all larger than their \(p3\) counterpart, which also indicates the stronger overall dissipation at lower order. Figure 3 plots the solenoidal dissipation \(\epsilon_{s}\), which is a measure of numerical accuracy and ability to resolve small vortices scales [55]. Results of solenoidal dissipation at \(p1\) are apparently lower that those at \(p3\) regardless of the simulation methodologies. It is clear that the vorticity-carrying small scales are not well resolved on the lower solution polynomial order because the sharpness of resolved derivatives is severely reduced, though the DOFs are the same. The pressure dilation dissipation \(\epsilon_{d}\), plotted in Figure 4 with the y-axis scale stretched, is supposed to be two-orders of magnitude smaller than the total dissipation rate since the Mach number of this testcase is so low. According to the DNS results, the very low level of pressure dilation dissipation \(\epsilon_{d}\), compared with the total dissipation rate, confirms the incompressible flow assumption though the simulation is compressible. This feature of the pressure dilation dissipation could serve as some evaluation of the error of numerical method when simulating incompressible flows via a compressible solver. In Figure 4, the \(\epsilon_{d}\) at \(p1\) deviates from the reference at the initial time moment, while the \(p3\) solution agrees well with the reference until the flow becomes under-resolved. The peak values of \(\epsilon_{d}\) decrease as the solution order increases. Similar phenomenon is observed in [56] and is reported to converge to the ideal solution with grid refinement. Figure 5 shows the evolution of modeling dissipation terms, \(\epsilon_{sgs}\) and \(\epsilon_{a}\). Later peaks and higher levels of SGS modeling dissipation are observed at \(p3\), which result from the improved resolution of the velocity gradients. However, for the DB model, later peak but lower level of modeling dissipation is observed at \(p3\). This opposite behavior of the DB model is due to its inherent sensitiveness to dilation term, as defined in Equation (28). It can be inferred from Figure 4 that at \(p1\) solution order, the error between the dilation term and ideal zero is larger than \(p3\), leading to higher level of \(\epsilon_{a}\). This decrease of precision caused by larger error of dilation at lower solution polynomial order was also reported in [37] for weakly compressible flow. The 3D energy spectra at time \(t=9\) are plotted on a log-log scale in Figure 6. At this time, the slope of the energy spectra reaches the Kolmogorov value -5/3 and the dissipation rate reaches peak value. A reference solution via pseudo-spectral method on a \(512^{3}\) grid [50] is plotted for comparison. The effects of the solution polynomial order is evident. It can be observed that the \(p1\) simulations hardly recover the reference slope and deviates at very low wavenumber, \(p3\) simulations produce more reasonable energy spectra. The \(p3\) results also deviate from the reference at high wavenumber, which is normal in consideration of the insufficient resolution. To sum up, simulations at \(p3\) order produce more accurate results, better resolution of small scales and significantly lower numerical error though the simulation is under-resolved. Higher order representation of the solution are also superior at resolving energy of the small scales. The explicit SGS models and the DB artificial viscosity model, though built in different manner, all benefit from more accurate resolution of velocity gradients produced by higher order of solution polynomial.