## 4 Results and Discussion
### Homogeneous Isotropic Turbulence  
The freely decaying homogeneous isotropic turbulence (HIT) is a fundamental case for investigation of the physical properties of turbulence and an ideal configuration for measuring the ability of numerical method to reproduce the fundamental properties of turbulence. The simulation is conducted in a triple periodic cube \(\Omega=[0,2\pi)^{3}\) where the velocity field is initially generated from a given spectrum via the approach of Rogallo [57]. The energy spectrum is expressed as follows:  
\[E(k)=\frac{16}{3}\sqrt{\frac{2}{\pi}}M_{10}^{2}\frac{k^{4}}{k_{0}^{5}}e^{-2k^{ 2}/k_{0}^{2}}\,, \tag{50}\]  
where \(k\) is the wavenumber, \(M_{10}\) the turbulent Mach number at \(t=0\), \(k_{0}\) the wavenumber at the peak of the energy spectrum, taken as \(k_{0}=8.0\) in this work. The physical duration of the computation is set to \(5\tau\), where \(\tau\) is the initial large-eddy-turnover time. The initial density and pressure fields are set uniformly, which means that no fluctuations of any thermodynamic quantities is in the initial field. The methodologies of computation is identical to the TGV case, namely, a \(p1\) solution polynomial on a finer mesh of \(60^{3}\) elements and a \(p3\) solution polynomial on a coarse mesh of \(30^{3}\) elements.  
In addition to the total volume-averaged kinetic energy \(E_{k}\), its time derivative \(\epsilon_{total}\) as well as the various components of \(e^{\prime}\), more statistical quantities of concern are defined and addressed as follows.  
The turbulent Mach number \(Ma_{t}\) and Taylor Reynolds number \(Re_{\lambda}\) are defined as follows [51]:  
\[Ma_{t}=\frac{\sqrt{3}u_{rms}}{\langle c\rangle}\,, \tag{51}\]  
\[Re_{\lambda}=\frac{u_{rms}\lambda\langle\rho\rangle}{\langle\mu\rangle}\,, \tag{52}\]  
where  
\[u_{rms} =\sqrt{\langle\frac{u^{2}+v^{2}+w^{2}}{3}\rangle}\,, \tag{53}\] \[\lambda =\sqrt{\frac{u_{rms}^{2}}{\langle(\partial u/\partial x)^{2}\rangle }}\,, \tag{54}\]  
is the turbulent fluctuating velocity and the Taylor microscale, respectively, \(c\) stands for speed of sound, \(\langle\cdot\rangle\) denotes a volume average over the computational domain at a fixed time instant.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Discretization** & **DOFs** & **Model** & **Runtime(s)** & **Ratio** \\ \hline \multirow{4}{*}{\(30^{3}\) elements, \(p3\)} & \multirow{4}{*}{120} & iLES & 3443 & 1.0 \\  & & DB & 3935 & 1.14 \\  & & Smagorinsky & 4631 & 1.35 \\  & & similarity & 4823 & 1.40 \\ \hline \multirow{4}{*}{\(60^{3}\) elements, \(p1\)} & \multirow{4}{*}{120} & iLES & 3686 & 1.07 \\  & & DB & 4246 & 1.23 \\ \cline{1-1}  & & Smagorinsky & 5197 & 1.51 \\ \cline{1-1}  & & similarity & 5200 & 1.51 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Runtime for 1000 iterations of TGV problem at \(120^{3}\) DOFs.  
Figure 10: Time evolution of the dissipation \(\epsilon_{sgS}\) for explicit LES and \(\epsilon_{a}\) for LES with artificial viscosity of HIT simulation, \(Ma_{10}=0.5\), \(120^{3}\) DOFs. (**a**) \(p1\); (**b**) \(p3\).  
Figure 9: Time evolution of the solenoidal dissipation \(\epsilon_{s}\) of HIT simulation, \(Ma_{t0}=0.5\), \(120^{3}\) DOFs. (**a**) \(p1\); (**b**) \(p3\).