Comparative statistics of selected subgrid-scale models in large-eddy simulations of decaying, supersonic magnetohydrodynamic turbulence

Large-eddy simulations (LES) are a powerful tool in understanding processes that are inaccessible by direct simulations due to their complexity, for example, in the highly turbulent regime. However, their accuracy and success depends on a proper subgrid-scale (SGS) model that accounts for the unresolved scales in the simulation. We evaluate the applicability of two traditional SGS models, namely the eddy-viscosity (EV) and the scale-similarity (SS) models, and one recently proposed nonlinear (NL) SGS model in the realm of compressible magnetohydrodynamic (MHD) turbulence. Using 209 simulations of decaying, supersonic (initial sonic Mach number $M_s\approx 3$) MHD turbulence with a shock-capturing scheme and varying resolution, SGS model, and filter, we analyze the ensemble statistics of kinetic and magnetic energy spectra and structure functions. Furthermore, we compare the temporal evolution of lower- and higher-order statistical moments of the spatial distributions of kinetic and magnetic energy, vorticity, current density, and dilatation magnitudes. We find no statistical influence on the evolution of the flow by any model if grid-scale quantities are used to calculate SGS contributions. In addition, the SS models, which employ an explicit filter, have no impact in general. On the contrary, both the EV and NL models change the statistics if an explicit filter is used. For example, they slightly increase the dissipation on the smallest scales. We demonstrate that the nonlinear model improves higher-order statistics already with a small explicit filter, i.e., a three-point stencil. The results of, e.g., the structure functions or the skewness and kurtosis of the current density distribution are closer to the ones obtained from simulations at higher resolution. In addition, no additional regularization to stabilize the model is required. We conclude that the nonlinear model with a small explicit filter is suitable for application in more complex scenarios when higher-order statistics are important.

Figure 1

Kinetic (left) and magnetic (right) energy spectra at initial time $t=0T$ and after two turnover times $t=2T$ of free decay. The lines correspond to the median over all 19 realizations and the shaded areas (if not hidden by the linewidth) show the interquartile range. The kinetic energy spectrum has been calculated based on the Fourier transform of $\sqrt{\rho }\mathit{u}$. The insets show a magnification of the configurations in the intermediate wave-number range at $t=2T$.

Figure 2

Temporal evolution of the ensemble median (over 19 different realizations) of the spatial mean kinetic energy, magnetic energy, vorticity magnitude, current density magnitude, and dilatation magnitude. The variations in terms of interquartile ranges are illustrated in the insets.

Figure 3

Temporal evolution of the skewness and kurtosis of the distributions of current density (left column), vorticity (center column), and dilatation (right column) magnitude. The lines indicate the median over all 19 realizations. The shaded areas correspond to the interquartile ranges (IQR). For clarity, they are only shown for ILES-128, ILES-512, NL-F3, and NL-F5 as the IQRs of similar lines are virtually identical, e.g., the lines of ILES-128, EV-GS, NL-GS, SS-F3, and SS-F5.

Figure 4

Probability density function of the stacked current density magnitude, i.e., each bin contains the sum over all 19 realizations, at $t=1T$. The insets are magnifications of the indicated regions. The top panel shows the original raw data. The bottom panel details the right tail. It has been normalized by the respective mean value of each configuration in order to highlight resolution-independent features.

Figure 5

Second-order longitudinal velocity structure function of one arbitrary realization at $t=1T$.

Figure 6

Second-order versus third-order longitudinal velocity structure function illustrating extended self-similarity of the same realization as in Fig. 5. The best power-law fit (blue dotted line) to the ILES-512 simulation has an index of $0.694\left(0\right)$.

Figure 7

Second-order longitudinal velocity structure functions normalized to the third-order structure function scaled by the best-fit exponent. The lines indicate the median over all 19 realizations at $t=1T$. Variations as measured by the interquartile range are $<9\%$.

Figure 8

Scaling exponents ${\zeta }_p$ of the longitudinal velocity structure functions of order $p$ at $t=1T$. The lines indicate the median over all 19 realization and the error bars illustrate the interquartile range. For reference, the theoretical scaling ${\zeta }_p^{\mathrm{SL}}$ derived by She and Leveque [39] is shown in the left panel, and the reference simulation (ILES-512) is drawn in all panels.