## 1 Introduction
For simulations of turbulent flows, implicit LES using the FR approach on canonical testcases such as isentropic vortex advection [26], the Taylor-Green vortex, turbulent channel flow and transitional flow [6] has presented satisfying results, while explicit SGS models may be still in need for under-resolved flows, where the aliasing error can impact the robustness [27]. For simulation of compressible turbulence via high order methods, another computational challenge is to resolve shock waves. Shock-capturing schemes, which are mostly relied on to represent shock waves, can be generally classified into two types. The first approach involves a nonlinear limiter to detect shock waves and control oscillations near them, such as total variation bounded (TVB) type slope limiters [28, 29], moment limiters [30, 31, 32] and the weighted essentially non-oscillatory (WENO) type schemes [33, 34]. Another approach is adding to the governing equations an artificial diffusion term, which is expected to smear the shocks at discontinuities over a numerically resolvable scale and vanish in the smooth region. The advantages of the artificial diffusion term is the simple formulation, ease of implementation, and low computational cost. In the context of LES, an adaptive approach was proposed by [PERSON] et al. [35] to handle shocks and turbulence simultaneously by employing an artificial bulk viscosity and an artificial shear viscosity. A simple modification was proposed by [PERSON] et al. [36], which replaced the strain rate tensor based artificial bulk viscosity by a dilation based counterpart. The modified version by [PERSON] et al. [36] also applied a switch function to further localize the bulk viscosity to regions surrounding the shocks. Recently a simplified form of the artificial bulk viscosity model was presented by [PERSON] et al. [37] and was shown to achieve good balance between accuracy and robustness. In this paper, this simplified form, referredto as dilation-based (DB) model, is adopted to take advantage of its analytical form in governing equations. An important issue about LES method is that the computations are not well resolved, in other word, under-resolved because the scales smaller than the resolution, which is associated to Nyquist wavenumber of the grid, cannot be captured [38]. Thus for LES of compressible turbulence the following essential under-resolved features may arise [5]. First, for flows free from shocks, the turbulent structures under subgrid scales (SGS) are accounted for by implicit or explicit SGS models, which may be originally designed for incompressible flows and then directly extended to variable-density cases. This extension is expected to be sufficient for their respective original purpose, though the underlying dynamic mechanism about the nonlinear transfer of kinetic energy between the scales of motion may be altered by the compressibility effects. Second, for flows with shocks or other sharp features [39], additional numerical dissipation is required for stability purposes, which reduces the range of well-resolved scales in LES [40]. Such shock-capturing methods may also bring unphysical dissipation into the smooth turbulent regions and thus should be handled carefully. Moreover, for under-resolved computation, the behavior of high order methods is not as clear as that for the well resolved cases because the order of convergence is defined as cell size tending to zero, while under-resolved computation means relatively larger cell size [41]. To sum up, despite the tremendous works on LES of compressible turbulence via high order methods, the relative roles of the SGS models, the shock-capturing schemes and the high order spatial discretization methods on dissipation property and resolving ability remain unclear. In this study, we focused on the performance of the DB artificial viscosity in the context of LES on compressible turbulence via high order flux reconstruction method and compared the DB model with several typical explicit SGS models as well as implicit LES.