## 5 Conclusions  
In this paper, the ability of the DB artificial viscosity model to simulate under-resolved compressible turbulent flows is investigated in high order flux reconstruction framework. The results are compared with several explicit SGS models and iLES. A detailed derivation of the dissipation rate due to modeling terms, including the Laplacian artificial viscosity model or the explicit SGS model, is conducted. The total dissipation rate of kinetic energy \(\epsilon_{total}\) is thus decomposed into several components which are discussed in detail. The effects of the order of solution polynomial, the SGS models and the DB model on the dissipative property and resolving ability in the simulation are investigated thoroughly. The Taylor-Green vortex (TGV) problem at \(Re=1600\) and freely decaying homogeneous isotropic turbulence (HIT) at \(Ma_{t0}=0.5\) are simulated at 120 DOFs with two sets of meshes and solution polynomial orders: a coarse mesh of \(30^{3}\) elements with \(p3\) solution polynomial and a finer mesh of \(60^{3}\) elements with \(p1\) solution polynomial. The DB method is further validated in two compressible problems, the compressible TGV problem and the compressible channel flow problem, with \(p3\) solution polynomial and relatively coarse meshes. The results of TGV simulations show that the fourth-order-accurate \(p3\) FR method matches the reference solution better than the \(p1\) counterpart at the same low DOFs. The \(p1\) method is limited by its excessive dissipation and low resolution at small scales and leads to over dissipation at high resolvable wavenumbers. Higher order representationof the solution are superior at resolving energy of the small scales. The explicit SGS models and the DB artificial viscosity model all benefit from more accurate resolution of velocity gradients produced by higher order of solution polynomial. Another benefit brought by the higher order scheme is a general improvement in model dissipation. The dissipation introduced by the DB model at \(p3\) behaves in a similar manner to some explicit SGS models, and compensates the dissipation when the smallest scales emerge, while the iLES alone is incapable of producing sufficient dissipation. However, it should be noticed that the DB model is sensitive to the dilation term and brings excessive dissipation at \(p1\) solution polynomial as the error of dilation term is larger. The HIT simulations show that the initial oscillation of flow field leads to excessive dissipation of DB model, which can be damped at higher solution polynomial order. The initial field also leads to oscillation of the pressure dilation dissipation, which is unaffected by the order of solution polynomial order or simulation methodologies. After the oscillation, lower order solution causes larger error of the pressure dilation dissipation, which leads to over dissipation of the DB model. This also explains the damped dissipation of the DB model at higher solution polynomial order. Dissipation of the Smagorinsky model and the Similarity model are not as sensitive to the solution order as the DB model. The results of spectrum and \(Re_{\lambda}\) indicate that at \(p1\) solution order the numerical dissipation is excessive regardless of the models applied, and the viscosity dominates on a larger range of scales. The \(p3\) simulations produce more accurate results on the whole thanks to the lower numerical dissipation and better resolution on small scales. However, future works still need to be conducted on detailed DNS results of the components of \(\epsilon_{total}\) derived by this paper. Both above test cases show that the iLES results via the fourth-order accurate FR method generally match the reference solution well at the given under-resolved DOFs, but they are insufficiently dissipative when the smallest scales emerge in the flow. The explicit SGS models and the DB model all bring additional dissipation to the computation but behave in a different manner.