Implicit large eddy simulation (ILES) has provided many computer simulations with an efficient and effective model for turbulence. The capacity for ILES has been shown to arise from a broad class of numerical methods with specific properties producing nonoscillatory solutions using limiters that provide these methods with nonlinear stability. The use of modified equation has allowed us to understand the mechanisms behind the efficacy of ILES as a model. Much of the understanding of the ILES modeling has proceeded in the realm of incompressible flows. Here, we extend this analysis to compressible flows. While the general conclusions are consistent with our previous findings, the compressible case has several important distinctions. Like the incompressible analysis, the ILES of compressible flow is dominated by an effective self-similarity model (Bardina, J., Ferziger, J. H., and Reynolds, W. C., 1980, “Improved Subgrid Scale Models for Large Eddy Simulations
,” AIAA Paper No. 80–1357; Borue, V., and Orszag, S. A., 1998, “Local Energy Flux and Subgrid-Scale Statistics in Three Dimensional Turbulence
,” J. Fluid Mech., 366, pp. 1–31; Meneveau, C., and Katz, J., 2000, “Scale-Invariance and Turbulence Models for Large-Eddy Simulations
,” Annu. Rev. Fluid. Mech., 32, pp. 1–32). Here, we focus on one of these issues, the form of the effective subgrid model for the conservation of mass equations. In the mass equation, the leading order model is a self-similarity model acting on the joint gradients of density and velocity. The dissipative ILES model results from the limiter and upwind differencing resulting in effects proportional to the acoustic modes in the flow as well as the convective effects. We examine the model in several limits including the incompressible limit. This equation differs from the standard form found in the classical Navier–Stokes equations, but generally follows the form suggested by Brenner (2005, “Navier-Stokes Revisited
,” Physica A, 349(1–2), pp. 60–133) in a modification of Navier–Stokes necessary to successfully reproduce some experimentally measured phenomena. The implications of these developments are discussed in relation to the usual turbulence modeling approaches.
