Optimal large-eddy simulation of forced Burgers equation

Abstract

To explore the properties of optimal large eddy simulation (LES) formulations, they have been applied to the case of forced one-dimensional Burgers equation with a Fourier cutoff filter, rather than three-dimensional turbulence for which this approach was developed. This simplified model problem allows more complex models to be evaluated than was possible in three-dimensional turbulence. Further, in “Burgers turbulence,” the small scales consist of shocks, and the evolution of Fourier filtered shocks can be constructed with high accuracy. Thus, the lower bound on the modeling error in this case is known to be very small, if not zero, which makes it possible to correctly interpret the observed errors in the optimal LES models. Optimal LES models were used in Burgers equation LES, and it was found that the solution exhibits shocks similar to the exact solution but that the Gibbs phenomenon that exists in the exact filtered solution is gradually eliminated. This also produced poor results for the high wavenumber spectrum. In contrast, previous optimal LES studies in isotropic turbulence yielded very good results for the spectra. The differences and similarities in performance on Burgers equation and three-dimensional turbulence are instructive for characterizing the properties of optimal LES models.