Similarities in the Development of Numerical Schemes and Sub-Grid Modeling in Large-Eddy Simulations

Abstract

Schemes for the discretization of spatial derivatives commonly employed in the numerical solution of Partial Differential Equations present intrinsic errors that become more important as the wave number content of solution field approaches the Nyquist criteria based on the mesh spacing. In many situations, this can be overcome by simply refining the mesh, so that the wavelength of the structures becomes much larger than the mesh spacing, and the discretization errors become again negligible. However, in some other cases, like in Large-Eddy Simulations of highly turbulent flows, the cost per discretization element is so high that further mesh refinement is prohibitive. In this case, it is more appropriate to work towards understanding and improving the numerical schemes, so that the wider possible range of the spectrum is accurately resolved, and the fewest possible number of degrees of freedom is needed to provide a satisfactory solution. By analyzing the similarities between the problems faced by numerical schemes and the challenges of sub-grid modeling in Large-Eddy Simulations (LES), an alternative for the numerical simulation of turbulence in the context of Large-Eddy Simulation is developed, that accumulates two main functionalities: represent the interaction between unresolved and resolved scales, while keeping the discretization errors at acceptable levels. The proposed scheme is of advective nature and has been applied in several test cases, ranging from simple one-dimensional convection of a passive scalar, to more complex turbulent flows. As a result, a better understanding of the role of discretization errors in Large-Eddy Simulation was obtained.