Variational Germano identity applied to the numerical simulation of multiscale partial differential equations.

摘要

The simulation of turbulence is a challenging problem due to the cost of resolving all length scales that arise in a flow. This problem is addressed in engineering applications using large eddy simulation (LES), a class of numerical algorithms designed to only resolve a coarse representation of the exact solution.; In the LES of a multiscale system the effect of the unresolved scales on the resolved scales is represented by a sub-grid model, which is a functional of only the resolved scales. Most popular subgrid models contain parameters which must be fine-tuned for a particular flow and numerical method. Furthermore, the performance of these models is sensitive to the precise value of the parameters. In this dissertation, the variational Germano identity (VGI) is derived to automatically determine the parameters of an arbitrary subgrid model. In addition, a new class of LES models based on a dynamic generalization of the variational multiscale method is developed.; The VGI derived in this dissertation is inspired by its filtered counterpart (derived by M. Germano in 1991). It is based on requiring the numerical solution to be optimal in a user-defined metric. Experiments, comparing filtered and variational forms of the Germano identity and a range of subgrid models, are performed on several systems including the incompressible Navier Stokes, Burgers, and compressible Navier-stokes equations. The results of these simulations are compared to solutions obtained using well-resolved, but computationally expensive, direct numerical simulations. It is concluded that the VGI is more robust than the filtered form and that it is able to adjust to a definition of an optimal solution.; The VGI is also applied to a new class of LES models developed in this study. These models, which are based on the variational multiscale concept, better represent the interscale transfer of energy within the resolved scales by allowing for two independent viscosities: one which acts on all resolved scales and another which acts only on the fine resolved scales. These models are tested for, Burgers equation and the three dimensional Navier Stokes equations, and are found to be more accurate than their single viscosity counterparts.