Optimal LES formulations for isotropic turbulence

It is shown that there is an abstract subgrid model that is in all senses ideal. An LES using the ideal subgrid model will exactly reproduce all single-time, multi-point statistics, and at the same time will have minimum possible error in instantaneous dynamics. The ideal model is written as an average over the real turbulent fields whose large scales match the current LES field. But this conditional average cannot be computed directly. Rather, the ideal model is the target for approximation when developing practical models, though no new practical models are presented here. To construct such models, the conditional average can be formally approximated using stochastic estimation. These optimal formulations are presented, and it is shown that a relatively simple but general class of one-point estimates can be computed from two-point correlation data, and that the estimates retain some of the statistical properties of the ideal model. To investigate the nature of these models, optimal formulations were applied to forced isotropic turbulence. A variety of optimal models of increasing complexity were computed. In all cases, it was found that the errors between the real and estimated subgrid force were nearly as large as the subgrid force itself. It is suggested that this may also be characteristic of the ideal model in isotropic turbulence. If this is the case, then it explains why subgrid models produce reasonable results in actual LES while performing poorly in a priori tests. Despite the large errors in the optimal models, one feature of the subgrid interaction that is exactly represented is the energy transfer to the subgrid scales by each wavenumber.