In the past decades, the necessity for numerical quality in Direct Numerical Simulations (DNS) and especially Large-Eddy Simulations (LES) of turbulent flows, has been recognized by many researchers. In a properly resolved Direct Numerical Simulation, the smallest resolved scales are located far into the dissipation range. Since these scales have only a very small energy-content in comparison with the largest resolved scales in the flow, they are often considered to have a negligible influence on the mean flow statistics. In a Large-Eddy Simulation, however, where only the most important large scale structures are resolved, the smallest resolved scales are part of the inertial subrange such that they contain relatively more energy than those in the dissipation range. Hence, the smallest resolved scales in Large-Eddy Simulation are not negligible and have a significant influence on the evolution of the LES-flow. The accuracy with which these small scales are described is therefore expected to be important. In order to reduce the computational costs, it is highly desirable in LES to resolve as much scales as possible on a given computational grid. In order to accomplish this, the numerical method requires sufficient accuracy for all scales. This ensure that the magnitudes of the discretization errors remain smaller than the magnitude of the modeled unresolved scales of motion. However, if the accuracy of the numerical method cannot be guaranteed, the amount of resolved small-scale structures must be reduced in order to control the discretization errors. Hence, in order to resolve the same amount of scales as before, a much finer computational grid is required. This is often prohibitively expensive for most three-dimensional LES computations of industrial applications.

The current research focuses on the development of  a family of Dynamic Finite Difference approximations, which succeed in minimizing the instantaneous global discretization error on the solution during the calculation. This strategy allows to obtain always a (nearly) optimal numerical method that corresponds to the flow properties related to the spectral content, at that time. The approach implies that the intrinsic characteristics of the developed Dynamic Finite Difference method vary during the simulation in such a way that the global numerical error is always minimized. 

The newly developed Dynamic Finite Difference Approximations have been successfully applied to Large-Eddy Simulation of the Taylor-Green Vortex flow at Re=1500. An illustration of the Taylor-Green Vortex flow at Re=3000 is given in the movie below.

Results on the good numerical performance of the dynamic finite difference schemes  can be found in my Ph.D. dissertation, or in my papers. 

Illustration:

Direct Numerical Simulation (384³) of the Taylor-Green Vortex flow at Reynolds number Re=3000 

(click  to watch movie)