Numerical analysis and phenomenology of homogeneous, isotropic turbulence generated by higher order models of turbulence

摘要

Turbulence appears in many processes in the nature and it is connected with many engineering, biophysical and climate applications. Therefore, the accurate, efficient and reliable simulation of turbulent flows is an essential difficulty in many current applications. Fundamental and universal (i.e. mathematical) insights into fluid structures will enable such simulations. To that end, we apply the phenomenology of homogeneous, isotropic turbulence to a family of Large Eddy Simulation (LES) models, the so-calledfamily of Approximate Deconvolution Models (ADM). We establish that the models themselves have an energy cascade with two asymptotically different inertial ranges. Delineation of these gives insight into the resolution requirements of using ADM. A correct prediction of a 3D turbulent flow means getting the energy balance and rotational structures correct, i.e., it means (in the large) matching the energy and helicity statistics. Thus, we consider the prediction of energy and helicity statistics of the family of Approximate Deconvolution Models of turbulence. We show that the family of ADM has a helicity cascade that it is linked to its energy cascade and predicted correctly over the large/resolved scales. Turbulent flows are very rich in scales and to be able to capture all of them, we need to use a very fine mesh. Unfortunately, even with the amazing development of the computer power, we are not able to perform such simulations. Thus, many numerical regularization (aiming to truncate the small scales) have been explored in computational fluid dynamics. We investigated one of such regularization, called the Time Relaxation Model (TRM). We apply the phenomenology of homogeneous, isotropic turbulence to understand how the time relaxation term, by itself, acts to truncate solution scales and to use this understanding to give insight into coefficient selection. We also study the stability and convergence analysis of a finite element discretization of TRM. Next we complement this with an experimentalstudy of the convergence rates and of the effect the time relaxation term has on the large scales of a flow near a transitional point.