Turbulent velocity field reconstruction using four-dimensional variational data assimilation

Important progress in computational fluid dynamics has been made recently by applying the data assimilation (DA) techniques. In this thesis, we apply the four-dimensional variational approach to rebuild the small scales of the velocity fields of fully developed three-dimensional turbulence, given a time sequence of measurement data on a coarse mesh of grid points. In this problem, we deal with new challenges since the flow is governed by the processes of nonlinear vortex stretching and forward energy cascade, which are absent in two-dimensional flows that have been investigated so far. Two different models are presented to examine their effects on the reconstruction quality: the Navier-Stokes equations as the model and when the large eddy simulations are applied as the model (Smagorinsky model). The investigations examine different statistics of the reconstructed fields. The results show that the agreement improves over time within the optimization horizon, where the rebuilt fields tend to the DNS target. Reasonable agreements are accomplished between the optimal initial fields and the target data. To assess the quality of the reconstruction of non-local structures, minimum volume enclosing ellipsoids are introduced, which enables us to perform quantitative comparisons for the geometry of non-local structures. The rebuilding of non-local structures with strong vorticity, strain rate and subgrid-scale energy dissipation gives satisfactory results. A small misalignment between the MVEE's axes can be obtained; structures in the rebuilt fields are reproduced with sizes smaller by a small percentage to what exists in the target field; the locations of the MVEE are different on average by around 20% of the axes lengths. Both Navier-Stokes equations and filtered Navier-Stokes equation (as models for the data) show satisfactory reconstruction. The imperfect model (Smagorinsky model) demonstrate the capability for recovering the target fields quicker in most of the presented examinations.