Reconstructed turbulent fields using 4D variational data assimilation: reproduction of instantaneous structures.

We look into a turbulent system where incomplete velocity measurement data over a time period are available. Assuming that the dynamical evolution of the system is governed by the Navier-Stokes (NS) equation, we use the 4D variational data assimilation, which is a valuable tool for recovering missing information about a system from what is directly measurable to reconstruct the initial state of the system, such that the evolution of the system matches the measurement data. We formulate the problem as an optimization problem, where the initial field is taken as the control variable. The goal is to find the optimal control variable to minimize the differences between the measurement and the velocity field evolved from the reconstructed initial field, subject to the constraint imposed by the NS equation. The reconstructed fields are compared with direct numerical simulation (DNS) data. We look into the difference between the instantaneous variations of the reconstructed and the DNS field. The examples of the averaged geometries show that the differences decrease with time within the optimization horizon. Also, the instantaneous high vorticity structures for the reconstructed fields are reproduced with good agreement with the DNS field.