Eulerian time-marching in Vortex-In-Cell (VIC) method: reconstruction of multiple time-steps from a single vorticity volume and time-resolved boundary condition

Abstract

A data assimilation approach is proposed to enhance the dynamic range of the Vortex-In-Cell (VIC) method by simulating future- and past- instances. The VIC method mainly considers a vorticity field from which velocity and acceleration fields are calculated through Poisson equations, respectively bounded by prescribed conditions. In addition, a vorticity time derivative is also available by the vorticity transport equation. The proposed approach focuses on such already available data, i.e., the vorticity and its time derivative fields, for simulating additional instances and getting feedbacks from the corresponding measurement instances, e.g., particle image velocimetry (PTV). However, the self-simulated flow field can be depleted due to a lack of incoming information, which is out of the reconstruction domain at the source instance. To supply that kind of information and thus sustain the simulation, boundary conditions of the simulated instances are required and considered. As a result, the proposed approach can gather corrections from multiple PTV instances while optimizing a single vorticity volume and time-resolved boundary conditions. Since the boundary grid points are much smaller in number than that of the whole volume, one can expect an increased dynamic range. A former work, VIC# (Jeon et al. 2018), which supplements additional constraints and coarse-grid approximation to VIC+ (Schneiders and Scarano 2016), is selected as a 3D method to which the proposed 4D approach is applied. Two explicit Eulerian time-marching methods are tested as a simulation scheme: the forward Euler and the Runge-Kutta methods. A numerical assessment is conducted using the synthetic PTV data, whose ground truth is known, and returns reconstruction qualities based on the velocity and the identified vortical structures. Other practical features regarding convergence and computation complexity are also reported. To visually verify an improvement by the proposed approach, two kinds of time-resolved Shake-the-Box (STB) measurements, which were acquired in high-speed systems, are processed and discussed.