Fine scale reconstruction (VIC#) by implementing additional constraints and coarse-grid approximation into VIC+

Abstract This study proposes a method that complements Vortex-In-Cell plus (VIC+) (Schneiders and Scarano, Exp Fluids 57:139, 2016), a data assimilation technique that reconstructs a dense flow field from sparse particle tracks. Here, the focus is on the treatment of boundary conditions. In the VIC+ method, the choice of boundary conditions significantly affects a large part of the inner domain through their role as Dirichlet boundary conditions of the Poisson equations. By nature, there are particle tracks on one side of the boundaries, and often, due to experimental limitations, the track density is low, just close to the boundaries. This lack of data near the boundaries leads to a poor iterative update of the boundary condition for VIC+. Overall, the VIC+ method tends to be sensitive about the specific choice of the initial conditions, including the inner domain and the boundaries. Without prior flow information, a large padded volume has been proposed to achieve stable and reliable convergence, at the cost of a large number of additional unknowns that need to be optimized. The present method pursues the following concepts to resolve the above issues: use of the smallest possible padding size, reconstruction starting with “all zero” initial conditions, and progressive correction of the boundary conditions by considering the continuity law and the Navier–Stokes equation. These physical laws are incorporated as additional terms in the cost function, which so far only contained the disparity between PTV measurements and the VIC+ reconstruction. Here, the Navier–Stokes equation allows an instantaneous pressure field to be optimized simultaneously with the velocity and acceleration fields. Moreover, the scale parameters in VIC+ are redefined to be directly computed from PTV measurement instead of using the initial condition, and new scaling factors for the additional cost function terms are introduced. A coarse-grid approximation is employed in order to both improve reconstruction stability and save computation time. It provides a subsequent finer-grid with its low-resolution result as an initial condition while the interrogation volume slightly shrinks. A numerical assessment is conducted using synthetic PTV data generated from the direct numerical simulation data of forced isotropic turbulence from the Johns Hopkins Turbulence Database. Improved reconstructions, especially near the volume boundary, are achieved while the virtues of VIC+ are preserved. As an experimental assessment, the existing data from a time-resolved water jet is processed. Two reconstruction domains with different sizes are considered to compare the boundary of the smaller domain with the inside of the larger one. Visible enhancements near the boundary of the smaller domain are observed for this new approach in time-varying flow fields despite the limited input from PTV data. Graphical abstract