Enhancing Data Assimilation through Omnidirectional Integration for Pressure Reconstruction in Noisy 2D Isotropic Turbulent Flow Observations

Numerical data assimilation plays a vital role in achieving optimal forecasts of dynamical processes by integrating theoretical model-based predictions with experimental observations. In our study, we focus on fluid mechanics and employ the forward Navier-Stokes solver as the theoretical model. The goal is to minimize the discrepancy between the computed velocity field and the measured velocity values using an adjoint-based data assimilation approach. To quantify this discrepancy, we utilize a cost function, which is minimized through the adjoint method, commonly employed in four-dimensional variational (4DVar) data assimilation. To validate our approach, we test the algorithm using 2D isotropic turbulent flow data with added noise to mimic real experimental conditions. The framework (Abassi et al. 2022) serves as a basis for applying data assimilation to noisy data, allowing us to reconstruct the velocity and pressure field under different pressure boundary conditions. Choosing an appropriate pressure boundary condition is a critical aspect of the process, as it ensures a divergence-free flow throughout. However, traditional Poisson solvers do not yield satisfactory pressure reconstructions when dealing with noisy data. To address this challenge, we implement the parallel ray omnidirectional integration technique developed by Liu et al. in 2016 to compute the pressure at the boundary at each time step. This approach allows us to impose Dirichlet boundary conditions for the pressure, thereby improving the accuracy of pressure computations. The validation of our algorithm using 2D turbulent noisy data will pave the way for extending the framework to handle 3D data. Ultimately, our objective is to establish a robust data assimilation method that can augment Time Resolved Tomo PIV measurements, enabling more accurate forecasting and analysis of complex fluid dynamics phenomena.