The impact of Omnidirectional Integration on the pressure solver in subdomain data assimilation for incompressible flows.

The divergence-free projection in incompressible Navier-Stokes solvers is essential for the fractional-step method in incompressible flow simulations. When reconstructing velocity and pressure fields in a subdomain through data assimilation, challenges arise particularly with unknown boundary conditions, where the projection step changes accordingly. Assuming an orthogonal projection would lead to zero normal pressure gradient at boundaries, which is compensated with a large domain, often requiring extra costly computation and grid storage even when the region of interest is small. Additionally, using Neumann boundary conditions can degrade performance with noisy velocity measurements. Liu and Moreto (2020) demonstrated that Omnidirectional Integration (ODI) can achieve Dirichlet boundary conditions and as a result significantly outperforms Neumann boundary conditions in ensuring pressure reconstruction accuracy. This study integrates ODI into a 2D incompressible Navier-Stokes solver by taking advantage of the pressure Dirichlet boundary conditions enabled by ODI. This method demonstrates that the improved projection provides results for both velocity and pressure closest to the true divergence-free flow.