Adjoint-based data assimilation in a subdomain using omnidirectional-integration-enabled pressure Dirichlet boundary conditions

Solving the pressure Poisson equation within the Navier–Stokes solver for incompressible flows with a subdomain presents challenges, particularly due to the selection of boundary conditions. Typically, assumptions such as a large fluid domain with zero normal pressure gradient on the boundaries are often employed. However, this approach of using a larger domain exceeds the inherently needed, and often relies on inaccurate assumptions, especially when the focus is on a smaller subdomain. Moreover, when Neumann boundary conditions obtained from experimental data that inevitably includes noise are used, the accuracy of pressure reconstruction deteriorates. This issue is especially prevalent in the context of data assimilation where observational data is integrated into a numerical model using gradient-based optimization to enhance the model accuracy. To overcome the aforementioned difficulties, this study introduces a novel technique that utilizes the Omnidirectional Integration/Green's Function Integral (ODI/GFI) method to compute appropriate Dirichlet boundary conditions for pressure fields within an in-house two-dimensional Navier–Stokes solver. An adjoint-based framework for data assimilation is adopted for the reconstruction of velocity and pressure fields in a subdomain based on sparse observations. The method is validated with 1000 statistically independent realizations (50 base flows each coupled with 20 different noise distributions) of error-embedded two-dimensional decaying isotropic turbulence flows at a Reynolds number of Re = 200, thus enabling detailed statistical comparisons. The validation test results clearly demonstrate that the ODI/GFI method significantly outperforms the conventional Neumann boundary condition approach in providing not only accurate pressure predictions, but also improved accuracy of the velocity and the vorticity calculations. This improvement is evidenced by the comparison of a variety of metrics including the cost function, the instantaneous error distribution, the probability density function, the error spectrum, the standard deviation of the error and the time variation of flow quantities during the computation process of the data assimilation. The successful demonstration of the capability of the new ODI/GFI method in handling error-embedded instantaneous data in a subdomain immersed in a turbulent flow field provides a promising path for innovation in computation in data assimilation in particular and computational fluid dynamics in general.