Derivation and Numerical Simulation of the Navier–Stokes Equations: From Chicken vs Godzilla to Computational Singularity

This project presents the derivation of the Navier-Stokes equations from first principles (Newton's Second Law and the Reynolds Transport Theorem) and their numerical implementation as a 2D Computational Fluid Dynamics (CFD) solver in Python. The solver utilizes the Chorin Projection Method and a Gauss-Seidel iteration for the Poisson pressure equation. It is validated by comparing the dimensional and dimensionless forms of the equations. The solver's ability to handle complex geometries is demonstrated using custom image-based obstacles. Furthermore, the drag coefficients (Cd) of these geometries are calculated across varying fluid densities to evaluate hydrodynamic efficiency. Finally, the numerical stability envelope of the explicit solver is analyzed, revealing two distinct modes of computational failure: the diffusive explosion at Re ≤ 24 (driven by extreme viscosity) and the advective explosion at Re ≥ 765 (driven by non-linear advection instabilities). The localized nature of these numerical singularities is mapped and discussed in the context of future applications in physical oceanography.