Bifurcations, stability and numerical analysis turbulent flow bidimensional

This study investigates the dynamics of two-dimensional incompressible turbulent flow through a rigorous mathematical and numerical framework. Using the Navier-Stokes equations as a foundation, we explore the transition from laminar to turbulent regimes, supported by energy estimates derived from Sobolev spaces and the Local Uniqueness Theorem. A new theorem is proposed that establishes sufficient conditions for the stability of turbulent flow solutions under critical Reynolds number thresholds. Weak solutions are analyzed via the Galerkin method, offering insight into long-term turbulent behavior. A numericalscheme combining the Finite Element Method (FEM) and spectral analysis is developed to study complex bifurcations and flow instabilities. Eigenvalue problems arising from linearized systems are formulated to evaluate flow stability, and simulations confirm the transition to turbulence through energy spectrum and kinetic energy evolution analyses. The results validate the robustness of the proposed method and reinforce the relevance of viscosity and energy dissipation mechanisms in turbulent structures. Thestudy bridges theoretical and computational approaches, contributing to the understanding of turbulence phenomena in confined domains.