This talk presents a novel and efficient approach to solving incompressible Navier-Stokes flows by combining a projection scheme with the Axial Green Function Method (AGM). AGM employs one-dimensional Green functions tailored for axially split differential operators, enabling the resolution of intricate multidimensional challenges. Our methodology integrates the projection method with a predictor-corrector mechanism, thereby ensuring stable and accurate velocity corrections. By transforming complex differential equations into simpler one-dimensional integral equations along axis-parallel lines within the flow domain, a notable enhancement in computational efficiency is achieved. A significant innovation of our approach is the use of axial Green functions that have been specifically designed for the reaction-diffusion ordinary differential operator. This enables the effective handling of discrete-time derivatives and viscous terms in the momentum equation. The flexibility of constructing axis-parallel lines at will allows for a detailed analysis of critical flow regions and even permits a random distribution of these lines, thereby enhancing adaptability. The efficacy of our methodology is validated through numerical examples involving benchmark flow scenarios, such as lid-driven cavity flow and flow past an obstacle, which illustrate convergence, adaptability to arbitrary domain geometries, and potential applicability to three-dimensional flow problems.