We introduce a new approach to solving incompressible Navier–Stokes flow. This method combines a projection scheme with the Axial Green Function Method (AGM). Based on the Kim and Moin methods, our methodology employs a predictor–corrector mechanism to achieve stable and accurate velocity corrections. Using axial Green functions, we transform complex differential equations into simpler one-dimensional integral equations. These are strategically placed along a minimal axis-parallel lines, known as axial lines, within the flow domain. This transformation makes computation and analysis more efficient from a numerical viewpoint. A significant innovation in our approach is using one-dimensional axial Green functions tailored explicitly for the reaction–diffusion ordinary differential operator. These functions efficiently handle the discrete-time derivative and viscous terms of the momentum equation. Furthermore, our approach allows for the arbitrary construction of axis-parallel lines, facilitating analysis near critical flow regions and even enabling the random distribution of these lines. We validate our proposed method through numerical examples, demonstrating the convergence of numerical solutions, the effectiveness of arbitrarily constructed axis-parallel lines, and the potential extension of our method to three-dimensional flow problems. Additionally, this study provides a robust and adaptable alternative way to solve incompressible Navier–Stokes flows, proving its effectiveness in practical applications such as Tesla valves.
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