Application of the Curvilinear Coordinate Method for the Numerical Solution of the Navier–Stokes Equations in Domains with Complex Boundaries

In this paper, the coordinate transformation method is applied to the Navier–Stokes equations expressed in terms of the stream function and vorticity formulation. An elliptical grid generator is used to construct an orthogonal curvilinear grid within an irregular domain of complex geometry, mapping the physical region onto a computational square domain. The developed algorithm is capable of generating both orthogonal and general curvilinear grids. The finite-difference scheme of the Navier–Stokes system in arbitrary orthogonal curvilinear coordinates is then solved numerically on this grid using the alternating direction method. Numerical simulations of the Roach problem are conducted at low Reynolds numbers and on grids of varying resolutions. The obtained results are compared with the reference studies of Napolitano and Orlandi, showing satisfactory agreement with the data reported by 16 other research groups. Overall, the proposed method enables efficient numerical simulation of laminar flows in domains with complex geometry. The developed approach provides high accuracy and stability and can be effectively used for the numerical analysis of applied fluid dynamics problems. Furthermore, the methodology described in this work may serve as a foundation for future studies focused on improving computational efficiency and expanding the applicability of curvilinear grid techniques in modern fluid dynamics.