Potential flow over undulated cylinders

This paper presents a general potential flow solution for an n-family of undulated cylinders. The formulation represents a methodological advancement beyond the classical flow over a circular cylinder by introducing a systematic and straightforward perturbation procedure that enforces boundary conditions on undulated cylindrical geometries. The resulting model yields closed-form analytical expressions for the stream function, enabling direct evaluation of the flow field, surface velocities, and surface pressure coefficients for cylinders whose radii contain arbitrary undulation amplitudes and wave numbers. Sample results demonstrate that the method accurately captures the geometric influence of surface undulations on streamline curvature, local velocity gradients, and pressure distributions. Odd and even undulation modes produce distinct asymmetric and symmetric pressure patterns, respectively, while the undulation amplitude acts as a scaling parameter that intensifies local surface pressure fluctuations. The solution remains fully analytical and preserves the linear structure of potential flow theory, allowing efficient exploration of a broad class of geometries without resorting to numerical simulations. Together, these results establish a versatile analytical foundation for studying inviscid flow around geometrically complex n-family of undulated cylinders and for guiding the design and analysis of robotic swimmers and the control of bluff body flows in aerodynamic, and other bio-inspired engineering applications involving undulated surfaces.