Classical Poiseuille flow through rectangular channels exhibits intricate perturbation dynamics when interacting with sinusoidally modulated walls. Prior studies have demonstrated that the spatial structure of these perturbations is highly sensitive to wall geometry, particularly the wavenumber k of the wavy surfaces. Building on this foundation, the present two-dimensional numerical study investigates five geometrically distinct configurations, characterized by the normalized channel half-depth α=kd, spanning from highly undulated walls (α=3.2π) to near-planar walls (α=0.2π), with a fixed small amplitude A set to be 20 times smaller than the channel half-depth d. The results show that wall-normal velocity perturbations gradually develop a forward tilt as the relative importance of convective transport increases. This structural change occurs over an O(1) range of the viscous length scale θ, with its development depending on channel geometry. The study also examines penetration depth using two threshold-based criteria and compares the numerical results with linear theory. Good agreement is obtained for shallow and intermediate channels, while deeper channels show systematic deviations from linear predictions. Overall, the work provides a systematic numerical characterization of how wall-induced perturbations evolve across geometry and flow conditions in Poiseuille flow over sinusoidal surfaces.
Singha et al. (2026) studied this question.