Indicators of Velocity Profile Distortion in Unsteady Laminar Pipe Flow

Unsteady laminar pipe flows, prevalent in oscillating liquid columns, physiological pulsatile systems, and hydraulic transients, exhibit instantaneous velocity distributions that deviate significantly from the classical Hagen–Poiseuille parabolic profile. This deviation is governed by the interaction of inertial effects, viscous diffusion, phase lags, and localized flow reversal. Despite extensive documentation of these phenomena, a systematic framework for quantifying such morphological deviations remains largely undeveloped. This study proposes a suite of physically grounded indicators to quantify velocity profile distortion. These metrics characterize complementary aspects of non-Poiseuillean behavior, specifically: centerline curvature variation, profile flattening, energetic divergence, radial peak migration, reverse-flow fraction, and the emergence of inflection points. The methodology employs an analytical reconstruction of the local velocity field through a generalized Fourier–Bessel solution of the linearized Navier–Stokes equations, calibrated against experimental free-surface motion in a vertical U-tube undergoing damped oscillations. The application of these indicators demonstrates consistent correlations between geometric, energetic, and topological features during acceleration, deceleration, and flow-reversal phases. The findings indicate that no single parameter adequately captures the complexity of unsteady fields, but the integrated framework provides a robust classification of instantaneous flow regimes. This approach delineates the limitations of Poiseuille-based assumptions and establishes a quantitative foundation for advanced unsteady friction models.