Drag and lift forces on partially submerged boulders on rigid beds in supercritical flow: a laboratory study

Understanding the hydraulic resistance and stability of boulders is essential for river engineering, river morphology, and hydraulic structure design. This study quantifies hydrodynamic forces acting on a cubical block resting on a smooth, rigid bed under near-critical to supercritical flow conditions, with bed slopes of 0.4-−1.3% and Froude numbers ( Fr ) in the range 1.2 ≤ Fr ≤ 2.8 . Time-resolved force measurements were used to derive drag and lift coefficients ( CD and CL ), and their fluctuations. The drag coefficient decreases systematically with increasing relative submergence ( h / H ), while the lift coefficient follows a nonlinear asymptotic recovery, shifting from negative toward positive values as h / H increases. Smoothing the block edges reduces drag by 20–30% and lift by 10–20%, whereas block rotation (perpendicular vs. 45 ^∘ ) produces only minor differences (within ± 5% ). The drag coefficient exhibits a consistent trend with Fr , and the lift coefficient becomes tightly organized when expressed as a function of h / H , from which empirical relationships are derived for partially submerged transitional flow. Standing waves, linked to upstream flow deceleration, are associated with increased unsteadiness of drag and lift forces and higher mean values of CD and CL . These results clarify how geometry, flow regime, and free-surface instabilities control CD and CL under near-critical and supercritical conditions, providing a force-based understanding of boulder–flow interactions on smooth, rigid beds.