Three-dimensional flow dynamics and vibration characteristics of flexible multi-stepped cylinders

The vortex-induced vibrations (VIVs) of a flexible multi-stepped cylinder that models a buoyancy-moduled riser are investigated using 3D direct numerical simulations at Reynolds number Re = 100 across a reduced velocity range of Ur = 5.9–38.3. The structure has aspect ratio L/d = 50 and three modules (D/d = 2) arranged in a staggered pattern with 50% coverage. Space–time fields, variational mode decomposition, force–velocity phase diagnostics isolate power-delivering components from harmonic content and correlate them with wake organization. A regime map emerges with increasing Ur: (regime I) a standing 1st-mode cross-flow lock-in with 2nd-mode in-line; (regime II) dominance of 1st-mode cross-flow vibration, accompanied by a detuned second harmonic in the in-line direction and step-induced modulation; (regime III) transition to the 2nd-mode with antinodes localized at outer modules and spanwise asymmetry; (regime IV) dominance of 3rd-mode cross-flow and 6th-mode in-line vibrations with traveling-wave components. These regime transitions and spectral broadening are driven by step-centered couplings between the wake and cylinder. At the steps, filaments of streamwise vorticity intensify, injecting traveling-wave patches into standing-wave envelopes and mediating axial energy redistribution as Ur increases. Large-diameter modules concentrate root mean square lift, mean drag, and peak amplitudes, whereas small-diameter sections exhibit lower-magnitude and narrow-band response. The power-input phase spans the module regions in regimes I and II, localizes onto the mode-2 antinode-bearing modules in regime III, and shifts toward near-quadrature over modules in regime IV. The net energy remains cross-flow dominated, with in-line harmonics acting mainly as a secondary modulation.