Geometrical Effects of Free-End Nozzles on Wave Reflection in Self-Oscillating Flexible Nozzles

Fluid-conveying elastic shells, such as vascular tracts, urinary tracts, oil-conveying pipes, and flexible nozzles of cephalopods, exhibit unsteady flow rates that are highly coupled to wave dynamics propagating along the shell membrane. However, previous studies have generally focused on both-end-clamped configurations, primarily simulating venous pipes, but lack understanding of one-end-free, one-end-clamped systems (i.e., flexible nozzles) where distinctly different wave dynamics occur. This study analyzes wave dynamics in flexible nozzles as functions of averaged structural stiffness, aspect ratio, and stiffness distribution (i.e., tapered geometry) using simultaneous measurement of both solid deformation and internal flow. The nozzles are fed with a top-hat velocity profile having low turbulence intensity (<3%) generated by a flow stabilizer comprised of a contraction section and honeycomb mesh. 3D resin printing is used to fabricate nozzles with varying geometries and controlled stiffness. Stereo imaging using two high-speed cameras enables measurement of both solid deformation and internal flow at 1 kHz sampling rate through three-dimensional particle tracking velocimetry (3D PTV). Red fluorescent particles are distributed on the nozzle wall for displacement tracking, while polyamide particles are seeded for flow visualization, both made visible through index-matching techniques. With increasing jet Reynolds number and decreasing structural stiffness, various modes of self-oscillatory behavior emerge at critical jet velocities. We calculate the traveling-to-standing wave ratio, S, which quantifies wave reflection strength at the nozzle tip. S increases with nozzle taper ratio (converging geometry) and decreases with nozzle length. The pulsatile power ratio (PPR) is also significantly influenced by wave reflection at the free end. The workload on the flow generator is minimized at PPR minima (conversely, hydrodynamic impulse is maximized), which are primarily governed by the traveling-to-standing wave ratio and nozzle aspect ratio. These insights are potentially valuable for the design of medical devices or biological systems with fluid-filled compliant tubes.