Model reveals dual-mode vibrations in pipes under internal and external flows

A nonlinear numerical study finds that a cantilever pipe conveying fluid can develop two distinct vibration patterns when internal flow interacts with vortex shedding from an external cross-flow. The model reproduces periodic responses combining the first and second bending modes, first under a 1:3 internal resonance and then under a post-flutter condition created by high-speed internal flow. Although these coupled responses remain periodic in the simulations, internal flow can substantially alter their amplitudes and modal composition, adding complexity to the design of flexible pipes used in settings such as deep-sea mining systems.

The research, published in Frontiers in Mechanical Engineering, focuses on a slender cantilever pipe resting on an elastic foundation while carrying steady flow. A uniform external flow crosses the pipe perpendicular to its plane of vibration, producing periodic vortices, while the internal flow moves along the pipe’s axis. Both fluids are modeled as inviscid, irrotational and incompressible. The pipe material is represented by a Kelvin-Voigt viscoelastic model, which includes structural stiffness and damping. This configuration captures a fundamental engineering challenge: the fluid inside a pipe changes its effective stiffness and can destabilize it, while the surrounding flow adds a time-dependent lifting force that excites the structure.

Cantilever fluid-conveying systems differ from pipes supported at both ends because their boundary conditions introduce non-conservative behavior and can permit flutter at sufficiently high internal-flow velocities. The study therefore considered two interacting mechanisms. At selected internal-flow velocities and foundation stiffnesses, the ratio between the first and second natural frequencies approaches 1:3, creating an internal resonance in which vortex excitation can involve both modes. At higher internal-flow velocities, the pipe enters a post-flutter state. The researchers examined how that instability couples with external vortex-induced vibration, rather than treating resonance and flutter as separate phenomena.

The researchers derived the governing equations with a Hamiltonian variational principle, combining the kinetic energies of the pipe and its internal fluid, the strain energy of the solid and the work of external forces. Geometric nonlinearity enters through nonlinear curvature in the displacement field, while nonlinear inertia is included by treating the pipe axis as inextensible. That assumption means the longitudinal displacement is determined by the transverse motion rather than being independent. The resulting equations couple the pipe’s transverse displacement to a dimensionless wake variable representing vortex shedding, allowing the model to calculate fluid-structure feedback instead of prescribing a fixed sinusoidal force.

The wake variable follows a van der Pol-type oscillator equation. Its parameters represent the nonlinear damping characteristics of the fluid oscillator and the strength of coupling between the wake and the pipe. The vortex-shedding frequency is calculated using the Strouhal number, a dimensionless measure linked to the cross-sectional configuration and Reynolds number; the study notes an experimentally observed value of 0.2 in the subcritical flow regime. This wake-oscillator framework is intended to reproduce the changing lift and drag forces that occur when vortex shedding synchronizes with structural motion, a condition commonly called lock-in.

To solve the coupled equations, the researchers expressed both pipe displacement and wake motion in a modal expansion consistent with the fixed-free boundary conditions. They then applied a second-order Galerkin truncation, producing four ordinary differential equations: two for pipe vibration and two for wake dynamics. Time integration used a fourth-order Runge-Kutta method, while a harmonic balance calculation retaining five odd harmonics provided an independent check for periodic responses. Frequency analysis of the time-domain solutions was used to identify response frequencies, and changes in external or internal flow velocity generated amplitude bifurcation curves, vibration spectra and deformation modes.

The two numerical approaches agreed closely along the amplitude-flow-velocity bifurcation curve, with a small deviation near the resonance peak. The researchers attributed that discrepancy to the larger-amplitude region, where nonlinear curvature and inertia produce stronger higher-order harmonics than can be fully represented by the finite harmonic balance calculation. The time-domain integration captured both transient and steady-state behavior, making it suitable for responses that might not be strictly periodic. Within the modeled 1:3 internal-resonance case, the pipe exhibited frequency lock-in over an external-flow range of 1.14 to 1.448 meters per second. The response frequency departed from the first natural frequency by approximately 5 percent and was within a range of about 20 percent of that frequency.

The lock-in region did not coincide exactly with the location of the largest amplitude peak. Nonlinear stiffness hardening shifted the resonance peak toward higher external-flow velocities and produced an asymmetric synchronization band, broadening it on the high-velocity side. The calculated results therefore indicate that the flow velocities associated with the strongest resonance and the broad range of frequency synchronization need not be identical. This distinction matters because a pipe can continue to exhibit organized vortex-induced vibration even where its amplitude response is no longer centered on the principal resonance peak. The modeled system consequently shows that synchronization is not described adequately by amplitude alone.

Internal resonance created a second route to multimodal behavior. By adjusting the internal-flow velocity and foundation stiffness, the researchers arranged for the ratio between the first and second natural frequencies to approach 1:3. Under vortex excitation, the pipe responded in both modes, although the first-mode component remained more prominent because the relevant third-harmonic contribution was relatively small. As external-flow velocity increased, the second-mode response was activated and exhibited a softening nonlinearity. This means the internal and external flows can cooperate through modal coupling: the external vortex force supplies the excitation, while internal-flow-induced stiffness changes make the first and second bending frequencies participate in the same response.

At higher internal-flow velocities, the cantilever system became unstable through flutter rather than simple resonance. A supercritical Hopf bifurcation occurred as the internal velocity reached 18 meters per second, changing an attenuated response into a limit cycle. The resulting post-flutter motion had second-order modal characteristics, consistent with the earlier literature cited by the authors. When the vortex-shedding frequency was near the first natural frequency, the internal-flow-induced second-mode flutter and the external-flow-induced first-mode vibration coexisted. The combined response remained periodic and its amplitude-flow-velocity curve resembled that of vortex-induced vibration alone, but spectra and deformation modes contained substantial components from both the first and second bending modes.

Foundation stiffness altered the likelihood and severity of this post-flutter coupling. Increasing foundation stiffness raised the system’s critical flutter velocity and reduced the vibration amplitude, while a softer foundation produced lower flutter thresholds and larger coupled oscillations. In one comparison at an internal velocity of 18.7 meters per second, the pipe coupled internal post-flutter motion with external vortex-induced vibration for a foundation stiffness of 100 newtons per meter, whereas the 200-newton-per-meter case showed vortex-induced vibration without the same internal-flow dynamic instability. The findings are numerical predictions for an idealized pipe model rather than measurements from an operating offshore structure. They identify how internal flow, external vortex shedding, modal resonance and post-flutter can combine within the model, while leaving the response of full-scale pipes to separate assessment.

Subject of Research: Nonlinear modal coupling and post-flutter in cantilever fluid-conveying pipes

Article Title: Nonlinear vortex-induced vibration of cantilever fluid-conveying pipes under internal flow-induced modal interaction and post-flutter

Article References: Wang, C., & Ren, Y. (2026). Nonlinear vortex-induced vibration of cantilever fluid-conveying pipes under internal flow-induced modal interaction and post-flutter. Frontiers in Mechanical Engineering, 12, Article 2001438. https://doi.org/10.3389/fmech.2026.2001438

Image Credits: AI Generated

DOI: 10.3389/fmech.2026.2001438

Keywords: fluid-conveying pipes, vortex-induced vibration, internal resonance, post-flutter, frequency lock-in, bimodal vibration, fluid-structure interaction, nonlinear dynamics, cantilever structures, offshore mining, Nonlinear, vortex-induced