[Paper Review] Phase-lag predicts nonlinear response maxima in liquid-sloshing experiments

Bastian Bäuerlein, Kerstin Avila|arXiv (Cornell University)|Nov 5, 2020

Fluid Dynamics Simulations and Interactions38 references33 citations

TL;DR

This study demonstrates that phase-lag between the liquid's center of mass and external forcing is the key predictor of nonlinear resonance maxima in sloshing experiments. Using stereoscopic PIV and center-of-mass tracking, the authors show that at resonance, the response consistently lags by 90°, confirming the theoretical 90°-phase-lag criterion and revealing that this phase relationship remains invariant even under complex wave dynamics such as breaking and run-up.

ABSTRACT

Mass-spring models are essential for the description of sloshing resonances in engineering. By experimentally measuring the liquid's centre of mass in a horizontally oscillated rectangular tank, we show that low-amplitude sloshing obeys the Duffing equation. A bending of the response curve in analogy to a softening spring is observed, with growing hysteresis as the driving amplitude increases. At large amplitudes, complex wave patterns emerge (including wave-breaking and run up at the tank walls), competition between flow states is observed and the dynamics departs progressively from Duffing. We also provide a quantitative comparison of wave shapes and response curves to the predictions of a multimodal model based on potential flow theory (Faltinsen & Timokha 2009) and show that it systematically overestimates the sloshing amplitudes and the hysteresis. We find that the phase-lag between the liquid's centre of mass and the forcing is the key predictor of the nonlinear response maxima. The phase-lag reflects precisely the onset of deviations from Duffing dynamics and - most importantly - at resonance the sloshing motion always lags the driving by 90{\deg} (independently of the wave pattern). This confirms the theoretical 90{\deg}-phase-lag criterion (Cenedese & Haller 2020).

Motivation & Objective

• To identify reliable predictors of nonlinear resonance maxima in liquid sloshing under large-amplitude excitation.
• To quantify deviations between experimental data and multimodal potential flow models (Faltinsen & Timokha, 2009) in high-amplitude sloshing regimes.
• To test the validity of the 90°-phase-lag criterion in real experimental systems with complex wave patterns.
• To improve experimental measurement fidelity by using the liquid’s center of mass instead of single-point surface height measurements.
• To assess the limitations of current multimodal models and mass-spring (Duffing) models in capturing dissipation and nonlinear dynamics.

Proposed method

• Conducted horizontal oscillation experiments in a rectangular water tank using a controlled frequency and amplitude sweep.
• Employed stereoscopic particle image velocimetry (PIV) to measure in-plane velocity fields and reconstruct the liquid’s center of mass motion.
• Tracked the center of mass displacement as the primary response variable, reducing data scatter from flow state variability.
• Compared experimental response curves (amplitude vs. excitation frequency) with predictions from the multimodal potential flow model (Faltinsen & Timokha, 2009) using the first three modes.
• Calculated phase-lag between the tank’s driving motion and the center-of-mass response across all excitation conditions.
• Applied the Duffing equation to model low-amplitude dynamics and assessed its validity through observed softening behavior and hysteresis.

Experimental results

Research questions

• RQ1 Does the 90°-phase-lag criterion hold in experimental sloshing systems with complex wave patterns such as breaking and run-up?
• RQ2 How do multimodal potential flow models (Faltinsen & Timokha, 2009) compare quantitatively to experimental data at increasing driving amplitudes?
• RQ3 Can the liquid’s center of mass serve as a more robust and less scattered response metric than single-point surface elevation measurements?
• RQ4 What role does phase-lag play in predicting the onset of nonlinear resonance maxima, especially when wave morphology and dissipation vary?
• RQ5 Why do multimodal models systematically overestimate sloshing amplitudes and hysteresis, and is this due to inadequate damping modeling?

Key findings

• At resonance, the liquid’s center of mass consistently lags the driving force by exactly 90°, regardless of wave pattern complexity, confirming the theoretical 90°-phase-lag criterion.
• The phase-lag between the forcing and the center-of-mass response precisely predicts the onset of nonlinear deviations from Duffing dynamics and the location of resonance maxima.
• Multimodal potential flow models (Faltinsen & Timokha, 2009) systematically overestimate both sloshing amplitudes and hysteresis, especially at higher driving amplitudes.
• Low-amplitude sloshing follows the Duffing equation with softening behavior (negative cubic nonlinearity), evidenced by bending of the response curve and increasing hysteresis with amplitude.
• The use of center-of-mass displacement as a response metric reduces experimental scatter compared to single-point surface elevation measurements, particularly in regimes with coexisting flow states.
• Deviations between experiment and model increase significantly at large amplitudes, suggesting that the models’ single-mode damping treatment is insufficient and that accurate dissipation modeling remains a key challenge.