Regime Maps for Sloshing in Horizontal Cylindrical Tanks Under Vertical Acceleration

TL;DR

This work addresses gravity-dominated vertical sloshing in a horizontal cylindrical tank subjected to vertical excitation near the principal parametric resonance $\omega_f \approx 2\,\omega_{1,0}$. It introduces a data-driven framework that combines stabilized high-speed imaging, multiscale POD (mPOD) for spectral modal decomposition, prototype-based labeling, and a physics-informed SVM to construct a nondimensional regime map across fill levels. The results reveal a spectrum of regimes from stable waves to pure and mixed longitudinal modes, including wave-breaking and mode competition, with instability tongues broadly aligning with the Mathieu equation predictions but shifted by viscous and geometric effects. The regime maps provide a predictive tool for sloshing-induced loads and mixing, supporting design and operation of horizontal fuel tanks and cryogenic storage systems in aerospace contexts.

Abstract

Vertical sloshing in partially filled fuel tanks can significantly impact vehicle stability and structural integrity, particularly under harmonic accelerations near twice the sloshing natural frequency. In this regime, parametric resonance may arise, with nonlinear free-surface dynamics driving large-amplitude waves, interface break-up, and severe sloshing-induced mixing. In this work, we identify and characterize the distinct sloshing regimes associated with the lowest-frequency parametric instability, specifically when the external forcing frequency approaches twice the lowest natural frequency. Experiments were conducted in a transparent cylindrical tank with diameter D = 134.5 mm and length L = 336.3 mm. This work presents a data-driven approach for regime identification and classification that relies solely on high-speed video recordings and circumvents the need for interface tracking. The method combines prototype-based data labeling with dimensionality reduction via multiscale proper orthogonal decomposition (mPOD) and automatic kernel-based classification. The results are summarized in a dimensionless regime map across three fill ratios, where stable waves, longitudinal and transverse mode shapes, and mode-competition regimes are distinguished. The developed map provides a predictive tool for assessing sloshing-induced loads, supporting structural and operational optimization of fuel systems.

Figures (16)

• Figure 1: Experimental setup integration in the SHAKESPEARE sloshing table facility. The sloshing cell comprises a PMMA horizontal cylindrical test section with an aspect ratio of $L/R = 5.0$, with aluminum cylindrical end domes. A high-speed camera acquires the free-surface displacement ($\eta$) under vertical accelerations ($a_z$), and the dynamic conditions are controlled through an optical displacement sensor and a triaxial accelerometer ($\boldsymbol{a}(t) = \left(a_x(t), a_y(t), a_z(t)\right)$).
• Figure 2: Flowchart of the proposed algorithm.Step 1: acquired image sequences are pre-processed to correct for unwanted vibrations and tank tilt. Step 2: the mPOD is used to analyze the acquired snapshots between $t \in [0, 60]\,$ seconds. Data is then truncated to the first $n_R = 30$ mPOD modes. Step 3: a reference prototype library ($\boldsymbol{\Phi}_{ref}$) is created, and each truncated test case ($\tilde{\boldsymbol{D}}$) is projected into it to identify dominant patterns over time ($\boldsymbol{A}_{proj} \rightarrow \boldsymbol{z}_i$). Finally, unsupervised $k$-means clustering is used to label the data, and a non-linear multiclass SVM determines continuous decision boundaries between sloshing regimes (Step 4).
• Figure 3: mPOD analysis for $H_l/D = 0.50$ at the forcing condition $A_f = 39.7mm$ and $f_f = 1.85Hz$, performed over the full image acquisition interval $t \in [-5.0,\,100.0]$ seconds. Spatial modes $\boldsymbol{\phi}_{r}$, temporal coefficients $\boldsymbol{\psi}_{r}$, and their normalized frequency spectra $|\widehat{\boldsymbol{\psi}}_{r}|$ are presented for the three leading modes. The sloshing signal starts at $t = 0.0\s$ and the dominant mode shape $\boldsymbol{\phi}_1$ has a single-node antisymmetric longitudinal structure with a subharmonic response.
• Figure 4: Wavelet scalograms (time–frequency representation) obtained from the complex Morlet transform (bandwidth 6.0, center frequency 2.0) of the temporal coefficients of the three most energetic POD modes ($\boldsymbol{\psi}_1$, $\boldsymbol{\psi}_2$ and $\boldsymbol{\psi}_3$) in Fig. \ref{['fig:nat_freq_POD_modes']}, the two load-cell force signals ($F_{z_1}$ and $F_{z_2}$), and the accelerometer ($a_z$) over the interval $t \in [-5.0, 90.0]$ seconds. The sloshing signal is initiated at $t = 0.0s$. The frequency axis is normalized by the forcing frequency $f_f = 1.85\Hz$, and wavelet power is represented on a logarithmic scale using $P_{\log}(t,f) = 10\log_{10}(|W(t,f)|^2)$.
• Figure 5: Wavelet ridge diagnostics for $H_l/D = 0.50$ at $A_f = 39.7mm$ and $f_f = 1.85Hz$. Columns correspond to the three signals ($\boldsymbol{\psi}_1$, $F_{z_1}$, $F_{z_2}$). Top row: ridge frequency $f_r(t)/f_f$ identified as $\arg\max_{f\in\mathcal{B}}\,P(t,f)$; the dashed vertical lines indicate ridge jumps. Bottom row: logarithmic ridge power $\ln P_r(t)$ with least-squares fit $\ln P_r(t)=m_{\ln P_r}\,t+P_0$ over the ring-down window $[67.5, 90.0]$ (reported $m_{\ln P_r} = - 2\zeta_{1,0} \omega_{d_{1,0}}$ and $R^2$ shown in each panel).