Experimentally Resolving Gravity-Capillary Wave Evolution in Vessels of Unknown Boundary Conditions

Abstract

The geometries of surface wave modes are determined by the highly nontrivial interplay of capillarity and wetting effects at the boundaries of their domain. Aside from idealised scenarios, this commonly leads to unknown boundary conditions, thereby hindering theoretical formulation and experimental analysis. To address this problem, we introduce Extracted Mode Tracking (EMT), a data-analysis framework to obtain instantaneous amplitude and phase content of axisymmetric surface-wave modes from spatio-temporal measurements. This approach uses unsupervised machine learning techniques to extract a basis of wave modes directly from collected data; the spatial profiles require no prior theoretical modelling, and so the issue of unknown boundary conditions is circumvented. Time-resolved mode amplitudes are reconstructed by geometric fitting at each recorded time-step, and the success is evaluated by a spectral signal-to-noise quantifier. Capabilities and limitations of EMT are systematically benchmarked on synthetic datasets, finding strong resilience against noise, improved accuracy over alternative methodologies, and the ability to operate with restricted domains which poses significant merit for use in experimental systems with limited measurement field-of-view. Finally, we conduct a Faraday-wave experiment in a regime highly sensitive to boundary effects in order to further validate the method, and demonstrate the observational access to nonlinear wave-dynamics enabled by EMT. These results establish EMT as a general tool for analysing wave mode dynamics of axially-symmetric fluid interface systems, and open pathways for quantitative studies of nonlinear mode-interactions, stability, and turbulence.

Figures (11)

• Figure 1: Cross section of the fluid system The dynamical object of interest is the interfacial surface that forms between two immiscible fluids of differing densities. The horizontal domain of the interface is a disk bounded by impermeable horizontal walls at radius $r=r_0$. The interface is composed of dynamic fluctuations $\eta(t,r,\theta)$ around a static background $\eta_{\mathrm{s}}(r)$ which may have curvature as it approaches $r_0$ due to the formation of a meniscus. We define $z=0$ at the static contact line $\eta_{\mathrm{s}}(r_0)$, which is at distances $h_\mathrm{b}$, $h_\mathrm{t}$ from the horizontal floor and ceiling respectively.
• Figure 2: Faraday wave experiment Panel a depicts our experimental setup. A cylindrical vessel (i) containing the two-fluid system is mounted to a bespoke shaking platform. The platform is suspended by springs and glides freely in the $z$ direction along pneumatic air bearings (ii). A voice coil actuator (iii) provides a sinusoidal acceleration at a frequency $\omega_{\mathrm{d}}=6.2(2\pi)\mathrm{Hz}$ with amplitude $|a_z|=2.53(4)m s^{-2}$. This is measured via a platform-mounted accelerometer (presented in panel c for one experimental run). Light from a backlit chequerboard pattern (iv) passes through the fluid-fluid interface where it reflects off a mirror (v) and passes to the high-speed camera (vi). This recorded data can be used to reconstruct the interface fluctuation height $\eta(t,r,\theta)$ via Fourier Transform Profilometry. Panel b displays $\eta_{\mathrm{rms}}$: the root-mean-square of $\eta$ over the spatial domain and time intervals of $4\pi/\omega_{\mathrm{d}}$, over the course of the same experiment as that presented in panel c. One observes an exponential growth of $\eta_{\mathrm{rms}}$ above the initial noise level at around $20s$ after the actuator starts oscillating. Around 12 seconds later, $\eta_{\mathrm{rms}}$ stabilises as the system enters a nonequilibrium steady-state. The steady-state persists until the driving is switched of and $\eta_{\mathrm{rms}}$ rapidly decays back to equilibrium.
• Figure 3: Mode extraction The wave mode extraction protocol is illustrated for the example of the $m=4, \omega = 3.1(2\pi)\mathrm{Hz}$ mode in data from the experiment detailed in Section \ref{['experimental_setup_subsection']}. Data is collected on the evolution of $\eta(t,r,\theta)$, a snapshot of which is displayed in panel a. Fourier transforming over $\theta$, and fixing an azimuthal number, one obtains $\eta_m (t,r)$ - the real component of which is displayed in panel b. The PSD of this quantity is calculated during the steady state (panel c). Frequencies at which peaks are found in the PSD correspond to the frequencies at which modes are oscillating. Data of $\Tilde{\eta}_m(\omega',r)$ is band-pass filtered in a narrow frequency domain $\omega'\in[\omega - \mathrm{ENBW}, \omega + \mathrm{ENBW}]$, shaded in yellow, then passed to a truncated SVD algorithm to find a principal component proportional to the radial function of the wave mode. The radial function for $\omega = 3.1(2\pi)\mathrm{Hz}$ was extracted with an EVR of 1 to 6 significant figures, and is displayed in panel d. After extracting the radial profiles for each other mode found in $\mathrm{PSD}[\eta_m]$, one can track the evolution of each by a geometric fitting at each recorded time frame. The complex amplitudes $\xi_{m,\omega}$ obtained have the same time resolution as the original recording, and panel e displays this for our example mode. To confirm how successfully the mode was tracked, one can examine the PSD of the obtained amplitude. The majority of the signal should be present at $\omega'=\omega$, which is confirmed visually in panel f, and quantitively by the signal-to-noise ratio of $\mathrm{SNR}=140$ (to 3 significant figures) as defined in Eq. (\ref{['eqn:SNR_def']}).
• Figure 4: Tracking accuracy vs noise level Synthetic data of $\eta(t,r,\theta)$ is constructed from synthetic $m=4$ modes at dimensionless frequencies $\omega\sqrt{r_0/g}=1.25, 3.75, 6.25$ and equal envelope amplitudes $|\overline{\xi}_{m,\omega}|/r_{0}=0.5$. Gaussian noise is added with varying standard deviation $\sigma_N$. The data is processed using both the extracted mode tracking (EMT) and discrete Hankel transform (DHT) methods and, to quantify the precision, signal-to-noise-ratios (SNR) are calculated for extracted modes found at frequencies corresponding to the synthetic modes. For each mode, SNR is plotted against $\sigma_N$ with the 95% confidence interval from 50 repetitions shaded in the same colour. We see the precision of both methods fall with increasing noise level. We also see the EMT method provides a consistently stronger signal than the DHT method as a result of the ridge regression.
• Figure 5: Radial profile extraction accuracy vs temporal resolution The upper panel displays the Explained Variance Ratio (EVR) of the truncated SVD used in the mode extraction procedure, and the lower panel displays the absolute radial correlation between the extracted radial profile $R_{m,\omega}^{(\mathrm{Ext})}$ with that of the corresponding synthetic mode $R_{m,\omega}^{(\mathrm{Syn})}$ present in the data. The former quantity informs us on the success of the SVD and the latter informs us on how accurate the result is. Both quantities are plotted as a function of the number of points in time considered $N_t$ for different noise levels $\sigma_N$. The same trend is observed across both panels.