Swirling against the forcing: evidence of stable counter-directed sloshing waves in orbital-shaken reservoirs

TL;DR

This paper investigates the surface wave behavior in a cylindrical container shaken elliptically, revealing stable counter-directed swirling waves and validating findings with an inviscid asymptotic model.

Contribution

It provides experimental evidence and a theoretical model for stable counter-directed swirling waves in orbital-shaken reservoirs, highlighting the influence of orbit aspect ratio.

Findings

Existence of stable counter-directed swirling waves

Experimental validation of wave behavior

Model predictions align with observed phenomena

Abstract

We study the free surface response in a cylindrical container undergoing an elliptic periodic orbit. For small forcing amplitudes and deep liquid layers, we quantify the effect of orbit's aspect ratio onto the surface dynamics in the vicinity of the fluid system's lowest natural frequency. We provide experimental evidences of the existence of a frequency range where stable swirling can be either co- or counter-directed with respect to the container's direction of motion. Our findings are successfully predicted by an inviscid asymptotic model, amended with a heuristic damping.

Equations

$$\frac{{\text{d}}^2{\mathbf{X}}_0}{\text{d}{t}^2}=\{\begin{array}{l}\left(-f_x\cos \Omega t\cos \theta -f_y\sin \Omega t\sin \theta \right){\mathbf{e}}_r,\\ \left(f_x\cos \Omega t\sin \theta -f_y\sin \Omega t\sin \theta \right){\mathbf{e}}_{\theta },\end{array}$$

$$\frac{\text{d}A}{\text{d}t}=-\left(\sigma +\text{i}\lambda \right)A+\text{i}\mu {\alpha }_{{}_A}f+\text{i}\nu |A{|}^{2}A+\text{i}\xi |B{|}^{2}A,$$

$$\frac{\text{d}B}{\text{d}t}=-\left(\sigma +\text{i}\lambda \right)B+\text{i}\mu {\alpha }_{{}_B}f+\text{i}\nu |B{|}^{2}B+\text{i}\xi |A{|}^{2}B.$$

Full text

††preprint: APS/123-QED

The problem of liquid sloshing pertains to many aspects of daily life, ranging from mundane wine tasting to more pragmatic issues such as liquid spilling (Mayer and Krechetnikov, 2012) and transport safety (Faltinsen and Timokha, 2009). A proper predictive understanding and modelling of the sloshing hydrodynamics at stake is therefore essential in the design process of liquid tanks and in implementing control systems of vehicles (Ibrahim, 2005).

In this regard, the case of orbital sloshing in partially filled circular cylinders represents one of the archetypal sloshing systems (Abramson, 1966). Previous experimental studies have described its close-to-resonance dynamics for either circular or purely longitudinal shaking, casting light on a rich variety of wave regimes attracting interest to dynamicists over the last decades (Hutton, 1963; Ockendon and Ockendon, 1973; Miles, 1984a, b).

For circular orbits, the system responds with a swirling wave always co-directed with the container motion (Reclari et al., 2014). This well-defined hydrodynamics, often simply modelled by a one-degree-of-freedom Duffing oscillator (Ockendon and Ockendon, 1973; Bongarzone et al., 2022a), is advantageously exploited, for example in biology, in the design of bioreactors, where the container is shaken so as to mix the liquid, prevent sedimentation and enhance gas transfer, hence providing a suitable oxygenation to the growing cell population (Klöckner and Büchs, 2012). In the case of longitudinal forcing, the standing wave solution may undergo close-to-resonance a symmetry-breaking, with clock- and anti-clockwise swirling waves equally probable, or completely lose regularity showing an irregular and chaotic alternance of planar and swirling motions (Hutton, 1963; Royon-Lebeaud et al., 2007). Such a configuration finds a close mechanical analogy in the resonant motion of a forced spherical pendulum (Miles, 1984b), a four degrees-of-freedom system that has been widely studied in the context of order-to-chaos transitions (Miles, 1984a; Tritton, 1986) for its similarities with the Lorentz’s problem (Lorenz, 1963).

Surprisingly however, no experimental studies devoted to the more generic case of elliptic orbits have been reported so far in the sloshing literature. Yet, existing theoretical analyses of this forcing condition brought out interesting features of the resonant liquid response that depend on the orbit’s ellipticity. In particular, a recent inviscid theory Faltinsen et al. (2016) suggested the counter-intuitive existence, under resonant elliptic forcing, of stable swirling waves that propagate in the direction opposite to the forcing direction. Moreover, the theory anticipated that such counter-waves may exist even for quasi-circular orbits and travel with a smaller amplitude than co-directed waves. This, if confirmed, would further enrich the variety of observable dynamical sloshing regimes and possibly open new room in this rich dynamic system field.

For moderately large-size containers, the use of inviscid hydrodynamic models is well accepted (Ibrahim, 2005), but in real sloshing problems, waves are always subjected to a non-vanishing viscous dissipation. Hence, the counter-swirling wave predicted by the inviscid model Faltinsen et al. (2016), being intrinsically disfavored by the forcing direction, is likely to be more sensitive to damping than co-swirling solutions, and it is currently unclear whether such a solution can actually arise in a real-life lab-experiment.

With this Letter, we aim at providing a joint experimental and theoretical characterization of the free liquid surface response for a generic, elliptic periodic container trajectory, so as to bridge the gap between the two diametrically opposed shaking conditions previously discussed. Specifically, we intend to identify the range of external control parameters, i.e. driving frequency, amplitude and orbit aspect ratio, for which stable counter-directed swirling waves do occur, and assess the extent of the forcing regime where inviscid models break down.

To this end, we used a Plexiglas circular cylindrical container of total height $50\,\text{cm}$ and internal radius $R=0.086\,\text{m}$, filled to a depth $h=0.15\,\text{m}$ with water: density \rho=1000\,\text{kg\,m{}^{-3}}, surface tension \gamma=0.0725\,\text{N\,m{}^{-1}} and dynamic viscosity \mu=0.001\,\text{kg\,m{}^{-1}$\,s${}^{-1}}.

The gravity acceleration is denoted by $g$. The container is fixed on a double-axes linear motion actuator (Aerotech pro165LM + pro225LM), which imposes along the x and y axes two sinusoidal forcings of angular frequency $\overline{\Omega}$ and amplitudes $\overline{a}_{x}$ and $\overline{a}_{y}$, that are $\pi/2$-phase shifted with respect to each other. The fluid motion is recorded with a digital camera (Nikon D850) coupled with a Nikon 60mm f/2.8D lens and operated in slow mode with an acquisition frequency of 120fps. The camera optical axis is aligned with the x axis (see also Fig. 1(a)). A LED panel is placed behind the container so as to provide a back illumination for a better optical contrast.

In the moving reference frame, any planar elliptic-like shaking can be represented by the following equations describing the motion acceleration of the container axis parametrized in polar coordinates ($r$, $\theta$),

[TABLE]

where $f_{x}=f=\overline{a}_{x}\Omega^{2}/R$ and $f_{y}=\alpha f=\overline{a}_{y}\Omega^{2}/R$ are the non-dimensional major- and minor-axis driving acceleration components, respectively, and $\Omega=\overline{\Omega}/\sqrt{g/R}$ the non-dimensional driving angular frequency. The bar symbol refers to the dimensional quantities. Note that the minor-to-major-axis component ratio, $\alpha=\overline{a}_{y}/\overline{a}_{x}=f_{y}/f_{x}$, has been introduced. A value $0<\alpha<1$ refers to elliptic orbits, whereas the two limiting cases $\alpha=0$ and $\alpha=1$ correspond, respectively, to longitudinal and circular shaking conditions.

With this experimental campaign we intend to study the free surface response in the vicinity of the lowest natural frequency $\omega_{0}=\overline{\omega}_{0}/\sqrt{g/R}=\sqrt{k\tanh{\left(kh/R\right)}}=1.3547$ (with wavenumber $k=1.8412$) (Lamb, 1993), for varying orbit’s aspect ratios $\alpha$ and forcing amplitudes $\overline{a}_{x}$. In particular, we aim at triggering a counter-directed wave, if such a solution can arise in our experiments, and study how its existence depends on the forcing parameters.

In a typical experiment, the amplitude $\overline{a}_{x}\in\left[1,3\right]\,\text{mm}$ and ellipticity $\alpha\in\left[0.10.95\right]$ are fixed, while frequencies are swept up- and downward within the (dimensionless) range $\Omega/\omega_{0}\in[0.82,1.21]$. The increment between two consecutive steps in the frequency sweep is $0.0217$. Each frequency step consists in two parts: the container undergoes first an harmonic elliptic forcing that is in the anti-clockwise direction for 150 oscillation periods and then in the clockwise direction for another 150 oscillation periods. Two movies are then recorded at each step so as to monitor the free surface response to both clockwise and anti-clockwise forcing. Switching the direction of the tank’s trajectory induces a flow perturbation that can be enough to produce a counter-directed wave if the latter is an admissible configuration for the system. To ensure that the steady-state regime is established at each step and for each container direction, the camera is triggered only after 100 cycles so that it only records the last 50 oscillation periods.

The procedure to analyze the free surface response is extensively described in Marcotte et al. (2023) and illustrated here in Fig. 1(b). Briefly, we build from each movie an image $I(y)=[I_{t_{1}}(y),I_{t_{2}}(y),...]$ where $I_{t_{i}}(y)$ is the intensity profile along the vertical axis $Y(t_{i})=y$ on the frame $i$ corresponding to time $t_{i}$, with $Y(t_{i})=0$ being the vertical middle axis between the edges of the container image (represented by $Y(t_{i})=R$ and $Y(t_{i})=-R$). The resulting image, as illustrated in Fig. 2, displays a periodic dark pattern that represents the free surface response to the harmonic forcing. The free surface appears as the darkest feature on each frame, so that the intensity profile along a vertical line at a given time $t_{i}$ represents the vertical extension of the free surface in this direction.

The usefulness of the resulting image $I(y)$ is threefold: (i) it allows to detect irregular dynamics. This corresponds to the absence of any stable wave-amplitude for a given set of forcing parameters, and is easily identified by the time-varying envelope modulating the free surface oscillations, see Fig. 2(a). (ii) For a regular response, $I(0)$ enables one to measure the amplitude of the front contact line in the azimuthal direction $\theta=0$. (iii) The comparison of the profiles along two vertical directions that are mirror-symmetric with respect to the vertical middle axis, e.g. $I(-R/2)$ and $I(R/2)$, makes possible to determine the propagation direction of the wave and to compare it with the container’s motion direction.

Fig. 2 displays two intensity profiles as a function of time along the vertical middle axis $(Y=0)$ for same forcing amplitude $\overline{a}_{x}$ and ellipticity $\alpha$, but for two different forcing frequencies $\Omega/\omega_{0}$. Those images show that depending of the forcing parameters, the amplitude of the free surface oscillations can be either irregular, Fig. 2(a)-(b), or stationary, Fig. 2(c)-(d). In the analysis of the close-to-resonance dynamics, we therefore use the profile $I(0)$ to identify the irregular regime.

The intensity profile $I(0)$ also provides the amplitude $\delta(\theta=0,t)$ of the swirling wave at the front wall of the container, i.e. at the azimuthal coordinate $\theta=0$, such as defined in Fig. 1(b). Indeed, due to the back lighting, the intensity signal corresponding to the front contact line appears darker than the one due to the back contact line, so that informations associated with the latter can be filtered out by a proper thresholding of profile $I(0)$. On the resulting binarized image, the maximal and minimal heights of the final periodic pattern correspond then to the peaks and troughs of the swirling wave at the front wall along $\theta=0$. The amplitude of the wave (in pixel) is thus experimentally retrieved as half the difference between the height of the top and of the bottom envelopes enclosing its oscillations, displayed in in Fig. 2(d) as red lines, and converted into millimeters by using a scale put on the front wall of the container. Note that in this procedure, we neglect the variation of the pixel size that can occur along the container motion, the camera being fixed. This is justified by the very small forcing amplitude ($1\,\text{mm}\leq\overline{a}_{x}\leq 3\,\text{mm}$) with respect to the distance between the camera and the front wall of the container ($1\,\text{m}$). The error related to the variation of the pixel size is therefore of the order of $0.1\%$, i.e. negligible compared to the typical dispersion of our measurements.

To detect the direction of propagation of the wave, we compare for each movie the intensity profiles along two vertical directions that are mirror-symmetric with respect to the vertical middle axis of the container. Fig. 3 shows composite images that each consists in the superposition of $I(R/2)$ and $I(-R/2)$ into a composite RGB image, where gray areas correspond to pixels where $I(R/2)$ and $I(-R/2)$ have the same intensity, while red (resp. blue) areas correspond to the part of $I(-R/2)$ (resp. $I(R/2)$) that do not overlap with $I(R/2)$ (resp. $I(-R/2)$). Thus, a red (resp. blue) peak preceding a blue (resp. red) peak corresponds to a wave traveling from the left (resp. right) to the right (resp. left) hand-side of the front wall of the container, i.e. in the anti-clockwise (resp. clockwise) direction. The propagation direction of the wave can then be determined and compared to the direction of the container motion. In Fig. 3, the dynamics associated with two different aspect ratio $\alpha=0.5$ and $\alpha=0.95$ (quasi-circular orbit) are compared for the same forcing frequency and amplitude. For each $\alpha$, the right and left hand-side signals $I(R/2)$ and $I(-R/2)$ are superposed to each other for two motion configurations, namely an anti-clockwise followed by a clockwise container trajectory. In the case of the anti-clockwise tank’s motion, Fig. 3(a)-(b), the swirling wave travels in the same direction as the container, but the change of container’s motion direction induces a flow perturbation sufficient to produce a robust counter-directed wave, if the latter corresponds to a system’s stable solution. We indeed observe in the case of $\alpha=0.5$ that the wave, though of smaller amplitude, is still traveling from the left to the right hand-side of the container’s front wall despite the reverse of direction in the tank trajectory. This appears glaring in Fig. 4, where the two series of free surface snapshots show how the wave’s direction of rotation remains unchanged despite the reversal of the container’s direction of motion. On the contrary, Fig. 3(c)-(d), the wave switches direction for the large ellipticity $\alpha=0.95$ and is therefore co-directed with the forcing for both container motion directions. These results provide the first experimental evidence for the existence of counter-swirling solutions, and validate this procedure as suitable to trigger and identify stable counter-directed waves.

To assess the extent of the validity of an inviscid, and thus simplified, hydrodynamic model to predict resonant counter-swirling in a lab-scale experiment, we aim at comparing our experimental data to quantitative predictions provided by the model formalized in Marcotte et al. (2023) and recalled in the Supplementary Materials (Sup, b). This weakly nonlinear model has been extensively compared with Faltinsen et al. (2016) for both purely circular Bongarzone et al. (2022b) and longitudinal Marcotte et al. (2023) shaking conditions and it has been shown to provide consistent results. Briefly, we assume an inviscid, irrotational and incompressible flow (as common in sloshing theory applied to lab-scale containers (Faltinsen and Timokha, 2009; Ibrahim, 2005)) and a small forcing acceleration $f\sim\epsilon^{3}$, with $\epsilon$ a small parameter $\ll 1$. This last assumption is justified by the fact that close to the resonance $\Omega\approx\omega_{0}$, even a small forcing will induce a large system response. Our model consists then in a multiple time-scale weakly nonlinear (WNL) analysis Whitham (1974), whose lowest order asymptotic solution is described by the following system of amplitude equations governing the close-to-resonance interaction of two counter-propagating waves of complex amplitudes $A=|A|e^{\text{i}\Phi_{A}}$ $\left(-\right)$ and $B=|B|e^{\text{i}\Phi_{B}}$ $\left(+\right)$ and oscillating in time-space as $\cos{\left(\Omega t\mp\theta\right)}$, such that the free surface elevation $\delta\left(0,t\right)$, e.g. measured at the lateral wall, $r=R$, and at an azimuthal coordinate $\theta=0$, reads $\delta\left(0,t\right)/R=2\left(|A|\cos{\left(\Omega t+\Phi_{A}\right)}+|B|\cos{\left(\Omega t+\Phi_{B}\right)}\right)$,

[TABLE]

The parameter $\lambda=\Omega-\omega_{0}$ represents a frequency detuning with respect to the exact resonance, whereas $\alpha_{{}_{A}}=\left(1+\alpha\right)/2$ and $\alpha_{{}_{B}}=\left(1-\alpha\right)/2$ are two auxiliary orbit parameters. The values of $\mu$, $\nu$ and $\xi\in\mathbb{R}$ are given in Supplementary Material (Sup, b).

In contradistinction with Marcotte et al. (2023), here we account for an heuristic viscous damping coefficient, $\sigma$, introduced a posteriori in (2a)-(2b) and whose value is estimated according to equation (17) of Raynovskyy and Timokha (2018). For our setup, the damping associated with $\omega_{0}$ amounts to $\sigma=0.0055$.

We now compare, in Fig. 5, our measurements to the asymptotic model (2a)-(2b). At small ellipticity, e.g. $\alpha$ close to $0.10$, the amplitude response curve is similar to that induced by a longitudinal forcing (Royon-Lebeaud et al., 2007; Marcotte et al., 2023) except that the planar wave solution no longer exists, owing to the preferential direction of motion, and that the co- and counter-rotating waves are no more equally probable, with the counter-wave exhibiting a slightly lower amplitude. By increasing the value of $\alpha$, the counter-wave displays a decreasing amplitude and the range of frequency for which irregular motion occurs shrinks down and ultimately vanishes (Faltinsen et al., 2016). In this context, irregular means that both the co- and counter-swirling solutions are unstable and the system exhibits irregular and chaotic patterns switching between co- and, at a small ellipticity, counter-swirling dynamics alternating transient intervals of nearly-planar motions (see Supplementary Movies (Sup, a)). As $\alpha$ approaches $1$, the admissible frequency range associated for counter-waves reduces and it is eventually suppressed, whereas the frequency range associated with co-directed swirling widens and covers all of the resonance frequency range at $\alpha=0.95$, i.e. approaching the limiting case of a circular trajectory ($\alpha=1$) (Reclari et al., 2014; Bongarzone et al., 2022a). We also observe a decrease in the wave amplitude at $\overline{a}_{x}=3\,\text{mm}$ for $\alpha\geq 0.5$, occurring just before the jump-down frequency and which can be tentatively attributed to increasing nonlinear effects overlooked by the WNL model.

The experimental steady-state wave amplitudes are in a good quantitative agreement with the WNL analysis for all $\overline{a}_{x}$ and $\alpha$ values explored, hence proving the validity of the inviscid analysis in our regime of operation. The only major limitation is intrinsic to the use of a simple phenomenological damping. As the latter does not depend on the wave amplitude, it cannot accurately predict the phase-lag between forcing and the system response (Bäuerlein and Avila, 2021). This translates in an imprecise estimation of the jump-down frequency occurring above resonance and of the frequency range associated to the counter-swirling, that appears slightly overestimated.

To conclude, we have investigated the sloshing dynamics in the vicinity of the first harmonic resonance, i.e. $\Omega/\omega_{0}\approx 1$, for container’s elliptic orbits. The amplitude-response curves at different forcing amplitudes were examined versus the orbit aspect ratio, $0<\alpha<1$. Our experiments demonstrated the existence of a significant frequency range associated with stable counter-swirling, up to surprisingly high orbit’s aspect ratio.

Our findings have been successfully rationalized by the asymptotic model formalized in Marcotte et al. (2023) starting from the full hydrodynamic system supplemented with a simple heuristic damping coefficient. The close-to-resonant sloshing dynamics for any container’s elliptic-like orbit is therefore well represented by four degrees of freedom only. This suggests that a simple generalization of the forcing condition in the problem of the resonantly forced spherical pendulum (Miles, 1984a) could provide a suitable mechanical analogy for this entire class of sloshing dynamics, thus offering further room in this archetypical low degrees-of-freedom class of dynamical systems.

Perspectives include the experimental characterization of the effect of variations in the fluid depth and the WNL model extension to a viscous framework (Bongarzone et al., 2022b). Although the latter presently hinges on the subtle modelling of the moving contact line dynamics, such an extension is desirable, as it would enable one to better quantify the overall system dissipation and to predict the viscous streaming experimentally observed in orbital shaken containers (Bouvard et al., 2017).

This work was founded by the Swiss National Science Foundation under grants 178971 and 200341.