Physics of a strongly oscillating axisymmetric air-water interface with a fixed boundary condition

In this work, we experimentally investigate the physics of a strongly oscillating, millimeter-sized, axisymmetric air-water interface with a fixed contact line boundary condition. Many previous studies focused on the regime of small oscillation amplitude, e.g., $R=d/D\ll 1$, where $d$ is the oscillation amplitude and $D$ is the characteristic size of the air-water interface. The current investigation instead focuses on a less-studied oscillation regime with large $R$ that is up to 0.33. The dynamic oscillations induce different steady streaming patterns, such as a low-speed streaming vortex or a fast-speed streaming jet. The steady streaming jet, in particular, was not much studied previously and is the major focus of this work. The streaming jet is only generated when the oscillating interface exhibits the higher-order axisymmetric oscillation modes with a large oscillation amplitude, which correspond to the regime with large $R$ and large Weber number (We). The streaming jet has a larger Reynolds number $[\mathrm{Re}\sim \mathcal{O}(100\left)\right]$ than the typical streaming motions induced by an oscillating interface $[\mathrm{Re}\sim \mathcal{O}(1\left)\right]$. In addition, the streaming jet has a high ratio of the streaming velocity versus the oscillatory velocity, which suggests a high efficiency in generating steady streaming motion. The dynamic velocity and vorticity field of the streaming jet in both the initiation stage and the quasisteady stage are presented, which demonstrates that the streaming jet onset process is a consequence of vorticity generation, transportation, and accumulation happening at the oscillating interface. As a zero-mass-flux jet, the streaming jet is sustained by entraining fluid mass flux from the circumferential regions through a process similar to the classical Stokes drift. It is further found that a streaming jet can be induced by an oscillating elastic no-slip membrane as well, when the oscillation has large $R$ and We. The extraordinary characters of the streaming jet can be employed in many engineering applications.

Figure 1

Schematic drawing of an oscillating free-slip air-water interface with a fixed contact line boundary condition. The contact line of the air-water interface is fixed at the edge of a surface cavity by applying superhydrophobic and superhydrophilic material coating in different regions of the surface cavity. The static and dynamic curvature of the interface is controlled by regulating the static and dynamic pressure inside the surface cavity. The dashed line in panel (b) marks the location of the experiment measurement plane.

Figure 2

Schematic experimental setup where a high-speed camera and a continuous laser were employed to conduct high-resolution digital particle image velocimetry (DPIV) in the central plane of the axisymmetric air-water interface [marked by the dashed line in Fig. 1]. The Cartesian coordinate system and the gravitational direction are also shown.

Figure 3

Snapshots of the dynamic free-slip surface at five equally spaced phases ($\varphi$) over the period of one oscillation cycle (T). The static interface was either flat [(a), (b), and (c)] or bulged-out [(d), (e), and (f)]. The modulation frequency was 20 Hz [(a) and (d)], 30 Hz [(b) and (e)], and 50 Hz [(c) and (f)].

Figure 4

The phase-varying profiles of a bulged-out interface oscillating at (a) 20 Hz, (b) 30 Hz, and (c) 50 Hz. The dynamic oscillations resemble the different modes of the Bessel function. The oscillation-amplitude peaks at 30 Hz among the selected frequencies.

Figure 5

Flow pathlines (in 1 s) showing the steady streaming motions induced by the different dynamic oscillations, with the same panel labeling as Fig. 3. The outmost boundary of the oscillating interface is highlighted by the blue dashed lines. The observed streaming patterns include small-scale streaming vortex and large-scale streaming jet as well as a transitional streaming vortex in between.

Figure 6

Mean velocity contour associated with an oscillating bulged-out interface (a) 20 Hz, (b) 30 Hz, and (c) 50 Hz. The 30-Hz case has the fastest streaming velocity. The streamlines (gray arrow curves) demonstrate the different streaming patterns and entrainment processes.

Figure 7

Phase diagram of the different streaming patterns based on the nondimensional amplitude ($R=\frac{d}{D}$) and the Weber number $(\mathrm{We}=\frac{\rho {\left(fd\right)}^{2}D}{\gamma })$. Circle: streaming vortex; cross: streaming jet. Gray dashed line: an approximate boundary of streaming pattern change.

Figure 8

Snapshots of the synchronized particle movement and velocity and vorticity contour in the initiation process of the streaming jet. The three snapshots are in (a) the 1st oscillation cycle, (b) the 5th oscillation cycle, and (c) the 10th oscillation cycle. The accumulation of the ring-shaped vorticity along the symmetric axis corresponds to the formation of the jet.

Figure 9

Synchronized time series of the vertical velocity $\left[V\right(t\left)\right]$ and the transverse vorticity $\left[{\omega }_z\left(t\right)\right]$ measured at the $(x,y)$ location of (7.7 mm, 10 mm), which is along the symmetry axis and far away from the oscillating interface. The streaming jet is created and sustained because the accumulated $\overline{{\omega }_z}$ induces a steady vertical velocity following the Biot-Savart law.

Figure 10

Phase-averaged velocity vector contour at four equally spaced sample phases. The contour color shows the velocity magnitude (unit is m/s) and the vector arrows shows the velocity direction.

Figure 11

Phase-averaged vertical velocity $\tilde{v}$ measured at (a) $y=5\mathrm{mm}$, (b) $y=13\mathrm{mm}$, and (c) $y=20\mathrm{mm}$. The near-field velocity exhibits features of a steep traveling wave whereas the far-field velocity reflects a steady current.

Figure 12

Panel (L): Flow pathlines near a 30-Hz oscillating bulged-out interface over the period one oscillation cycle. In the circumferential entrainment regions [e.g., labeled as (a)–(d)], the flow pathlines are closed-loop circles. In the jetting region, the flow pathlines are vertical straight lines. Panel (R): Phase-plane plots for the phase-averaged velocity vector $(\tilde{u},\tilde{v})$ at locations labeled as (a)–(e) in panel (L). Blue circles: $(\tilde{u},\tilde{v})$; red dots: $(\overline{u},\overline{v})$; arrows: visual representation of the time-averaged velocity vector $(\overline{u},\overline{v})$. The dashed arrows in panels (a)–(d) are amplified 10 times, whereas the solid arrow in panel (e) is not amplified.

Figure 13

Flow pathlines over the period of three oscillation cycles for a 50-Hz oscillating no-slip elastic membrane under (a) low-power modulation and (b) high-power modulation.

Figure 14

Time-averaged velocity contour for the 50-Hz oscillating no-slip elastic membrane under (a) low-power modulation and (b) high-power modulation. Note that the contour scales are different in the two panels.

Figure 15

Phase-averaged vertical velocity $\tilde{v}$ measured at (a) $y=5\mathrm{mm}$ and (b) $y=13\mathrm{mm}$ for the streaming jet created by the high-power modulation. In panel (a), $\tilde{v}$ is approximately symmetric. In panel (b), features of steady streaming jet are observed. In addition, $\tilde{v}$ also exhibits features of strong oscillatory motion in regions both within and out of the streaming jet.