From flying wheel to square flow: Dynamics of a flow driven by acoustic forcing

Acoustic streaming designates the ability to drive quasisteady flows by acoustic propagation in dissipative fluids and results from an acoustohydrodynamics coupling. It is a noninvasive way of putting a fluid into motion using the volumetric acoustic force and can be used for different applications such as mixing purposes. We present an experimental investigation of a kind of square flow driven by acoustic streaming, with the use of beam reflections, in a water tank. Time-resolved experiments using particle image velocimetry have been performed to investigate the velocity field in the reference plane of the experiments for six powers: 0.5, 1, 2, 4, 6, and 8 W. The evolution of the flow regime from almost steady to strongly unsteady states is characterized using different tools: the plot of time-averaged and instantaneous velocity fields, the calculation of presence density maps for vortex positions and for the maximal velocity and vorticity crest lines, and the use of spatiotemporal maps of the waving observed on the jets created by acoustic streaming. A transition is observed between two regimes at moderate and high acoustic forcing.

Figure 1

Sketch of the experimental setup (top view at midheight of the domain). The origin of the Cartesian frame is set at the middle of the square investigation domain. The acoustic beam axis describes a square trajectory at midheight of this domain $(z=0$ mm) by reflecting at the center of each lateral glass wall. A glass lid on top of the cavity avoids the presence of a free surface.

Figure 2

Time-averaged velocity field and colored map of velocity magnitude obtained by 2D PIV in the $z=0$ horizontal plane of the investigation domain: A, acoustic jets; B, inertial jets; C, center area; and D, corner areas. The flow is generated by acoustic streaming at an electrical power of $P=6$ W. The acoustic beam enters the measurement volume from the left with a ${45}^{\circ }$ angle through a circular opening depicted as a thick black line. It then reflects successively on three sidewalls before leaving the measurement volume through the same opening following the path depicted in Fig. 1.

Figure 3

Three-dimensional view of the steady velocity field obtained by numerical simulation for 1 W power, showing the 3.3-mm ${\mathrm{s}}^{-1}$ isosurface of velocity magnitude. The velocity field is also depicted in two planes: the symmetry plane $\left(z=0\right)$, with the colored map of velocity magnitude, and the $y=0$ plane, with the colored map of the vertical velocity $w$ to give indications on the velocity field outside the symmetry plane. For the vertical $y=0$ plane, only the in-plane velocity components $(u$ and $w$, but not $v)$ are given by the arrows to make the visualization of the vertical motion easier.

Figure 4

Colored maps for the time-averaged velocity magnitude in the $z=0$ horizontal plane at different electrical powers $P$: (a) $P=0.5$ W, (b) $P=1$ W, (c) $P=2$ W, (d) $P=4$ W, (e) $P=6$ W, (f) $P=8$ W, (g) $P=0.01$ W, (h) $P=0.1$ W, and (i) $P=1$ W. (a)–(f) Experimental results. (g)–(i) Numerical simulations.

Figure 5

Time series of the velocity magnitude colored map in the $z=0$ horizontal plane for the $P=8$ W experiment: (a) $t=t_{\mathrm{start}}$, (b) $t=t_{\mathrm{start}}+2\text{s}$, (c) $t=t_{\mathrm{start}}+4\text{s}$, (d) $t=t_{\mathrm{start}}+6\text{s}$, (e) $t=t_{\mathrm{start}}+8\text{s}$, (f) $t=t_{\mathrm{start}}+10\text{s}$, (g) $t=t_{\mathrm{start}}+12\text{s}$, and (h) $t=t_{\mathrm{start}}+14\text{s}$. Here $t_{\mathrm{start}}$ is the beginning of the time series and not the beginning of the PIV recording. The following abbreviations are used: CoV, corner vortex; CeV, central vortex; WAJ, waving acoustic jet; and CoF, corner filament.

Figure 6

(a)–(f) Density maps of the presence of velocity minima in the center of the measurement domain for the different electrical powers: (a) $P=0.5$ W, (b) $P=1$ W, (c) $P=2$ W, (d) $P=4$ W, (e) $P=6$ W, and (f) $P=8$ W. Gray color stands for no minimum found. (g) Graph of the mean number of velocity minima per PIV frame with respect to the power. Instantaneous velocity magnitude fields for the $P=0.5$, 2, 4, and 8 W experiments illustrate the position and the displacement of the detected minima. Red circles correspond to the velocity minima in the current frame, while the white crosses show the position of the velocity minima for the previous ten frames, as tails emphasizing the displacement of the minima (see [41] for a dynamic visualization).

Figure 7

(a)–(f) Spatiotemporal diagrams showing the velocity maximum crest shift with respect to the crest of the time-averaged flow for the bottom left acoustic jet and at different electrical powers $P$: (a) $P=0.5$ W, (b) $P=1$ W, (c) $P=2$ W, (d) $P=4$ W, (e) $P=6$ W, and (f) $P=8$ W. The same color scale is chosen for all the cases. (g) Relative positions of the mean velocity crest extracted from the time-averaged velocity field and of an instantaneous velocity crest. (h) Shift of the instantaneous velocity crest shown in (g) with respect to the mean velocity crest. The shift is colored relative to its sign defined in (g): in red for an instantaneous positive shift toward the corner side and in blue for a negative shift toward the center side.

Figure 8

Presence density maps of the crest lines of the velocity maxima associated with the inertial and acoustic jets for (a) $P=0.5$ W, (b) $P=1$ W, (c) $P=2$ W, and (d) $P=8$ W. Gray color stands for no crest found over the 6000 frames.

Figure 9

Normalized spatiotemporal maps of the shift of the velocity maxima crest lines: (a) $P=1$ W, (b) $P=4$ W, (c) $P=6$ W, and (d) $P=8$ W. The curvilinear abscissa is normalized by the length of the acoustic jet $L_{\mathrm{ac}\mathrm{jet}}$ and the time by $L_{\mathrm{ac}\mathrm{jet}}/V_{\max }$, where $V_{\max }$ is the maximal velocity on the acoustic jet.

Figure 10

Presence density maps of the crest lines of the vorticity maxima associated with the acoustic jets: (a) $P=0.5$ W, (b) $P=1$ W, (c) $P=2$ W, (d) $P=4$ W, (e) $P=6$ W, and (f) $P=8$ W. Gray color stands for no crest found over the 6000 frames.

Figure 11

Close-up of the spatiotemporal maps showing the shift of the crest lines of the velocity maxima: (a) $P=1$ W and (b) $P=8$ W. The four acoustic jets are considered, as they appear along the acoustic beam; hence, shown, from top to bottom, are the bottom left, the bottom right, the top right, the top left, and again the bottom left acoustic jets. Gray color stands for no crest found at a given space-time location.

Figure 12

Graph summarizing the changes of flow regime with increasing acoustic forcing.