Near-field acoustic streaming jet

A numerical and experimental investigation of the acoustic streaming flow in the near field of a circular plane ultrasonic transducer in water is performed. The experimental domain is a parallelepipedic cavity delimited by absorbing walls to avoid acoustic reflection, with a top free surface. The flow velocities are measured by particle image velocimetry, leading to well-resolved velocity profiles. The theoretical model is based on a linear acoustic propagation model, which correctly reproduces the acoustic field mapped experimentally using a hydrophone, and an acoustic force term introduced in the Navier-Stokes equations under the plane-wave assumption. Despite the complexity of the acoustic field in the near field, in particular in the vicinity of the acoustic source, a good agreement between the experimental measurements and the numerical results for the velocity field is obtained, validating our numerical approach and justifying the planar wave assumption in conditions where it is a priori far from obvious. The flow structure is found to be correlated with the acoustic field shape. Indeed, the longitudinal profiles of the velocity present a wavering linked to the variations in acoustic intensity along the beam axis and transverse profiles exhibit a complex shape strongly influenced by the transverse variations of the acoustic intensity in the beam. Finally, the velocity in the jet is found to increase as the square root of the acoustic force times the distance from the origin of the jet over a major part of the cavity, after a strong short initial increase, where the velocity scales with the square of the distance from the upstream wall.

Figure 1

Experimental setup, view from the side. The origin of the Cartesian frame is set at the middle of the transducer plane surface: $x$ axis coincides with the propagation direction, and the $y$ and $z$ axes are, respectively, horizontal and vertical. The overall length of the aquarium is 900 mm.

Figure 2

Plots of the phase isocontours of the acoustic wave in the near field at (a) $x=100$ mm (0.$36L_f)$ and (b) $x=150$ mm (0.$55L_f)$ and in the far field at (c) $x=274$ mm $\left(L_f\right)$ and (d) $x=549$ mm $\left(2L_f\right)$.

Figure 3

Velocity vector maps in the $xy$ horizontal plane including the acoustic beam axis. (a) Experimental measurements with the PIV technique for $P=4$ W. (b) Numerical computations for $f_{\mathrm{ac}\max }=1.5\mathrm{N}{\mathrm{m}}^{-3}$. The velocity fields for the two other considered values of $P$ can be found in Moudjed [22].

Figure 4

(a) Longitudinal profiles of the experimental axial velocity along the acoustic beam axis for $P=2$ W (blue solid lines), 4 W (pink solid lines), and 8 W (green solid lines); the profiles obtained by the corresponding numerical calculations are also plotted as red dashed lines for $f_{\mathrm{ac}\max }=0.725$, 1.5, and 2.$9\mathrm{N}{\mathrm{m}}^{-3}$. These three numerical profiles are normalized and plotted in (b) with blue solid lines, pink dashed lines, and green dotted lines, respectively; the normalized acoustic intensity along the beam axis is also plotted as a black line (analytical expression [23]).

Figure 5

Horizontal transverse profiles of the normalized experimental acoustic intensity (black dashed lines), the normalized experimental velocity at $P=2$ W (blue open squares), 4 W (pink open circles), and 8 W (green stars), and the normalized numerical velocity calculated with $f_{\mathrm{ac}\max }=1.5\mathrm{N}{\mathrm{m}}^{-3}$ (black solid lines). These profiles are plotted at a distance $x=50$, 100, 150, and 200 mm from the transducer. The black dotted-dashed vertical lines indicate the acoustic source diameter. Note that these plots are focused on the jet region, so the location of the lateral walls $\left(\right|y|/d_s\approx 3.2)$ is not represented.

Figure 6

Longitudinal velocities along the acoustic beam axis plotted as a function of the expression appearing under the radical sign in Eq. (5). Four cases are presented corresponding to our simulations performed for $f_{\mathrm{ac}\max }=0.725$, 1.5, and 2.$9\mathrm{N}{\mathrm{m}}^{-3}$ and the simulation of Kamakura et al. [18]. Two characteristic variations of the velocity are obtained: an initial quadratic increase, as depicted by the heavy black dashed line and heavy black dotted-dashed line obtained from Eq. (9) for $f_{\mathrm{ac}\max }=0.725\mathrm{N}{\mathrm{m}}^{-3}$ and for Kamakura et al. [18], respectively, and a square-root increase, as depicted by the heavy black solid line obtained from Eq. (5). On the different numerical profiles, the black solid diamonds indicate the points located at the distance $d_s/2$ from the upstream wall, which give the transition between the two characteristic variations. The $f_{\mathrm{ac}\max }^{1/2}$ dependence in these different zones is also shown.

Figure 7

(a) Sketch showing a two-dimensional axisymmetric view of the region close to the upstream wall. Mass conservation is performed in a control volume between the inflow (radial velocity considered as linear in the boundary layer along the upstream wall) and the outflow (constant axial component for $0\le r\le d_s/2)$. Sketched streamlines are represented with black lines. (b) Radial profiles of the axial velocity obtained numerically for $f_{\mathrm{ac}\max }=0.725\mathrm{N}{\mathrm{m}}^{-3}$ and given at different distances $x^{\prime }$ from the upstream wall.

Figure 8

Dimensionless longitudinal velocity $U=u/{(f_{\mathrm{ac}\max }{d}_s/2\rho )}^{1/2}$ along the acoustic beam axis plotted as a function of the dimensionless distance to the upstream wall $X^{\prime }=x^{\prime }/(d_s/2)$ for the different experiments. A collapse of the curves is observed and the two characteristic variations of the velocity are shown: $U=X^{\prime 2}$ (heavy black dashed line) for $X^{\prime }<1$ and $U=X^{\prime 1/2}$ (heavy black solid line) for $X^{\prime }>1$.