Scaling and dimensional analysis of acoustic streaming jets

Abstract

This paper focuses on acoustic streaming free jets. This is to say that progressive acoustic waves are used to generate a steady flow far from any wall. The derivation of the governing equations under the form of a nonlinear hydrodynamics problem coupled with an acoustic propagation problem is made on the basis of a time scale discrimination approach. This approach is preferred to the usually invoked amplitude perturbations expansion since it is consistent with experimental observations of acoustic streaming flows featuring hydrodynamic nonlinearities and turbulence. Experimental results obtained with a plane transducer in water are also presented together with a review of the former experimental investigations using similar configurations. A comparison of the shape of the acoustic field with the shape of the velocity field shows that diffraction is a key ingredient in the problem though it is rarely accounted for in the literature. A scaling analysis is made and leads to two scaling laws for the typical velocity level in acoustic streaming free jets; these are both observed in our setup and in former studies by other teams. We also perform a dimensional analysis of this problem: a set of seven dimensionless groups is required to describe a typical acoustic experiment. We find that a full similarity is usually not possible between two acoustic streaming experiments featuring different fluids. We then choose to relax the similarity with respect to sound attenuation and to focus on the case of a scaled water experiment representing an acoustic streaming application in liquid metals, in particular, in liquid silicon and in liquid sodium. We show that small acoustic powers can yield relatively high Reynolds numbers and velocity levels; this could be a virtue for heat and mass transfer applications, but a drawback for ultrasonic velocimetry.

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

“…It must also be said that the numerical solutions obtained at the three low acoustic powers corresponding to Figs • turn and are redirected along this new wall. As expected from former observations [36], in this inertia-dominated regime, the acoustic jet's width and path match those of the acoustic beam. Let us recall that this was attributed to a longer momentum transverse diffusion time compared to the travel time along the jet.…”

“…The isosurface displays the same principal features corresponding to the inertial and acoustic jets (here merged in a single isosurface due to the choice of the isosurface value). In the volume, the shape of the acoustic jets is a slightly expanding cylinder, because they keep the shape of the acoustic forcing, as formerly observed in a different configuration [36]. The inertial jets shape is characteristic of an impinging wall jet, elongated along the wall in the direction of impingement, but also with global recirculations out of the symmetry plane, symmetrically on both sides of this plane.…”

“…1) after a 280-mm travel distance. Because previous studies [34,36] carefully measured a 274-mm Fresnel length for this transducer, the acoustic field is known to have transited from the near field to the far field when it enters the investigation volume, making the jet flow structure simpler [37]. In the investigation domain, the acoustic beam impinges at the center of the first lateral glass wall with a 45…”

“…These latter instabilities are unable to develop due to the extremely stabilizing effect of the acoustic forcing. Indeed, previous studies [34,[36][37][38] showed that the acoustic jet local expansion quite accurately follows the width of the acoustic beam. The acoustic field has therefore a forcing effect as well as a confining effect, which both contribute to the stabilization of the acoustic jet compared to a free jet.…”

“…In contrast to that, our former experimental observations have shown the existence of a ubiquitous and very rich dynamics [34,35], whose development is worth studying. These water experiments were made using ultrasounds in the megahertz range in several configurations [34,[36][37][38].…”