Phenomenology of laminar acoustic streaming jets

TL;DR

This paper addresses long-range Eckart acoustic streaming jets by modeling the acoustic force with attenuation in a cylindrical cavity, enabling a consistent description beyond the near-field. It develops region-specific scaling laws that connect on-axis velocity to forcing and diffraction, including an inertia-dominated acceleration, a velocity peak with viscous corrections, a diffraction/attenuation-driven downstream decay, and a far-field self-similar decay when the beam is largely attenuated. A key finding is that attenuation controls the momentum-flux buildup length scale $L_\alpha$ and saturates the jet momentum to $M$, while neglecting attenuation leads to large errors in velocity and jet spreading. The results offer practical design rules for optimizing Eckart streaming jets in industrial contexts, such as metallurgy or 3D beam shaping, and provide a framework applicable to other beam-driven flow phenomena.

Abstract

This work identifies the physical mechanisms at play in the different flow regions along an Eckart acoustic streaming jet by means of numerical simulation based on a novel modeling of the driving acoustic force including attenuation effects. The flow is forced by an axisymmetric beam of progressive sound waves attenuating over a significant part of a closed cylindrical vessel where the jet is confined. We focus on the steady, axisymmetric and laminar regime. The jet typically displays a strong acceleration close to the source before reaching a peak velocity. At further distances from the transducer, the on-axis jet velocity smoothly decays before reaching the opposite wall. For each of these flow regions along the jet, we derive scaling laws for the on-axis velocity with the magnitude of the acoustic force and the diffraction of the driving acoustic beam. These laws highlight the different flow regimes along the jet and establish a clear picture of its spatial structure, able to inform the design of experimental or industrial setups involving Eckart streaming jets.

Figures (11)

• Figure 1: Sketch of the computational domain in the $\left( x_d, r_d \right)$ plane, where $x_d$ and $r_d$ refer to the dimensional axial and radial coordinates, respectively. The domain is a closed cylindrical cavity filled with a Newtonian fluid of density $\rho$ and kinematic viscosity $\nu$. It is fitted with a plane circular transducer (gray) at $x=0$ that radiates an axisymmetric beam-shaped acoustic field (green). The sound waves of wavelength $\lambda$ are emitted with an initial power $P_{ac}$ and travel at the phase speed $c$. The pressure amplitude decays at a rate $\alpha$ along the wave propagation path. The attenuation of the sound beam creates a body force that drives a flow of velocity field $\bm{u_s}$. The symmetry axis of the cylindrical container and the flow field is the $r=0$ line, and all boundaries are impermeable walls that completely absorb the sound waves.
• Figure 2: Discretization of the fluid domain with elements of polynomial degree $N_p = 1$. The main lobe of the unattenuated acoustic beam is represented by the green-shaded area, of approximate radius given from Eq. \ref{['eq:beam_radius']}. The red dots located at $\left( x, r \right) = \left( 400, 15 \right)$, $\left( 1000, 70 \right)$ and $\left( 1400, 20 \right)$, are the points where the time series of $\bm{u}$ and $p$ are recorded to monitor the convergence towards a steady state.
• Figure 3: Effects of acoustic force attenuation in the expression of $\widetilde{ \bm{I} }$ (Eqs. \ref{['eq:Rayleigh_integral_all']}) on steady flows computed for $Gr_{ac} = 10^3$. The cases of attenuated ($N/L=0.003$) and unattenuated ($N/L=0$) $\bm{\widetilde{I}}$ are plotted with solid and dashed lines, respectively. (a) Longitudinal profiles of the on-axis velocity $u_x$, with a detailed view close to the transducer shown in the inset. (b) Longitudinal profiles of the jet radius $R_{jet}$ defined as $u_x \left( x, R_{jet} \right) = 0.5 \, u_x \left(x, 0 \right)$. The approximate beam radius (Eq. \ref{['eq:beam_radius']}) is shown as a black dotted curve. The purple and red vertical lines locate the Fresnel distance $L_F$ and the force attenuation distance $L_{\alpha}$, respectively. In (b), the vertical blue line locates the on-axis velocity peak position $x_p$ in the attenuated $\bm{\widetilde{I}}$ case.
• Figure 4: Evolution of the momentum flux accumulated by the jet with the distance $x$ from the transducer for $Gr_{ac} = 10^3$. The numerical results obtained when attenuation is accounted for ($N / L = 0.003$, Table \ref{['tab:dimensionless_parameters_values']}) or not ($N / L = 0$) in $\widetilde{ \bm{I} }$ (Eqs. \ref{['eq:Rayleigh_integral_all']}) are represented by the filled and empty symbols, respectively. The numerical results are overlayed to the profiles of the injected momentum flux $M_{th}$, given by Eq. \ref{['eq:axial_momentum_theoretical']} and evaluated for $N / L = 0.003$ (solid blue line) and for $N / L \rightarrow 0$ (dashed blue line). The red vertical line locates the force attenuation length $L_{\alpha} = L / \left( 2 N \right)$. The horizontal black dashed line indicates the maximum momentum flux $M$ (from Lighthill1978, Eq. \ref{['eq:dimensionless_parameters']}). A detailed view of the momentum profiles, closer to the transducer, is shown in the inset.
• Figure 5: Typical steady velocity field obtained for $Gr_{ac} = 5 \times 10^3$. Top: Map of velocity magnitude together with streamlines. The approximate beam radius, defined by Eq. \ref{['eq:beam_radius']}, is displayed in blue, and the inset on the right focuses on the flow region close to the transducer. Bottom: longitudinal profile of $u_x$ on the jet axis on which the Fresnel distance $L_F = 1.22 / \left( 4 \, S \right)$, the on-axis velocity peak position $x_p$ and the acoustic force attenuation distance $L_{\alpha} = L / \left( 2 \, N \right)$ are reported. The right plot focuses on the region framed in black in the left plot.