Acoustic streaming

From Wikipedia, the free encyclopedia

Acoustic streaming is a steady current forced by the absorption of high amplitude acoustic oscillations.

This phenomenon can be observed near sound emitters, or in the standing waves within a Kundt's tube. It is the less-known opposite of sound generation by a flow.

There two situations where sound is absorbed in its medium of propagation:

  • during propagation [1]. The attenuation coefficient is α = 2ηω2 / 3ρc3, following Stokes' law (sound attenuation). This effect is more intense at elevated frequencies and is much greater in air (where attenuation occurs on a characteristic distance α − 1~10 cm at 1 Mhz) than in water (α − 1~100 m at 1 Mhz). In air it is know as the Quartz wind.
  • near a boundary. Either when sound reaches a boundary, or when a boundary is vibrating in a still medium. A wall vibrating parallel to himself generates a shear wave, of attenuated amplitude within the Stokes oscillating boundary layer. This effect is localised on an attenuation length of characteristic size δ = (η / ρω)1 / 2 whose order of magnitude is a few microns in both air and water at 1 MHz.

[edit] Origin: a body force due to acoustic absorption in the fluid

Acoustic streaming is a non-linear effect [2]. We can decompose the velocity field in a vibration part and a steady part ui = vj + Ui. The vibration part due to sound, while the steady part is acoustic streaming velocity. The average velocity in time is \overline{u_i}=U_i, and the Navier–Stokes equations implies for the acoustic streaming velocity:

\overline{\rho}\frac{\partial U_i}{\partial t}+\overline{\rho} U_j \frac{\partial U_i}{\partial x_j}=-\frac{\partial \overline{p}}{x_i}+\eta \frac{\partial^2 U_i}{\partial x_j^2}-\frac{\partial}{\partial x_j}(\overline{\rho v_i v_j} ).

The steady streaming originates from a steady body force f_i=-{\partial}(\overline{\rho v_i v_j} )/{\partial x_j} that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence -\overline{\rho v_i v_j}. The Reynolds stress is depends on the amplitude of sound vibrations, and the body force reflects diminutions in this sound amplitude.

We see that this stress is non-linear (quadratic) in the velocity amplitude. It is non vanishing only where the velocity amplitude varies. If the velocity of the fluid oscillate because of sound as εcos(ωt), the quadratic non-linearity generates a steady force proportional to \overline{\epsilon^2\cos^2(\omega t)}=\epsilon^2/2.

[edit] References

  1. ^ see video on http://www.lmfa.ec-lyon.fr/perso/Valery.Botton/acoustic_streaming_bis.html (French)
  2. ^ Sir James Lighthill 1978 "Acoustic streaming", 61, 391, Journal of Sound and Vibration