Acoustic streaming

Acoustic streaming is a steady flow in a fluid driven 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 are two situations where sound is absorbed in its medium of propagation:

• during propagation. The attenuation coefficient is $\alpha=2\eta\omega^2/(3\rho c^3)$, 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 $\alpha^{-1}$~10 cm at 1 MHz) than in water ($\alpha^{-1}$~100 m at 1 MHz). In air it is known 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 itself generates a shear wave, of attenuated amplitude within the Stokes oscillating boundary layer. This effect is localised on an attenuation length of characteristic size $\delta=[\eta/(\rho\omega)]^{1/2}$ whose order of magnitude is a few micrometres in both air and water at 1 MHz.

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

Acoustic streaming is a non-linear effect. We can decompose the velocity field in a vibration part and a steady part ${u}=v+\overline{u}$. The vibration part $v$ is due to sound, while the steady part is the acoustic streaming velocity (average velocity). The Navier–Stokes equations implies for the acoustic streaming velocity:

$\overline{\rho}{\partial_{t} \overline{u}_i}+\overline{\rho} \overline{u}_j {\partial_{j} \overline{u}_i}=-{\partial \overline{p}_{i}}+\eta {\partial^2_{jj} \overline{u}_i}-{\partial_j}(\overline{\rho v_i v_j} ).$

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 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 oscillates because of sound as $\epsilon\cos(\omega t)$, the quadratic non-linearity generates a steady force proportional to $\scriptstyle \overline{\epsilon^2\cos^2(\omega t)}=\epsilon^2/2$.

Order of magnitude of acoustic streaming velocities

Even if viscosity is responsible for acoustic streaming, the value of viscosity disappears from the resulting streaming velocities in the case of near-boundary acoustic steaming.

The order of magnitude of streaming velocities are:

• near a boundary (outside of the boundary layer):

$U \sim -{3}/{(4\omega)} \times v_0 dv_0/dx,$

with $v_0$ the sound vibration velocity and $x$ along the wall boundary. The flow is directed towards decreasing sound vibrations (vibration nodes).

• near a vibrating bubble of rest radius a, whose radius pulsates with relative amplitude $\epsilon=\delta r/a$ (or $r=\epsilon a \sin( \omega t)$), and whose center of mass also periodically translates with relative amplitude $\epsilon'=\delta x/a$ (or $x=\epsilon' a \sin( \omega t/\phi)$). with a phase shift $\phi$

$\displaystyle U \sim \epsilon \epsilon' a \omega \sin \phi$

• far from walls $U \sim \alpha P/(\pi \mu c)$ far from the origin of the flow ( with $P$the acoustic power, $\mu$ the dynamic viscosity and $c$ the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of $P$.