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
α
=
2
η
ω
2
/
(
3
ρ
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
α
−
1
~10 cm at 1 MHz) than in water (
α
−
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
δ
=
[
η
/
(
ρ
ω
)
]
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
+
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:
ρ
¯
∂
t
u
¯
i
+
ρ
¯
u
¯
j
∂
j
u
¯
i
=
−
∂
p
¯
i
+
η
∂
j
j
2
u
¯
i
−
∂
j
(
ρ
v
i
v
j
¯
)
.
The steady streaming originates from a steady body force
f
i
=
−
∂
(
ρ
v
i
v
j
¯
)
/
∂
x
j
that appears on the right hand side. This force is a function of what is known as the Reynolds stresses in turbulence
−
ρ
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
ϵ
cos
(
ω
t
)
, the quadratic non-linearity generates a steady force proportional to
ϵ
2
cos
2
(
ω
t
)
¯
=
ϵ
2
/
2
.
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
∼
−
3
/
(
4
ω
)
×
v
0
d
v
0
/
d
x
,
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
ϵ
=
δ
r
/
a
(or
r
=
ϵ
a
sin
(
ω
t
)
), and whose center of mass also periodically translates with relative amplitude
ϵ
′
=
δ
x
/
a
(or
x
=
ϵ
′
a
sin
(
ω
t
/
ϕ
)
). with a phase shift
ϕ
U
∼
ϵ
ϵ
′
a
ω
sin
ϕ
far from walls
U
∼
α
P
/
(
π
μ
c
)
far from the origin of the flow ( with
P
the acoustic power,
μ
the dynamic viscosity and
c
the celerity of sound). Nearer from the origin of the flow, the velocity scales as the root of
P
.
